Daniele Regoli tesi

Daniele Regoli tesi
Alma Mater Studiorum · Università di Bologna
Dottorato di ricerca in Fisica
ciclo XXIII
settore scientifico disciplinare FIS/02
The relation between Geometry and
Matter in Classical and Quantum Gravity
and Cosmology
Daniele Regoli
relatore: Alexandre Kamenchtchik
coordinatore: Fabio Ortolani
esame finale 2011
Quaerendo invenietis
J. S. Bach
Contents
Introduction
1
I. Cosmology and Dark Energy models
7
1. Cosmology: the basics
1.1. Accelerated expansion . . . . . . . .
1.1.1. ΛCDM: ups and downs . . . .
1.1.2. The case −1 < w < −1/3 . .
1.1.3. The case w < −1: the Big Rip
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2. Two-field cosmological models
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. The system of equations for two-field cosmological models . . .
2.2.1. Reconstruction of the function of two variables, which in
turn depends on a third parameter . . . . . . . . . . . .
2.2.2. Cosmological applications, an evolution “Bang to Rip” .
2.3. Analysis of cosmological models . . . . . . . . . . . . . . . . . .
2.3.1. Model I . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2. Model II . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Two-field models and cosmic magnetic fields
3.1. Introduction to the content of the paper [2] . . . . . . . . . . .
3.2. Cosmological evolution and (pseudo)-scalar fields . . . . . . .
3.3. Interaction between magnetic and pseudo-scalar/phantom field
3.4. Generation of magnetic fields: numerical results . . . . . . . .
3.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Phantom without phantom in a PT-symmetric background
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4.1. PT-symmetric Quantum Mechanics: a brief introduction . . . . . 45
4.2. PT-symmetric oscillator . . . . . . . . . . . . . . . . . . . . . . . 47
4.3. Introduction and resume of the paper [3] . . . . . . . . . . . . . . 50
v
Contents
4.4.
4.5.
4.6.
4.7.
Phantom and stability . .
Cosmological solution with
Cosmological evolution . .
Conclusions . . . . . . . .
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classical
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phantom field
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5. Cosmological singularities with finite non-zero radius
5.1. Introduction and review of paper [4] . . . . . . . . .
5.2. Construction of scalar field potentials . . . . . . . .
5.3. The dynamics of the cosmological model with α = 12
5.4. Conclusions . . . . . . . . . . . . . . . . . . . . . .
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II. Loop Quantum Gravity and Spinfoam models
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Introduction
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Non-perturbative Quantum Gravity: some good reasons to consider
this way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6. Canonical Quantum Gravity: from ADM formalism to Ashtekar
variables
6.1. Hamiltonian formulation of General Relativity . . . . . . . . . . .
6.2. The tetrad/triad formalism . . . . . . . . . . . . . . . . . . . . .
6.3. The Ashtekar-Barbero variables . . . . . . . . . . . . . . . . . . .
6.4. Smearing of the algebra . . . . . . . . . . . . . . . . . . . . . . .
7. Loop Quantum Gravity
7.1. The program . . . . . . . . . . . . . . . . . . . . . .
7.2. The kinematical Hilbert space Hkin . . . . . . . . . .
G
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7.3. The Gauss constraint. Hkin
D
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7.4. The vector/diffeomorphism constraint. Hkin
7.5. Quantization of the algebra and geometric operators .
7.6. The scalar constraint . . . . . . . . . . . . . . . . . .
7.7. Concluding remarks . . . . . . . . . . . . . . . . . . .
8. The
8.1.
8.2.
8.3.
spinfoam approach to the dynamics
Sum-over-histories from hamiltonian formulation
Path integral discretization: BF theory . . . . .
Spinfoam models for Quantum Gravity . . . . .
8.3.1. The Barrett-Crane model . . . . . . . .
8.3.2. The EPRL model . . . . . . . . . . . . .
8.4. Spinfoam: a unified view . . . . . . . . . . . . .
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Contents
8.5. Spinfoams as a field theory: Group Field Theory . . . . . . . . . . 133
9. A proposal for the face amplitude
9.1. Introduction and resume of the content
9.2. BF theory . . . . . . . . . . . . . . . .
9.3. Three inputs . . . . . . . . . . . . . . .
9.4. Face amplitude . . . . . . . . . . . . .
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Conclusion
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Acknowledgements
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Bibliography
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vii
Introduction
Cosmology and Quantum Gravity are the two main research areas this thesis is
about.
Cosmology is the study of the universe as a dynamical system. It is a rather
peculiar chapter of Physics for more than one reason. The main one, I believe, is
that we have but one universe (someone may not agree on this) and there is no
way to do experiments on the universe, in the proper sense of the word. We can
at best observe it, and that’s what astronomers do very well. Another reason,
at least for myself, is on purely intellectual basis: Cosmology is one of the areas
(if not the area) of Science that most touches in deep the questions Science was
born for. One more argument, which is somehow mixed with the preceding one,
is that Cosmology is in a certain sense a rather young science: Einstein’s General
Relativity (1916) is the first theoretical framework that has allowed a scientific
and mathematical approach to the problem of the study of the universe as a
single system. Before that, questions like ‘where does the universe come from?’
and ‘what is the fate of the universe?’ where mere philosophy, with no scientific
attempt to be answered1 .
Since its birth, Cosmology has taken gigantic steps, both on the phenomenological and theoretical viewpoints. The Friedmann models (see section 1) are the
first theoretical models comprehending the dynamical expansion of the universe,
together with the Big Bang initial singularity, both compatible with observations
and accepted from the scientific community2 .
I dare say that expansion and initial moment of the universe have somehow
given birth (for what it is possible) to two macro lines of research. Observations
of the Cosmic Microwave Background have indeed fostered the research on the
nature of the universe in its very beginning. We talk then about “early-time
Cosmology”, referring to the study of primordial perturbations amplified by some
1
Maybe it is superfluous to remark that they are not answered even nowadays. The point are
not the answers, which will probably never come, but the scientific and rigorous attempt
to give them.
2
The Big Bang, being a singularity of the theory, cannot be observed, strictly speaking. What
is accepted, is the existence, some 15 billion years ago, of a phase of the universe in which
everything (spacetime itself) was contracted in a tiny and extremely hot bubble. But on
the nature of that bubble, on its real size and so on, the debate is very far from being
closed.
1
Introduction
(quantum) mechanism (e.g. inflation), giving birth to the large-scale structures
we observe nowadays, such as galaxies, clusters and so on. On the other hand
we have the discovery of the cosmic acceleration [6, 7], that has guided many
theoreticians in the study of cosmological models able to reproduce this (and
others to it related) experimental evidence.
In the (first part of the) present thesis I focus on the second of this problem, the one regarding the entire evolution of the universe, its expansion, its
acceleration and so on; while I will not touch the cosmological perturbations or
amplification mechanisms such as inflation.
Quantum Gravity is not a self-consistent theory, yet. It is a work in progress
attempt to find a coherent set of tools in order to formulate some kind of predictions at physical scales where both Quantum Mechanics and General Relativity
should hold. And we know that taken as they are, Quantum Mechanics and
General Relativity cannot be both valid. Thus, we cannot properly speak of a
theory, since a theory is something self-contained (at least at some degree) that
has the possibility of making scientific predictions.
Loop Quantum Gravity is one of these attempts, so one usually says that it
is a candidate for Quantum Gravity. The keystone of Loop Quantum Gravity, namely the feature that identifies it among other candidates, is to quantize
(canonically) the metric tensor (or, better, an appropriate manipulation of the
metric tensor) without assuming the existence of a somehow fixed classical background, around which fluctuates the “quantum part” of the metric. This is known
as background independence.
Since its birth in 1986, LQG has reached amazing and important results, the
main of which – I dare say – is the suggestive prediction of the ‘granular’ nature
of space: LQG indeed predicts the existence of a minimum length in space.
However, much is still to be done; and the main drawback, which is actually
common to all the candidates Quantum Gravity theories, is the lack (more or
less total) of testable predictions.
A decade ago a new “spin-off” theory was born in the framework of LQG:
spinfoam theory. It is an attempt to define quantum gravitational amplitudes
between space-geometries (given by LQG) in terms of a sum-over-histories. The
histories are called ‘spinfoams’, a kind of bubble-like representation of spacetime
at fundamental level. Many problems are still open, and it is a very stimulating
field of research, particularly for young researchers, since there are entire lines
still to be explored and thousands of bridges with other theoretical frameworks
still to be built.
2
Introduction
The main results of the research here presented are the following:
• we analyzed in depth two-field cosmological models, with one scalar and
one phantom scalar field. The procedure of reconstruction of two-field potentials reproducing a dynamical evolution (specifically an evolution with a
crossing of the phantom divide line) has been worked out: we have discovered that two-field models have a huge freedom with respect to single-field
models, namely there is an infinite number of potentials which give, with
a specific choice of initial conditions, the wanted dynamics for the model.
Moreover, a thorough analysis of the phase space of two specific two-field
models is carried out, showing that inside a single model, varying the initial conditions on the fields, qualitatively different families of evolutions
are present, with different cosmological singularities as well.
• In line with the preceding result, we have coupled our two-field models with
cosmic magnetic fields, which are experimentally observed quantities. We
have shown the sensitivity of the amplification of the magnetic fields to the
change of underlying cosmological model, but keeping the same background
evolution of the universe. This, in principle, is a way to experimentally
discriminate between cosmological models with different potentials.
• The greatest problem of phantom scalar fields is quantum instability. We
have shown that in the framework of P T symmetric Quantum Theory
(somehow adapted to Cosmology) this instability can be cured: we have
an effective phantom scalar field, with stable quantum fluctuations.
• We analyzed one-field cosmological models comprehending a peculiar version of the Big Bang and Big Crunch singularity, namely singularity with
finite and non-zero radius. We performed a detailed analysis of the phase
space and reveal the presence of different classes of evolutions of the universe.
• In the framework of spinfoam theory for Quantum Gravity, we proposed
a fixing of the face amplitude of the spinfoam sum, somewhat neglected
until now, motivated essentially by a form of “unitarity” of gravitational
evolution.
I list here the publications in which the above said research has been collected
(chronologically from the latest) :
• E. Bianchi, D. Regoli and C. Rovelli, “Face amplitude of spinfoam quantum
gravity” Classical and Quantum Gravity CQG 27, 185009 (2010)
(arXiv:1005.0764 [gr-qc]).
3
Introduction
• A. A. Andrianov, F. Cannata, A. Y. Kamenshchik and D. Regoli, “Phantom Cosmology based on P T -symmetry”, International Journal of Modern
Physics D IJMPD 19, 97 (2010).
A. A. Andrianov, F. Cannata, A. Y. Kamenshchik and D. Regoli, “Cosmology of non-Hermitian (C)P T -invariant scalar matter”, J. Phys. Conf.
Ser. 171, 012043 (2009).
• F. Cannata, A. Y. Kamenshchik and D. Regoli, “Scalar field cosmological models with finite scale factor singularities”, Physics Letters B 670
(2009) 241-245 (arXiv:0801.2348v1 [gr-qc]).
• A. A. Andrianov, F. Cannata, A. Y. Kamenshchik and D. Regoli, “Twofield cosmological models and large-scale cosmic magnetic fields”, Journal
of Cosmology and Astroparticle Physics (JCAP) 10 (2008) 019
(arXiv:0806.1844v1 [hep-th]).
• A. A. Andrianov, F. Cannata, A. Y. Kamenshchik and D. Regoli, “Reconstruction of scalar potentials in two-field cosmological models”, Journal of
Cosmology and Astroparticle Physics (JCAP) 02 (2008) 015
(arXiv:0711.4300 [gr-qc]).
This thesis thus is intended to pursue two targets:
• to provide, for the fields of interest, a sufficient framework of notions and
results that are the current background on which research is performed
and are necessary to understand (or closely related to) the subject of the
following goal
• to resume and explain the results of the research I have been working on
during this three-year PhD course, under the supervision of Dr. Alexander Kamenshchik.
Gravity is the main character of the play. Matter is the other one. ‘Matter’
of course in a cosmological sense, i.e. everything that is embedded in spacetime.
Talking in field theory jargon, we have nothing but fields: the gravitational field
(the metric) and all the others: the matter fields (including those whose (quantum) excitations are the well-known elementary particles, and possibly others,
which should model/explain such things as the dark matter, dark energy and,
generically, other types of exotic matter/energy). All living on the spacetime
manifold.
We know that if we stay away from the quantum regime, General Relativity
is the stage on which our characters play3 . GR, in the intuitive rephrasing by
3
4
There are of course many proposals to modify somehow GR, e.g. the Hořava-Lifshitz proposal [9], the f (R) models [10, 11], and others, but I will not consider them here at all.
Introduction
Wheeler [8], is simply encoded in the following aphorism-like sentence: matter
teaches spacetime how to curve, spacetime teaches matter how to move. The
“how”, is encoded in Einstein’s equation and in general in the Einstein-Hilbert
action for GR.
Classical Cosmology is precisely the study of this interplay between spacetime
geometry and matter, on the scale of the entire universe. The idea is to build
models with various kinds of matter, possibly encoded in terms of fields, that are
able to reproduce the observed data on the dynamical evolution of the universe.
More specifically for what concerns this thesis, the key empirical observation is
the cosmic acceleration [6, 7], and the possibility of a kind of “super-acceleration”,
that has to be somehow explained theoretically, which we tried to, for our (little)
part.
When one enters the quantum regime, attention must be paid. We know that
there is a typical scale, the Planck scale, under which we expect both GR and
QM to be valid. However there is no accepted framework for this scale. No
complete Quantum Gravity theory exists, yet. Actually, it is the Holy Grail of
contemporary theoretical physics.
The second (shorter) part of the present thesis is entirely focused on one
specific candidate of Quantum Gravity: loop quantum gravity. I will have a
more appropriate occasion to discuss this thoroughly II, but for the moment
let me stress that I believe it fundamental to study what happens to spacetime
under the Planck scale in order to properly catch the nature of the relation
between geometry and matter. I think Quantum Gravity is going to revolutionize
our concept of the interplay between gravity and matter as much as Einstein’s
theory has drastically changed the idea of matter fields top of a passive, fixed
background. My personal contribution in this respect, is in the framework of
spinfoam models for quantum gravity: a kind of path-integral formulation of
quantum gravity, based and founded on the results of canonical Loop Quantum
Gravity.
The thesis is organized as follows:
• Part I is dedicated to classical Cosmology: it starts with an introductory
chapter intended to provide the necessary concepts of classical Cosmology;
this is followed by four ‘research’-chapters, each dedicated to one of the four
publications about Cosmology listed before (or, equivalently, see [1],[3],[2],
[4]). The second of these four chapters, namely the one dedicated to “P T phantom Cosmology”, includes also a short review of P T -symmetric Quantum Mechanics, whose ideas play a key role in the research explained in
(what remains of) that chapter.
• Part II is focused on loop quantum gravity and spinfoam models: here
I dedicate more time to properly introduce the basics of these theories.
5
Introduction
Namely, four entire chapters are devoted to a review of fundamental concepts of this theoretical framework. Indeed loop quantum gravity stands
on much more advanced and sophisticated basis than classical Cosmology,
and, since the work I have done is about a rather technical aspect of spinfoam models, I thought it absolutely necessary to properly review all the
underlying framework, trying to be as self contained as possible.
The last chapter of this second part resumes the results and explains the
content of the paper [5], about a proposal for fixing the face amplitudes of
spinfoam models.
A brief conclusion 9.4 is intended to give a global view ex post on all the issues
discussed.
6
Part I.
Cosmology and Dark Energy
models
7
1. Cosmology: the basics
Cosmology is the study of the universe as a dynamical system. At a classical
level the tools to handle this kind of study are given by the Einstein’s General
Relativity theory, in nuce by equations
1
8πG
Rµν − gµν R = 4 Tµν ,
2
c
(1.1)
where Rµν is the Ricci tensor, gµν is the metric tensor, R = g µν Rµν is the scalar
curvature and Tµν is the stress-energy tensor. As for the constants, G is Newton’s
gravitational constant and c is the speed of light1 .
Of course, to (hope to) resolve these equations – in order to catch the dynamics
of the universe, encoded in the solution gµν (x) – one has to know the source
term Tµν point by point in spacetime. This can be achieved only by drastic
exemplifications on the nature of the source and of the spacetime. Namely one
assumes a rather strict – but also very reasonable at this large scale – symmetry:
roughly speaking the assumption is homogeneity and isotropy of space. More
precisely one assumes that the spacetime 4d manifold admits a foliation is 3d
spaces that are homogeneous and isotropic, i.e. one admits the existence of a
family of observers that see uniform space-like surfaces. This can be rephrased
in a more suggestive and simple way by saying that we exclude the existence
of special regions in space. This is known as cosmological principle and, as we
will see in a moment, it drastically exemplifies equations (1.1). Of course this
principle is just a device through which one tempts to extract physics from a
general model which would be by far too complicated. One can relax in many
ways this principle, for example in studying cosmological perturbations or by
admitting anisotropies in some ways [98]. However, for our purposes, it won’t
be necessary to abandon this principle.
The assumption of such a principle results in the following form of the line
element in spacetime (the Friedmann-Lemàitre-Robertson-Walker line element,
or simply FLRW):
ds2 = dt2 − a2 (t)dl2 ,
(1.2)
1
The reader can find detailed informations in every good textbook on General Relativity. Let
me just say that for general GR framework I really like the book by Hawking and Ellis [12]
while for classical Cosmology (and for much more) I find very useful the book by Landau
and Lifshitz [13].
9
1. Cosmology: the basics
where
dr2
− r2 (dθ2 − sin2 θdφ2 ) .
(1.3)
1 − kr2
Recall that k is the curvature parameter and equals 1, 0 or -1 for a closed, openflat and open-hyperbolic universe respectively. In all what follows we shall take
k = 0, i.e. we assume a (spatially-)flat universe. This is in accordance with all
the observational data2 .
In equation (1.2) the only dynamical variable is the scale factor a(t), the only
dynamical degree of freedom in classical (uniform) Cosmology. As is clear from
(1.2), it represents the ‘size’ of the universe at a certain moment, and thus it
gives information on the expansion or contraction of the universe itself.
Recall that for a non-dissipative fluid one has Tµν = (p + ε)uµ uν − pgµν with
p and ε the pressure and the energy density of the fluid, respectively, and uµ =
dxµ (s)/ds is the 4-velocity of the fluid. With this in mind Einstein’s equations
become much more simple than (1.1), namely
2
k
ȧ
(1.4)
+ 3 2 = 8πGε ,
3
a
a
2aä + a2 + k
= −8πGp .
(1.5)
a2
dl2 =
These are the well-known Friedmann equations. Actually one usually replaces
the second of these equations with the simpler requirement of conservation of
energy, i.e.
ȧ
ε̇ = 3 (p + ε) ,
(1.6)
a
and thus uses equations (1.5) and (1.6) as the set of two independent equations
that governs the dynamics of the universe3 .
Solving the system (1.5), (1.6) for a dust-like fluid (a fluid with zero pressure,
which is a good approximation for visible matter like galaxies, clusters of galaxies
and so on) one obtains the well-known Friedmann models, summarized in figure
1.1.
Remark. These results, I want to stress it, are striking. Doing research in
this field one is often ‘obliged’ to investigate technical (and often exotic) details
somewhat losing the importance of the Friedmann models: with a couple of
amazingly simple equations one is able to obtain a qualitative (past and future)
2
Actually it is true that k = 0 is coherent with all the data, but some argue this is only
a proof that the space is locally flat, and still it may have a positive or negative global
curvature. I am not going to discuss this (important) issue here.
3
Intuitively, one for determining the source (say ε) in terms of the scale factor, and then the
other to determine the scale factor itself.
10
a @tD
k=-1
k=0
k=+1
t
Figure 1.1.
evolution of the universe. Of course these results must be corrected in order to
fit observations introducing different kind of matter/energy, must be generalized
to embrace anisotropies or inhomogeneities and must be supported by some
quantum gravity theory to investigate what physics is inside the initial (and
possibly final) singularity. But much of classical Cosmology is already inside
these simple models.
In what follows we shall make a huge usage of the Hubble parameter
h(t) =
ȧ(t)
,
a(t)
(1.7)
in terms of which the Friedmann equations (1.5), (1.6) become4
k
,
a2
ε̇ = −3h(ε + p) ,
h2 = ε −
and, for the flat case,
h2 = ε ,
ε̇ = −3h(ε + p) .
(1.8a)
(1.8b)
and thus can be written in the compact form
3
ḣ = − (ε + p) .
2
4
(1.9)
I use here, and everywhere in what follows about Cosmology, the convention 8πG/3 = 1.
11
1. Cosmology: the basics
For sake of completeness, let me recall the form of the curvature in terms of
the Hubble parameter:
k
2
(1.10)
R = −6 ḣ + h + 2 .
a
which is important to understand how h and a determines curvature singularities.
1.1. Accelerated expansion
Up till here we have talked about a universe filled with galaxies, and we have
seen its large scale evolution, which is an expansion starting from the Big Bang
singularity and ending eventually either in a future singularity after a contraction
(the closed case) or in an endless cold expansion.
However it is now well-known that this is not the case. The cosmic expansion
is accelerated [6, 7]. This is the empirical observation that acted like a spark
for the explosion of a large amount of cosmological models trying to incorporate
this effect [14]-[19], [20, 21], [22]-[32] in the decade after it was first discovered
(1998). The first part of this thesis lays in this line.
Now I shall introduce the general ideas and the framework for these models
as well as the technical jargon necessary to understand them.
Take a generic cosmic fluid, and write its equation of state as follows
p = wε ,
(1.11)
The w parameter is crucial to discriminate between different kind of expansions,
as well as of fluids, of course. Indeed, Friedmann equations tell us that in order
to have acceleration (ä > 0) one has to have
1
.
(1.12)
3
Such a fluid is generically known as Dark Energy (DE) [14]-[18]. The w = −1
(constant) case is the ΛCDM model of Cosmology5 , and is actually the best
candidate model to fit observational data, as well as the simplest DE model.
As is well-known, at a sufficiently large time scale (when the baryonic and dark
matter contribution becomes negligible) the evolution of ΛCDM is de Sitter-like.
There are some motivations to investigate DE models different from the simple
ΛCDM. Some more practical and some more “philosophical”. Let me spend some
words on this issue. The not interested reader of course can skip the entire
following subsection.
w<−
5
Obviously we are here talking of the DE side of the story. In every decent cosmological
model there must be also a baryonic sector as well as a dark matter sector, which, when
talking specifically about DE, are often understood.
12
1.1. Accelerated expansion
1.1.1. ΛCDM: ups and downs
Let me remark one thing: ΛCDM is in accordance with all observational data.
This, as contemporary theoretical physics teaches us, does not prevent theoreticians to investigate alternatives, if there are some motivations supporting this
research.
A practical (and I think more common-sense) motivation is the following: it is
true that observations do not exclude ΛCDM, but there are other possibilities not
excluded as well. Moreover, some observations indicate a best fit for w which is
less then -1 6 (we shall see that this is not a painless difference), and contemplate
the possibility of “crossing” this benchmark w = −1, thus with a non-constant
value of w. In my humble opinion I think this is enough for researchers to explore
alternatives, even if exotic; obviously keeping clear in mind that there is a good
candidate model in accordance with data.
Theoretical, or “philosophical”, motivations have been put out as well (I draw
arguments mainly by [56]). They are generically of two types. The first is the
‘coincidence problem’: observations tell us that the dark energy density is about
2.5 times bigger than baryonic + dark matter(dama) density (about 74% for
DE and 26% for the rest). Since the baryonic and dark matter contributions
dilute in time (as a−1/3 ) while the cosmological constant does not, it will eventually dominate the evolution of the universe and its density should be much
greater than the observed + dama sector, starting from some cosmic era on.
The argument is that it is not likely that we are just in that era in which the
DE density is comparable to the observed+dama one, it would be too much of
a coincidence. Think of it as a kind of cosmological principle in time. Thus,
the argument claims, ΛCDM must be wrong. This is a rather poor argument,
in my point of view. First, because probability arguments are to be taken very
carefully, always. Second, because it is quite clear that in a universe dominated
by the cosmological constant, matter and thus us, wouldn’t exist. This is a kind
of anthropic (counter)argument, but – I think – of a very light and reasonable
kind.
The second type is based on QFT reasonings: the cosmological constant (i.e.
the DE in ΛCDM) is a kind of vacuum energy; QFT ‘predicts’ the existence of
a energy of the vacuum, but if we compare the QFT calculation with the (tiny)
observed value for Λ they differ by 120 orders of magnitude. So ΛCDM must be
wrong. For this argument there are counter-arguments as well: it is true that
QFT predicts a vacuum energy, but it is actually a huge amount of energy, and
it is not at all observed. If QFT vacuum energy was really there, any region of
space with a quantum field would have a huge mass, and it would certainly be
observed. The Casimir effect reveals only the effect of a difference (or, better, a
6
See [46]-[52] and [53]-[55].
13
1. Cosmology: the basics
change) in vacuum energy. Thus it is likely that there would be some unknown
mechanism in QFT that prevents the vacuum energy to gravitate, or protect it
from huge radiative corrections, rather than ΛCDM is ‘wrong’.
Thus, I do think it is worthwhile exploring alternatives of ΛCDM, but mainly
for practical and observational reasons. And still I consider the ΛCDM model
as a good standpoint for classical Cosmology.
1.1.2. The case −1 < w < −1/3
Let us briefly sketch the behavior of this class, i.e. the class of models with state
parameter w of DE type but greater than -1. The calculations starting from the
evolution equation (1.9) are straightforward
h(t) =
2
.
3(w + 1)(t − t0 )
(1.13)
with a Big Bang initial singularity, in the sense that
a(t) −−−→
0,
+
(1.14)
t→t0
indeed
a(t) = (t − t0 )2/3(w+1) ,
ȧ =
3w+1
2
(t − t0 )− w+1 .
3(w + 1)
(1.15)
But differently from the Friedmann models – where ȧ is increasingly high going
towards the initial instant – here ȧ goes to zero as well.
One could as well tempt to analyze cases where w varies with time. One
remarkable example is the so-called Chaplygin gas [26], characterized by the
following state equation
A
(1.16)
p=− ,
ε
with A a positive constant. This fluid gives a sort of interpolation between a
dust-like era and a de Sitter era.
1.1.3. The case w < −1: the Big Rip type singularity
I usually call the evolution of the models with w constant and less than -1
“super-acceleration”. Indeed they are characterized by
that is
14
ḣ > 0 ,
(1.17)
ȧ2
ä
> 2 .
a
a
(1.18)
1.1. Accelerated expansion
The dynamical evolution reads
a(t) = (t − t0 )−2/3|w+1| ,
ȧ =
3w+1
2
(t − t0 )− w+1 .
3|w + 1|
(1.19)
Notice that now the exponents are both negative, thus
a −−−→
∞,
+
t→t0
ȧ −−−→
−∞ .
+
(1.20)
t→t0
These features – scale factor and its velocity both infinite at a finite instant –
define the Big Rip singularity, which is typical of the super-accelerated models [57, 58]. Sometimes, the fluid responsible of such an exotic evolution is
dubbed phantom energy [36]-[45].
I hope it is now clear to the reader that the benchmark w = −1 (somehow
represented by the ΛCDM model) discriminates between dynamical evolutions
qualitatively different: cosmic acceleration with a Big Bang and a cold infinity
expansion versus a “super-acceleration” with a different cosmological singularity.
The line w = −1 has deserved a name, the phantom divide line.
Remark (the phantom stability problem). Let me anticipate here the big drawback of the phantom energy. A kind of energy with state parameter w < −1
violates all the energy conditions prescribed by General Relativity (see e.g. [12]).
Indeed, when a phantom energy is modeled by a scalar field (see section 2.1) it
requires a negative kinetic term. Its hamiltonian becomes, obviously, unbounded
and such a field is plagued by manifest quantum instability: its quantum fluctuations grow exponentially with time [59, 60]. This is actually the main critic done
to the use of such things as phantom fields. And, I have to admit, it is a quite
strong and reasonable critic. However, ghost fields like these, were studied long
before the phantom field was first introduce; and in chapter 4 (or equivalently
in [3]) I will describe a mechanism that could fix this plague, paying the price of
enlarging the framework of Quantum Theory to its P T symmetric version.
15
2. Two-field cosmological models
This chapter is devoted to reviewing and explaining the content of [1]. The
idea is to work with two scalar fields, one of which phantom, and try with this
model to reproduce an evolution from a Big Bang to a Big Rip singularity. The
procedure of reconstruction of the model starting by a specif evolution is carried
out in detail. Then, the dynamical phase space of two of such models is studied,
analyzing the dynamical classes included in it by varying the initial conditions
on the fields.
2.1. Introduction
As we have seen in the previous section, the discovery of cosmic acceleration
[6, 7] has stimulated the construction of a class of dark energy models [14][19], [20, 21], [22]-[32] describing this effect. In what follows we shall deal with
cosmological models based on scalar fields, i.e. the matter content of the universe
is modeled by means of scalar fields. Notice that the cosmological models based
on scalar fields were considered long before the observational discovery of cosmic
acceleration [33]-[35].
According to some authors, the analysis of observations permits the existence
of the moment when the universe changes the value of the parameter w from
w > −1 to w < −1 [46]-[52]. This transition is called “crossing of the phantom
divide line”. The most recent investigations have shown that the phantom divide
line crossing is still not excluded by the data [53]-[55].
It is easy to see that the standard minimally coupled scalar field cannot give
rise to the phantom dark energy, because in this model the absolute value of
energy density is always greater than that of pressure, i.e. |w| < 1. A possible
way out of this situation is the consideration of the scalar field models with the
negative kinetic term, i.e. phantom field models, as I will briefly review in a
moment. Thus, the important problem arising in connection with the phantom
energy is the crossing of the phantom divide line. The general belief is that while
this crossing is not admissible in simple minimally coupled models its explanation
requires more complicated models such as “multifield” ones or models with nonminimal coupling between scalar field and gravity (see e.g. [61]-[66]).
Some authors of [1] described the phenomenon of the change of sign of the
17
2. Two-field cosmological models
kinetic term of the scalar field implied by the Einstein equations[67, 68]. It
was shown that such a change is possible only when the potential of the scalar
field possesses some cusps and, moreover, for some very special initial conditions
on the time derivatives and values of the considered scalar field approaching
to the phantom divide line. At the same time, two-field models including one
standard scalar field and a phantom field can describe the phenomenon of the (de)phantomization under very general conditions and using rather simple potentials
[69]-[72].
In the paper under consideration [1] we have focused to the drastic difference
between two- and one-field models.
The reconstruction procedure of the (single) scalar field potential models is
well-known [73], [74, 75], [76], [77], [78, 79], [80] and [81]-[84]. Let me recapitulate
it briefly.
If the matter is represented by a spatially homogeneous minimally coupled
scalar field, then the energy density and the pressure are given by the formulæ
1
ε = φ̇2 + V (φ) ,
2
1
p = φ̇2 − V (φ) ,
2
where V (φ) is a scalar field potential. Friedmann equations thus read
h2 = = 12 φ̇2 + V (φ) ,
dV
φ̈ + 3hφ̇ +
=0,
dφ
(2.1)
(2.2)
(2.3)
(2.4)
Where the second is clearly the Klein-Gordon equation on a FRLW background.
Combining these equations we have
V =
and
ḣ
+ h2 ,
3
(2.5)
2
φ̇2 = − ḣ .
(2.6)
3
Equation (2.5) represents the potential as a function of time t. Integrating
equation (2.6) one can find the scalar field as a function of time. Inverting this
dependence we can obtain the time parameter as a function of φ and substituting
the corresponding formula into equation (2.5) one arrives to the uniquely reconstructed potential V (φ). It is necessary to stress that this potential reproduces
a given cosmological evolution only for some special choice of initial conditions
on the scalar field and its time derivative [67, 76]. Below I shall show that in
18
2.1. Introduction
the case of two scalar fields one has an enormous freedom in the choice of the
two-field potential providing the same cosmological evolution. This freedom is
connected with the fact that the kinetic term has now two contributions.
Notice that equation (2.6) immediately implies one thing: in order to have (at
least a phase of) super-acceleration (which, I recall, is defined by ḣ > 0) with an
energy density produced by a scalar field, one has to have a negative sign of the
kinetic term. This is even clearer if one take the specific Hubble parameter
h(t) =
2
,
3(w + 1)t
(2.7)
i.e. the Hubble parameter for a fluid with w = constant, and tries to reconstruct
the scalar field model. Integrating (2.6) one has
s
4
φ(t) = ±
ln t ,
(2.8)
9(w + 1)
which clearly implies w > −1. Had we chose a scalar field with a negative kinetic
term, we would have been able to reproduce the w < −1 case.
In order to examine the problem of phantom divide line crossing we shall be
interested in the case of one standard scalar field and one phantom field ξ ,
whose kinetic term has a negative sign. In this case the total energy density and
pressure will be given by
1
ε = φ̇2 −
2
1
p = φ̇2 −
2
1 ˙2
ξ + V (φ, ξ) ,
2
1 ˙2
ξ − V (φ, ξ) .
2
(2.9)
(2.10)
The relation (2.5) expressing the potential as a function of t does not change in
form, but instead of equation (2.6) we have
2
φ̇2 − ξ˙2 = − ḣ .
3
(2.11)
Now, one has rather a wide freedom in the choice of the time dependence of one of
two fields. After that the time dependence of the second field can be found from
equation (2.11). However, the freedom is not yet exhausted. Indeed, having two
representations for the time parameter t as a function of φ or ξ, one can construct
an infinite number of potentials V (φ, ξ) using the formula (2.5) and some rather
loose consistency conditions. It is rather difficult to characterize all the family of
possible two-field potentials, reproducing given cosmological evolution h(t). In
the following, I describe some general principles of construction of such potentials
and then consider some concrete examples.
19
2. Two-field cosmological models
2.2. The system of equations for two-field
cosmological models
The system of equations, which we study contains (2.5) and (2.11) and two
Klein-Gordon equations
∂V (φ, ξ)
=0 ,
∂φ
∂V (φ, ξ)
ξ¨ + 3hξ˙ −
=0 .
∂ξ
φ̈ + 3hφ̇ +
From equations (2.12) and (2.13) we can find the partial derivatives
∂V (φ,ξ)
∂ξ
(2.12)
(2.13)
∂V (φ,ξ)
∂φ
and
as functions of time t. The consistency relation
V̇ =
∂V (φ, ξ)
∂V (φ, ξ) ˙
(t)φ̇ +
(t)ξ
∂φ
∂ξ
(2.14)
is respected.
Before starting the construction of potentials for particular cosmological evolutions, it is useful to consider some mathematical aspects of the problem of
reconstruction of a function of two variables in general terms.
2.2.1. Reconstruction of the function of two variables,
which in turn depends on a third parameter
Let us consider the function of two variables F (x, y) defined on a curve, parameterized by t. Suppose that we know the function F (t) and its partial derivatives
as functions of t:
F (x(t), y(t)) = F (t) ,
(2.15)
∂F
∂F (x(t), y(t))
=
(t) ,
∂x
∂x
∂F (x(t), y(t))
∂F
=
(t) .
∂y
∂y
These three functions should satisfy the consistency relation
Ḟ (t) =
∂F
∂F
(t)ẋ +
(t)ẏ .
∂x
∂y
(2.16)
(2.17)
(2.18)
As a simple example we can consider the curve
x(t) = t, y(t) = t2 ,
20
(2.19)
2.2. The system of equations for two-field cosmological models
while
F (t) = t2 ,
(2.20)
∂F
= t2 ,
∂x
∂F
=t,
∂y
(2.21)
(2.22)
and equation (2.18) is satisfied.
Thus, we would like to reconstruct the function F (x, y) having explicit expressions in right-hand side of equations (2.15)– (2.17). This reconstruction is not
unique. We shall begin the reconstruction process taking such simple ansatzes
as
F1 (x, y) = G1 (x) + H1 (y) ,
(2.23)
F2 (x, y) = G2 (x)H2 (y) ,
(2.24)
F3 (x, y) = (G3 (x) + H3 (y))α .
(2.25)
The assumption (2.23) immediately implies
∂G1
∂F1
=
,
∂x
∂x
(2.26)
∂F1
∂H1
=
.
∂y
∂y
(2.27)
Therefrom one obtains
Z
x
G1 (x) =
Z
H1 (y) =
Hence
Z
F1 (x, y) =
x
y
∂F1
(t(x0 ))dx0 ,
∂x0
(2.28)
∂H1
(t(y 0 ))dy 0 .
0
∂y
(2.29)
∂F1
(t(x0 ))dx0 +
0
∂x
Z
y
∂H1
(t(y 0 ))dy 0 .
0
∂y
(2.30)
For an example given by equations (2.19)–(2.22) the function F1 (x, y) is
Z x
Z y
Z x
Z yp
2 0
0
0
0
02
0
F1 (x, y) =
t (x )dx +
t(y )dy =
x dx ±
y 0 dy 0 .
(2.31)
Explicitly

1

 x3 + 2y 3/2 , x > 0 ,
F1 (x, y) = 3

 1 x3 − 2y 3/2 , x < 0 .
3
(2.32)
21
2. Two-field cosmological models
Similar reasonings give for the assumptions (2.24),and (2.25) correspondingly
Z Z 1 ∂F
1 ∂F
F2 (x, y) = exp
(t(x))dx +
(t(y))dy ,
(2.33)
F ∂x
F ∂y
Z
F3 (x, y) =
dx
α
α
Z
1
1
dy
∂F
∂F
−1
−1
Fα
(t(x)) +
Fα
(t(y))
. (2.34)
∂x
α
∂y
For our simple example (2.19)–(2.22) the functions F2 (x, y) and F3 (x, y) have
the form
F2 (x, y) = xy ,
(2.35)

α
1 3/α
3/2α


, x>0,
x
+
2y

3
F3 (x, y) = (2.36)
α

1 3/α

3/α
3/2α

x (−) 2y
, x<0.
3
Thus, we have seen that the same input of “time” functions (2.20)–(2.22) on the
curve (2.19) produces quite different functions of variables x and y.
Naturally, one can introduce many other assumptions for reconstruction of
F (x, y). For example, one can consider linear combinations of x and y as functions of the parameter t and decompose the presumed function F as a sum or a
product of the functions of these new variables.
Now I present a way of constructing the whole family of solutions starting
from a given one. Let us suppose that we have a function F0 (x, y) satisfying all
the necessary conditions. Let us take an arbitrary function
t(x)
,
(2.37)
f (x, y) = f
t(y)
which depends only on the ratio t(x)/t(y). We require also
f (1) = 1 ,
f 0 (1) = 0 ,
(2.38)
i.e. the function reduces to unity and its derivative vanishes on the curve
(x(t), y(t)). Then it is obvious that the function
t(x)
F (x, y) = F0 (x, y)f
(2.39)
t(y)
is also a solution. This permits us to generate a whole family of solutions,
depending on a choice of the function f . Moreover, one can construct other
solutions, adding to the function F (x, y) a term proportional to (t(x) − t(y))2 .
22
2.2. The system of equations for two-field cosmological models
2.2.2. Cosmological applications, an evolution “Bang to
Rip”
To show how this procedure works in Cosmology, we consider a relatively simple
cosmological evolution, which nevertheless is of particular interest, because it
describes the phantom divide line crossing. Let us suppose that the Hubble
variable for this evolution behaves as
h(t) =
A
,
t(tR − t)
(2.40)
where A is a positive constant. At the beginning of the cosmological evolution,
when t → 0 the universe is born from the standard Big Bang type cosmological
singularity, because h(t) ∼ 1/t. Then, when t → tR , the universe is superaccelerated, approaching the Big Rip singularity h(t) ∼ 1/(tR − t). Substituting
the function (2.40) and its time derivative into equations. (2.11) and (2.5) we
come to
2A(2t − tR )
,
(2.41)
φ̇2 − ξ˙2 = − 2
3t (tR − t)2
V (t) =
A(2t − tR + 3A)
.
3t2 (tR − t)2
(2.42)
For convenience let me choose also the parameter A as
A=
Then,
tR
,
3
(2.43)
h(t) =
tR
,
3t(tR − t)
(2.44)
V (t) =
2tR
.
9t(tR − t)2
(2.45)
and
Let us consider now a special choice of functions φ(t) and ξ(t) used already1
in [69]-[72],
r
4
tR − t
φ(t) = − arctanh
,
(2.46)
3
tR
√
2
t
ξ(t) =
ln
.
(2.47)
3
tR − t
1
Notice that the origin of two scalar fields has been associated in [69]-[72] with a non-hermitian
complex scalar field theory and there a classical solution was found as a saddle point in
"double" complexification. I will discuss in more detail the link between non-hermiticity
and phantom fields in chapter 4.
23
2. Two-field cosmological models
The derivatives of the potential with respect to the fields φ and ξ could be found
from the Klein-Gordon equations (2.12) and (2.13):
√
2 2tR
∂V
= ξ¨ + 3hξ˙ =
,
(2.48)
∂ξ
3 t(tR − t)2
√
∂V
tR
= −φ̈ − 3hφ̇ = −
.
(2.49)
∂φ
t(tR − t)3/2
We can obtain also the time parameter as a function of φ or ξ:
tR
,
cosh (−3φ/4)
(2.50)
tR
√
.
exp(−3ξ/ 2) + 1
(2.51)
t(φ) =
t(ξ) =
2
Now we can make a hypothesis about the structure of the potential V (ξ, φ):
V2 (ξ, φ) = G(ξ)H(φ) .
(2.52)
Applying the technique described in the subsection A, we can get G(ξ):
Z 1 ∂V
ln G(ξ) =
(t(ξ))dξ
(2.53)
V ∂ξ
√
Z
9t(ξ)(t(ξ) − tR )2 2
2tR
=
dξ
2tR
3 t(ξ)(tR − t(ξ))2
√
= 3 2ξ ,
and
√
G(ξ) = exp(3 2ξ) .
(2.54)
To find H(φ) one can use the analogous direct integration, but we preferred to
implement a formula
V (t(φ)
H(φ) =
,
(2.55)
G(ξ(t(φ)))
which gives
2tR
H(φ(t)) = 3 ,
(2.56)
9t
and hence,
2
(2.57)
H(φ) = 2 cosh6 (−3φ/4) .
9tR
Finally,
V2 (ξ, φ) =
24
√
2
6
cosh
(−3φ/4)
exp(3
2ξ) .
9t2R
(2.58)
2.2. The system of equations for two-field cosmological models
Here we have reproduced the potential studied in [69]-[72].
Making the choice
V1 (ξ, φ) = G(ξ) + H(φ),
(2.59)
we derive
√
√
√
√
1
2
1
V1 (ξ, φ) = 2 − e−3ξ/ 2 + 3 2ξ + 2e3ξ/ 2 + e6ξ/ 2
3tR
3
3
4
2
sinh (−3φ/4) + sinh (−3φ/4) − 1
4
+ ln sinh (−3φ/4) .
+
sinh2 (−3φ/4)
(2.60)
Now, we can make another choice of the field functions φ(t) and ξ(t), satisfying
the condition (2.41):
√
2
t
φ(t) =
ln
,
(2.61)
3
tR − t
r
4
t
ξ(t) = arctanh
.
(2.62)
3
tR
The time parameter t is a function of fields is
t(φ) =
tR
√
,
exp(−3φ/ 2) + 1
(2.63)
tR
.
(2.64)
tanh (3ξ/4)
Looking for the potential as a sum of functions of two fields as in equation
(2.59) after lengthy but straightforward calculations one comes to the following
potential:
h
V1 (ξ, φ) = 3t22 1 + 23 sinh4 (3ξ/4) + 3 sinh2 (3ξ/4) − 3 sinh21(3ξ/4)
R
√
√
+2 ln sinh2 (3ξ/4) + 23 exp(−3φ/ 2) − 3 2φ
√
√ −2 exp(3φ/ 2) − 31 exp(φ/ 2) .
(2.65)
t(ξ) =
2
Similarly for the potential designed as a product of functions of two fields
(2.52) we obtain
V2 (ξ, φ) =
√
2
2
2
2φ) .
sinh
(3ξ/4)
cosh
(3ξ/4)
exp(−3
9t2R
(2.66)
One can make also other choices of functions φ(t) and ξ(t) generating other
potentials, but I shall not do it here, concentrating instead on the qualitative
and numerical analysis of two toy cosmological models described by potentials
(2.65) and (2.58).
25
2. Two-field cosmological models
2.3. Analysis of cosmological models
It is well known [85] that for the qualitative analysis of the system of cosmological
equations it is convenient to present it as a dynamical system, i.e. a system of
first-order differential equations. Introducing the new variables x and y we can
write

φ̇ = x ,




 ξ˙ = y ,
q
2
2
(2.67)
,
ẋ = −3 sign(h) x x2 − y2 + V (ξ, φ) − ∂V

∂φ

q


 ẏ = −3 sign(h) y x2 − y2 + V (ξ, φ) − ∂V .
2
2
∂ξ
Notice that the reflection
x → −x ,
y → −y ,
t → −t
(2.68)
transforms the system into one describing the cosmological evolution with the
opposite sign of the Hubble parameter. The stationary points of the system
(2.67) are given by
x=0, y=0,
∂V
∂V
=0,
=0.
∂φ
∂ξ
(2.69)
2.3.1. Model I
In this subsection I shall analyze the cosmological model with two fields – standard scalar and phantom – described by the potential (2.65). For this potential
the system of equations (2.67) reads












φ̇ = x ,
ξ˙ = y ,
ẋ = −3 sign(h) x
√
+ 23t22
R
q
x2
2
−
y2
+ 9t22
2
R
6
√
sinh2 (3ξ/4) cosh6 (3ξ/4)e−3
√
−3 2φ
2φ
sinh2 (3ξ/4) cosh (3ξ/4)e
,


q

√
2

2


ẏ = −3 sign(h) y x2 − y2 + 9t22 sinh2 (3ξ/4) cosh6 (3ξ/4)e−3 2φ


R
√

5

(3ξ/4) 1
2

+ sinh(3ξ/4)3tcosh
cosh
(3ξ/4) + sinh2 (3ξ/4) e−3 2φ .
2
3
(2.70)
R
It is easy to see that there are stationary points
φ = φ0 , ξ = 0 , x = 0 , y = 0 ,
26
(2.71)
2.3. Analysis of cosmological models
where φ0 is arbitrary. For these points the potential and hence the Hubble
variable vanish. Thus, we have a static cosmological solution. We should study
the behavior of our system in the neighborhood of the point (2.71) in linear
approximation:


φ̇ = x ,


 ξ˙ = y ,
(2.72)
ẋ = 0 ,

√


 ẏ = + ξ2 e−3 2φ0 .
4t
R
One sees that the dynamics of φ in this approximation is frozen and hence we
can focus on the study of the dynamics of the variables ξ, y. The eigenvalues of
the corresponding subsystem of two equations are
√
λ1,2
e−3φ0 /
=∓
2tR
2
.
(2.73)
These eigenvalues are real and have opposite signs, so one has a saddle point in
the plane (ξ, y) and this means that the points (2.71) are unstable.
One can make another qualitative observation. Freezing the dynamics of ξ
independently of φ, namely choosing y = 0, ξ = 0, which implies also ẏ = ξ¨ = 0,
one has the following equation of motion for φ:
φ̈ + 3hφ = 0 .
(2.74)
Equation (2.74) is nothing but the Klein-Gordon equation for a massless scalar
field on the Friedmann background, whose solution is
√
2
ln t ,
(2.75)
φ(t) =
3
and which gives a Hubble variable
h(t) =
1
.
3t
(2.76)
This is an evolution of the flat Friedmann universe, filled with stiff matter with
the equation of state p = ε. It describes a universe born from the Big Bang
singularity and infinitely expanding. Naturally, for the opposite sign of the
Hubble parameter, one has the contracting universe ending in the Big Crunch
cosmological singularity.
Now, I describe some results of numerical calculations to have an idea about
the structure of the set of possible cosmological evolutions coexisting in the
model under consideration. We have carried out two kinds of simulations. First,
we have considered neighborhood of the plane y, ξ with the initial conditions on
27
2. Two-field cosmological models
y
4
2
Ξ
-4
-2
2
4
-2
-4
Figure 2.1.: An example of section of the 4d phase space obtained with numerical
calculations. We can see a series of trajectories corresponding to
different choices of the initial conditions. Every initial condition in
the graphic is chosen on the dashed “ellipse” centered in the origin
and is emphasized by a colored dot. A “saddle point like” structure
of the set of the trajectories clearly emerges.
the field φ such that φ(0) = 0, φ̇(0) = 0 (see figure 2.1). The initial conditions for
the phantom field were chosen in such a way that the sum of absolute values of
the kinetic and potential energies were fixed. Then, running the time back and
forward we have seen that the absolute majority of the cosmological evolutions
began at the singularity of the “anti-Big Rip” type (figure 2.2). Namely, the
initial cosmological singularity were characterized by an infinite value of the
cosmological radius and an infinite negative value of its time derivative (and
also of the Hubble variable). Then the universe squeezes, being dominated by
the phantom scalar field ξ. At some moment the universe passes the phantom
divide line and the universe continues squeezing but with ḣ < 0. Then it achieves
the minimal value of the cosmological radius and an expansion begins. At some
moment the universe undergoes the second phantom divide line crossing and its
expansion becomes super-accelerated culminating in an encounter with a Big
Rip singularity. Apparently this scenario is very different from the standard
cosmological scenario and from its phantom version Bang-to-Rip, which has
played a role of an input in the construction of our potentials. The second
procedure, which we have used is the consideration of trajectories close to our
initial trajectory of the Bang-to-Rip type. The numerical analysis shows that
this trajectory is unstable and the neighboring trajectories again have anti-Big
Rip – double crossing of the phantom line – Big Rip behavior described above.
28
2.3. Analysis of cosmological models
h
1.0
0.5
t
-100
-80
-60
-40
-20
0
-0.5
-1.0
Figure 2.2.: Typical behavior of the Hubble parameter for the model I trajectories. The presence of two stationary points (namely a maximum followed by a minimum) indicates the double crossing of the phantom
divide line. Both the initial and final singularities are characterized
by a Big Rip behavior, the first is a contraction and the second is
an expansion.
29
2. Two-field cosmological models
However, it is necessary to emphasize that a small subset of the trajectories of
the Bang-to-Rip type exist, being not in the vicinity of our initial trajectory.
2.3.2. Model II
In this subsection we shall study the cosmological model with the potential
(2.58). Now the system of equations (2.67) looks like


φ̇ = x ,




ξ˙ = y ,


q

√
2

y2
6

 ẋ = −3 sign(h) x x2 − 2 + 9t22 cosh (3φ/4) exp{3 2ξ}
R √
5
(2.77)
1
− t2 cosh (3φ/4) sinh(3φ/4)e3 2ξ ,


R
q

√

y2
6
x2
2


cosh
(3φ/4)
exp{3
ẏ
=
−3sign(h)y
−
+
2ξ}
2

2
2
9tR

√
√



+ 23t22 cosh6 (3φ/4)e3 2ξ .
R
Notice that the potential (2.58) has an additional reflection symmetry
V2 (ξ, φ) = V2 (ξ, −φ) .
(2.78)
This provides the symmetry with respect to the origin in the plane (φ, x). The
system (2.77) has no stationary points. However, there is an interesting point
φ=0, x=0,
(2.79)
which freezes the dynamics of φ and hence, permits to consider independently
the dynamics of ξ and y, described by the subsystem
(
ξ˙ = y ,
q 2
√
√
√
(2.80)
ẏ = −3 sign(h) y − y2 + 9t22 e3 2ξ + 23t22 e3 2ξ .
R
R
Apparently, the evolution of the universe is driven now by the phantom field and
is subject to super-acceleration.
In this case the qualitative analysis of the differential equations for ξ and y,
confirmed by the numerical simulations gives a predictable result: being determined by the only phantom scalar field the cosmological evolution is characterized by the growing positive value of h. Namely, the universe begins its
evolution from the anti-Big Rip singularity (h = −∞) then h is growing passing
at some moment of time the value h = 0 (the point of minimal contraction of
the cosmological radius a(t)) and then expands ending its evolution in the Big
Rip cosmological singularity (h = +∞).
30
2.4. Conclusions
Another numerical simulation can be done by fixing initial conditions for the
phantom field as ξ(0) = 0, y(0) = 0 (see figure 2.3). Choosing various values of
the initial conditions for the scalar field φ(0), x(0) around the point of freezing
φ = 0, x = 0 we found two types of cosmological trajectories:
1. The trajectories starting from the anti-Big Rip singularity and ending in the
Big Rip after the double crossing of the phantom divide line. These trajectories
are similar to those discussed in the preceding subsection for the model I.
2. The evolutions of the type Bang-to-Rip.
Then we have carried out the numerical simulations of cosmological evolutions,
choosing the initial conditions around the point of the phantomization point with
the coordinates
φ(0) = 0 ,
√
2
,
x(0) =
3
4
1
ξ(0) = arctanh √ ,
3
2
√
2
y(0) =
.
(2.81)
3
This analysis shows that in contrast to the model I, here the standard phantomization trajectory is stable and the trajectories of the type Bang-to-Rip are
not exceptional, though less probable then those of the type anti-Big Rip to Big
Rip.
2.4. Conclusions
In the paper [1] we have considered the problem of reconstruction of the potential
in a theory with two scalar fields (one standard and one phantom) starting with
a given cosmological evolution. It is known ( see e.g. [73, 76]) that in the case
of the only scalar field this potential is determined uniquely as well as the initial
conditions for the scalar field, reproducing the given cosmological evolution.
Changing the initial conditions, one can find a variety of cosmological evolutions,
sometimes qualitatively different from the “input” one (see e.g. [76]). In the case
of two fields the procedure of reconstruction becomes much more involved. As
we have shown here, there is a huge variety of different potentials reproducing
the given cosmological evolution (a very simple one in the case, which we have
explicitly studied here). Every potential entails different cosmological evolutions,
depending on the initial conditions.
It is interesting that the existence of different dynamics of scalar fields corresponding to the same evolution of the Hubble parameter h(t) can imply some
31
2. Two-field cosmological models
x
20
15
Φ
1
-1
-10
-20
Figure 2.3.: An example of section of the 4D phase space for the model II obtained with numerical calculations. We can see a family of different
trajectories corresponding to different choices of the initial conditions. Every initial condition in the graphic is chosen on the dashed
“ellipse”. The origin represents the point of freezing.
observable consequences connected with the possible interactions of the scalar
fields with other fields. In this case, the time dependence of the scalar fields considered above can directly affect physically observable quantities. Indeed, some
results in this direction are presented in chapter 3, devoted to the explanation
of the paper [2].
32
3. Two-field models and cosmic
magnetic fields
The present chapter is intended to present the content of the paper [2]. Starting
from the models studied in the preceding chapter, we couple one of the fields to a
cosmic electromagnetic field, whose existence is experimentally confirmed. The
idea is to try and find a discrimination between cosmological models presenting
the same background evolution, by means of the effects of the coupling with other
(observable) fields.
3.1. Introduction to the content of the paper [2]
In the preceding chapter I have shown the procedure of reconstruction of the
potential in two-field models. It was shown that there exists a huge variety of
potentials and time dependences of the fields realizing the same cosmological
evolution. Some concrete examples were considered, corresponding to the evolution beginning with the standard Big Bang singularity and ending in the Big
Rip singularity [57, 58].
One can ask oneself: what is the sense of studying different potentials and
scalar field dynamics if they imply the same cosmological evolution? The point
is that the scalar and phantom field can interact with other fields and influence
not only the global cosmological evolution but also other observable quantities.
One of the possible effects of the presence of normal and phantom fields could
be their influence on the dynamics of cosmic magnetic fields. The problem
of the origin and of possible amplification of cosmic magnetic fields is widely
discussed in the literature [86, 87, 88]. In particular, the origin of such fields can
be attributed to primordial quantum fluctuations [89, 90, 91] and their further
evolution can be influenced by hypothetic interaction with pseudo-scalar fields
breaking the conformal invariance of the electromagnetic field [92]-[96], [97].
In the paper under consideration we analyzed the evolution of magnetic fields
created as a result of quantum fluctuations, undergoing the inflationary period
with unbroken conformal invariance and beginning the interaction with pseudoscalar or pseudo-phantom fields after exiting the inflation and entering the Big
Bang expansion stage, which is a part of the Bang-to-Rip scenario described in
33
3. Two-field models and cosmic magnetic fields
the preceding chapter. We used different field realizations of this scenario and
we shall see how the dynamics of the field with negative parity influences the
dynamics of cosmic magnetic fields.
What follows is so organized: in section 3.2 I very briefly recall the Bang-toRip scenario and the two models that will be considered (the same of chapter 2;
in section 3.3 I introduce the interaction of the fields (phantom or normal) with
an electromagnetic field and write down the corresponding equations of motion;
in section 3.4 I describe the numerical simulations of the evolution of magnetic
fields and present the results of these simulations; the last section 3.5 is devoted
to concluding remarks.
3.2. Cosmological evolution and (pseudo)-scalar
fields
I shall consider a spatially flat Friedmann universe with the FLRW metric (1.2).
Notice that the physical distance is obtained by multiplying dl by the cosmological radius a(t). We would like to consider the cosmological evolution characterized by the following time dependence of the Hubble variable h(t), where,
I recall, “dot” denotes the differentiation with respect to the cosmic time t1 :
h(t) =
tR
.
3t(tR − t)
(3.1)
I called this kind of scenario Bang-to-Rip in section 2.2.2: at small values of t the
universe expands according to power law: a ∼ t1/3 while at t → tR the Hubble
variable explodes and one encounters the typical Big Rip type singularity. (The
factor one third in (3.1) was chosen for calculation simplicity).
In the paper we are considering [2] we kind of continued the work presented in
the preceding chapter, or equivalently in [1]. Specifically we considered precisely
the two models of the preceding chapter (see sections 2.3.1, 2.3.2) and coupled
them with a magnetic field.
We have seen in chapter 2 that, analyzing the Friedmann equation (1.8a)2 ,
for two-field models there is huge variety of potentials V (φ, ξ) realizing a given
1
2
Cfr. equation (2.44).
I use here, in accordance with [1, 2], the following system of units ~ = 1, c = 1 and 8πG = 3.
In this system the Planck mass mP , the Planck length lP and the Planck time tP are equal
to 1. Then when we need to make the transition to the “normal”, say, cgs units, we should
simply express the Planck units in terms of the cgs units. In all that follows we tacitly
assume that all our units are normalized by the proper Planck units. Thus, the scalar field
entering
as an argument into the dimensionless expressions should be divided by the factor
p
mP /tp .
34
3.2. Cosmological evolution and (pseudo)-scalar fields
evolution, in contrast to models with one scalar field. Moreover, besides the
freedom in the choice of the potential, one can choose different dynamics of the
fields φ(t) and ξ(t) realizing the given evolution. For simplicity I repeat here
the main ingredients of the two models of 2.3.1, 2.3.2: The first potential is (cfr.
with (2.66))
√
2
(3.2)
VI (ξ, φ) = 2 cosh6 (−3φ/4) exp(3 2ξ) .
9tR
and the fields
r
tR − t
4
φ(t) = − arctanh
,
3
tR
ξ(t) =
√
2
t
ln
.
3
tR − t
(3.3)
(3.4)
(The expression for the potential should be multiplied by the factor m
pP /(clP ),
while the expressions for the fields φ(t) and ξ(t) should be multiplied by mP /tP .
For the relation between Planck units and cgs ones see e.g. [98]). If we would
like to substitute one of these two fields by the pseudo-scalar field, conserving
the correct parity of the potential, we can choose only the field φ because the
potential VI is even with respect to φ, but not with respect to ξ. In what follows
I shall call the model with the potential (3.2), the pseudo-scalar field (3.3) and
the scalar phantom (3.4) model I.
Consider another potential (cfr. with (2.58))
VII (ξ, φ) =
with the fields
√
2
2
2
2φ) .
sinh
(3ξ/4)
cosh
(3ξ/4)
exp(−3
9t2R
(3.5)
√
2
t
φ(t) =
ln
,
3
tR − t
(3.6)
r
4
t
ξ(t) = arctanh
.
3
tR
(3.7)
This potential is even with respect to the field ξ. Hence the model II is based
on the potential (3.5), the pseudo-phantom field (3.7) and the scalar field (3.6).
They will be the fields with the negative parity which couple to the magnetic
field.
35
3. Two-field models and cosmic magnetic fields
3.3. Post-inflationary evolution of a magnetic
field interacting with a pseudo-scalar or
pseudo-phantom fields
The action of an electromagnetic field interacting with a pseudo-scalar or pseudophantom field φ is
Z
√
1
S=−
d4 x −g(Fµν F µν + αφFµν F̃ µν ) ,
(3.8)
4
where α is an interaction constant and the dual electromagnetic tensor F̃ µν is
defined as
1
(3.9)
F̃ µν ≡ E µνρσ Fρσ ,
2
where
√
1
µνρσ ,
(3.10)
Eµνρσ ≡ −g µνρσ , E µνρσ ≡ − √
−g
with the standard Levi-Civita symbol
µνρσ = [µνρσ] , 0123 = +1 .
(3.11)
Variating the action (3.8) with respect to the field Aµ we obtain the field
equations
∇µ F µν = −α∂µ φF̃ µν ,
(3.12)
∇µ F̃ µν = 0 .
(3.13)
The Klein-Gordon equation for the pseudo-scalar field is
∇µ ∇µ φ +
∂V
= −αFµν F̃ µν .
∂φ
(3.14)
The Klein-Gordon equation for the pseudo-phantom field (which is the one that
couples with the magnetic field in the model II) differs from equation (3.14) by
change of sign in front of the kinetic term. In what follows we shall neglect the
influence of magnetic fields on the cosmological evolution, i.e. we will discard
the electromagnetic coupling in equation (3.14).
If one wants to rewrite these formulæ in terms of the three-dimensional quantities (i.e. the electric and magnetic fields) one can find the expression of the
electromagnetic tensor in a generic curved background, starting from a locally
flat reference frame — in which it is well known the relation between electromagnetic fields and F — and using a coordinate transformation. It is easy to
36
3.3. Interaction between magnetic and pseudo-scalar/phantom field
see that we have, for the metric ((1.2)):

F µν

0 −aE1 −aE2 −aE3
1  aE1
0
B3
−B2 
 .
= 2

aE2 −B3
0
B1 
a
aE1
B2
−B1
0
(3.15)
~ and B
~
The field equations (3.12), (3.13) and (3.14) rewritten in terms of E
become
~ ·E
~ = −α∇φ
~ ·B
~ .
∇
(3.16a)
~ −∇
~ × (aB)
~ = −α[∂0 φ(a2 B)
~ − ∇φ
~ × (aE)]
~ ,
∂0 (a2 E)
(3.16b)
~ −∇
~ × (aE)
~ =0,
∂0 (a2 B)
(3.16c)
~ ·B
~ =0.
∇
(3.16d)
For a spatially homogeneous pseudo-scalar field equations (3.16a) and (3.16b)
look like
~ ·E
~ =0,
∇
(3.17a)
~ −∇
~ × (aB)
~ = −α∂0 φ(a2 B)
~ .
∂0 (a2 E)
(3.17b)
~ from (3.16c)
Taking the curl of (3.17b) and substituting into it the value of E
we obtain
~ + h(t)∂0 (a2 B)
~ −
∂02 (a2 B)
~
α
∆(3) (a2 B)
~ × (a2 B)
~ =0,
− ∂0 φ∇
2
a
a
(3.18)
where ∆(3) stands for the three-dimensional Euclidean Laplacian operator.
Let me introduce
~ x, t)
F~ (~x, t) ≡ a2 (t)B(~
(3.19)
and its Fourier transform
1
F~ (~k, t) =
(2π)3/2
Z
~
e−ik·~x F~ (~x, t)d3 x .
(3.20)
~ is an observable magnetic field entering into the expression for
Here the field B
the Lorentz force. The field equation for F~ (~k, t) is
¨
˙
F~ (~k, t) + h(t)F~ (~k, t) +
" #
2
k
iα ~
− φ̇k× F~ (~k, t) = 0 ,
a
a
(3.21)
37
3. Two-field models and cosmic magnetic fields
where “dot” means the time derivative. This last equation can be further sim√
plified: assuming ~k = (k, 0, 0) and defining the functions F± ≡ (F2 ± iF3 )/ 2
one arrives to
#
" 2
k
k
± α φ̇ F± = 0 ,
(3.22)
F̈± + hḞ± +
a
a
where I have omitted the arguments k and t.
Assuming that the electromagnetic field has a quantum origin (as all the fields
in the cosmology of the early universe [99]) the modes of this field are represented
by harmonic oscillators. Considering their vacuum fluctuations responsible for
their birth we can neglect the small breakdown of the conformal symmetry and
treat them as free. In conformal coordinates (η, ~x) such that the Friedmann
metric has the form
ds2 = a2 (η)(dη 2 − dl2 ) ,
(3.23)
the electromagnetic potential Ai with the gauge choice A0 = 0, ∂j Aj = 0 satisfies
the standard harmonic oscillator equation of motion
Äi + k 2 Ai = 0 .
(3.24)
√
2k, while the
Hence the initial amplitude of the field A
behaves
as
A
=
1/
i
i
p
initial amplitude of the functions F is k/2. The evolution of the field F
during the inflationary period was described in [97], where it was shown that the
growing solution at the end of inflation is amplified by some factor depending
on the intensity of the interaction between the pseudo-scalar field and magnetic
field.
Here we are interested in the evolution of the magnetic field interacting with
the pseudo-scalar field after inflation, where our hypothetic Bang-to-Rip scenario takes place. More precisely, I would like to see how different types of
scalar-pseudo-scalar potentials and field dynamics providing the same cosmological evolution could be distinguished by their influence on the evolution of
the magnetic field. We do not take into account the breaking of the conformal
invariance during the inflationary stage and all the effects connected with this
breakdown will be revealed only after the end of inflation and the beginning of
the Bang-to-Rip evolution. This beginning is such that the value of the Hubble
parameter, characterizing this evolution is equal to that of the inflation, i.e.
h(t0 ) =
tR
= hinflation ' 1033 s−1 .
3t0 (tR − t0 )
(3.25)
In turn, this implies that we begin evolution at the time moment of the order of
10−33 s.
38
3.4. Generation of magnetic fields: numerical results
I shall consider both the components F+ and F− and we shall dwell on the
scenarios I and II described in the preceding section 3.2. Anyway, our assumption regarding the initial conditions for equation (3.22) can be easily modified in
order to account for the previous possible amplification of primordial magnetic
fields as was discussed by [97]. Thus, all estimates for the numerical values of the
magnetic fields in today’s universe should be multiplied by some factor corresponding to the amplification of the magnetic field during the inflationary stage.
Hence, our results refer more to differences between various models of a post
inflationary evolution than of the real present values of magnetic fields, whose
amplification might be also combined effect of different mechanism [86, 87, 88],
[92]-[96], [89, 90, 91], [97].
3.4. Generation of magnetic fields: numerical
results
In this section I present the results of numerical simulations for the two models
I and II. In these models, the equations of motion for the modes F± (k, t) (3.22)
reads3 :
" 2/3
1/3 #
t
−
t
t
−
t
tR Ḟ±
R
R
±α
k Φ̇ F± = 0 ,
(3.26)
+ k2
F̈± +
3t(tR − t)
t
t
where Φ stands for the scalar field φ in the model I and for the phantom ξ in
the model II, so that
√
√
2
tR
2
tR
Φ̇I = φ̇ = √
; Φ̇II = ξ˙ = √
.
(3.27)
3 t tR − t
3 t(tR − t)
Equation (3.26) is solved for different values of the wave number k and the
coupling parameter α. (The parameter α has the dimensionality inverse with
respect to that of the scalar field; the wave number k has the dimensionality of
inverse length; the time tR = 1017 s). Qualitatively let me remark that in (3.22)
the coupling term influence becomes negligible after some critical period. After
that the magnetic fields in our different scenarios evolve as if the parameter
α in (3.22) had been put equal to zero. Indeed, it can be easily seen that the
interaction term vanishes with the growth of the cosmological radius a. Then the
3
The reader can easily verify that this equation is obtained imposing the normalization
a|today = 1, where today-time is taken to be near the crossing of the phantom divide
line, i.e. at t ' tR /2. This implies in turn that at the beginning of the “Bang-to-Rip”
evolution the cosmological radius is a(t0 ) ' 10−17 .
39
3. Two-field models and cosmic magnetic fields
distinction between the two models is to be searched in the early time behavior
of the field evolution.
Noting that in both our models I and II the time derivative Φ̇ is positive, by
inspection of the linear term in equation (3.22) we expect the amplification to be
mainly given for the mode F− provided the positive sign for α is chosen; so we
will restrict our attention on F− . We can also argue that the relative strength
of the last two terms in the left-hand side of (3.26) is crucial for determining
the behavior of the solution: when the coupling term prevails (I remark that
we are talking about F− so this term is negative in our models) then we expect
an amplification, while when the first term dominates we expect an oscillatory
behavior. For future reference it is convenient to define
1/3
tR − t
k
k
=
,
(3.28)
A(t; Φ) ≡
t
a(t)αΦ̇
αΦ̇
which is just the ratio between the last two terms in the left-hand side of equation (3.26).
Indeed our numerical simulations confirm these predictions. Let us consider
the model I with α = 1(tP /mP )1/2 and k = 10−55 lP−1 , where lP is the Planck
length. Such a value of the wave number k corresponds to the wave length of
1kpc at the present moment. We obtain an early-time amplification of about
2 orders of magnitude, with the subsequent oscillatory decay. Notice that the
parameter A in this model at the beginning is very small: this corresponds to
the dominance of the term proportional to Φ̇ and, hence, to the amplification
of the field F− . At the time scale of the order of 1051 tP , where tP is the Planck
time, this regime turns to that with big values of A where the influence of the
term proportional to Φ̇ is negligible.
For the same choice of the parameters α and k in the model II the amplification is absent.
In figure 3.1 you see the time dependence of the function A for the model I
for the values of α = 1(tP /mP )1/2 and k = 10−55 lP−1 chosen above. The figure 3.2
manifests the amplification of the magnetic field in the model I.
Naturally the effect of amplification of the magnetic field grows with the coupling constant α and diminishes when the wave number k increases. In figure 3.3
are displayed the results for the case of α = 100(tP /mP )1/2 , which is admittedly
extreme and possibly non realistic, but good for illustrative purposes. Here the
amplification is more evident and extends for a longer time period.
In [2], we also tried to make some estimates of the cosmic magnetic fields in
the universe today, using the correlation functions. The correlation function for
the variable F is defined as the quantum vacuum average
Gij (t, ~x − ~y ) = h0|Fi (t, ~x)Fj (t, ~y )|0i
40
(3.29)
3.4. Generation of magnetic fields: numerical results
A
1.4
1.2
1.0
0.8
0.6
0.4
0.2
2. ´ 1051
4. ´ 1051
6. ´ 1051
8. ´ 1051
t
1. ´ 1052
Figure 3.1.: Plot of the ratio A for the model I, with the parameter choice
α = 1(tP /mP )1/2 , k = 10−55 lP−1 . It can be easily seen that at a time
scale of order 1051 tP the ratio becomes greater than 1.
ÈFÈ
2.5 ´ 10-26
2. ´ 10-26
1.5 ´ 10-26
1. ´ 10-26
5. ´ 10-27
1. ´ 1052
2. ´ 1052
3. ´ 1052
t
4. ´ 1052
Figure 3.2.: Plot of the time evolution of the absolute value of the complex field
F (given in Planck units) in model I with the parameter choice
α = 1(tP /mP )1/2 , k = 10−55 lP−1 . The behavior, as said above in the
text, consists in an amplification till a time of order 1051 tP , after
which the oscillations begin.
41
3. Two-field models and cosmic magnetic fields
ÈFÈ
6. ´ 1043
4. ´ 1043
2. ´ 1043
2. ´ 1054
4. ´ 1054
6. ´ 1054
8. ´ 1054
t
1. ´ 1055
Figure 3.3.: Plot of the time evolution of the absolute value of the complex field
F (given in Planck units) in model I with the parameter choice
α = 100(tP /mP )1/2 , k = 10−55 lP−1 . The behavior, consists in an
amplification till a time of order 1054 tP , after which the oscillations
begin.
and can be rewritten as
Gij (t, ~x − ~y ) =
Z
d3 k −i~k·(~x−~y)
e
Fi (t, ~k)Fj∗ (t, ~k).
(2π)3
Integrating over the angles, we come to
Z 2
k dk sin(k|~x − ~y |)
Gij (t, ~x − ~y ) =
Fi (t, k)Fj∗ (t, k).
2π 2 k|~x − ~y |
(3.30)
(3.31)
To estimate the integral (3.31) we notice that the main contribution to it comes
from the region where k ≈ 1/|~x − ~y | (see e.g.[98]) and it is of order
1
|Fi (t, 1/L)|2 ,
L3
where L = |~x − ~y |. In this estimation the amplification factor is
F (1/L) ,
p
1/L (3.32)
(3.33)
where the subscript i is not present since we have taken the trace over polarizations.
42
3.5. Conclusions
Now we are in a position to give numerical values for the magnetic fields at
different scales in the model I for different values of the coupling parameter α.
These values (see Table 3.1) correspond to three values of the coupling parameter
α (1,10 and 100 (tP /mP )1/2 ) 4 and to two spatial scales L determined by the
values of the wave number k. We did not impose some physical restrictions on
the value of α. It is easy to see that the increase of α implies the growth of the
value of the magnetic field B.
Let me stress once again that we ignored the effects of amplification of the
magnetic fields during inflation to focus on seizable effects during evolution.
Table 3.1.: The Table displays the values of the magnetic field B
corresponding to the chosen values of α and k. The length L
refers to the present moment when a = 1.
p
p
p
α = 1 tP /mP α = 10 tP /mP α = 100 tP /mP
k = 10−55 lP−1 (L = 1kpc)
B ∼ 10−67 G
B ∼ 10−60 G
B ∼ 100G
k = 10−50 lP−1 (L = 10−2 pc)
B ∼ 10−55 G
B ∼ 10−49 G
B ∼ 1013 G
Finally notice that our quantum “initial” conditions correspond to physical
magnetic fields which for presented values of k are Bin ∼ k 2 /a2 is equal to
10−34 G for k = 10−55 lP−1 and Bin ∼ 10−24 G for k = 10−50 lP−1 .
3.5. Conclusions
We have seen that the evolution of the cosmic magnetic fields interacting with
a pseudo-scalar (pseudo-phantom) field is quite sensitive to the concrete form of
the dynamics of this field in two-field models where different scalar field dynamics
and potentials realize the same cosmological evolution.
The sensitivity of the evolution of the magnetic field with respect to its helicity is confirmed, given the sign of the coupling constant α and that the Φ is
a monotonic function of time (as it is really so in our models). We gave also
some numerical estimates of the actual magnetic fields up to the factor of amplification of such fields during the inflationary period. The toy model of the
Bang-to-Rip evolution studied in this paper [2], cannot be regarded as the only
4
It is useful to remark that at this length scales values of α less than 1 make the coupling
with the cosmological evolution negligible.
43
3. Two-field models and cosmic magnetic fields
responsible for the amplification of cosmic magnetic fields implying their present
observable values. It rather complements some other mechanisms acting before.
However, the difference between cosmic magnetic fields arising in various models
(giving the same expansion law after the inflation) is essential. It may provide
a discriminating test for such models.
44
4. Phantom without phantom in
a PT-symmetric background
This chapter is devoted to presenting the content of the paper [3]. In nuce, the
idea is that if we put ourselves in the framework of P T symmetric Quantum
Theory we can have an effective phantom field which is stable to quantum fluctuations. “Effective” in the usual sense, i.e. the real field is a standard scalar
field, but becomes a phantom field once we ‘sit’ on a (specific) classical solution.
Before going into the details of the model presented in [3], let me briefly introduce the key ideas and concepts of P T symmetric Quantum Mechanics.
4.1. PT-symmetric Quantum Mechanics: a brief
introduction
P T symmetric Quantum Mechanics was born in 1998 in the seminal paper [100]
by Carl Bender and Stefan Boettcher. The motivating idea is very simple: how
to make sense of non-hermitian hamiltonians. Indeed it had been known for
long (since late 50’s) that some non-hermitian hamiltonians come out naturally
in some systems [101]-[105], but everybody where rather skeptical about their
physical sense.
However, since the 80’s, there where hints in the following direction: some
non-hermitian hamiltonians do have a real and bounded spectrum [106, 107].
You guess the point: Bender and collaborators rigorously proved since the very
beginning of the theory in 1998 that it is not necessary to require hermiticity
in order to have a real and bounded spectrum. We can weaken that axiom
of QM with the more physical requirement: the hamiltonian operator must be
P T symmetric.
A P T transformation is a combined transformation made of parity
(
P
x → −x
p → −p
(4.1)
45
4. Phantom without phantom in a PT-symmetric background
and time reversal
thus


x → x
T p → −p

 i → −i
(4.2)


 x → −x
PT p → p

 i → −i
(4.3)
we can think of it as a spacetime reflection.
P T symmetric hamiltonians do have real and bounded spectrum, thus they
have the usual physical interpretation as operators whose eigenvalues are the
energy levels of the system under investigation1 .
As Gell-Mann said (and as QM teaches us) “everything that is not forbidden is
compulsory”. This is the sort of principle that guided – together with the need of
dealing with non-hermitian hamiltonians – the flourishing of the P T symmetric
approach in the last decade or so.
Remarkably Bender and collaborators proved also another (even more) fundamental result [110]: in P T symmetric QM it is possible to have unitary evolution.
This is not at all a trivial result. Indeed the self-adjointness of the hamiltonian
is ‘responsible’ not only for the reality of the energy levels, but also – and I dare
say most importantly – for the unitarity of time evolution, i.e. the conservation
of probability in time. Indeed in standard QM unitarity is completely due to the
hermeticity of the hamiltonian2 : the evolution operator is U (t) = e−iHt and is
unitary as long as H = H † . I am not going in details in this respect, I only say
that it has been proved that it can be defined a proper inner product with respect
to which P T symmetric QM does have a unitary evolution [110, 111, 112].
Remark. One key point must be stressed: non-hermitian hamiltonians classically mean complex forces. This implies that we have to consider complex dynamical variables x and p. This is completely acceptable in the quantum world,
once you understand that this only means that the position and momentum operators are not observables in P T symmetric quantum theory. Observables must
have real spectrum, this has to be true indeed in any theory.
However, this is a problem in the corresponding classical limit. One has to
include somehow complex valued trajectories. Here, I must admit, I see a drawback of this approach. Frankly, I am not aware of any reasonable interpretation
1
Actually it is only a sub-class of P T symmetric operators that have this property. This
sub-class is said to have an un-broken P T symmetry. See [108, 109] for more details.
2
I am using here hermeticity and self adjointness as synonimous. Strictly speaking, they are
not, because of domain subtleties.
46
4.2. PT-symmetric oscillator
about it, and it seems to me that in the literature this point is quite often
underestimated.
One more thing to say is that, at present, no empirical clear and definitive
evidence of the existence of systems with complex hamiltonian has been found.
It is far beyond the scope of this thesis to discuss the details of this approach,
let alone the many insights that have been reached in these years in many respects. However, in the following section I present a class of very well studied
and archetypal complex P T symmetric hamiltonians, which I find particularly
instructive and also very useful to understand the mechanism that plays a central
role in the model studied in [3].
4.2. PT-symmetric oscillator
Take the hamiltonian
H = p2 + x2 (ix) ,
(4.4)
this is complex for 6= 0 and P T symmetric3 . This is a generalization of the standard harmonic oscillator. We shall call this class the P T symmetric oscillator,
as is sometimes done in the literature.
I will review here mainly the classical analysis of the hamiltonians (4.4), i.e.
the study of the trajectories of a point particle subject to the complex potential
x2 (ix) . Notice that the richness of the model arises precisely from considering
it in the complex x plane, as it is natural to do since we have complex forces.
The equation of motion is
d2 x
= 2i(2 + )(ix)1+ ,
dt2
(4.5)
that can be integrated, e.g. using the energy first integral H = E, to give
p
1 dx
= ± E + (ix)2+ ,
2 dt
(4.6)
where E is the constant value of the energy of the particle. We set for simplicity
E = 1; obviously this does not spoil the generality of the following considerations.
Let us analyze separately the cases = 0, 1, 2.
3
For < 0 this hamiltonian is said to have a broken P T symmetry, and the reality of the
spectrum is not guaranteed anymore [108, 109]. Thus we restrict to positive (or null) value
of .
47
4. Phantom without phantom in a PT-symmetric background
ℑ(x)
−1
+1
ℜ(x)
Figure 4.1.: Qualitative plot of the trajectories in the complex x plane of the
system with hamiltonian (4.4) with = 0.
=0
This is the standard harmonic oscillator. It has two turning points x = −1, +1,
both on the real axis. The trajectory depends on the choice of x(0). Here we
can choose an initial condition wherever we want on the complex x plane. The
solutions are all periodic of period 2π, and are all oscillating. If we start on the
real axis, then – as we know from the standard oscillator – we remain on the
real axis, otherwise we oscillate on ellipses on the complex plane (see figure 4.1).
Notice that the trajectories are P invariant (reflection with respect to the origin)
T invariant (reflection with respect to the real axis) and P T invariant (reflection
about the imaginary axis).
=1
In this case there are three turning points, namely the solution of the equation
ix3 = 1 ,
(4.7)
i.e. x− = e−5iπ/6 , x+ = e−iπ/6 and x0 = i (see figure 4.2). In this case we
have two classes of trajectories: periodic and non-periodic. Actually the nonperiodic trajectories are a null-measure subset of all the trajectories. Namely,
trajectories starting in x(0) = ie
x with x
e real and greater than or equal to 1, goes
up to to infinity on the imaginary axis. In other words, trajectories starting at
the x0 turning point or somewhere upper but always on the imaginary axis, are
48
4.2. PT-symmetric oscillator
ℑ(x)
x0
x−
x+
ℜ(x)
Figure 4.2.: Qualitative plot of the trajectories for the case = 1.
not periodic. All the other are periodic and have the same period, by virtue of
Cauchy’s theorem. Notice that the trajectories are manifestly P T -invariant, but
P and T symmetries taken separately are lost.
=2
This is the most interesting case. Notice that the hamiltonian (4.4) now reads
H2 = p2 − x4 ,
(4.8)
which is actually a real but unbounded hamiltonian. Being it real we can analyze
it on real trajectories, and we know that the system is completely unstable: the
particle will roll down the maximum in x = 0 going away at infinity. The energy
is doomed to fall to −∞. However, we can also expand our horizon and see
what happens in the complex plane. There are four (complex) turning points,
solutions of x4 = −1
±iπ/4
±3iπ/4
x±
, x±
,
(4.9)
1 = e
2 = e
two above and two under the real axis (see figure 4.3). The trajectories are
here divided in three families: one encircling the top turning points x+
1,2 , one
around the bottom turning points x−
and
one
stuck
on
the
real
axis
–
again
of
1,2
null measure on the whole set. The latter is of course the one we talked about
previously. Indeed it is unstable, in the sense that it is not periodic and goes to
infinity. So, here you are: while on standard basis you look to hamiltonian (4.8)
49
4. Phantom without phantom in a PT-symmetric background
ℑ(x)
x+
2
x+
1
ℜ(x)
x−
2
x−
1
Figure 4.3.: Qualitative plot of the trajectories for the case = 2
and think ‘unstable, good for nothing’, in the complex plain you’d better bet on
the periodicity/stability of the same system described by (4.8).
This last example shows the key point: a manifest instability can be ‘absorbed’
by the ‘complexity’. Or better (but less suggestively), what is manifestly unstable in a “real” world, might not be so in a “complex” world.
With these brief examples I end this introduction to P T symmetric QM.
Actually, this was more a presentation of the ideas behind P T symmetry rather
than of the theory itself. Indeed there is a huge amount of work, both in the
foundations of the theory and in more border-line issues. The interested reader
is encouraged to look to the very good reviews by Bender [108, 109].
4.3. Introduction and resume of the paper [3]
As I pointed out at the beginning of the chapter, complex (non-hermitian) hamiltonians with P T symmetry have been vigorously investigated in Quantum Mechanics and Quantum Field Theory [106, 107]. A possibility of applications to
quantum cosmology has been pointed out in [113]. In [3] we mainly focused
attention on complex field theory. We explored the use of a particular complex
scalar field lagrangian, whose solutions of the classical equations of motion provide us with real physical observables and well-defined geometric characteristics.
In the said paper we proposed a cosmological model inspired by P T sym-
50
4.4. Phantom and stability
metric Quantum Theory, choosing potentials so that the equations of motion
have classical phantom solutions for homogeneous and isotropic universe. Meanwhile quantum fluctuations have positive energy density and thus ensuring the
stability around a classical background configuration.
We considered the complex extension of matter lagrangians requiring the reality of all the physically measurable quantities and the well-definiteness of geometrical characteristics. It is worthwhile to underline here that we considered
only real space-time manifolds. Attempts to use complex manifolds for studying
the problem of dark energy in cosmology can be found here [114, 115, 116].
In [3], we start with the model of two scalar fields with positive kinetic terms.
The potential of the model is additive. One term of the potential is real, while
the other is complex and P T symmetric. We find a classical complex solution
of the system of the two Klein-Gordon equations together with the Friedmann
equation. The solution for one field is real (the “normal” field) while the solution
for the other field is purely imaginary, realizing classically the phantom behavior.
Moreover, the effective lagrangian for the linear perturbations has the correct
potential signs for both the fields, so that the problem of stability does not arise.
However, the background (homogeneous Friedmann) dynamics is determined by
an effective action including two real fields one normal and one phantom. As a
byproduct, we notice that the phantom phase in the cosmological evolution is
inevitably transient. The number of phantom divide line crossings, (i.e. events
such that the ratio w between pressure and energy density passes through the
value −1) can be only even and the Big Rip never occurs. The avoidance of Big
Rip singularity constitutes an essential difference between our model and wellknown quintom models, including one normal and one phantom fields[117, 118].
The other differences will be discussed in more detail later.
What follows is so organized: in sections 4.4 and 4.5 the cosmological model
we considered is described, together with a brief explanation of the “interplay”
between P T symmetric Quantum Mechanics and two-field models; in section 4.6
I present the results of the qualitative analysis and of the numerical simulations
for the dynamical system under consideration; conclusions and perspectives are
presented in the last section 4.7.
4.4. Phantom and stability
As I pointed out at the end of section 1.1.3, the great problem of phantom fields
is the quantum instability [59, 60]. Their hamiltonian is unbounded from below,
thus its quantum fluctuations grow exponentially. This should remind the reader
of the P T oscillator case 4.2, which indeed I will again discuss, in a slightly more
specific context. The idea is to exploit the P T symmetric framework to “go
51
4. Phantom without phantom in a PT-symmetric background
around” (in the complex plane) this kind of instability.
We shall study the flat Friedmann cosmological model described by the (usual)
FLRW metric (1.2).
Let us consider the matter represented by scalar fields with complex potentials. Namely, we shall try to find a complex potential possessing the solutions
of classical equations of motion which guarantee the reality of all observables.
Such an approach is of course inspired by the quantum theory of the P T symmetric non-hermitian hamiltonians, whose spectrum is real and bounded from
below. Thus, it is natural for us to look for lagrangians which have consistent
counterparts in the quantum theory.
Let me elucidate how the phantom-like classical dynamics arises in such lagrangians. For this purpose, at risk of being repetitive, choose the one-dimensional
P T symmetric potential of an-harmonic oscillator V (2+) (q) = λq 2 (iq) , 0 < ≤
2 which has been rapidly analyzed above (4.2). For illustration, let me take here
the most interesting case of = 2, V (4) (q) = −λq 4 . As said above, the classical dynamics for real coordinates q(t) offers the infinite motion with increasing
speed and energy or, in the quantum mechanical language, indicates the absence
of bound states and unboundness of energy from below. However, just there is a
more consistent solution which, at the quantum level, provides the real discrete
energy spectrum, certainly, bounded from below. It has been proven, first, by
means of path integral [106] and, further on, by means of the theory of ordinary differential equations [119]. In fact, this classically “crazy” potential on a
curve in the complex coordinate plane generates the same energy spectrum as
a two-dimensional quantum an-harmonic oscillator V (4) (~q) = λ(q14 + q24 )/4 with
real coordinates q1,2 in the sector of zero angular momentum [106]. Although
the superficially unstable an-harmonic oscillator is well defined on the essentially complex coordinate contour any calculation in the style of perturbation
theory (among them the semi-classical expansion) proceeds along the contour
with a fixed complex part (corresponding to a "classical" solution) and varying
unboundly in real direction. In particular, the classical trajectory for V (4) (q)
with keeping real the kinetic, (q̇)2 and potential, −λq 4 (t) energies (as required
by its incorporation into a cosmological scenario) can be chosen imaginary,
q = iξ,
ξ¨ = −2λξ 3 ,
ξ˙2 = C − λξ 4 ,
C>0
which obviously represents a bounded, finite motion with |ξ| ≤ (C/λ)1/4 . Such
a motion supports the quasi-classical treatment of bound states with the help of
Bohr’s quantization. Evidently, the leading, second variation of the lagrangian
around this solution, q(t) = iξ(t) + δq(t) gives a positive definite energy,
1
L(2) = p(t)δ q̇(t) − H, H = p2 (t) + 12ξ 2 (t)(δq(t))2
4
52
4.4. Phantom and stability
realizing the perturbative stability of this an-harmonic oscillator in the vicinity of
imaginary classical trajectory. It again reflects the existence of positive discrete
spectrum for this type of an-harmonicity. However the classical kinetic energy
−ξ˙2 is negative, i.e. it is phantom-like.
In a more general, Quantum Field Theory setting let us consider a nonhermitian (complex) lagrangian of a scalar field
1
L = ∂µ φ∂ µ φ∗ − V (φ, φ∗ ) ,
2
(4.10)
with the corresponding action,
Z
S[φ, g] =
√
1
d x −g L + R(g) ,
6
4
(4.11)
where g stands for the determinant of a metric g µν and R(g) is the scalar curvature term and the Newton gravitational constant is as usual normalized to 3/8π
to simplify the Friedmann equations.
We employ potentials V (Φ, Φ∗ ) satisfying the invariance condition
(V (Φ, Φ∗ ))∗ = V (Φ∗ , Φ) ,
(4.12)
(V (Φ, Φ∗ ))∗ = V (Φ, Φ∗ ) ,
(4.13)
while the condition
is not satisfied. This condition represents a generalized requirement of (C)P T symmetry.
Let’s define two real fields,
1
φ ≡ (Φ + Φ∗ ) ,
2
χ≡
1
(Φ − Φ∗ ) .
2i
(4.14)
Then, for example, such a potential can have a form
V (Φ, Φ∗ ) = Ṽ (Φ + Φ∗ , Φ − Φ∗ ) = Ṽ (φ, iχ) ,
(4.15)
where Ṽ (x, y) is a real function of its arguments. In the last equation one can
recognize the link to the so called (C)P T symmetric potentials if to supply the
field χ with a discrete charge or negative parity. When keeping in mind the
perturbative stability we impose also the requirement for the second variation
of the potential to be a positive definite matrix which, in general, leads to its
P T symmetry [69, 70].
Here, the functions φ and χ appear as the real and the imaginary parts of the
complex scalar field Φ, however, in what follows, we shall treat them as independent spatially homogeneous variables depending only on the time parameter
t and, when necessary, admitting the continuation to complex values.
53
4. Phantom without phantom in a PT-symmetric background
4.5. Cosmological solution with classical
phantom field
It appears that among known P T symmetric hamiltonians (lagrangians) possessing the real spectrum one which is most suitable for our purposes is that
with the exponential potential. It is connected with the fact that the properties
of scalar field based cosmological models with exponential potentials are well
studied[74, 76, 120, 121]. In particular, the corresponding models have some
exact solutions providing a universe expanding according to some power law
a(t) = a0 tq . We shall study the model with two scalar fields and the additive potential. Usually, in cosmology the consideration of models with two scalar fields
(one normal and one phantom, i.e. with the negative kinetic term) is motivated
by the desire of describing the phenomenon of the so called phantom divide line
crossing. At the moment of the phantom divide line crossing the equation of
state parameter w = p/ε crosses the value w = −1 and (equivalently) the Hubble variable h has an extremum. Usually the models using two fields are called
“quintom models” [69]-[72]. As a matter of fact the phantom divide line crossing
phenomenon can be described in the models with one scalar field, provided some
particular potentials are chosen[67, 68] or in the models with non-minimal coupling between the scalar field and gravity [122]. However, the use of two fields
make all the considerations more simple and natural. In the framework we are
considering here, the necessity of using two scalar fields follows from other requirements. We would like to implement a scalar field with a complex potential
to provide the effective phantom behavior of this field on some classical solutions
of equations of motion. Simultaneously, we would like to have the standard form
of the effective Hamiltonian for linear perturbations of this field. The combination of these two conditions results in the fact the background contribution
of both the kinetic and potential term in the energy density, coming from this
field are negative. To provide the positivity of the total energy density which
is required by the Friedmann equation (1.8a) we need the other normal scalar
field. Thus, we shall consider the two-field scalar lagrangian with the complex
potential
φ̇2 χ̇2
+
− Aeαφ + Beiβχ ,
(4.16)
L=
2
2
where A and B are real, positive constants. This lagrangian is the sum of
two terms. The term representing the scalar field φ is a standard one, and
it can generate a power-law cosmological expansion [74, 76, 120, 121]. The
kinetic term of the scalar field χ is also standard, but its potential is complex.
Notice that the exponential potential has been analyzed in great detail in [123,
124]. The most important feature of this potential is that the spectrum of the
54
4.5. Cosmological solution with classical phantom field
corresponding Hamiltonian is real and bounded from below, provided correct
boundary conditions are assigned.
Inspired by this fact we are looking for a classical complex solution of the
system, including two Klein-Gordon equations for the fields φ and χ:
φ̈ + 3hφ̇ + Aαeαφ = 0 ,
(4.17)
χ̈ + 3hχ̇ − iBβeiβχ = 0 ,
(4.18)
and the Friedmann equation
h2 =
φ̇2 χ̇2
+
+ Aeαφ − Beiβχ .
2
2
(4.19)
The classical solution which we are looking for should provide the reality and
positivity of the right-hand side of the Friedmann equation (4.19). The solution
where the scalar field φ is real, while the scalar field χ is purely imaginary
χ = iξ,
ξ real ,
(4.20)
uniquely satisfies this condition. Moreover, the lagrangian (4.16) evaluated on
this solution is real as well. This is remarkable because on homogeneous solutions
the lagrangian coincides with the pressure, which indeed should be real.
Substituting the equation (4.20) into the Friedmann equation (4.19) we shall
have
φ̇2 ξ˙2
h2 =
−
+ Aeαφ − Be−βξ .
(4.21)
2
2
Hence, effectively we have the Friedmann equation with two fields: one (φ) is
a standard scalar field, the other (ξ) has a phantom behavior, as we pointed
out above. In the next section we shall study the cosmological dynamics of the
(effective) system, including (4.21), (4.17) and
ξ¨ + 3hξ˙ − Bβe−βξ = 0 .
(4.22)
The distinguishing feature of such an approach rather than the direct construction of phantom lagrangians becomes clear when one calculates the linear
perturbations around the classical solutions. Indeed the second variation of the
action for the field χ gives the quadratic part of the effective lagrangian of perturbations:
1 ˙ 2
Lef f = δχ
− Bβ 2 eiβχ0 (δχ)2 ,
(4.23)
2
where χ0 is a homogeneous purely imaginary solution of the dynamical system under consideration. It is easy to see that on this solution, the effective
lagrangian (4.23) will be real and its potential term has a sign providing the
55
4. Phantom without phantom in a PT-symmetric background
stability of the background solution with respect to linear perturbations as the
related Hamiltonian is positive,
1
(2)
Hef f = δπ 2 + Bβ 2 e−βξ0 (δχ)2 ,
2
δπ ⇔ δ χ̇.
(4.24)
Let us list the main differences between our model and quintom models, using
two fields (normal scalar and phantom) and exponential potentials[117, 118].
First, we begin with two normal (non-phantom) scalar fields, with normal kinetic terms, but one of these fields is associated to a complex (P T symmetric)
exponential potential. Second, the (real) coefficient multiplying this exponential
potential is negative. Third, the background classical solution of the dynamical
system, including two Klein-Gordon equations and the Friedmann equation, is
such that the second field is purely imaginary, while all the geometric characteristics are well-defined. Fourth, the interplay between transition to the purely
imaginary solution of the equation for the field χ and the negative sign of the
corresponding potential provides us with the effective lagrangian for the linear
perturbations of this field which have correct sign for both the kinetic and potential terms: in such a way the problem of stability of the our effective phantom
field is resolved. Fifth, the qualitative analysis of the corresponding differential
equations, shows that in contrast to the quintom models in our model the Big
Rip never occurs. The numerical calculations confirm this statement.
In the next section we shall describe the cosmological solutions for our system
of equations.
4.6. Cosmological evolution
First of all notice that our dynamical system permits the existence of cosmological trajectories which cross the phantom divide line. Indeed, the crossing point
is such that the time derivative of the Hubble parameter
3
ḣ = − (φ̇2 − ξ˙2 )
2
(4.25)
˙ at t = tP DL provided the
is equal to zero. We always can choose φ̇ = ±ξ,
values of the fields φ(tP DL ) and ξ(tP DL ) are chosen in such a way, that the
general potential energy Aeαφ − Be−βξ is non-negative. Obviously, tP DL is the
moment of the phantom divide line crossing. However, the event of the phantom
divide line crossing cannot happen only once. Indeed, the fact that the universe
has crossed phantom divide line means that it was in effectively phantom state
before or after such an event, i.e. the effective phantom field ξ dominated over
the normal field φ. However, if this dominance lasts for a long time it implies
56
4.6. Cosmological evolution
that not only the kinetic term −ξ˙2 /2 dominates over the kinetic term φ̇2 /2 but
also the potential term −B exp(−βξ) should dominate over A exp(αφ); but it is
impossible, because contradicts to the Friedmann equation (4.21). Hence, the
period of the phantom dominance should finish and one shall have another point
of phantom divide line crossing. Generally speaking, only the regimes with even
number of phantom divide line crossing events are possible. Numerically, we
have found only the cosmological trajectories with the double phantom divide
line crossing. Naturally, the trajectories which do not experience the crossing at
all also exist and correspond to the permanent domination of the normal scalar
field. Thus, in this picture, there is no place for the Big Rip singularity as well,
because such a singularity is connected with the drastically dominant behavior
of the effective phantom field, which is impossible as was explained above. The
impossibility of approaching the Big Rip singularity can be argued in a more
rigorous way as follows. Approaching the Big Rip, one has a growing behavior
of the scale factor a(t) of the type a(t) ∼ (tBR − t)−q , where q > 0. Then the
Hubble parameter is
q
(4.26)
h(t) =
tBR − t
and its time derivative
ḣ(t) =
q
.
(tBR − t)2
Then, according to equation (4.25),
1 ˙2 1 2
q
ξ − φ̇ =
.
2
2
3(tBR − t)2
(4.27)
Substituting equations (4.27), (4.26) into the Friedmann equation (4.21), we
come to
q
q
=
+ Aeαφ − Be−βξ .
2
(tBR − t)
3(tBR − t)2
In order for this to be satisfied and consistent, the potential of the scalar field φ
should behave as 1/(tBR − t)2 . Hence the field φ should be
φ = φ0 −
2
ln(tBR − t) ,
α
(4.28)
where φ0 is an arbitrary constant. Now substituting equations (4.26) and (4.28)
into the Klein-Gordon equation for the scalar field φ (4.17), the condition of the
cancellation of the most singular terms in this equation which are proportional
to 1/(tBR − t)2 reads
2 + 6q + Aα2 exp(αφ0 ) = 0 .
(4.29)
57
4. Phantom without phantom in a PT-symmetric background
This condition cannot be satisfied because all the terms in the left-hand side of
equation (4.29) are positive. This contradiction demonstrates that it is impossible to reach the Big Rip.
Now I would like to describe briefly some examples of cosmologies contained in
our model, deduced by numerical analysis of the system of equations of motion.
In figure 4.4 a double crossing of the phantom divide line is present. The evolution starts from a Big Bang-type singularity and goes through a transient phase
of super-accelerated expansion (“phantom era”), which lies between two crossings
of the phantom divide line. Then the universe undergoes an endless expansion.
In the right plot I present the time evolution of the total energy density and of
its partial contributions due to the two fields, given by the equations
1
εφ = φ̇2 + Aeαφ ,
2
1
εξ = − ξ˙2 − Beβξ ,
2
ε = εφ + εξ ,
which clarify the roles of the two fields in driving the cosmological evolution.
The evolution presented in figure 4.5 starts with a contraction in the infinitely
remote past. Then the contraction becomes super-decelerated and turns later in
a super-accelerated expansion . With the second phantom divide line crossing
the “phantom era” ends; the decelerated expansion continues till the universe
begins contracting. After a finite time a Big Crunch-type singularity is encountered. From the right plot we can clearly see that the “phantom era” is indeed
characterized by a bump in the (negative) energy density of the phantom field.
In figure 4.6 the cosmological evolution again begins with a contraction in the
infinitely remote past. Then the universe crosses the phantom line: the contraction becomes super-decelerated until the universe stops and starts expanding.
Then the "phantom era" ends and the expansion is endless.
In figure 4.7 the evolution from a Big Bang-type singularity to an eternal expansion is shown. The phantom phase is absent. Indeed the phantom energy
density is almost zero everywhere.
4.7. Conclusions
As was already said many times, the data are compatible with the presence of
the phantom energy, which, in turn, can be in a most natural way realized by
the phantom scalar field with a negative kinetic term. However, such a field
suffers from the instability problem, which makes it vulnerable. Inspired by the
development of P T symmetric Quantum Theory we introduced the P T symmetric two-field cosmological model where both the kinetic terms are positive, but
58
4.7. Conclusions
h
Ε
4
10
3
5
2
t
1
2
3
4
5
1
-5
t
1
2
3
4
5
Figure 4.4.: (left) Plot of the Hubble parameter representing the cosmological
evolution. The evolution starts from a Big Bang-type singularity
and goes through a transient phase of super-accelerated expansion
(“phantom era”), which lies between two crossings of the phantom
divide line (when the derivative of h crosses zero). Then the universe
expands infinitely. (right) Plots of the total energy density (blue),
and of the energy density of the normal field (purple) and of the
phantom one (green).
h
Ε
3
10
2
5
1
t
t
-4
2
-2
-4
2
-2
4
4
-1
-5
-2
-10
Figure 4.5.: (left) The evolution starts with a contraction in the infinitely remote
past. At the first phantom divide line crossing the contraction becomes super-decelerated and turns in a super-accelerated expansion
when h crosses zero. The second crossing ends the “phantom era”;
the decelerated expansion continues till the universe begins contracting. In a finite time a Big Crunch-type singularity is reached. (right)
Plots of the total energy density (blue), and of the energy density
of the normal field (purple) and of the phantom one (green).
59
4. Phantom without phantom in a PT-symmetric background
h
Ε
150
2
100
t
0.2
-0.4 -0.2
0.4
0.6
50
0.8
-2
t
-4
-0.4 -0.2
-50
-6
-100
-8
-150
0.2
0.4
0.6
0.8
Figure 4.6.: (left) The cosmological evolution begins with a contraction in the
infinitely remote past. With the first phantom divide line crossing
the contraction becomes super-decelerated until the universe stops
(h = 0) and starts expanding. With the second crossing the “phantom era” ends and the expansion continues infinitely. (right) Plots
of the total energy density (blue), and of the energy density of the
normal field (purple) and of the phantom one (green).
h
Ε
200 000
400
150 000
300
200
100 000
100
50 000
t
t
-0.004
-0.002
0.002
0.004
-0.004
-0.002
0.002
0.004
Figure 4.7.: (left) Evolution from a Big Bang-type singularity to an infinite expansion, without any crossing of the phantom divide line. This
evolution is thus guided by the “normal” field φ. (right) Plots of the
total energy density (blue), and of the energy density of the normal
field (purple) and of the phantom one (green). Notice that the energy density of the phantom field (green) is very close to zero, thus
the total energy density is mainly due to the standard field.
60
4.7. Conclusions
the potential of one of the fields is complex. We studied a classical background
solution of two Klein-Gordon equations together with the Friedmann equation,
when one of this fields (normal) is real while the other is purely imaginary. The
scale factor in this case is real and positive just like the energy density and the
pressure. The background dynamics of the universe is determined by two effective fields – one normal and one phantom – while the lagrangian of the linear
perturbations has the correct sign of the mass term. Thus, so to speak, the
quantum normal theory is compatible with the classical phantom dynamics and
the problem of instability is absent.
As a byproduct of the structure of the model, the phantom dominance era is
transient, the number of the phantom divide line crossings is even and the Big
Rip singularity is avoided.
61
5. Cosmological singularities
with finite non-zero radius
The present chapter is intended to present the results of [4]. This is kind of
off-topic, with respect to the preceding work, but not at all by far. Indeed the idea
is to study the reconstruction of a single scalar field model, able to reproduce a
specific dynamical evolution, which seemed to us particularly interesting, since
it presents some kind of mild version of the Big Bang singularity. Dynamical
analysis is performed on the model and its phase space is divided into classes of
qualitative different cosmic evolutions. Here there is no crossing of the phantom
divide line, nor phantom fields.
5.1. Introduction and review of paper [4]
We have repeatedly said that the discover of cosmic acceleration [6, 7] has stimulated the research of new cosmological models. This development of modeldesigning art has revealed cosmological evolutions possessing various types of
singularities, sometimes very different from the traditional Big Bang and Big
Crunch. The most popular between them is, perhaps, the Big Rip cosmological singularity [57, 58] arising in super-accelerating models driven by some
kind of phantom matter. Other types of singularities are sudden singularities [125, 126, 127], Big Brake [76], and so on [128]-[131]. Here we would like
to study the singularities which are close to the known Big Bang, Big Crunch
and Big Rip singularities, but arising at finite values of the cosmological factor (different from zero and infinity as well). Similar singularities were recently
considered in [132, 133, 134].
In the paper under consideration we construct potentials which can drive the
cosmological evolution towards (or from) such singularities. Combining qualitative and numerical methods we study the set of possible cosmological histories
in the suggested models to show that the presence of such singularities in a
cosmological model under consideration depends essentially on initial conditions
and that the same model can accommodate qualitatively different cosmological
scenarios.
The structure of what follows is: in section 5.2 we construct some potentials
63
5. Cosmological singularities with finite non-zero radius
corresponding to evolutions with “mild” singularities. In the third section 5.3 we
analyze their dynamics. The conclusion 5.4 is devoted to an interpretation of
the obtained results.
5.2. Construction of scalar field potentials
Let me repeat here very briefly the idea of the reconstruction of potentials for
cosmological models (cfr. section 2.1).
We shall consider flat Friedmann models with the metric (1.2) The Hubble
parameter h(t) ≡ ȧ/a satisfies the Friedmann equation (1.8a) and also its ‘manipulation’ (1.9).
If the matter is represented by a spatially homogeneous minimally coupled
scalar field, then the energy density and the pressure are given by the formulæ
(2.1) and (2.2), respectively; while equations (2.5) and (2.6) give the expression
of the potential and of φ̇ as functions of time. Integrating equation (2.6) one
can find the scalar field as a function of time. Inverting this dependence we can
obtain the time parameter as a function of φ and substituting the corresponding
formula into equation (2.5) one arrives to the uniquely reconstructed potential
V (φ). It is necessary to stress that this potential reproduces a given cosmological
evolution only for some special choice of initial conditions on the scalar field and
its time derivative.
It is known that the power-law cosmological evolution is given by the Hubble
parameter h(t) ∼ 1t . We shall look for a “softer” version of the cosmological
evolution given by the law
S
(5.1)
h(t) = α ,
t
where S is a positive constant and 0 < α < 1. At t = 0 a singularity is present,
but it is different from the traditional Big Bang singularity. Indeed, integrating
we obtain
S 1−α
a(t)
ln
=
t .
(5.2)
a(0)
1−α
If t > 0 the right-hand side of equation (5.2) is finite and hence one cannot
have a(0) = 0 in the left-hand side of this equation, because it would imply a
a(t)
contradiction, making ln a(0)
divergent. Hence a(0) > 0, while
S
ȧ = a(0) α exp
t
S 1−α
t
1−α
−−→ ∞ .
t→0
(5.3)
This type of singularity can be called “mild” Bing Bang singularity because the
cosmological radius is finite (and non-zero) while its time derivative, the Hubble
64
5.3. The dynamics of the cosmological model with α =
1
2
variable and the scalar curvature are singular. It is interesting to note that
when t → ∞ both a(t) and ȧ(t) tend to infinity, but they do not encounter any
cosmological singularity because the Hubble variable and its derivatives tend to
zero.
Let us reconstruct the potential of the scalar field model, producing the cosmological evolution (5.1) using the technique described above. Equation (2.6)
gives
r
α+1
2
(5.4)
αS t− 2 .
φ̇ = ±
3
We shall choose the positive sign, without loosing generality. Integrating, we get
r
1−α
2t 2
2
αS
,
(5.5)
φ(t) =
3
1−α
up to an arbitrary constant. Inverting the last relation we find
t(φ) =
3
2αS
1/2
2
! 1−α
1−α
φ
.
2
(5.6)
Hence, using equation (2.5) we obtain
V (φ) = q
S2
αS
,
4α −
1−α
q
2(α+1)
1−α
3 1−α
3 1−α
φ
3
φ
2αS 2
2αS 2
(5.7)
This potential provides the cosmological evolution (5.1) if initial conditions compatible with equations (5.4) and (5.5) are chosen. Naturally, there are also other
cosmological evolutions, generated by other initial conditions, which will be studied in the next section.
5.3. The dynamics of the cosmological model
with α = 21
In order to achieve some simplification of calculations we shall consider a particular model, namely the one with the choice α = 12 . In this case
a(t) = a(0)e2S
and
√
t
,
16S 4
32S 4
√
√
V (φ) =
−
.
( 3φ/2)4 3( 3φ/2)6
(5.8)
(5.9)
65
5. Cosmological singularities with finite non-zero radius
The Klein-Gordon equation reads
s
√
√
16S 4
32S 4
32 3S 4
32 3S 4
1 2
φ̇ + √
− √
− √
+ √
=0.
φ̈ + 3φ̇ sign(h)
2
( 3φ/2)4 3( 3φ/2)6 ( 3φ/2)5 ( 3φ/2)7
(5.10)
This equation is equivalent to the dynamical system


φ̇ = x,

q 2

4
4
ẋ = −3x sign(h) x2 + (√16S
− 3(√32S
(5.11)
3φ/2)4
3φ/2)6
√ 4
√ 4


3S
3S
32
32

+√
− √
.
( 3φ/2)5
( 3φ/2)7
The qualitative analysis of dynamical systems in cosmology was presented in
detail in [85].
First of all, let us notice that the system has two critical points: φ = ± √23 , x = 0.
We consider the linearized system around the point with the positive value of φ:
ϕ̇ = x,
(5.12)
ẋ = −3x sign(h) √43 S 2 + 32 · 3S 4 ϕ,
√
where ϕ ≡ φ − 2/ 3.
The Lyapunov indices for this system are (for h > 0)
√
λ1 = −8 3S 2 ,
√
λ2 = 4 3S 2 .
(5.13)
(5.14)
For negative h, corresponding to the cosmological contraction, the signs of λ1
and λ2 are changed. The eigenvalues are real and have opposite signs, hence both
the critical points are saddles. The universe being in one of these two saddle
points means that it undergoes a de Sitter expansion or contraction, according
to the sign of h, with the value of h = h0 given by
4S 2
h0 = √ .
3
(5.15)
For each saddle point there are four separatrices which separate four classes of
trajectories in the phase plane x, φ corresponding to four types of cosmological
evolutions.
In order to simplify the study of the dynamics let us note that the potential
is an even function of the scalar field φ and that the saddle points are also
symmetrical with respect to the x axis. Thus, it is sufficient to consider only
one of this saddle points. We shall carry out our qualitative analysis taking
66
5.3. The dynamics of the cosmological model with α =
V (φ)
1
2
I
III
II
φ
IV
Figure 5.1.: Plot of the potential V (φ) as given by equation (5.9). Here only the
positive φ-axis is shown, which is enough to understand the behavior thanks to the parity of the function. We have also delineated
the four different dynamical behaviors of the scalar field, clearer to
understand taking into account the phase portrait (see figure 5.2).
α
x
I
IV
β
φ
II
III
δ
γ
Figure 5.2.: Positive φ-axis phase portrait for the dynamical system (5.11) with
h ≥ 0. The four separatrices of the saddle point (α, β, γ, δ) individuate four regions (I,II,III,IV ) with different behaviors of the
trajectories, as explained in the text.
67
5. Cosmological singularities with finite non-zero radius
into account both figure 5.1, giving the form of the potential, and figure 5.2,
representing the phase portrait in the plane φ, x.
First let us consider trajectories which begin at the moment t = 0, when
the initial value of the scalar field is infinite, its potential is equal to zero and
the time derivative of the scalar field is infinite and negative. In terms of the
figure 5.1 it means that we consider the motion of the point beginning at the
far right on the slope of the potential hill and moving towards the left (i.e.
towards the top of the hill) with an infinite initial velocity. Such a motion for
h > 0 describes a universe born from the standard Big Bang singularity. Further
details of this evolution depend on the asymptotic ratio between absolute values
of φ̇ and φ at t → 0. If this ratio is smaller than some critical value then the
scalar field does not reach the top of the hill and at some moment it begins
to roll down back to the right. During this process of rolling down the scalar
field increases, the potential is decreasing and the velocity φ̇ becomes positive
and increasing. However the universe expansion works as a friction and at some
moment its influence becomes dominant causing an asymptotic damping to zero
of the velocity. The universe expands infinitely with h(t) → 0. In the phase
portrait (figure 5.2) such trajectories populate the region II. This region is
limited by the separatrices β and γ. The first one corresponds to the positive
(for h > 0) eigenvalue λ2 , while γ corresponds to λ1 . If the ratio φ̇/φ has the
critical value, then the scalar field reaches asymptotically the top of the hill of
the potential, meaning that the universe becomes asymptotically de Sitter: in
the phase portrait it is nothing but the curve γ.
When the ratio introduced above is larger than the critical one we encounter a
different regime. In this case the scalar field passes with non-vanishing velocity
the top of the hill and begins to roll down in the abyss on the left. The velocity
is growing, but the potential becomes negative and at some moment the total
energy density of the scalar field vanishes together with the Hubble parameter h:
this means that the universe starts contracting. This contraction provides the
growing of the absolute value of the velocity of the scalar field and the kinetic
term again becomes larger then the potential one. Moreover, both the terms in
the Klein-Gordon equation increase the absolute value of φ̇. One can easily show
that the regime in which the time derivative φ̇ becomes equal to −∞ at some
finite value of φ is impossible, because it implies a contradiction between the
asymptotic behavior of different terms in equation (5.10). Thus, the universe
tends to the singularity squeezing to the state with the value of φ equal to zero
and an infinite time derivative φ̇. To understand which kind of singularity the
universe encounters, we need some detail about the behavior of the scalar field.
Let us suppose that, approaching the singularity at some moment t0 , the scalar
field behaves as
φ(t) = φ0 (t0 − t)µ ,
(5.16)
68
5.3. The dynamics of the cosmological model with α =
1
2
where 0 < µ < 1. Then the first and second time derivative are
φ̇(t) = −µφ0 (t0 − t)µ−1 , φ̈(t) = µ(µ − 1)φ0 (t0 − t)µ−2 .
(5.17)
The potential behaves as
V =−
2048S 4
.
81φ60 (t0 − t)6µ
(5.18)
To have the Hubble variable well defined we require that the kinetic term is larger
than the absolute value of the negative potential term (5.18), i.e. 2µ − 2 ≤ −6µ,
or µ ≤ 14 . Now two opposite cases may hold: (i) the friction term in the KleinGordon equation could dominate the potential term or (ii) the opposite situation.
For (i) to be the right case, one should require 2µ − 2 < −7µ or µ < 92 . In this
situation the asymptotic behavior of the second time derivative of φ should be
equal to that of the friction term, or, in other words µ−2 = 2µ−2, that is µ = 0,
which obviously is not relevant. Thus we have to consider the range 29 < µ ≤ 41 .
In this case the potential term should be equal to the second time derivative of
φ, which implies:
1
µ=
(5.19)
4
and
r
S
φ0 = 4
.
(5.20)
3
Substituting the values of µ and φ0 into the expression for h(t), we obtain
S
h(t) = − √
.
t0 − t
(5.21)
Thus, we see that the singularity we are approaching is of the “mild” Big Crunch
type.
In the phase portrait (figure 5.2) these trajectories occupy the region III
limited by the separatrices γ and δ. The cosmological evolutions run from the
Big Bang singularity to the mild Big Crunch one. However, the figure 5.2 is not
sufficient to describe the complete behavior of the universe under consideration,
because at some moment the Hubble variable changes sign and we should turn
to the figure 5.3, giving the phase portrait for the contracting universe h < 0.
Note that increasing the velocity φ̇ with which the scalar field overcomes the top
of the hill, implies increasing the moment of time when the point of maximal
expansion of the universe is reached. The limiting case in which this moment
tends to infinity corresponds to the separatrix δ.
The third regime begins from the mild Big Bang singularity, when the scalar
field is equal to zero and its time derivative is infinite and positive. In figure 5.1
69
5. Cosmological singularities with finite non-zero radius
x
φ
Figure 5.3.: Phase portrait for the dynamical system under discussion (equation (5.11)) with h ≤ 0, i.e. describing evolutions of the universe
characterized by contraction. As for the h-positive case, we can see
the four separatrices of the saddle point and the corresponding four
regions of different dynamical behaviors of the trajectories.
that situation is represented by the point climbing from the abyss to the top
of the hill. If the velocity term is not high enough, at some moment the field
stops climbing and rolls back down. During this fall the Hubble variable changes
sign and the universe ends its evolution in the mild Big Crunch singularity. The
corresponding trajectories belong to the region IV of our phase plane, bounded
by the separatrices δ and α. The universe has its finite life time between the
mild Big Bang and the mild Big Crunch singularities. The situation when the
scalar field arrives exactly to the top of the hill and stops corresponds to the
separatrix α.
The fourth set of cosmological trajectories is generated by the scalar field
climbing from the abyss and overcoming the top of the hill with the subsequent
infinite expansion: the scalar field is infinitely growing and the Hubble parameter
tends to zero. These trajectories occupy the region I and our original cosmological evolution (5.1) belongs to this family. Naturally there are other four classes
of cosmological evolutions, which can be easily obtained by inverting the time
direction.
70
5.4. Conclusions
5.4. Conclusions
Let me sum up the results. Wishing to describe a cosmological evolution beginning from the singularity characterized by a finite and non-zero initial (or
final) cosmological radius and an infinite value of the scalar curvature due to
the infinite value of the Hubble parameter, we have constructed a scalar field
potential, providing such an evolution. Then, using the methods of qualitative
analysis of the differential equations, we have shown that the proposed model
accommodates four different classes of cosmological evolutions, depending on
initial conditions. Numerical simulations have confirmed our predictions.
The main results of this work are: (i) the realization of a concrete cosmological model with a scalar field, where finite cosmological radius singularities are
present and (ii) the complete description of all the possible evolutions of this
model depending on the initial conditions. It is important to remark that, given
the fixed scalar field potential, one has different types of evolutions encountering
different kinds of singularities.
Let us note, that there are some studies [125, 126, 127], [128]-[131], [132,
133, 134] devoted to the general analysis of various kinds of new cosmological
singularities. In this work we were not looking for an exhaustive classification of
different possible cosmological models possessing some kind of singularities, but
rather we wanted to study in a complete way a particular cosmological model,
having some interesting properties.
71
Part II.
Loop Quantum Gravity and
Spinfoam models
73
Introduction
The title of the present thesis talks about the “relation between Geometry and
Matter”. Just to be clear from the very beginning: I will not talk about the
coupling of matter with gravity in the Loop Quantum Gravity framework. Even
though some work has been done in this respect1 , this is not quite the sense we
had in mind when we thought about that title. Let me spend some words about
it.
Gravity, or the gravitational field, is indeed a rather peculiar object in physics
and – I dare say – in Nature. This is true even at the classical level. This
has become apparent when Einstein discovered the particular “position” of the
gravitational field with respect to all the other (matter) fields. Indeed the gravitational field is a field like all the others, with its dynamical equations and so on.
But it is the background on which all the other fields live as well. All the matter fields are defined using the metric g in their equations, they live and “play”
on a certain metric manifold – the “game rules” – which “is” the gravitational
field itself. Moreover, there is interplay between them, since Einstein’s equations
tell us that the game rules depend on how the matter plays and viceversa, in a
recursive/non-linear interplay. This duality of being both a dynamical field and
the background spacetime on which the fields live is at the heart of the peculiar
nature of Gravity.
This is a classical picture. When we quantize the matter fields in the Quantum
Field Theory framework, we somewhat forget this relation, and assume a fixed
flat background, without gravity. We can try at best to do Quantum Field
Theory on curved background, with all its subtleties, still relying on that critical
and peculiar relation I was talking about; and, most importantly, we still need
a background spacetime metric on top of which our matter fields live.
But now a question urges: what if we quantize the gravitational field? We
can argue that this quantization is doomed to revolutionize once again the way
in which we think about matter fields living on a spacetime, just as General
Relativity revolutionized the concept of fixed spacetime. Indeed Loop Quantum
Gravity gives a picture of space, and of spacetime, which is purely combinatorial
and relational, where the very concept of “event” looses its meaning: totally
1
See e.g. [135, 136, 137] for matter coupling in the canonical LQG approach, and [139]-[143]
for the spinfoam approach.
75
Introduction
different from the smooth metric manifold to which we are accustomed. The
loss of a background is crucial in the way we think matter fields.
In this precise sense I think that the study of Quantum Gravity is fundamental
to understand the “relation between Geometry and Matter”.
The following is a brief review of the main aspects of Loop Quantum Gravity
and of spinfoam theory, the latter being an attempt to understand the quantum
dynamics of the gravitational field. After this review I will present the research
work I have done together with Eugenio Bianchi and Carlo Rovelli, during my
period of research in the Marseille Quantum Gravity group. It is about a rather
technical aspect of the spinfoam approach to quantum gravitational dynamics.
Non-perturbative Quantum Gravity: some good
reasons to consider this way
Loop Quantum Gravity is a form of canonical quantization of General Relativity.
We’d better say an attempt in this direction, since the target is still far from being
reached. However, there are quite amazing and interesting results that should not
be under-estimated. I will briefly review the standard approach to the canonical
quantization of gravity and then pass to a review of the fundamental concepts
of LQG. But first let us stress the two main ingredients that has permitted LQG
to become what it is:
• background independence,
• focus on connection (rather than metric).
A few words on each of these concepts are due, before diving into the theory.
Quantization of gravity means quantization of the gravitational field, the metric
tensor field g(x) = gµν (x)dxµ ⊗ dxν , i.e. the geometry of spacetime. One should
naively expect that such a quantization would bring to some form of “quantized geometry”, meaning a quantization of distances, volumes and such. This
should happen at a very small distance scale, of course. Dimensionally speaking,
we know that this scale should be the Planck length, which is of the order of
10−33 cm. Moreover, Quantum Mechanics has taught us that linear superpositions of (quantum) states must be taken into account, in order to give a proper
(correct) picture of nature. These two (quite speculative) reasonings are here
used just to introduce the concept of background independence. Background
independence is a technical way of saying nothing more than this: if one seeks
for a complete quantization of the metric, one should quantize the metric itself
and not its small perturbations around a fixed background geometry. In fact, the
76
Non-perturbative Quantum Gravity: some good reasons to consider this way
latter is what is done in the perturbative (or background-dependent) approach
to Quantum Gravity: to assume the following splitting
0
gµν (x) = gµν
(x) + hµν (x) ,
(5.22)
0
where gµν
is treated as a classical field, a background metric, and hµν are its
quantum fluctuations, i.e. it is what one tries to quantize. This is obviously a
legitimate procedure if one wants to analyze the behavior of a (self-interacting)
0
spin-2 particle on a (generally curved) gµν
background. The problem is the following: this splitting breaks down at high energies. Namely this kind of theory
has been proved to be UV-unrenormalizable [144]. Background-dependence supporters believe that this unrenormalizability is a hint of a more fundamental
theory still to be found. This is a chance, indeed. Background-independence
supporters instead believe that this is just a hint of a non-proper way to face the
problem, namely to believe that the splitting (5.22) should be valid at all energy
scales. Indeed in the regime we are interested in (namely the Planck scale) it
is reasonable to argue that the fluctuations won’t be small; thus it would be
a nonsense (read ‘wrong’) to assume that the geometry (along with the causal
0
structure) is determined by gµν
alone. We do not know what is the fabric of
spacetime at the Planck scale, but – as argued above – it is likely to be some
quantum superposition, some grain-like structure, no more capable (in general)
to be described by a smooth field + small fluctuations. In this respect background independence is both more conservative – because after all makes use of
the “old” covariant approach – and more radical – since it starts from the belief
that ‘pure’ quantum spacetime should be deeply different from that of QFT.
Let us come to the second issue: the focus on connection. In GR one usually
takes the metric as the “main character” of the play. It is seen as the fundamental
field of the theory. In LQG instead, the metric tensor is more like a secondary
player. Actually it is no news at all that GR can be recast in the language
of differential forms and tetrad field (we will soon see in what sense), and the
language of tetrads – as opposed to that of metric tensor – leads naturally to
put the focus on connections. At a classical level, everything is equivalent2 , but
for what concerns quantization, the choice of variables will prove to be crucial,
and this is one of the key points of the LQG approach.
2
Actually there is one (to my knowledge) difference: the tetrad formulation is “an extension”
of standard GR. We shall see in section 6.2 in what sense.
77
6. Canonical Quantum Gravity:
from ADM formalism to
Ashtekar variables
The present chapter is dedicated to review the basics of canonical quantization of
gravity. Everything is well-known and established, so I will skip tedious calculations and try to present only the key passages of this subject, especially the ones
which turn to be important in the LQG approach1 .
6.1. Hamiltonian formulation of General
Relativity
The starting point is the well known Einstein-Hilbert action:
Z
√
1
S[gµν ] =
d4 x −gR .
16πG
(6.1)
In the following I shall always set 16πG = 1 for simplicity, and I will recover the
constants when to stress the importance of physical scales.
In order to perform the canonical (i.e. hamiltonian) analysis, one needs to
know which are the conjugated variables to the metric and then perform the
Legendre transform. Indeed the hamiltonian formulation looses the manifest covariance, separating time from space – so to say – in order to find the conjugate
momenta. To this aim, one assumes that it is possible to foliate the spacetime
manifold (M from now on) into spacelike three dimensional hypersurfaces. This
amounts to say that M is diffeomorphic to Σ × R. This assumptions is actually
not that restrictive: a theorem by Geroch [145, 146] proves that it has to be so if
M is globally hyperbolic, namely if it has no causally disconnected regions. Obviously we are far from saying that we are defining an absolute time coordinate!
One can choose each timelike direction as “time”. This amounts to say that there
will be an infinity of diffeomorphisms φ : M → Σ × R, each inducing a time
1
I refer the reader to the good textbook by Baez and Muniain [147] for a thorough and
somewhat LQG-oriented presentation of this subject.
79
6. Canonical Quantum Gravity: from ADM formalism to Ashtekar variables
Nn
∂τ
N
Σ
Figure 6.1.: Splitting of ∂τ into normal and tangential components with respect
to Σ.
coordinate τ = φ∗ t on M (namely the pullback of the t coordinate in R). This
splitting procedure is known as the Arnowitt-Deser-Misner decomposition [148].
Now we decompose the timelike (coordinate) vector field ∂τ into its components tangential and normal to Σ (figure 6.1):
~ ;
∂τ = N n + N
(6.2)
~ belongs to the tangent bundle T Σ of Σ; n is the unit
the shift vector (field) N
normal to Σ (i.e. g(n, v) = 0 ∀v ∈ T Σ , g(n, n) = −1) and its coefficient N is
called the lapse function. Using the following identities2
~ ) = Na N a → g0a = Na ,
g0a N a = g(∂τ , N
(6.3)
the expression of the spacetime line element in terms of these variables is:
g00 = g(∂τ , ∂τ ) = −N 2 + gab N a N b ;
ds2 = gµν dxµ dxν = −(N 2 − Na N a )dt2 + 2Na dtdxa + gab dxa dxb .
(6.4)
Notice that the induced g-metric on Σ is given by
qµν = gµν − nµ nν ,
(6.5)
(which has, correctly, q00 = 0) and thus we will talk of qab as the metric on Σ.
The object qνµ = δνµ − nµ nν is indeed the projector on Σ
(
0; v normal to Σ ,
µ ν
µ
µ
qν v = v − g(n, v)n =
v; v tangent to Σ .
We want to use these variables qab , Na , N , to rewrite the Einstein-Hilbert action (6.1). Namely, defining the extrinsic curvature
1
Kab = Ln qab = N −1 (q̇ab −3 ∇a Nb −3 ∇b Na ) ,
2
2
(6.6)
From now on latin indices from the beginning of the alphabet (a, b, c. . . ) will denote spacelike
indices ranging from 1 to 3.
80
6.1. Hamiltonian formulation of General Relativity
where Ln stands for the Lie derivative along the n direction (thus the extrinsic
curvature contains informations on how Σ is embedded in M) and 3 ∇ is the
covariant derivative compatible with qab , it is a matter of algebra to find
√
(6.7)
L = R −g = q 1/2 N R = q 1/2 N (3 R + Tr(K 2 ) − (TrK)2 ) .
The momenta conjugate to the 3-metric are
pab =
∂L
= q 1/2 (K ab − TrKq ab ) ,
q̇ab
(6.8)
and there are no momenta for N and N a (they are identically zero). We are now
able to perform the Legendre transform and write down the action (6.1) as
Z
−1/2 ab
ab
π )
S[π , qab , Na , N ] =
dt d3 x π ab q̇ab + 2Nb ∇(3)
a (q
R×Σ
+N q 1/2 [R(3) − q −1 πab π ab + 12 q −1 π 2 ] , (6.9)
where 3 R is the scalar curvature of qab and π = π ab qab . Notice that the absence
of momenta for the lapse function and the shift vector means they carry no
dynamics, namely that their equation of motion are the following constraint
equations:
−1/2 ab
V b (qab , π ab ) = −2∇(3)
π )=0
(6.10)
a (q
and
S(qab , π ab ) = −q 1/2 [R(3) − q −1 πab π ab + 12 q −1 π 2 ] = 0 ,
(6.11)
usually referred to as the vector constraint and the scalar constraint, respectively.
With this symbol convention the action (6.9) becomes simply
Z
ab
S[π , qab , Na , N ] =
dt d3 x π ab q̇ab − Nb V b − N S .
(6.12)
R×Σ
We can also readily identify the hamiltonian density
H(π ab , qab , Na , N ) = Nb V b (π ab , qab ) + N S(π ab , qab ) .
(6.13)
Notice that the hamiltonian (6.13) is a combination of constraints, i.e. it vanishes identically on solutions, which is a standard feature of generally covariant
systems.
The symplectic structure is very easily found to be
a b
δd) δ(x − y)
{pab (x), qcd (y)} = δ(c
(6.14)
and zero otherwise.
81
6. Canonical Quantum Gravity: from ADM formalism to Ashtekar variables
Imagine we want to (canonically) quantize this system. Firstly we should
find a Hilbert space in which to represent the algebra (6.14). Notice that our
configuration space is the space of all the 3-metric on the space manifold Σ. We
shall call this space Met(Σ). Thus what we are looking for is something like
L2 (Met(Σ), µ) with respect to a suitable measure µ defined on Met(Σ). This is
the first great problem of our program of quantization. It is extremely difficult
to define such a measure! Indeed this space in not even a vector space, since
Einstein equations are non-linear.
Moreover, there are other difficulties. Namely We have to impose the constraints (6.10), (6.11). They’re expression in terms of configuration variables
and momenta is extremely unfit to quantization, they are non-polynomial and
√
contain nasty factors like q creating creates all sorts of ambiguities.
By the way, these kind of difficulties are much the same as the ones of YangMills QFT. In that case, however, the perturbative approach (namely to ‘linearize’ the equations of motion) proves to be successful, i.e. the Yang-Mills
theory is renormalizable.
6.2. The tetrad/triad formalism
We have some more work to do in order to have a suitable form to quantize. In
this section we shall talk a bit about the introduction and the importance of the
tetrad/triad formalism. Let’s for a moment forget the ADM spacetime splitting
and work covariantly in four dimensions. The tetrad field, or vierbein field, or
frame field is a rule that assigns to each point of spacetime an orthonormal local
inertial frame. Let’s do it carefully, since this is often a misleading concept.
You recall that on each spacetime point it is possible to choose a coordinate
system in which the Levi-Civita connection is zero (but not its derivatives!).
This set of coordinates is usually called Riemann normal coordinate system,
or “free fall” coordinates. Indeed they are just the mathematical expression
of the equivalence principle: this system embodies the possibilities of (locally)
“eliminating” the gravitational field by “falling” in it. Let us call ξ I , I = 0, 1, 2, 3
these coordinates. Obviously we can’t define in general a single flat coordinate
system on the whole manifold3 ; if so the manifold would be flat. Their key
feature is that they are orthonormal, that is
gIJ = ηIJ ,
(6.15)
locally, i.e. near a specific point: this expression is not true on all the manifold.
If we move to a “distant” event, it breaks down, in general. However we can
3
Or, better, these coordinates will be flat only in a small region of spacetime.
82
6.2. The tetrad/triad formalism
always change coordinates to another ξ I system, in which this flatness is true in
that point. The transformation matrix between these free fall coordinates-field
and a generic coordinate system, is what we call a tetrad4 , namely
eIµ (x) =
∂ξ I
(x) .
∂xµ
(6.16)
The important thing is that on each point we can put a tetrad eµI (x) which is
thus a tetrad field. There is a fundamental relation between the tetrad and the
metric:
ds2 = ηIJ dξ I dξ J = ηIJ eIµ (x)eJν (x)dxµ dxν = gµν dxµ dxν ,
(6.17)
which means
gµν (x) = eIµ (x)eJν (x)ηIJ .
(6.18)
Thus one can always “reconstruct” the metric from the tetrad. By the way,
the tetrad carries some redundancies: indeed the metric has 10 independent
components, while the tetrad 16. This is due to the fact that we can always
Lorentz-rotate a free fall system to get another equivalent free fall system:
eIµ (x) → eeIµ (x) = ΛIJ (x)eJµ (x) .
(6.19)
This is true whatever Lorentz transformation we attach at each particular point,
i.e. it is a Lorentz gauge transformation. Indeed the Lorentz group SO(3,1) is a
6-parameter group, which matches our counting of independent components.
This is the intuitive physical picture. Actually not so many words are necessary
to formally introduce the tetrads. Tetrads are simply a basis of orthonormal
vector fields. I.e. they are four vector fields eI such that g(eI , eJ ) = ηIJ . Every
vector can be expanded in a coordinate basis ∂µ thus
eI = eµI ∂µ ,
(6.20)
while, in general, the tetrad are not a coordinate basis.
There is however a high brow (but useful) way of saying these same things, in
the language of bundles. We are indeed picking a local isomorphism (sometimes
called a trivialization)
e : M × Rn → T M ,
(6.21)
which permits us to describe (again, locally) the tangent bundle T M as the
trivial bundle M × Rn (obviously for us n = 4, but this is a rather general
construction)5 .
4
Actually these are the components of a co-tetrad, since it is a 1-form; it is however common
to call it tetrad as well.
5
Notice that the existence of this local isomorphism is part of the definition of a differentiable
manifold, namely to locally look like Rn .
83
6. Canonical Quantum Gravity: from ADM formalism to Ashtekar variables
This trivialization is what high brow people call a frame field. Notice that,
seen from this perspective, everything sounds quite natural: a section of the
trivial bundle is just a function on M with value in Rn , and we can take a basis
of these sections ξI . Each section of the trivial bundle can thus be expanded as
s = sI ξI 6 . If we take e(ξI ) we obtain a basis of sections of the tangent bundle,
which can thus be expanded in a coordinate basis
e(ξI ) = eµI ∂µ .
(6.22)
The trivial bundle carries a metric structure, the Minkowski metric. Thus we
can rise and lower the flat indices with ηIJ . This is a quite powerful tool, but it
is useful only if we require
g(v, w) = η(e(v), e(w)) ,
(6.23)
namely that the local isomorphism sends η to g. The condition for this to be
fulfilled is simply
g(e(ξI ), e(ξJ )) = ηIJ
(6.24)
that is, we want orthonormal frame fields in order to work with the flat metric
in the trivial bundle and then simply “translate” the results in T M via e.
To get the expression of equation (6.18) between the metric g and the trivial
metric, we need the inverse frame field e−1 (the co-tetrad, which is usually called
tetrad as well, as argued above). Indeed
gµν = η(e−1 (∂µ ), e−1 (∂ν )) = η(eIµ ξI , eJν ξJ ) = ηIJ eJν eIµ .
(6.25)
Remark. An important remark is to be done now. We have a natural step
forward to do here: recognize that, along with the trivial vector bundle, we get –
by virtue of the invariance under (6.19) – a principal SO(3, 1)-bundle, sometimes
called the frame bundle. We can think of it as if we were attaching a copy of
SO(3, 1) to each point in spacetime; a section of this bundle (namely a choice of
a specific tetrad among the equivalent ones) is just a gauge choice.
Now I want to briefly show (or hint) that the whole General Relativity, and in
particular the Einstein-Hilbert action (6.1), can be seen as a theory of a tetrad
field eIµ (x) and connection. First of all, we have seen that there are two bundles
in the play: a vector bundle T M and a principal bundle with SO(3, 1)-fiber.
We have a well defined connection in T M, the Levi-Civita connection Γ, defined
6
Notice that here we are in the trivial bundle, which is not the tangent bundle, they are
only locally isomorphic (this is the same as saying that flat coordinates can be defined only
locally). This justifies our use of different kind of indices, capital latin letters, which are
sometimes called flat indices.
84
6.2. The tetrad/triad formalism
as the only torsionless connection compatible with the metric g: ∇g = 0, with
∇ = ∂ + Γ the covariant derivative associated with the parallel transport defined
by Γ. We can define a connection on the frame bundle as well, we shall call it
ω. It will define a covariant differentiation which will “act on” the flat indices,
i.e. it defines a parallel transport on the frame bundle. When we have objects
with ‘a leg in T M and one in the frame bundle’ we have the following “total”
covariant derivative.
I
Dµ vνI = ∂µ vνI − Γαµν vαI + ωµJ
vνJ .
(6.26)
Now, the requirement is that this covariant derivative has to be compatible with
the tetrad (in order to have scalar products invariant under parallel transport
both in the tangent and in the frame bundle), i.e.
Dµ eIν = 0 .
(6.27)
This requirement links together the Levi-Civita connection and the frame bundle
connection, namely
I
ωµJ
= eIν ∇µ eνJ ,
(6.28)
thus implying that the frame bundle connection ω is not arbitrary and contains
informations on the Levi-Civita connection as well. Such a connection is called
the spin connection, and we shall indicate it as ω(e) when we want to stress
that it is tetrad-compatible (namely that equation (6.28) holds). One can intuitively think that the Levi-Civita connection is the gauge potential due to the
diffeomorphism gauge freedom of general relativity, while the ω connection is
the gauge potential of SO(3, 1) gauge freedom.
The curvature is defined as usual
I
F IJ = dω IJ + ωK
∧ ω KJ ,
(6.29)
and it’s of course a 2-form. If we use the spin connection “definition” (6.28) we
have easily
IJ
Fµν
(ω(e)) = eIα eJβ Rαβµν ,
(6.30)
where R is the Riemann tensor constructed out of the metric defined by e
(through the usual relation (6.18)). This is a central result for our goal: the
Riemann tensor is nothing but the curvature of the spin connection. With this
machinery at hand, it is now a matter of calculations to show that the following
action
Z
1
eI ∧ eJ ∧ F KL (ω)
(6.31)
S[e, ω] = IJKL
2
M
has the same equations of motions of the Einstein-Hilbert action (6.1). Notice
that we have explicitly underlined that the action (6.31) is to be considered
85
6. Canonical Quantum Gravity: from ADM formalism to Ashtekar variables
with both e and ω as independent variables. Many of the readers have certainly
recognized the first-order or Palatini formulation of GR. The variation with
respect to the connection simply implies that ω is the spin connection on-shell.
The vanishing variation with respect to the tetrad gives Einstein equations. We
could have done the same thing with a second order (standard) formulation, in
which the action is the same as (6.31), but we take the curvature to be the one of
the spin-connection ab initio: again, the equation of motions for e are Einstein
field equations. However, it was our goal to stress that Einstein gravity can be
seen as a theory of ‘tetrads and connections as independent variables’.
Remark. Notice that the action (6.31) is actually an extension of the EinsteinHilbert gravity, in fact (6.31) is well defined also in the degenerate case where
the determinant of the metric is 0, i.e. when the metric is not invertible. In this
case the tetrad/triad formulation does not crash and still has something to say.
Let us now go back to our ADM splitting. We introduce the frame field
formalism, but we shall need it only on the spatial manifold Σ. Thus we call it
a triad field. The analogous of equation (6.18) shall be:
qab = eia ejb δij ,
(6.32)
where now we use i, j. . . to label the internal 3-space. We introduce also the the
densitized triads
1
(6.33)
Eia = abc ijk ejb ekc ,
2
and
1
Kab Ejb δ ij .
(6.34)
Kai = p
det(E)
One can check that
π ab q̇ab = 2Eia K̇ai .
(6.35)
Thus we now have, for the action (6.12)
S[Eja , Kaj , Na , N, N j ]
Z
=
h
dt d x Eia K̇ai
3
R×Σ
−Nb V b (Eja , Kaj )
− N S(Eja , Kaj ) − N i Gi (Eja , Kaj ) . (6.36)
where
Gi (Eja , Kaj ) = ijk E aj Kak .
Some remarks:
86
(6.37)
6.3. The Ashtekar-Barbero variables
• there are here three constraints the vector constraint V b imposing spatial diffeomorphism-invariance, the scalar constraint S imposing time diffinvariance and the Gauss constraint Gi , which is due to the redundancy in
using tetrads instead of metrics: namely we have a SO(3) gauge freedom
(all orthonormal frames in an euclidean 3d manifold are such up to rotations), which is the spatial analog of the Lorentz gauge freedom analyzed
above. The Gi constraint is there in order to impose physical states to be
SO(3)-invariant.
• This action is written in terms of densitized triads E and their conjugate
momenta, the K’s defined in (6.34). The latter are, essentially, a redefinition fo the extrinsic curvature. Thus here the connection is still in a second
plane: we shall need another change of variables in order to have it clearly
in the play.
6.3. The Ashtekar-Barbero variables
First of all, let us note that the spin connection ωaij (based on Σ) is a 1-form
with value in the Lie algebra so(3)7 Thus, we can expand ω on a basis of the
algebra, for example on the standard Pauli matrices τi = σi /2
ωaij = ωak (τk )ij ,
(6.38)
which (trivially) allows us to speak of ωai , with a single internal index i.
There is now one key change of variables that can be done now. We define a
new object as follows
Aia = ωai + γKai ,
(6.39)
where γ is a real arbitrary parameter, called the Immirzi parameter. It is easy
to see that A is still a connection. Indeed K i transforms as a vector under local
SO(3) transformations, while ω transforms as a connection, namely
g
ωi →
− g(x)ω i (x)g −1 (x) + gdg −1 (x) ,
g
Ki →
− g(x)K i (x)g −1 (x) ,
(6.40)
thus Ai transforms as a connection as well. This is known as the AshtekarBarbero connection 8 . The remarkable fact about this variable, is that its conjugate momentum is just the densitized triad, i.e.
{Aia (x), Ejb (y)} = γδji δab δ(x − y) ,
{Eia (x), Ejb (y)} = {Aia (x), Ajb (y)} = 0 ,
(6.41)
7
This is always the case: a connection on a principal G-bundle is always a 1-form with values
in g.
8
For more details on the Ashtekare-Barbero formulation of GR you can see [150]. For the
original Ashtekar’s contribution see [149].
87
6. Canonical Quantum Gravity: from ADM formalism to Ashtekar variables
The Einstein-Hilbert action becomes
Z
h
a
j
i
S[Ej , Aa , Na , N, N ] =
dt d3 x Eia Ȧia
R×Σ
−Nb V b (Eja , Aja )
− N S(Eja , Aja ) − N i Gi (Eja , Aja ) . (6.42)
with constraints
Vb (Eja , Aja ) = Eja Fab − (1 + γ 2 )Kai Gi ,
Eia Ejb ij k
i
Hb]j ,
k Fab − 2(1 + γ 2 )K[a
S(Eja , Aja ) = p
det(E)
Gi (Eja , Aja ) = Da Eia = ∂a Eia + ijk Aja E ak ,
(6.43a)
(6.43b)
(6.43c)
where Fab is the curvature of the Ashtekar-Barbero connection and Da is its
covariant derivative.
We have reached our goal: a formulation of gravity in terms of connection
and its conjugate momentum, the triad. It is nothing but a SO(3) gauge theory, plus constraints that implement space diffeomorphism invariance and time
diffeomorphism invariance. In a compact sentence we can say we have reduced
Einstein gravity to a background independent SO(3) Yang-Mills theory.
6.4. Smearing of the algebra
In order to give a geometric interpretation to the actual variables (Eja , Aib ) we
shall now smear them appropriately. Let us start from the densitized triad Eja (x).
This object is a 2-form, thus we shall smear it on a surface as follows
Z
Ei (S) =
Eia (x(σ))na d2 σ ,
(6.44)
S
b
c
∂x ∂x
where na = abc ∂σ
2 is the normal to the S surface. The object here defined,
1 ∂σ
Ei (S) is clearly the flux of the triad field across the surface, and is generally
referred to simply as ‘the flux’.
The Ashtekar-Barbero connection (6.39) is a one form, thus it is natural to
smear it along a one-dimensional path. Take a path γ with parametrization
xa (s) : [0, 1] → Σ and take Aa = Aia τi ∈ SU (2), with τi generators of SU (2).
The we can take the integral
Z
Z
A=
γ
88
0
1
Aia (x(s))τi
dxa (s)
ds .
ds
(6.45)
6.4. Smearing of the algebra
As one can see the triad field is identified on surfaces and its conjugate momentum, the connection, on paths. This is rather natural on a three dimensional
manifold Σ.
It is useful for future developments, to define the holonomy of the connection
along the path γ
Z hγ [A] = P exp
A
,
(6.46)
γ
where the P symbol stands for path ordering.
For future reference it is worth to introduce the so-called Holst action for GR.
We have seen that Einstein-Hilbert action (6.1) is equivalent to (6.31), which is
written in terms of tetrad fields and connections, in a first order formalism. We
can add to (6.31) a purely topological sector, which classically does not give any
contribution to the dynamics
Z
Z
1
1
I
J
KL
eI ∧ eJ ∧ F IJ (ω) .
(6.47)
S[e, ω] = IJKL e ∧ e ∧ F (ω) +
2
γ
The presence of the Immirzi parameter γ is of course arbitrary at this level. I
shall not go into details, however it can be proved that this action (6.47) is a 4d
manner to introduce the Ashtekar connection. Namely, introducing that topological sector amounts to shift the connection variable with the prescription (6.39).
See e.g. [150, 151, 153].
89
7. Loop Quantum Gravity
7.1. The program
The program of Loop Quantum Gravity is organized as follows:
(i) One starts with the canonical quantization of the fundamental variables
and choosing a representation reproducing the commutator algebra. The
program of Loop Quantum Gravity starts with choosing the connection
as the configuration space (position-like) variable. Next the kinematical
Hilbert space Hkin will be defined as the space of square integrable functions on the space of all the possible connections ψ(A), with respect to
some appropriate measure defined on the space, known as the AshtekarLewandowski measure. Notice that this Hilbert space is called kinematical
since it is not the actual Hilbert space of the theory, because we still have
to deal with the constraints, which – quite obviously – restrict this space
to one of its subspaces, the physical Hilbert space Hphys .
(ii) the following step is to deal with the constraints. The Gauss and Diffeomorphism constraints have a natural unitary action on Hkin , thus their
quantization is straightforward. Quite loosely the subspace thus obtained
is often indicated as Hkin as well. Sometimes, to stress the actual distincD
G
tion, one can find Hkin
⊆ Hkin
⊆ Hkin where the sup-scripts stand for
Diff constrained and Gauss constrained. Thus one obtains the space of
solutions to six of the nine constraint equations.
(iii) the problem and still an open issue of Loop Quantum Gravity, is of course
to find the space of solution of the scalar constraint, which actually drive
the time evolution of the system, i.e. the dynamics. Thus the physical
Hilbert space Hphys is still to be found, and also a proper scalar product
between physical states, i.e. quantum transition amplitudes, which are the
real clue of a every quantum theory.
We will now briefly sketch each of these three steps, i.e. the definition of the
kinematical Hilbert space with the Ashtekar-Lewandowski measure, the implementation of the Gauss and diffeomorphism constraints, and we shall discuss a
bit the problems inherent to the quantization of the scalar constraint.
91
7. Loop Quantum Gravity
I encourage the interested reader to have a look to [150, 151, 152, 153] for very
good reviews of the basics of LQG and to [154, 155] for more modern approaches
to this subject.
7.2. The kinematical Hilbert space Hkin
Firstly we define the space of cylindrical functions based on a graph γ denoted
Cylγ 1 .
A graph γ is a collection of simple paths which meet at most at their endpoints.
We call links l all these paths and denote with L their total number. We say
thatγ is in Σ when all the links (as paths) belongs to Σ themselves. Given a
graph γ ⊂ Σ and a smooth function f : SU (2)L → C, we define a cylindrical
function ψγ,f ∈ Cylγ as
ψγ,f [A] = f (hl1 [A], hl2 [A], . . . hlL [A]) .
(7.1)
You have to think these paths as the ones of section 6.4, i.e. as the natural
one-dimensional objects embedded in the spatial hypersurface Σ upon which the
connection A is integrated.
Next we define the space of cylindrical function on Σ as
[
Cyl =
Cylγ ,
(7.2)
γ⊂Σ
where the union means over all the graphs in Σ. Now for the measure. Let’s
define the Ashtekar-Lewandowski measure [157]:
Z Y
µAL (ψγ,f ) =
dhl f (hl1 , hl2 , . . . hlL ) ,
(7.3)
l⊂γ
where dh is the normalized Haar measure over SU (2). Recall that the Haar
measure is normalized to 1, thus µAL (1) = 1.
With this structure one can define the inner product in the space of Cyl:
Z Y
∗
0
0
hψγ,f | ψγ ,g i = µAL (ψγ,f ψγ ,g ) =
dhl f ∗ (hl1 . . . hlL ) g(hl1 . . . hlL ) . (7.4)
l⊂γ∪γ 0
Now we are able to define the kinematical Hilbert space Hkin as the Cauchycompletion of the space of cylindrical function in the Ashtekar-Lewandowski
1
I strongly recommend the reading of [156] for a very clear and suggestive presentation of
this subject.
92
G
7.3. The Gauss constraint. Hkin
measure. That this is indeed a Hilbert space is not obvious, and it was proved
by Ashtekar and Lewandowski [157].
Having defined properly the Hilbert space, we can now move to find a basis.
We shall use the Peter-Weyl theorem, which can be seen as a form of generalized
Fourier expansion. Namely, every function f ∈ L2 [SU (2), dµHaar ] can be written
as follows
Xp
0
j
2j + 1fjmm Dmm
(7.5)
f (g) =
0 (g) ,
j
where the D’s are nothing but the Wigner SU (2)-representation matrices, j is
the spin label and run from 0 to infinity in half integer steps, and the f (Fourier)
coefficients are given by
Z
p
j
mm0
= 2j + 1
fj
dµHaar f (g)Dmm
(7.6)
0 (g) .
SU (2)
The completeness relation reads
X
δ(gh−1 ) =
(2j + 1)Tr(Dj (gh−1 )) .
(7.7)
j
We can easily apply this to any cylindrical function
L
Y
X m ...m ,n ...n
p
j1
jL
1
L 1
L
Dm
ψγ,f [A] =
2ji + 1
fj1 ...j
(hl1 [A]) . . . Dm
(hlL [A])
1 n1
L nL
L
i=1
j1 ...jL
(7.8)
where the Fourier coefficient are obtained by taking the inner product (7.4) of the
ψ function with the tensor product of the Wigner matrices. Thus we
√ have found
j
a complete orthonormal basis of Hkin , namely – calling φjmm0 = 2j + 1Dmm
0
the normalized Wigner matrix – the functions
L
Y
φjmi i m0 ,
(7.9)
i
i=1
with the remark of taking all the possible graphs in Σ and all the values of spins
associated to the links of the graph.
G
7.3. The Gauss constraint. Hkin
Now we want to restrict the Hilbert space on its gauge invariant subspace, i.e.
the subspace in which the solutions of the Gauss constraint live. What is to do,
is simply to take only the states of Hkin which are SU (2) invariant. For this
93
7. Loop Quantum Gravity
purpose, let us try to understand how SU (2) gauge transformations act on the
basis (7.9).
It is indeed very easy to infer the result of a (finite) SU (2) gauge transformation on a general cylindrical function from the behavior of the holonomy:
g
hl [A] →
− g(x(0))hl [A]g −1 (x(1)) ,
(7.10)
where x(s) is the parametrized path. Usually, given a path/link l, one calls s(l)
the source of the path, i.e. the x(0) point, and t(l) the target x(1), thus writing
g
−1
the previous equation as: hl [A] →
− gs(l) hl [A]gt(l)
. Then, the action on a general
graph is
g
−1
−1
ψγ, f [A] = f (hl1 , . . . hlL ) →
− f (gs(l1 ) hl1 [A]gt(l
, . . . gs(lL ) hlL [A]gt(l
),
1)
L)
(7.11)
and we would like to find a basis of the space of cylindrical functions with such
a property. Let us denote with U (g) the operator that acts the transformation
(7.11), i.e. – writing it for the basis (7.9) –
−1
U (g)φjmm0 (hl ) = φjmm0 (gs(l) hl gt(l)
),
(7.12)
or, more generally,
U (g)
L
Y
φjmi i m0 (hli )
i
=
L
Y
−1
).
φjmi i m0 (gs(li ) hli gt(l
i)
(7.13)
i
i=1
i=1
Notice that the action of a gauge transformation is on the nodes only: We can
focus on the request of invariance on a single node, and then extend trivially the
results to every node of a graph. In order to make things as clear as possible
and also useful for future reference, let us take a 4-valent node n0 . Let us write
down in a smart way its generic state
X j
m1 m01 ...m4 m04
j4
1
ψγ, f [A] =
φm1 m0 (hl1 [A]) . . . φm4 m0 (hl4 [A]) Rj1 ...j
,
(7.14)
4
1
j1 ...j4
4
where R is all the rest of the factorization. l1 to l4 are the four links converging
at n0 . What we are going to do is to render invariant this small piece of graph,
that is the one sourrounding this node. The idea is to group average, i.e. to take
the action of a gauge transformation in n0 and to integrate over all the possible
such transformations:
Z
j1
j4
φm1 m0 (hl1 [A]) . . . φm4 m0 (hl4 [A]) →
dg φjm1 1 m0 (ghl1 [A]) . . . φjm4 4 m0 (ghl4 [A])
1
4
SU (2)
1
4
(7.15)
94
G
7.3. The Gauss constraint. Hkin
where we have assumed that n0 is the source for all the four links, but it would
be the same otherwise. It is clear that the result is invariant under the action
of U on the node n0 , since the Haar measure is such! Now, we can simply do
this on each node of a graph and obtain the desired invariant basis. But let
us investigate a bit more such a projection on the single node n0 : since the φ
functions are essentially Wigner matrices it is obvious that
φj (gh) = Dj (g)φj (h) ,
(7.16)
and then the projected state looks like
Z
j1
j4
dg Dm
(g)
.
.
.
D
(g)
φjm1 1 m0 (hl1 [A]) . . . φjm4 4 m0 (hl4 [A]) .
0
0
m4 m
1m
(7.17)
Let us give a name to this operator
Z
j1
j4
n0
dg Dm
Pm1 m0 ...m4 m0 =
(g)
.
.
.
D
.
(g)
0
m4 m0
1m
(7.18)
1
SU (2)
1
4
4
1
4
1
SU (2)
4
This is indeed a projection operator from the tensor product of the four representation spaces to its SU (2) invariant subspace, where the projection property
P n0 P n0 = P n0 is due to the Haar measure invariance, as it is easy to check. We
can write it as
P n0 : V j1 ⊗ V j2 ⊗ V j3 ⊗ V j4 → Inv[V j1 ⊗ V j2 ⊗ V j3 ⊗ V j4 ] ,
(7.19)
We can pick an orthonormal basis in Inv[V j1 ⊗ V j2 ⊗ V j3 ⊗ V j4 ] – let us call it
|ιi – and decompose P n0 as
X
P n0 =
|ιihι| .
(7.20)
ι
Obviously the dimension of the invariant subspace depends on the spins that
contribute to the node. The basis vectors |ιi, i.e. an orthonormal basis of the
invariant subspace of a tensor product of vector spaces, are usually known as
intertwiner operators or simply intertwiners.
We can do a very simple calculation just to explain how things works. Take a 3-valent
node with spins j1 , j2 j3 . we have to deal with V j1 ⊗ V j2 ⊗ V j3 , which is often written
in the sloppy way j1 ⊗ j2 ⊗ j3 . Then we decompose it into the sum of irreducible
representations, namely
j1 ⊗ j2 ⊗ j3 = (|j1 − j2 | ⊕ . . . j1 + j2 ) ⊗ j3 ,
(7.21)
95
7. Loop Quantum Gravity
and so on, which is of course the well-known Clebsh-Gordon condition. Take three
values, for example j1 = 1/2, j2 = 1, j3 = 3/2, then the decomposition reads
1
3
1 3
3
⊗1⊗ =( ⊕ )⊗ =1⊕2⊕0⊕1⊕2⊕3 .
2
2
2 2
2
(7.22)
The invariant subspace is, by definition, the 0 representation space. The dimension of
Inv[j1 ⊗j2 ⊗j3 ] is given by the multiplicity of the 0 representation in the decomposition.
It is not hard to see that for a 3-valent node there will be at most a one dimensional
invariant subspace, and this is the case when the ‘selection rule’
j3 ∈ {|j1 − j2 | . . . j1 + j2 }
(7.23)
holds. This means that for every 3-valent node that satisfy the selection rule (7.23),
the projector into the invariant subspace is unique up to normalization. Actually it is
just the (normalized) Wigner 3j-symbol [160]
j1 j2 j3
j1 ,j2 ,j3
hα1 , α2 , α3 | ιi = ια1 ,α2 ,α3 ∼
.
(7.24)
α1 α2 α3
Every node with valence more than 3 can be decomposed into 3-valent contractions.
For instance, a 4-valent intertwiner can be written as
viα1 ,α2 ,α3 ,α4 = ιαi 1 ,α2 ,β ιi,βα3,α4 ,
(7.25)
the label i is a spin representation, it ranges over the Clebsh-Gordon-allowed representations joining the four spins. Graphically is everything quite self-explanatory: you
can see the 4-valent intertwiner as
j3
j1
i
j4
j2
where i is sometimes said to be a virtual link, since it is a spin, but it is spans all the
allowed spins. Of course one can join the four different links in more than a way. In
this regard the following equality holds
j1
j3
j1
i
j2
j3
=
j4
X
j
j2 j1 j
(2j + 1)
j3 j4 i
(7.26)
j
j2
j4
I refer to the appendix of [151] for more details on SU (2) recoupling theory.
96
D
7.4. The vector/diffeomorphism constraint. Hkin
In general, for a v-valent node n, one can define with the same procedure of
group averaging, a projection operator P n , given by
X
∗
,
(7.27)
ιαmv1 ...mv ιαn1v...n
Pmn 1 ...mv ,n1 ...nv =
v
αv
where αv denotes the elements of the basis. This is clearly a generalization of
equation (7.20).
G
Thus, we have come to a result: an orthonormal basis of Hkin
is given by the
following
O
O
sΓ,{i},{j} [A] =
ιn ,
(7.28)
φjl (hl [A]) ·
l
n
where l runs over the links and n overs the nodes of the graph Γ. The · stands
for the contraction between the Wigner matrices and the intertwiner operators.
These elements are manifestly SU (2) invariant, thanks to the contraction with
the invariant intertwiner operators, and they are know as spin network states 2 .
For future reference, and also to understand better the relation between the
graph and the intertwiners defined on it, let us analyze a bit the situation of a
4-valent node.
D
7.4. The vector/diffeomorphism constraint. Hkin
G
,
We shall now deal with space diffeomorphisms. We proceed just as for Hkin
G
i.e. we first see how diffeomorphisms act on the vectors of Hkin , then we try
to “average” over all the possible actions, obtaining a diff-invariant subspace.
Here, however, one must be more careful, since diff-invariant functions surely
G
won’t be inside Hkin
, because the orbits of the action of diffeomorphisms are
not compact. It is just as if you want to constrain wave functions defined on a
cylinder ψ(θ, x), ψ ∈ L2 (S 1 × R), to p̂θ ψ = 0 and p̂x ψ = 0. This constraints say
simply that ψ cannot depend neither on x nor on θ, it is a constant. But while
the integration on θ gives no problem, since S 1 is compact, the integration over
x is not bounded, and the constrained states do not belong to L2 . However it is
always possible to choose a suitable dense subset Φ ⊂ L2 of test functions and
define the constrained states as distributions on this space. Then one gets the
Gelfand triple (see, e.g. [163]) Φ ⊂ L2 ⊂ Φ∗ , where Φ∗ is the ‘extension’ of L2
we are looking for.
If φ is a diffeomorphism of Σ, then its action on a cylindrical function ψγ,f
(7.1) is obvious
UD [φ]ψγ,f [A] = ψφ−1 γ,f [A] .
(7.29)
2
See [158, 159] for interesting details.
97
7. Loop Quantum Gravity
Then the vector constraint can be rephrased as
UD [φ]ψ = ψ ,
(7.30)
where, as we have just said, the solutions can be found only in distributional
sense, namely Cyl ⊂ Hkin ⊂ Cyl∗ , with ψ ∈ Cyl∗ the space of linear functionals
on Cyl. Now it is not worth to go on in technicalities: one simply average the
distributions on Cyl over all possible diffeomorphisms, thus obtaining only diffinvariant states. A look to (7.29) suggests that the resulting space is formed by
the equivalence classes of graphs under spatial diffeomorphisms. These graphs
are, mathematically speaking, knots 3 , and the spin network states after this
D
procedure are also known as s-knots. They are a basis of the space Hkin
.
Summarizing/simplifying: We have first obtained the kinematical Hilbert
space from a suitable definition of a measure on Cyl. Then we have seen that imposing the Gauss constraint amounts to insert intertwiner operators at each node
of the basis functions. Finally we have shown that the diff constraints simply
say that we have to take the equivalence classes of the base graphs (knots).
7.5. Quantization of the algebra and geometric
operators
We have been able to interpret and “solve” the Gauss constraint without actually
quantize its corresponding operator in equation (6.43). Here we shall face the
problem of quantizing the triad operator Eia and the connection operator, or,
better, their smeared version as introduced in section 6.4. It is rather simple:
we have defined the Hilbert space Hkin as (an opportune definition of) L2 (A),
thus on then spin-network basis (7.28), the connection operator shall act by
multiplication, i.e. – considering for simplicity the fundamental representation
he = D1/2 (he ) –
ĥγ [A]he [A] = hγ [A]he [A] .
(7.31)
The flux shall act by derivation
Z
δhe [A]
= ±i~γhe1 [A]τi he2 [A] .
Êi (S)he [A] = −i~γ d2 σ i
δAa (x(σ))
S
(7.32)
Some remarks: the flux should be seen as a surface (namely the S surface) (see
figure 7.1); the holonomy (i.e. the smeared connection) is represented by the
path e. If the surface S and the path e does non cross each other then the action
above is identically zero. If they do cross each other, then the path is “cut” in
3
A knot is an embedding of a circle in a 3d space, up to isotopies. See, e.g. [147] for details.
98
7.5. Quantization of the algebra and geometric operators
e1
S
e2
Figure 7.1.: The flux surface S ‘punctured’ by an holonomy e = e1 ∪ e2 .
two sub-paths e1 and e2 and the flux operator inserts an SU (2) generator among
them. The sign in front depends on the relative orientation of S and e. Notice
that recovering all the fundamental constants one gets an overall Planck length
squared lp2 for Êi (S) (replacing the ~),
r
G~
lp =
' 10−33 cm ;
(7.33)
c3
this will be important to understand the scales we are working with.
Now that we have understood how the fundamental variables of our theory
work, once quantized as operator living in our Hilbert space, we can turn to really
interesting questions about the structure that emerges from this framework of
canonical Quantum Gravity.
Actually it is quite simple to see that the area of a surface S embedded in our
space manifold Σ can be written in a amazingly simple way in terms of triad
variables, namely:
Z
q
1
2
(7.34)
A(S) =
dσ dσ Eia Ejb δ ij na nb ,
S
G
but in our quantum theory the triad is an operator that acts on Hkin
, what shall
become the area of a surface?
First of all let us understand how the product of two triads acts on holonomies.
With (7.32) is easy to see that
Êi (S)Êj (S)he [A] = −lp4 γ 2 he1 [A]τi τj he2 [A] .
(7.35)
When the two fluxes are contracted one obtains the Casimir operator of the
representation. In this case we have simply holonomies, i.e. fundamental representation, thus C 2 = τi τ i = − 34 112 . Notice that the Casimir commutes with all
99
7. Loop Quantum Gravity
j3
j1
S
SI
j2
Figure 7.2.: A surface S intersected by a generic spin network. You can see that
refining again the regularization does not change the result, since
each sub-cell has already at most one puncture.
the group elements: this fundamental feature allows us to “recompose” the path
e, namely
Êi (S)Ê i (S)he [A] = −lp4 γ 2 C 2 he [A] .
(7.36)
This means that the holonomy is an eigenstate of the square of fluxes! In a
generic spin-j representation one gets
Êi (S)Ê i (S)Dj (he [A]) = lp4 γ 2 j(j + 1)Dj (he [A]) ,
(7.37)
since the Casimir reads Cj2 = −j(j + 1)112j+1 .
Now let us turn back to the area (7.34). We have to write it in a way to
include the smeared version of the triad, the flux, in order to be able to act with
it on a general state vector. Thus we have to pick up a regularization of the area
(7.34): We decompose the surface S into N 2-cells, and write the integral as the
limit of the Riemann sum
A(S) = lim AN (S) ,
N →∞
with
AN (S) =
N
X
=
p
Ei (SI )E i (SI ) .
(7.38)
(7.39)
I=1
Summarizing: This amounts to divide S in N 2-cells, take the area of each of
them AN (S), and let N go to infinity. It is just a way to regularize the integral
over S. Next
q we have to deal with AN (S). We shall see what kind of action the
operator
100
Êi (SI )Ê i (SI ) does on a generic state vector ψΓ . The trick is to take
7.5. Quantization of the algebra and geometric operators
the regularization sufficiently fine so that each SI intersects once and only once
the embedded graph Γ. If so, we already know how this operator acts, namely
q
q
Êi (SI )Êj (SI ) ψΓ = γlp2 jp (jp + 1) ψΓ ,
(7.40)
where p denotes the single link of Γ that puncture that specific SI . Refining
again the decomposition will have no consequences, since at most we have no
intersection at all for some cells, but that gives a zero contribute to the area (see
figure 7.2). Thus, at the end,
q
X
(7.41)
Â(S) ψΓ =
γlp2 jp (jp + 1) ψΓ .
p∈S∪Γ
The great clues of this formula are:
• the area operator is discrete,
• its eigenfunctions are just the spin network states,
• a part from the Immirzi parameter (which is a free parameter of LQG,
even if there are some proposals to fix its value through the Black Hole
entropy calculation [161, 162]) the area eigenvalues are of the scale of the
Planck length squared, which is precisely what intuitively we expected.
We shall see in a moment what a powerful intuitive picture these two key points
provide. First, let us deal with the volume operator. We have to say that this
issue contains some technicalities, for instance there are at present two distinct
mathematically well defined volume operators. The distinctions come out in the
regularization process. We shall not discuss it here, we merely present the results
that are in agreement for both the versions, which – incidentally – are the truly
interesting part of the story. The volume operator has the same two remarkable
features of the area operator, namely it has discrete spectrum and it is diagonal
in the spin network basis, and is of order lp3 .
Now we are ready to argue the picture that quantization is suggesting us: space
geometry is discrete at the Planck scale. Each spin network is a polymer-like
excitation of space: volume excitation being dual to the nodes of a spin network
and area excitation dual to the links. The area excitation is proportional to the
spin quantum number, attached to every link of a spin network, while volume
ones are determined by the intertwiner space of the node to which it is dual.
This interpretation has also the power of being fully background independent,
space geometry is determined in a pure combinatorial way by the spin network
state, which thus can be seen as a quantum geometry of space. Something which
101
7. Loop Quantum Gravity
is obvious, but it is worth some words, is that this is not a built-in discretization,
as it for instance in the lattice formulation of Quantum Gravity: quantization
is telling us that space has a discrete nature, whose building blocks are of the
order of the Planck scale.
Of course one can rise many objections to all the structure we have built so far,
but it cannot be denied that this picture is extremely suggestive and powerful,
and this is at the heart of Loop Quantum Gravity itself.
7.6. The scalar constraint
One step remains to be analyzed in order to complete the program of (canonical)
Loop Quantum Gravity, i.e. to deal with the scalar constraint (6.43). Let us
recall its expression, in a smeared form
Eia Ejb ij k
i
Hb]j
k Fab − 2(1 + γ 2 )K[a
d3 x N p
det(E)
Σ
Z
S(N ) =
= S E (N ) − 2(1 + γ 2 )T (N ) .
(7.42)
(7.43)
The second line is a usual shorthand notation to separate the so-called euclidean
contribution
Z
Eia Ejb ij k
E
3
S (N ) =
d x Np
k Fab
(7.44)
det(E)
Σ
from the rest
Eia Ejb
i
d x Np
K[a
T (N ) =
Hb]j .
det(E)
Σ
Z
3
(7.45)
The non-linearity if this expression is really awkward, in order to be able to
quantize it properly. Notice that this is just the same problem one had before
the loop representation. And it is no mystery that this problem remains the real
big problem of canonical Quantum Gravity.
However, the rich structure we have built so far, surely has made possible many
important steps towards a better comprehension of the quantum scalar constraint
and (particularly) of its action on the kinematical state space. We shall not deal
here with the many complicated open issues in this respect [165, 166] but we
want at least to explain what has and what has not been achieved by now.
One important improvement has been put forward by Thiemann [164]. He
observed that, if one introduces
Z
e
(7.46)
K=
Kai Eia ,
Σ
102
7.6. The scalar constraint
then the following identities hold
e ,
Kai = γ −1 (Aia − Γia ) = γ −1 {Aia , K}
(7.47)
E
e = {S (1), V } ,
K
γ 3/2
Eia Ejb ijk
4
p
abc = {Aka , V } ,
γ
det(E)
with V =
(7.48)
(7.49)
Rp
det(E). This is actually very useful, since (7.43) becomes
Z
E
S (N ) =
i
d3 x N abc δij Fab
{Ajc , V } ,
(7.50)
and
Z
T (E) =
d3 x
N abc
ijk {Aia , {S E (1), V }}{Ajb , {S E (1), V }}{Akc , V } .
γ3
(7.51)
Now the trick proceeds by rewriting both the connection and the curvature in
terms of holonomies and so come to an expression that involves only the volume
and the holonomies. This expression can be transformed into an operator, since
we already know how ĥ and V̂ act on a generic state of our Hilbert space. We do
not list here all the passages, be sufficient to say that thanks to the well known
hea [A] ' 1 + εAia τi + O(ε2 ) ,
(7.52)
– where ea is a path along the xa direction – one can express both A and F in
terms of h. The integral in (7.43) must be regularized as for the area and volume
operator, thus the space must be decomposed into a cellular decomposition, a
triangulation, i.e. a collection of tetrahedra bound together. We give here the
regularized expression for the euclidean part of (7.43):
S E (N ) = lim
ε→0
X
I
NI ε3 a bcTr(Fa b(A){Ac , V }) =
lim
ε→0
X
I
abc
NI h
(hαIab [A] −
h−1
[A])h−1
[A]{h−1
[A], V
eIc
eIc
αIab
i
} , (7.53)
ε3 is the volume of a cell, αab is an infinitesimal loop in the ab-plane, around the
face of the I-th cell; while eIa is an infinitesimal path along the a-direction, along
an edge of the I-th cell (see figure 7.3). Notice that the dependence on the cell
scale disappears in terms of holonomies.
103
7. Loop Quantum Gravity
ec
αab
Figure 7.3.: One tetrahedron of the regularization cell-decomposition. One faceloop and one edge-path are shown.
l
l
p
Ĥ
m
q
∼
p
s
.
(7.55)
r
m
Figure 7.4.: Typical action of the hamiltonian operator on a node of spin network. In the framework we have sketched s = 1/2, but we shall see
later that links with generic spin label could be consistent as well,
and this is one of the (many) ambiguities in the quantization of the
scalar constraint.
Now we can formally ‘quantize’ this operator (7.53) by putting an hat over
the volume and the holonomies and see how it acts on a spin network:
h
i
X
−1
−1
Ŝ E (N ) = lim
NI abc (ĥαIab [A] − ĥ−1
[A])
ĥ
[A]{
ĥ
[A],
V̂
}
.
(7.54)
I
I
I
e
e
α
ε→0
I
ab
c
c
D
: it
This operator is well-defined and it is very easy to see how it acts on Hkin
has the property of the volume operator to act only on nodes of spin networks,
while the holonomies in (7.54) modifies the spin network by creating new links
in the fundamental representation (spin 1/2) (see figure 7.4).
(We have dealt only with the euclidean part, but a similar analysis can be
done for the T part of (7.43) as well, and the following results still hold.) This
way of acting is at the real clue of Loop Quantum Gravity with respect to the
quantization of the scalar constraint. For example, take all the spin network
states with no nodes: they are just Wilson loops, i.e. the trace of the holonomy
around a closed path. All the Wilson loops, irrespectively from the spin, are
solutions of the quantum scalar constraint equation! Actually this simple but
important fact is what started the interest in Loop Quantum Gravity: it provides
a set of explicit solutions of the quantum theory of gravity. Moreover, equation
104
7.6. The scalar constraint
(7.54) tells us that the action on nodes is rather peculiar: it creates special links,
sometimes called exceptional links. They are special in the sense that the new
nodes they carry are exactly of zero volume, thus being ‘invisible’ to a further
action of the hamiltonian operator. This allow the following picture: a generic
solution of the quantum scalar constraint is labeled by graphs with nodes of the
following kind
Ω
=α
+ ... + β
+ ... + γ
+ ... ,
(7.56)
also known as dressed nodes, i.e. with infinite superpositions of exceptional
edges. The Ω label stands for the collection of weights of the superposition.
Up till now the good points. Now let us spend some words for the big problems
still to be solved:
• the action of the hamiltonian operator that we have described above is
said to be ultra-local [167]. This feature has raised some concerns whether
this kind of theory is at all capable of reproducing general relativity in the
classical limit [167].
• there is a large degree of ambiguity in the definition of the quantum scalar
constraint! One of such ambiguities arises from using the holonomies in
the fundamental representation: we could have used any representation,
thus the action of the scalar operator would have been different, creating
links with arbitrary spin. This doesn’t affect the ultra-locality issue, but
certainly we would end up with an infinite set of dynamical theories, all
different. Another ambiguity is in the regularization scheme: the scalar
quantum operator is regularization-dependent!
• there is still a mathematical problem in analyzing thoroughly the limit in
(7.54) (and the analogous one for the T part of (7.43)), the limit must exist
and should give a well defined operator acting on the space of s-knots.
These kind of ambiguities have stimulated both the research into a better
comprehension of the scalar constraint itself, for example the Master constraint
approach [168]4 , and new paths of research, the most important of which is the
spinfoam formalism, to which we dedicate the next sections of this chapter.
4
See also the book by Thiemann [166] for a more self-contained and thorough presentation
of the entire scalar constraint problem.
105
7. Loop Quantum Gravity
7.7. Concluding remarks
I have tried in this chapter to review the very basics of LQG. Of course many
are the arguments I had to omit not to go beyond the goals I have in mind. Here
I show a schematic list of the most important issues I have not treated, but that
certainly deserve attention, in order to understand the global importance of such
theory:
• First of all the scalar constraints definitely deserves more attention. I refer
to [166].
• Black Hole entropy. The idea is to quantize a sector of the theory containing an isolated horizon and then to count the number of physical states
compatible with a given macroscopic area of the horizon. See [169] for
detailed calculations and [151] for discussions.
• Loop Quantum Cosmology. This is the cosmological sector of LQG, which
has rapidly developed since its birth in 2000. There are indeed quite interesting and stimulating results, particularly concerning the initial singularity of GR, i.e. precisely where we expected Quantum Gravity to
tell something. I refer to the review [174] and to the papers [170]-[173].
Recently also spinfoam calculations of cosmological problems have been
performed [175, 176, 177].
106
8. The spinfoam approach to the
dynamics
We have up till now described what “happens quantum mechanically” on spatial
hypersurfaces, and we have argued the difficulties in dealing with the evolution
of these quantum states in a timelike direction, difficulties connected with the
hamiltonian operator. However – with in mind the goal of some kind of sumover-histories formulation of the dynamics – we can take a quantum 3-geometry,
for simplicity a single spin network state (i.e. an eigenstate of area and volume)
|s0 i and imagine it evolving in a timelike direction. Imagine its “world-sheet”:
every node would sweep timelike paths, that we call edges, every link would
sweep faces and there will be some spacetime points in which two or three edges
meet, that is events in which, for instance, one edge split in two (or whatever)
or two or more edges converge in one, etc. . . We call these points vertices and
they are the points in which the original quantum state |s0 i changes by means
of the dynamics, i.e. by action of the hamiltonian operator 1 . Time evolution
has thus produced a 2-complex with vertices, edges and faces. This is a dressed
2-complex, since the coloring of the spin network will induce a coloring in the
2-complex as well: irreducible representations to faces and intertwiners to edges.
This dressed 2-complex is what is called a spinfoam.
The spinfoam approach to (Loop) Quantum Gravity began with the work of
Reiseberger and Rovelli [178, 179], which first put out the idea of defining quantum amplitudes as sum over histories starting from the hamiltonian formulation
(see next section for more details). Actually, work had already been done in this
respect (see for example [180]), but not in the LQG framework. The general
idea of a spinfoam derived from a path-integral discretization procedure (see
section 8.2), was instead proposed by Baez in the seminal paper [181] (in which
the name “spin foam” was first used) and in the lectures [182], which are still a
very good ‘beginner’s guide’ to spinfoam models, perhaps one of the best in the
literature. For other good, but a bit dated, reviews see for example [151, 152]; for
more up-to-date (but a bit more technical) resumes see instead [154, 155, 183].
1
thus we have already an idea about how two quantum states differing from a single vertex
– i.e. a action on a node by the hamiltonian operator – should be like, see section 7.6.
107
8. The spinfoam approach to the dynamics
Σ
s
v2
s′′
v1
j
s′
Σ′
ι
j
ι
Figure 8.1.: Example of spinfoam: evolution from a spin network s0 to s passing
through an intermediate spin network through two vertices, i.e. actions of the hamiltonian operators. You can see the edges (evolution
of nodes of s0 ), faces (evolution of links of s0 – one of the faces has
been colored as an example) and vertices.
8.1. Sum-over-histories from hamiltonian
formulation
We can now make a rather heuristic reasoning trying to “derive” a path integral formulation from the hamiltonian formulation, just as one does in standard
Quantum Mechanics. I want to stress again the heuristic approach of what follows, that is mainly taken by [151] (but you can find more detailed discussions
in [184]). In the next section we shall see a far more rigorous (and alternative)
definition of the sum over histories approach to the Quantum Gravity dynamics.
Rigorous proofs that the two derivations are actually the same thing have been
found only for the three dimensional case [185].
The spirit of the spinfoam approach is to try a Feynman-like procedure in
a gravitational and background independent context. In standard Quantum
Mechanics the Feynman idea is somehow summarized in the following expression
Z
0
hy, t | x, ti ∼
Dq eiS[q] .
(8.1)
q(t)=x
q(t0 )=y
On the left hand side we have the scalar product (transition amplitude) between
two different position eigenstates at two different times. The same amplitude
108
8.1. Sum-over-histories from hamiltonian formulation
can be written in the Heisenberg picture as hy | U (t, t0 ) | xi, with U (t, t0 ) evolution
operator form time t to t0 . In a gravitational context one should then give sense
to the following expression
Z
Dgµν eiSEH [g]
(8.2)
or, to be more specific,
0
A(g, g ) ∼
Z
g|t=1 =g 0
g|t=0 =g
Dgµν eiSEH [g] ,
(8.3)
which formally represents the transition amplitude from a space with metric g
to one with metric g 0 . The specific values of the time label are irrelevant for the
diffeomorphism invariance holds. There is one big trouble when approaching an
integral like the one in (8.3): we do not know a non perturbative definition of the
measure Dgµν and perturbatively we know that the theory is non-renormalizable
(and moreover in this case we should break the background independence).
In order to give a concrete significance to the expression (8.3) one usually
starts from the canonical formulation, which is just what is done for the actual
definition of path integrals in all the others fields. Let’s for a moment resume
what is the standard way to define the sum-over-paths procedure. The idea is
to take the time evolution operator e−iHt and to have it acting step by step (in
each step you have simply the action of H on a state, which you can calculate),
then taking the limit of this time step going to zero. This gives a mathematically
precise definition of the Feynman path integral representation of the propagator.
Unfortunately, we are not able to do the same thing in the Quantum Gravity
context. The problem is again the hamiltonian operator: we do not know how
to quantize it in a proper way, as was pointed out in 7.6. Thus, we cannot follow
this route pretending to obtain new rigorous insights.
However, we can still derive some basic properties that the Quantum Gravity
path integral should satisfy. We shall see that this will be enough to find at least
a form for Quantum Gravity transition amplitudes. Then, in the following section, we shall see an alternative way to follow, somewhat dual to the derivation
from the canonical formulation, which is called “spinfoam” approach.
Consider the integral (8.3): between which states the amplitude should be
computed? It should be computed between eigenstates of the three-geometry,
i.e. with states with a definite 3d metric: but these are nothing but the spinnetwork states. Thus what we would like to calculate is actually something
like
A(s, s0 ) = hs | s0 iphys = hs | A | s0 ikin .
(8.4)
109
8. The spinfoam approach to the dynamics
Let us try to explain thoroughly this last expression. |si is the usual spin-network
state (7.28), or, better, the s-knot states, i.e. with basis graph an equivalence
class of graphs under spatial diffs (see section 7.4). This is also the reason for
the subscript “kin”, to stress that the spin networks belong to the kinematical
Hilbert space Hkin , but not to the physical Hilbert space. So A (for ‘amplitude’)
represents the projector on the kernel of the hamiltonian operator Ĥ, i.e. the
projector onto the physical Hilbert space. If we assume for simplicity that the
hamiltonian Ĥ has a non-negative spectrum, then we could (formally) write
A = lim e−Ht ,
(8.5)
t→∞
indeed, if |ni is a basis that diagonalizes H (with eigenvalues En ), then
A = lim
X
t→∞
n
|nie−En t hn| =
X
n
δ0,En |nihn| ,
(8.6)
namely A projects onto the lowest-energy subspace, i.e. the kernel of Ĥ, if we
assume a non-negative spectrum.
Proceeding with these formal manipulations, we can also write
Y
R 3
A = lim
e−H(x)t = lim e− d x H(x)t ,
(8.7)
t→∞
t→∞
x
hence
A(s, s0 ) = lim hs | e−
R
d3 x H(x)t
t→∞
| s0 ikin .
(8.8)
Quite loosely, if we want the propagator to be 4d diff invariant, then the limit
is irrelevant, so
R1
R 3
A(s, s0 ) = hs | e− 0 dt d x H(x)t | s0 ikin .
(8.9)
P
Now we can split this expression by inserting identities in the form 11 = s |sihs|,
obtaining something like
X
R 3
R 3
A(s, s0 ) = lim
hs | e− d x H(x)dt | sN ikin hsN | e− d x H(x)dt | sN −1 ikin (8.10)
N →∞
s1 ...sN
. . . hs1 | e−
R
d3 x H(x)dt
| s0 ikin .
We can see then, that the transition amplitude between two spin-network stats,
can be expressed as a sum, as follows
X
A(s, s0 ) =
A(σ) ,
(8.11)
σ
110
8.2. Path integral discretization: BF theory
i.e. as a sum over histories σ of spin network. A history σ is a discrete sequence
of spin-network
σ = (s, sN , . . . , s1 , s0 ) .
(8.12)
Moreover the amplitude is just a product of the amplitudes between the single
steps in the history
Y
A(σ) =
Av (σ) ,
(8.13)
v
where we have labeled with v each of the above said single steps (this terminology
will be clarified in a moment). Now, to take a further step, let us recall that
the hamiltonian operator acts only on the nodes of the spin network. Thus the
single amplitude Av in (8.13) is non-vanishing only between spin-network that
differ at a node by the action of H.
A history of spin-networks σ is what is called a spinfoam, and it is exactly the
“world-sheet” of the spin network, as pointed out at the very beginning of this
section: it is the time evolution of a spin network state.
As a conclusion of this brief passage I want to recall again two things: 1. the
heuristic value of the former passages and 2. the fact that a rigorous derivation
of a sum-over-histories formula for Quantum Gravity transition amplitude can
be given in the three dimensional case [185] (and moreover it matches the result
of the following section).
8.2. Path integral discretization: BF theory
Now we would like to introduce the spinfoams formalism in a different (and
somewhat clearer) way. We shall try to “discretize” the path-integral itself. Or,
better, we shall discretize spacetime with triangulations à la Regge [186, 187],
and try to read out how the path integral of general relativity can be adapted
on this triangulation.
This is an approach somewhat dual to the hamiltonian derivation of the path
integral: there one take the physical inner product defined in terms of an evolution operator (the projector into the hamiltonian kernel) and “decompose” it
by inserting identity resolutions, de facto giving a discrete definition of the path
integral. The present approach is the other way around, i.e. one defines the amplitude as the path integral of (an appropriate form of) the General Relativity
action, and discretize this integral trying to give it a rigorous meaning. For a
good introduction to this approach see [182].
Remark. I point out from the very beginning that the discretization we are
imposing to define the path integral is not at all as the one of, say, relativity on
the lattice. Indeed, there one uses the lattice just as a regularization procedure,
111
8. The spinfoam approach to the dynamics
to be removed at the end by an appropriate continuous limit. Here instead,
quantization tells us that the fundamental theory is discrete! So we are justified
in our discretization procedure, and no continuous limit is to be done. Another
problem will be the triangulation dependence of our final results, but this is a
completely different issue.
We shall start with BF theory, which is actually a trivial theory, but it is up
till now the only theory in which the spinfoam approach (=discretization of path
integral) can be completely gone through.
To set up a general BF theory we need a principal bundle – let’s call it P
– with base space the spacetime manifold M, fiber a gauge Lie group G. The
basics fields in the theory are a connection A on P and an ad(P )-valued (n − 2)form B on M. Here ad(P ) is the associated vector bundle to P via the adjoint
action of the group G on its Lie algebra. However, we can forget for the time
being all these technicalities and just see how things work in calculations. The
lagrangian of BF theory is the defined as follows
L = Tr(B ∧ F ) ,
(8.14)
where, as usual, F is the curvature of the connection A. Now it easy to see that
this theory is (at least classically) trivial, indeed the equations of motion state
that
F = 0 ; dA B = 0 .
(8.15)
The first says that the manifold is flat, there are no local degrees of freedom.
Indeed the second says that the parallel transport rule is trivial. Now, take the
simplest path integral you can imagine for this theory, i.e. the partition function
Z
R
(8.16)
Z(M) = DADBei M Tr(B∧F )
Z
= DAδF ,
where we have integrated out the B field, using its equation of motion F = 0.
This expression has, by itself, no rigorous meaning. To give a sense to the above
formal expression the next step – as stated at the beginning of this section – is
the triangulation of the manifold M and the definition of a discretized version
of (8.16).
Let’s recall that a triangulation of a (sufficiently smooth) manifold is obtained
by a discretization by means of simplices, precisely n-simplices. Recall that in
the sum-over-histories framework, we argued that spacetime should be seen as
the dual to what we called a spinfoam, i.e. dual to spin networks world- sheets
(just as 3-volumes are dual to nodes of a spin network on space-hypersurfaces).
112
8.2. Path integral discretization: BF theory
This picture will somehow guide us in the discretization of the path integral,
indeed we shall define the discrete versions of B’s and A’s on the dual of the
triangulation.
We call ∆ such a triangulation. Now we take the so called dual 2-skeleton
of this triangulation. It is built in this way: put a vertex at the center of
each n-simplex; an edge intersecting each (n − 1)-simplex and one (polygonal)
face intersecting each (n − 2)-simplex. We could go on, defining dual volumes
intersecting (n − 3)-simplices, and so on till defining a dual n-complex to each
point (0-simplex) of ∆, thus building the dual triangulation ∆∗ . But we shall
need only its two dimensional subset, the skeleton.
∆
n-simplex
(n-1)-simplex
(n-2)-simplex
2-skeleton ⊂ ∆∗
vertex v
edge e
face f
For example, if n = 3, then the triangulation is made up by tetrahedra glued
along faces. For each tetrahedron we have one vertex of the dual skeleton, 4
edges (one for each triangle) and 6 faces (one for each edge of the tetrahedron).
∆
2-skeleton ⊂ ∆∗
tetrehedra
vertex v
triangle
edge e
edge (of a triangle)
face f
While, for n = 4 – the interesting case – we have
∆
2-skeleton ⊂ ∆∗
4-simplex
vertex v
tetrehedra
edge e
triangle
face f
Now we want to define our BF field theory on this dual skeleton, i.e. the
points on the manifold will be “replaced” by the vertices, edges and faces of the
dual skeleton.
A connection is a one form (precisely a g-valued one form), thus it is natural
to associate it to a n − 1 dimensional object in spacetime. And this is just what
we have called edge of the dual skeleton2 . Thus our connection is a prescription
to associate a group element to each edge of the skeleton (think of it as the
2
In this sense, the discretization defined on the dual of ∆ seems a rather natural choice.
113
8. The spinfoam approach to the dynamics
holonomy of A along that edge). If the connection has to be flat (as is the case
in BF theory) then we should want that
ge1f . . . geN f = 1 ,
(8.17)
where: g stands for a group element; ef are the edges that surround the face f ,
and there are N of them (i.e. we have called N the number of edges of the face
f 3 . This formula thus states that the holonomy around each face is the identity
and this is the flatness of the manifold. The B field is a g-valued 2-form on M
and we discretize it as a map assigning to each (n − 2)-simplex – i.e. to each
face f of the skeleton – a g element BfIJ .
Equation (8.16) in its discrete version, as just prescribed, becomes
Z Y
Z Y
Y
Y
iTr[Bf Uf ]
dge
dge
Z∆ (M) =
dBf e
=
δ(ge1 f . . . geN f ) ; (8.18)
e∈E
e∈E
f ∈F
f ∈F
where with E and F are denoted the set of all the edges and of all the faces of
the 2-skeleton, respectively. I have put a subscript to stress that this discrete
definition is, in general, triangulation -dependent.In these last equalities we have
performed the discretized version the integral over B. A remark on the expression
Tr[Bf Uf ]: Uf = ge1 f . . . geN f is the (discrete) holonomy around the f face; using
the identity Uf ' 11 + Ff where Ff ∈ g we get the exponent of (8.18).
The importance of this last formula (8.18) is that it has a precise and definite
meaning, and it is no more a purely formal prescription. It is actually the first
spinfoam path integral we encounter in this thesis. Here it is written in terms
of group variables. Now we shall see how to evaluate the integrals and pass to
the spin/intertwiner representation, more suitable for an intuitive grasp and for
a direct link wit spin networks, but completely equivalent.
Let us use the following decomposition of the group delta function (Peter-Weyl
decomposition)
X
δ(g) =
dim(ρ)Tr(ρ(g)) ,
(8.19)
ρ∈Irrep(G)
i.e. a sum over the all irreducible representations of the group of the character
of the representation weighted by its dimension. Thus we have
Z Y
X
Y
Z∆ (M) =
dge
dim(ρf )Tr(ρf (ge1 f . . . geN f )) ;
(8.20)
ρ:F →Irrep(G)
e∈E
f ∈F
where the sum is over all the manners to associate to the faces of the 2-skeleton
irreducible representation of the group (operation usually called “coloring”). We
3
Notice that the dual faces can have an arbitrary number of edges, it depends on how the
triangulation is done and on the manifold M.
114
8.2. Path integral discretization: BF theory
can go further. It is crucial the following identity:
Z
ιι∗
δ12
dg ρ1 (g) ⊗ ρ2 (g) =
dim(ρ1 )
(8.21)
if ρ1 ' ρ2 ∗ and 0 otherwise. Maybe it is useful to write this identity graphically
like this
ρ1
Z
g
dg
ρ1
=
ιι∗
dim(ρ1 )
ρ2
ρ2
ι
ι∗
ρ1
.
(8.22)
,
(8.23)
ρ2
A generalization of the above formula is the following:
ρ2
ρ1
g
Z
dg=
=
ρ1
ρ3
ρ3
ι
X
ι∗
ι
ρ1
ρ3
ρ2
ρ2
and so on, with arbitrary number of representations involved. It is important
to catch the meaning of these last formulæ. Actually it is quite simple: the left
hand side is a group averaging of the product of a certain number of representations, thus it belongs to the invariant subspace of the tensor product of those
representations. The right hand side is obviously a projector onto the invariant
subspace: the two sides are the same thing.
Let’s go back to our BF theory. Let’s make the things as simple as possible
first: n = 2, our spacetime is a surface. Thus it is triangulated by triangles and
the dual 2-skeleton is made by polygons. A look to figure 8.2 will certainly clarify
what’s going on. It is easy to see that each group element is shared by two faces
(which is the same thing of saying that each edge is common to two and only
115
8. The spinfoam approach to the dynamics
g e1 f
f
g e2 f
g e3 f
Figure 8.2.: (Part of the) triangulation of the manifold (dashed lines) and its
dual 2-skeleton (thick lines). One face f has been labeled with its
group elements gf ’s on the edges.
two faces). In our path integral formula (8.20) we thus see that each integration
actually concerns the product of two representations, precisely the two faces
that share the edge of that group element. Thus we can use formula (8.21)
which implies that the integral is non-vanishing only if all the representations
(i.e. on all the faces) are equal. The integral can then be evaluated and the final
result is
X
(2)
Z∆ (M) =
dim(ρ)χ(M) ,
(8.24)
ρ∈Irrep(G)
where χ(M) = |V| − |E| + |F| is the Euler characteristic of the manifold, and it
is a topological invariant quantity. For a manifold of genus g it equals 2 − 2g.
Thus our partition function for a 2d BF theory converges only for surfaces with
genus greater than 1 (thus it diverges for sphere and torus). See section 8.5 for
some discussions about divergences in spinfoam models.
Let’s step up to dimension 3. In this case the triangulation of M is made up
by tetrahedra. Again, it is worthwhile to have a look to figures 8.3 and 8.4. We
have shown an edge of the dual skeleton, which is shared by three faces. Thus
the typical integral will be now of the kind of the left hand side of equation
(8.23), and thus the integrations split the edge in the sum of intertwiners of the
kind shown in figure 8.4. Now, it is not difficult to see that the intertwiners of a
single tetrahedron combine to form a tetrahedron as well (a dual one) labeled by
the representations of its 6 edges and by the intertwiners on its vertices. Thus
116
8.2. Path integral discretization: BF theory
e4
e3
e1
f
e2
Figure 8.3.: A tetrahedron of a 3d triangulation is shown. The thick lines are
the four edges, one for each triangle. One (typical) face has been
colored, the one corresponding to the dotted line.
g
ι
ι∗
Figure 8.4.: (on the left) A tetrahedron of a triangulation of a 3d manifold. The
three thick lines represent actually a single edge, the one dual to
the triangle punctured, which is shared by three faces (one per each
side of the triangle). On the right figure, the integration over the g
group element has been evaluated by means of equation (8.23).
117
8. The spinfoam approach to the dynamics
we have an explicit formula for the partition function, i.e.
ι1
ρ2
ρ1
(3)
Z∆ (M) =
XX Y
ρ
ι
dim(ρf )
v∈V
f ∈F
ρ6
Y
ι2
ρ5
ι4
.
(8.25)
ρ3
ρ4
ι3
The first sum is over ρ : F → Irrep(G), i.e. a labeling of faces with Grepresentations, while the second on labeling of edges with intertwiners. The
graphic “atom” in this formula is just the contraction of four 3-valent intertwiners, and depends obviously on 6 spins. If G = SU (2), it is nothing but the
Wigner 6j-symbol. You can equivalently see this atom as a spin network (with 6
links and 4 nodes) in a tetrahedron-like pattern, with all the six group elements
put on the identity, which is often called evaluation of a spin network (recall
what a spin network is precisely, see equation (7.28)).
The trick is the same in 4d, mutatis mutandis. In this case one has 4 faces
that share an edge, thus we shall use the formula (8.23) with 4 representations.
In 4d the triangulation is made by a 4-simplex, and the integrations over the
groups gives the contraction of intertwiners in a 4-simplex pattern. The final
formula is
ι1
ρ1
ρ5
ι5
(4)
Z∆ (M)
=
XX Y
ρ
ι
f ∈F
dim(ρf )
ι2
ρ10
Y
.
ρ9
ρ6
ρ8 ρ 7
v∈V
(8.26)
ρ2
ρ4
ι4
ρ3
ι3
The atom in this formula is the contraction of five 4-valent intertwiners in a 4simplex like pattern. Incidentally, notice that the atom is precisely a 4-simplex
(its projection on 2 dimensions).
Notice that one can always split a 4-valent intertwiner into two 3-valent ones4 ,
thus one would end with a trivalent spin-network with 15 spins. If the gauge
group is SU (2), this is called the ‘15j-symbol’.
4
See section 7.3 and in particular equation (7.26).
118
8.2. Path integral discretization: BF theory
Remark. Take a 3d model of gravity, i.e. 3d Riemannian General Relativity.
Ponzano and Regge, in 1968, showed [188] that the SU (2)-BF path integral
(see equation (8.25)) has the correct semi-classical limit, in the following sense:
There is a discretized action (called the Regge action), which is a sum over all
the tetrahedra with which we have triangulated the 3d manifold of
X
SR =
le θe ,
(8.27)
e
i.e. the sum over the 6 edges of each tetrahedron of the product of the length
of that edge and the dihedral angle (the angle between the two normals to the
faces incident to that edge) – that is an approximation of the integral of the
Ricci curvature, i.e. an approximation of the Einstein-Hilbert action. Now, the
tetrahedral spin-network of equation (8.25), with gauge group SU (2), has the
following asymptotics for large spins
r
2
π
cos SR +
,
(8.28)
3πV
4
where V is the volume of the tetrahedron and le = je + 1/2.
This is wonderful, since for large spins – which means for scales much grater
than the Planck length – one exactly recovers something like eiS with the right
General Relativity action!5 This means that, at least in this simple and physically un-interesting case, our discretization procedure is consistent, and – reassured by this consistency – we feel more confident with the four dimensional
case as well.
The BF theory 3d model is commonly known as the Ponzano-Regge model.
Up till now, we have dealt only with manifold without boundary, i.e. – to use
an hamiltonian jargon – we have dealt only with vacuum-to-vacuum transition
amplitudes. In order to define something like hs | s0 iphys (see (8.4)), we have to
consider triangulation of manifold with boundary as well. It is quite natural,
but under this apparent simplicity many subtleties hide, so one must be careful.
The simple part is: take a manifold without boundary and cut it in a spacelike
direction. You get two manifold with spacelike boundary. Thinking in terms of
the spinfoam, that is in terms of the skeleton of the triangulation, the boundary
is made of links, results from the cutting of faces, and nodes, cut of edges. The
reader has surely recognized that we are actually doing the inverse procedure
with respect to world-sheet sweeping of a spinfoam from a spin-network (as in
the hamiltonian-to-spinfoam approach 8.1) , i.e. spinfoam sectioning.
5
Actually one gets just the real part of the exponential. This is a technicality, roughly
speaking the reason is that given the lengths of the edges of a tetrahedron, we still can
rotate and reflect the tetrahedron.
119
8. The spinfoam approach to the dynamics
The variables of the cut edges and faces will be variables of the boundary –
be them group elements or spin/intertwiner (or whatever), depending on the
representation we want to use – not to be summed/integrated over. All the construction is exactly the same, with some variables fixed, the boundary variables.
Now that we have the notion of spinfoam with boundary, we can take the
following picture: take a spinfoam (with or without boundary) and imagine to
surround each vertex v with a little 3-sphere. You will get an ensamble of little
bubbles – sometimes called atoms – each representing a little spinfoam formed
by a single vertex with a boundary ψv , which is a two dimensional graph with
links and nodes representing the vertex. Look to formula (8.25) or (8.26), this
bubble is just the symbol on the far right, the vertex amplitude6 . It is now clear
(if it wasn’t already) that a spinfoam amplitude is just a sum over labellings of
product of amplitudes: the face amplitude (about which we will have much more
to say) and, most important, the vertex amplitude.
This is quite often taken as a definition of a spinfoam amplitude.
The subtleties about boundary states arise in the Quantum Gravity context,
i.e. when one tries to define quantum transition amplitude for general relativity
and not for a generic G-BF theory. Indeed in that case, the boundary has to
represent a state of Hkin , the kinematical Hilbert space, i.e. SU (2) spin network
states. This will be a key point.
Before going on to explain what happens in the (interesting) case of gravity,
I want to focus on another little but useful issue in BF theory. I have said that
in formulæ (8.25), (8.26), the term on the far right is actually the amplitude of
a little spinfoam atom. What is like the amplitude of a spinfoam atom (i.e. the
vertex amplitude) written in the terms of integral over the group (in the sense of
equation (8.16)? To really understand this (and to really understand the vertex
amplitude in general), one has to think at what happen when you cut away with
a (n − 1)-sphere from the dual skeleton of an n-triangulation. Let us focus on
n = 3 and n = 4. Actually, it is far more easy to think a little about it and
catch it by oneself, rather than long and cumbersome (and useless) explanations.
I just give a few hints: take a tetrahedron, and draw the dual. There will be
one vertex, four edges departing and 6 faces. Cutting this dual will give another
(curved) tetrahedron graph (see figure 8.5), whose links are just the cut of the
six faces of the spinfoam, and whose nodes are the cut of the edges. In the same
manner, in a 4-simplex triangulation, when you cut away a 3 − sphere around
a vertex you get another 4-simplex whose edges and nodes are the cut of faces
and edges of the spinfoam.
6
Incidentally, this “bubble picture” is what suggested the name “spinfoam” in the early days
of this theory.
120
8.3. Spinfoam models for Quantum Gravity
e2
f1
f2
e1
f5
v
e3
f6
f3
f4
e4
Figure 8.5.: (on the left) atom ‘bubble’ of a 3d triangulation: dotted line for
the ∆-tetrahedron, thick lines for the edges of ∆∗ , colored lines
for the “cut” tetrahedron. On the right the same graph for a 4d
triangulation.
Thus, one can write the ampltiude simply as
Z Y
Z Y
Y
Y
−1
Av (Uij ) =
dgi
δ(gi Uij gj ) =
dge
δ(gef1 Ul≡f gef2 ) ,
i
i<j
e
(8.29)
f
where Uij (i, j = 1, . . . , 5; i < j) or Ul are the four(3d)/ten(4d) external variables attached to the links of the boundary graph, the notation is quite selfexplanatory. The integrals are of course over G. If one decompose the delta into
representations and performs the group integrals – just as we have done before
for the partition function – one ends precisely with
A3d
v (ρl , ιn ) = 6j(ρl , ιe ) ,
A4d
v (ρl , ιn ) = 15j(ρl , ιe ) ,
(8.30)
i.e. the vertex amplitude in the spin/intertwiner representations (i.e. if we take
as boundary variables irreducibles on links and intertwiners on nodes) is – as it
had to be – the tetrahedron-like or 4-simplex-like intertwiner contraction.
8.3. Spinfoam models for Quantum Gravity
Let us now come to the interesting part of the story: the spinfoam ‘machinery’
applied to General Relativity, rather than to the (trivial) BF theory. This is
121
8. The spinfoam approach to the dynamics
still a very open issue, thus everything that follows is in some degree work in
progress.
The first fundamental step is to recognize that GR can be cast into “BF theory + constraints”. Thus the general guideline will be to treat the path-integral
discretization à la BF theory, and then impose the constraints. It is not surprising that all the big troubles will be in that second step. But let us do everything
at its proper time.
8.3.1. The Barrett-Crane model
The first spinfoam model for quantum gravity is the Barrett-Crane model [189].
Take an SO(4)-BF theory in 4d. Its action can be written (see (8.14))
Z
S[B, ω] = BIJ ∧ F IJ (ω) ,
(8.31)
where F is the curvature of the connection ω. If we replace
1
BIJ = IJKL eK ∧ eL ,
2
(8.32)
we get precisely Einstein-Hilbert action, in the form of equation (6.31). Thus,
we could formally write the action for General Relativity as
Z
S[B, ω, φ, µ] =
BIJ ∧ F IJ (ω) + φIJKL B IJ ∧ B KL + µIJKL φIJKL . (8.33)
Let us check it: φ is a 0-form while µ is a 4-form, they both have no dynamics
and thus, classically, they just give constraints, namely
IJKL φIJKL = 0 ,
(8.34)
i.e. φIJKL = −φJIKL = −φIJLK = φKLIJ , for the variation with respect to µ,
while for φ, using (8.34)
IJ KL
µνρσ Bµν
Bρσ = e IJKL ,
(8.35)
IJ KL µνρσ
Bρσ . This last expression (8.35) contains 20 equawhere e = 4!1 IJKL Bµν
tions – often called simplicity constraints – one for each independent component
of φ. They constrain 20 of the 36 independent components of the B field. The
solutions to (8.35) are of two kinds
1
BIJ = ± IJKL eK ∧ eL ;
2
122
B IJ = ±eI ∧ eJ ;
(8.36)
8.3. Spinfoam models for Quantum Gravity
in terms of the 16 degrees of freedom of the tetrad field eIa . Only the first type of
solution gives the GR action (6.31)7 , while the second give a topological (trivial)
action. This must be remembered when to quantize the action
For the following, it is useful to rewrite the constraints (8.35) as
∗
Bµν
· Bµν = 0 ,
(8.37a)
∗
Bµν
· Bµσ = 0 ,
∗
Bµν
(8.37b)
· Bστ = ±2e
e,
(8.37c)
with µνστ all different, and with e = 4!ee µνστ dxµ ∧ dxν ∧ dxσ ∧ dxτ . The first two
constraints in these form are also known as diagonal and off-diagonal simplicity
constraints.
Remark. Notice that we have taken an SO(4) connection – or, better, its double
cover Spin(4) – thus this is an euclidean version of General Relativity. To
obtain the actual GR one should take an SL(2, C) (the double cover of SO(3, 1))
BF theory. The problem is the non compactness of SL(2, C) that requires special
care in handling the (divergent) integrals over the group. It is a general custom
in Quantum Gravity to investigate the simpler euclidean case first.
Equation (8.33) is our GR “BF + constraints” action. Now the goal is to
take the BF spinfoam model and analyze what kind of consequences have the
imposition of constraints and, firstly, to understand how to impose them. The
intuitive idea is that the constraints should restrict in some (interesting) sense
the spinfoam sum in (8.26). In particular the constraints must impose that on
the boundary of the spinfoam one gets the kinematical Hilbert space of LQG, and
so they must somehow restrict SO(4) to SU (2) on the boundary. However, the
imposition of the constraints is actually a rather cumbersome issue, to be dealt
with very carefully. I shall sketch here briefly the key points. The interested
reader is encouraged to read the review on Barrett-Crane model [191]. However,
the issue of constraints imposition will be gone through more carefully in the
case of the Engle, Pereira, Rovelli, Livine (EPRL) model (see section 8.3.2 for
details and references).
The idea is to take the following path integral
Z
Z
BC
∗
Z (M) = DADBδ[B − (e ∧ e) ] exp i
B ∧ F [A] .
(8.38)
M
Naively speaking, this means that one must restrict the integral to those configurations satisfying the delta function in (8.38). This turns out to be the case if
7
Up to an overall sign.
123
8. The spinfoam approach to the dynamics
one restricts the topological spinfoam sum (8.26) – with G = Spin(4) – to the socalled Spin(4) simple representations: recall that Spin(4) = SU (2)×SU (2), thus
you can label Spin(4) irreducible representations with couples of spins (j + , j − ):
simple representations are the ones with j + = j − .
The spinfoam partition function results in
BC
ρ12
ρ15
BC
Z∆
X Y
=
dim(ρf )
Y BC
v
simple ρ f
BC
ρ25
ρ14 ρ13
ρ35 ρ24
,
(8.39)
ρ23
ρ45
ρ34
BC
BC
where, as just said, the sum is over simple Spin(4) representations; BC is a fixed
intertwiner operator between (and depending only on) the 4 representations converging in it (following the 4-simplex pattern). The vertex amplitude ultimately
depends on 10 spins, and it is thus referred to as the 10j-symbol. Of course,
thanks to the ‘simplicity’ of the representations, we can rewrite this partition
function as a sum over SU (2) irreducible, using the following property of the BC
intertwiner
BC
BC
BC
=
ρ14 ρ13
ρ35 ρ24
ρ23
ρ34
BC
j12
15
ρ25
ρ45
ι∗1
ι1
ρ12
ρ15
X
ι5
ι1 ,...,ι5
∗
j12
∗15
ι2 ι∗5
j25
j23
j45
j34
ι4
BC
ι∗2
∗
j25
∗
∗
j14
j13
∗
∗
j24
j35
j14 j13
j35 j24
∗
j23
∗
j45
ι3
,
∗
j34
ι∗4
ι∗3
(8.40)
namely
ι∗1
ι1
j12
15
BC
Z∆
=
XY
Y X
(2jf + 1)2
j
f
v ι1 ,...,ι5
ι5
ι2 ι∗5
j25
j14 j13
j35 j24
j23
j34
ι3
ι∗2
∗
j25
∗
j14
∗
j35
j45
ι4
∗
j12
∗15
∗
j13
∗
j24
.
∗
j23
∗
j45
ι∗4
∗
j34
ι∗3
(8.41)
Notice that, according to (8.26), the face amplitude has been set equal to the
dimension of the face representation. Here this representation is simple, so
dim(j ⊗ j) = (2j + 1)2 .
124
8.3. Spinfoam models for Quantum Gravity
However, this happens to be a subtle point, since we know (e.g. from QFT)
that constraints change the integration measure of the path integral. Thus the
use of the topological face amplitude is not justified at all in dealing with Quantum Gravity spinfoam models. This is a key point, regarding the choice of the
face amplitude, and it will be treated more carefully in section 9.
We have sketched briefly the structure of the Barrett-Crane model. However
it must be said that this spinfoam model, despite all its success in the few
years after its first appearance [189], has been proved to be unfit to Quantum
Gravity [190]: the big problem is that the boundary states of the BC model are
only a small subgroup of the spin network states. This is due to the way by
which constraints are imposed.
This is the main reason for which much work has been done in order to ameliorate the model. There have been a few proposal starting from 2007. In the
following section we present the model which is considered the present day best
candidate to attempt doing calculations in Quantum Gravity.
8.3.2. The EPRL model
Here I present the Engle, Pereira, Rovelli, Livine proposal for the vertex amplitude of spinfoam models. I refer the reader to the original papers [192, 193, 194,
195] for a thorough discussion.
I don’t want to give here a rigorous (and cumbersome) derivation and justification of the model, I just want to stress the key points. We start with a
triangulation ∆ with the usual association of group elements to (dual) edges
etc. . . (see section 8.2), but we use now the General Relativity action in its Holst
form (6.47), which can be compactly written as
Z 1
∗
(8.42)
S[e, ω] =
(e ∧ e) + (e ∧ e) ∧ F (ω) .
γ
with ∗ the Hodge dual operator, namely
1
∗
FIJ
≡ IJKL F KL .
2
(8.43)
This action has the merit of including the Immirzi parameter γ of Loop Quantum
Gravity as a pre-factor of the topological sector, which has no consequences on
the equations of motion (they remain the Einstein field equations) but allows
the formulation in terms of Ashtekar-Barbero variables. The program is, as for
the Barrett-Crane model, to write then action as ‘BF + constraints’ then write
the BF spinfoam sum and only at the end apply the constraints. The difference
with respect to BC are in the starting action and (more important) in the way of
125
8. The spinfoam approach to the dynamics
imposing the constraints. Here we shall work directly in the Lorentz framework,
thus with gauge group SL(2, C) rather then SO(4).
Holst action (6.47), (8.42) can be written as
Z S[e, ω] =
1
B + B∗
γ
∧ F (ω) + constraints.
(8.44)
where the constraints must impose, as in BC, B = (e∧e)∗ . The discrete variables
shall be, as previously, Bf ∈ g associated to triangles/dual faces and ge associated
to tetrahedra/dual edges. We call Uf the holonomy around the face f , i.e. the
product of the group elements of the edges bounding f .
This time I shall give a more detailed derivation of the EPRL formula.
We begin by giving a discretized form of the simplicity constraints (8.37). We can
easily discretize the constraints in the form (8.37) as
Bf∗ · Bf = 0 ,
Bf∗ · Bf 0 = 0 ,
Bf∗
· Bf 0 = ±12V ,
(8.45a)
(8.45b)
(8.45c)
where: V is the volume of the simplex; in the second equations f and f 0 are faces
sharing an edge (equivalently: they are associated to triangles living on the same
tetrahedron) while in the third they are don’t (equivalently: they are attached to
triangles belonging to two distinct tetrahedra). Actually, the correct way to deduce
the discrete Bf variable from the 2-form Bµν is the following8
Z
Bf =
B.
(8.46)
f
Instead of requiring the last constraint (8.45c), we take the following closure constraint
Bf1 + Bf2 + Bf3 + Bf4 = 0 ,
(8.47)
for the four faces sharing an edge. It is easy to check that diagonal+off diagonal+closure is an equivalent system of constraints.
Remark. In the BC model the diagonal constraint implies the face representation to
be simple, while the off diagonal one implies the uniqueness (and the specific form) of
the BC intertwiner. Both constraints are imposed strongly.
8
Actually the situation is a little more tricky, I refer to the good paper by Engle, Pereira and
Rovelli [193] for a thorough explanation of this point.
126
8.3. Spinfoam models for Quantum Gravity
We have seen that these constraints admit more solutions than GR, precisely a trivial
topological sector. Now we exclude the trivial sector by imposing a slightly different
form of the constraints, i.e. we require that for each tetrahedron exist a vector nI such
that, for each triangle of the that tetrahedron, holds9
nI (Bf∗ )IJ = 0 .
(8.48)
This constraint is intended to replace the off diagonal simplicity constraint (8.45b).
Geometrically the nI represents a vector normal to the tetrahedron/edge.
Having cleared the meaning of the discrete versions of the simplicity constraints
(8.35), we rephrase them in a convenient way. The conjugate momenta to the holonomies
are
1
Jf = Bf + Bf∗ .
(8.49)
γ
Which, once inverted
Bf =
γ2
γ2 + 1
1 ∗
Jf − Jf ,
γ
thus we can reformulate the constraints as
1
2
∗
Cf f = Jf Jf 1 − 2 + Jf Jf = 0 ,
γ
γ
1
CfJ = nI J ∗ IJ + JfIJ = 0 .
γ
(8.50)
(8.51)
(8.52)
Now we choose a specific nI . A typical choice is nI = δI0 , that means that all the
tetrahedra are spacelike. In other words, we are selecting a specific SU (2) subgroup
of the full SL(2, C). Obviously this is a gauge choice, and must not influence physical
results. With this choice the constraint (8.52) becomes
1
1
1
Cfj = jkl Jfkl + Jf0j = Ljf + Kfj = 0 ,
2
γ
γ
(8.53)
where L are the generators of the SU (2) subgroup of SL(2, C) that leaves nI invariant;
K are the generators of boosts in the nI direction.
Let us now deal with the quantization of the constraints. In order to do that we must
identify a Hilbert space in which we define operators. Taking a single vertex bubble,
recall that the graph on its boundary γv naturally defines the following boundary
Hilbert space
L2 (SL(2, C)L ) ,
(8.54)
where L, N are the number of links and nodes of γv (cut from the spinfoam by the
3-sphere). We impose the constraints as follows.
9
Indeed the off diagonal simplicity constraints imply that the triangles of each tetrahedron
lie on a common hypersurface. If they are satisfied, there will be a direction nI orthogonal
to all the faces.
127
8. The spinfoam approach to the dynamics
The closure constraint (8.47) imposes SL(2, C)-invariance on this space, i.e. it implements the (usual) quotient
L2 (SL(2, C)L /SL(2, C)N ) .
(8.55)
The simplicity constraints (8.51), (8.53) are defined on faces, thus they act each on a
single copy of the group, namely on L2 (SL(2, C)). The diagonal simplicity constraint
(8.51) on this space reads
1
2
C1 1 − 2 + C1 “ = “0 ,
(8.56)
γ
γ
with C1 and C2 the Casimir operator of g, i.e.
C1 = J · J = 2(L2 − K 2 );
C2 = J ∗ · J = −4L · K .
(8.57)
The quotations mark means that we have to decide how to impose this constraint.
However notice that having expressed it in terms of Casimir operators eigenvalues, it
commutes with all the other operators, thus we can impose it strongly, i.e. requiring
it to annihilate physical states.
The constraint (8.53), on the other hand, is more subtle. The technique used in
[193], first proposed by Thiemann [196] is to pack them into a master constraint
X
Mf =
(C j )2 “ = “ 0 .
(8.58)
j
Classically it is of course equivalent to imposing the C j equal zero separately. The
power of this approach, is that Mf is now a combination of Casimirs
C1
C2
1
2
“=“0.
(8.59)
L 1+ 2 − 2 −
γ
2γ
2γ
Combining (8.56) and (8.59) we get the following (definitive) set of 2 constraints:
2
1
C2 1 −
+ C1 “ = “ 0 ,
(8.60)
γ
γ
C2 − 4γL2 “ = “ 0 .
(8.61)
Having done this, we now simply have to see what consequences these constraints have
on states of L2 (SL(2, C)). It is easy to see that, labeling with (p, k) (p real and k
half-integer) the SL(2, C)-irreducible representations, thus having the decomposition
2
L (SL(2, C)) =
X
(p,k)
H(p,k) ⊗ H(p,k) ,
H(p,k) =
∞
M
j0 ,
(8.62)
j 0 =k
– with j 0 in the sum denoting the SU (2) spin-j 0 irreducible – of SL(2, C)-irreducibles
into SU (2)-irreducibles, the constraints impose, for each face,
pf = γjf ,
128
kf = jf ,
(8.63)
8.3. Spinfoam models for Quantum Gravity
for some half-integer jf , and they restrict the decomposition to the lowest spin, j 0 =
k = j.
You see that the constraints have selected a SU (2) subgroup of SL(2, C)! Precisely
they tell two things: 1. the permitted SL(2, C) face labels are only the irreducibles
of the type (pf = γjf , kf = jf ) for some half-integer jf 10 ; 2. given the face label
(γjf , jf ) they select out the spin-j SU (2) irreducible. Let’s give a precise definition of
the embedding L2 (SU (2)) → L2 (SL(2, C)):
Y : j → j ⊂ H(γj,j) ⊂ L2 (SL(2, C)) ,
(8.64)
is the map that sends each SU (2) spin j irreducible in the lowest spin (i.e. – of course
– the spin j) irreducible inside (γj, j). Thus, the Y map takes each state in L2 (SU (2))
to a state of L2 (SL(2, C)). In terms of Wigner matrices it is really simple
Y
Dj −
→ Y Dj Y † = D(γj,j) .
(8.65)
This is very nice indeed, since we can define boundary variables to be SU (2) states,
and this is just what we expect to have, since on the boundary we want to put spin
networks.
Remark. I have done all the calculation for SL(2, C) since it is the “reality”. However, as I had occasion to say, it is sometimes useful to see what happens in the simpler
euclidean case, i.e. for SO(4). I do not repeat the discussion – which proceeds very
similarly – the result is the following: the simplicity constraints reduce the SO(4) irreducibles (jf1 , jf2 ) to the ones given by (γ+ jf , γ− jf ), jf ‘running’ over SU (2) irreducibles
and with [193]
|1 ± γ|
γ± =
.
(8.66)
2
Thus, in the euclidean SO(4) case we have an embedding of SU (2) in SO(4), quite
similarly to what happens in the lorentzian case.
We are almost done. We just have to see the consequences of what I have just said
in terms of concrete formulæ.
Let us focus on a single vertex. The idea is to take the BF vertex amplitude (8.29)
(or equivalently the 4-simplex spin network evaluation, i.e the graph on the far right
of (8.26)) and to restrict the boundary variables to satisfy the simplicity constraints,
in the form we have just said:
Z
Y
Y
−1
EP RL
Av
(Ul ) =
dgn
P (Ul , gs(l) gt(l)
),
(8.67)
SL(2,C)N n
10
f
Notice that this also implies that the effective sum over p, which should be an integral,
restricts to a sum.
129
8. The spinfoam approach to the dynamics
where we have no more a delta inside the face product, since now Ul is an SU (2)
element, but a ‘generalized’ delta, i.e.
X
P (U, g) =
(2j + 1)Tr(Dj (U )Y † D(γj,j) (g)Y ) ,
(8.68)
j
with U ∈ SU (2), g ∈ SL(2, C). Performing the SU (2) integrals, just as in the standard
BF theory case (8.2), one can easily find the spin/intertwiner representation of the
vertex amplitude done in the BF theory case.
One ends up with
RL
AEP
(jl , ιn )
v
=
XZ
dpn (kn /2)2 + p2n
Y
n
kn
Yιι(pnn ,kn ) × 15j((γjl , jl ); ι(pn ,kn ) ) ;
(8.69)
where Yιι(pne ,ke ) = hι(p,k) | Y | ιi. Putting together all the vertex amplitudes we get
RL
AEP
(jl , ι, n) =
∆
XY
j,ι
Af (jf )
Y
RL
AEP
(j, ι; jl , ιn ) ;
v
(8.70)
v
f
where the last parenthesis indicates that some simplices are cut by the boundary,
thus some spins and intertwiners will be fixed (and not summed) to the boundary
value, given in the left hand side11
Notice again that we have no clue about the face amplitude: should it be
the SU (2) dimension of the representation attached to that face (as in a SU (2)BF theory)? Should it be the SL(2, C)/SO(4) dimension (as in a SL(2, C)/SO(4)BF theory)? The imposing of constraint changes the measure of the path integral, and we have no control about that in this derivation.
8.4. Spinfoam: a unified view
After all this rather cumbersome (and quite chronological) model-making I would
like to give a more coherent and compact view of spinfoam approach and of its
relation with Loop Quantum Gravity. As pointed out in [154] it is time to provide
a “top-to-bottom” framework, in which set some properties we want these models
to satisfy and deduce from them specific models. This has (at least) the merit of
11
The partition function is obviously a particular case of this formula, i.e.
XY
Y
EP RL
RL
Z∆
=
Af (jf )
AEP
(j, ι) .
v
j,ι
130
f
v
(8.71)
8.4. Spinfoam: a unified view
clarifying the mess of technicalities that plague – in my opinion – the spinfoam
model-making approach.
Remark. What follows is just an attempt to give a unified and more coherent
framework of spinfoam approach, but it is not a well established and rigorous
“chapter” of Loop Quantum Gravity. You should take what follows as a reasoning
ex post on how we can insert all the models into a single framework. For this
section I mainly follow the paper [154].
First of all: what do we want from a spinfoam model? We want a way to
compute transition amplitudes
hs | A | s0 i ,
(8.72)
between Loop Quantum Gravity spin network states. A represents the projection
operator into the kernel of the hamiltonian operator (recall equation (8.4)). In a
covariant 4d context this is rephrased: take a 4d manifold with boundary state
ψ, we want to calculate
hA | ψi .
(8.73)
The boundary state ψ is typically formed by two spin networks (in terms of
graphs Γψ = Γψ1 ∪ Γψ2 ) the “initial” and “final” spin networks, but we can
be as generic as we want. This last expression is the amplitude associated to
the specific state ψ. The linear functional A (or, if you want, h0|A, but here
we enter in the subtle (and fascinating) issue of vacuum in Quantum Gravity,
maybe we will spend some words about it in the rest of the chapter) is the heart
of the spinfoam model, and must be thought of as an evolution operator. What
properties do we ask (8.73) to have?
• Superposition. Namely we want
hA | ψi =
X
A(σ) ,
(8.74)
σ
i.e. we want that the amplitude is expressed as a “sum over histories”. This
is what we argued with a rather heuristic touch in section 8. Obviously
the specific set over which to sum is at this level completely undetermined.
• Locality. This is a request on the single history-amplitude A(σ). We require
Y
A(σ) ∼
Av .
(8.75)
v
This amounts to ask that each history has an amplitude that is the product
of elementary amplitudes, or vertex amplitudes (the “atoms” we have been
talking about).
131
8. The spinfoam approach to the dynamics
• Local Lorentz invariance. Recall that classical general relativity, in tetrad
formulation, has a local Lorentz invariance, namely a SL(2, C) gauge invariance. However the boundary states know nothing of SL(2, C), they are
built up over SU (2) gauge invariance. Thus there must be some embedding from SU (2)-gauge invariant states to SL(2, C)-invariant ones. This
map, usually called simply f is what actually determines the spinfoam
amplitude.
Now we have the key ingredients. We have to focus on a single vertex, surrounded by a kinematical state ψI . We define
Av (I) = hAv | ψI i = h11 | f | ψI i ,
(8.76)
f = PSL(2,C) ◦ Y
(8.77)
with
the map that embeds the boundary state in L2 (SL(2, C)L ) and then projects
onto the SL(2, C)-invariant on the nodes, i.e. onto L2 (SL(2, C)L /SL(2, C)N ).
|ψI i stands for a state in the boundary Hilbert space of the vertex v, and I
denotes a set of quantum numbers (think of a spin network state).
Writing it a bit more explicitly
Z
Y
hAv | ψI i =
dgn h11|U(gn )Y |ψI i .
(8.78)
SL(2,C)N
n
ψI is a generic boundary state: it could be a spin network state |jl , ιn i giving
Av (jl , ιn ) or, in the holonomy(‘position’) representation, |Ul i giving Av (Ul ). h11|
stands for the SL(2, C) holonomy state where all the links are set to the identity
(it corresponds to the evaluation of the state on which it acts). U(g) is the action
of g ∈ SL(2, C) over the state Y |ψI i. In a suggestive way one could write
Z
Y
hAv | =
dgn h11|U(gn )Y ,
(8.79)
SL(2,C)N
n
as a functional over the kinematical Hilbert space of Loop Quantum Gravity. Of
course this definition gives the EPRL formula (8.69), when |ψI i = |jl , ιn i, and
gives exactly (8.67) in terms of holonomies.
Moreover, this framework is completely general, in the sense that it is able to
reproduce every spinfoam model that matches Loop Quantum Gravity kinematical Hilbert space on the boundary. Obviously various models can be reproduced
by a choice of the f function, i.e. of the Y map, which is, ultimately, what
determines the specific vertex amplitude, and – indeed – is itself completely determined by how we impose the constraints on the BF model (recall that there
are infinite SU (2) irreducibles inside each SL(2, C) irreducible).
132
8.5. Spinfoams as a field theory: Group Field Theory
Once got the vertex amplitude one get the total amplitude by
XY
hA | ψI i '
Av (I) ,
σ
(8.80)
v
notice that this sum is two-fold: fix a triangulation and you have to sum over the
labeling of internal faces and edges; then you have to sum over all the possible
triangulation in order to really catch all the possible “paths” bounded by ψI !
This make apparent the problem in dealing with triangulation dependence and
in how to weight different triangulations (the attentive reader was surely already
aware of this problem since the very beginning of the discretization stuff). I will
briefly review the topic (which is a totally open issue) in the following section.
Moreover here there is no hint about something attached to the faces of each
spinfoam. We know from the BF theory derivation that something there must
be, and that it very likely has to do with the dimension of the representations
attached to the faces (see chapter 9).
8.5. Spinfoams as a field theory: Group Field
Theory
The present section is a bit beyond the scope of this review of Loop Quantum
Gravity and spinfoam models, particularly since to understand the research I
did in this field (9) the argument of this section is not strictly necessary, nor – I
must admit – closely related.
However, I spent quite a long time in Marseille studying Group Field Theory,
and I seriously think it has more than a few chances of becoming a bridge
between Quantum Gravity in its spinfoam formulation and the prolific and wellestablished framework of Quantum Field Theory.
In a sense that will be clear – I do hope – from what follows, Group Field
Theory is precisely a way of putting the spinfoam models in the framework of
QFT, a very special kind of QFT.
GFT was first introduced by Boulatov in 1992 [197], and had, since then, a
very promising development as a Quantum Gravity framework. The nowadays
work in this topic is mainly to study the derivation of the spinfoam models
(particularly the EPRL model) in this framework; to study quantum corrections
and renormalization issues; control the sum over triangulations; all done with
QFT-like tools. For a panoramic of the present-day research see the following
papers [199, 198], while for good reviews of GFT see [200, 201, 202].
The very basic idea is the following: find a field theory whose Feynman diagrams are spinfoams. The natural ancestor of GFT are the well known matrix
133
8. The spinfoam approach to the dynamics
models: their Feynman diagrams are ribbon-like, which can be seen as dual to
a triangulated surface [203, 204].
A matrix model is a model whose action is something like
S[M ] =
Nλ
N
Tr(M 2 ) +
Tr(M 3 ) ,
2
3!
(8.81)
where M is a N × N matrix. I have taken a potential term of order 3, but of
course it is just an example.
Going directly to the quantum side of the story, let us try to calculate the
partition function for this action
Z
Z
X 1 N λ n
2
−S[M ]
−N
TrM
(TrM 3 )n .
(8.82)
Z = DM e
= DM e 2
n!
3!
n
What are the differences from standard QFT? There are (at least) two: 1. it is
non local, since our ‘fields’ Mij are here objects which depend on two ‘points’;
2. it is not a field theory, since the ‘space’ variables are discrete, i, j = 1, . . . , N .
We shall drop the non-field stuff in a moment; the true ‘news’ is the dropping
of locality, which might sound rather blasphemous, but recall that we are in a
completely combinatorial framework and these fields do not pretend to represent
causal propagating particles, but blocks of spacetime, as we shall see . Deriving
the propagator and interaction vertex of the theory is straightforward
Gijkl = δij δkl
i
j
k
l
j
Vijklmn = δjm δkl δmn
,
(8.83)
.
(8.84)
m
i
n
k
l
Now imagine to glue interaction vertices through propagators to obtain Feynman
diagrams, you will get – as hinted above – ribbon-like structures. If we take the
dual of each of these graphs you obtain precisely a triangulation of a 2-manifold.
In this sense each propagator represents a side of a triangle, while the vertex
represent the triangle itself, namely
G∼
134
,
V ∼
.
8.5. Spinfoams as a field theory: Group Field Theory
We can rephrase this by saying that each propagator is an edge and each vertex
a face of the dual 2-skeleton of a triangulation. This gives also a powerful tool
for calculation explicitly the partition function (8.82), indeed it has been proven
that it is given by
X
Z=
wg (λ)N 2−2g ,
(8.85)
g
where g is the genus of the triangulated surface and w is a weight depending on
the diagram symmetry factor. This is precisely the partition function of a 2d
BF theory (8.24)!
You guess the clue: going up in the number of indices – i.e. building tensor
models [205, 206] – we can have the hope of getting skeletons of 3d and 4d
manifolds, i.e. spinfoams.
Actually, this is not the case. Tensor models has been proven to fail in this
respect. Essentially: 1. they are too simple to incorporate even the 3d BF theory
case and 2. they produce both manifolds and pseudo-manifolds12 [205, 206].
The natural way to proceed, is to pass from matrices to fields, defined on an
appropriate domain space. In GFT – hence the name – this domain space is
chosen to be an appropriate number of copies of a compact group G
Z
Z
λ
1
2
2d
dg1 dg2 φ (g1 , g2 ) +
dg1 dg2 dg3 φ(g1 , g2 )φ(g2 , g3 )φ(g3 , g1 ) ,
S [φ] =
2 G2
3! G3
(8.86)
this is the GFT analogous of the matrix model (8.81). Instead of analyzing its
features, let us move on seeing the key features of a general GFT.
• φ is a (typically) real valued field on Gn , with G compact Lie group and n
will be the dimension of the ‘produced’ triangulated manifold. We require
a gauge invariance under the G-(right)action
φ(g1 , . . . , gn ) = φ(g1 g, . . . , gn g) ,
∀gi , g ∈ G .
(8.87)
• the GFT action (for a real valued field) has the (very) generic structure
S[φ] = T [φ] + λV [φ] ,
(8.88)
with a “kinetic” part which has no dynamical meaning, since at this stage
there is nothing as a time variable; and an interaction term which defines how the various group elements are attached with one another: the
interaction term is actually the ‘atom’ of hidden spinfoam model, and it
determines the kind of n-complexes by which the manifold is triangulated.
12
Sort of triangulation of manifold with singularities [207].
135
8. The spinfoam approach to the dynamics
• The Feynman diagrams are 2-complexes, whose combinatorial structure
is entirely given by the interaction term of S[φ]. They are interpreted
as dual to a n-discretization in terms of n-complexes (if the interaction
term is particularly simple – actually the only case I will consider – these
complexes are indeed simplices, and the discretization is a triangulation).
• The value of the Feynman diagram is seen as a spinfoam amplitude, the
amplitude associated with the discretization of that specific diagram.
• The GFT partition function
Z
Z
X
−S[φ]
Z = Dφ e
= Dφ e−T [φ] e−λV [φ] =
w(∆)λV (∆)A∆ ,
(8.89)
∆
which is the typical perturbative expansion in terms of Feynman diagram:
∆ denote equivalently the Feynman diagram or its associated discretization; V (∆) is the number of vertices of the diagram, w(∆) is its symmetry
factor and A∆ is the value of the diagram (the spinfoam amplitude).
Notice that this gives also a very natural way of dealing with triangulation dependence of spinfoam: each triangulation has a different spinfoam
amplitude and we simply sum them with the weight given by the coupling
and the symmetry factor.
• The correlation functions of GFT give the amplitude in going from one
state to another, i.e. the (most wanted) hs | A | s0 i
Z
X
V
A[Γ, ψ] =
w(∆)λ (∆)A∆ = Dφ Pψ (φ)e−S[φ] ,
(8.90)
∆|∂∆=Γ
where ψ is a state on the (boundary) graph Γ and P is some polynomial
function of φ that is able to render the specific graph Γ labeled by variables
so to reproduce ψ 13 .
Let us now review some GFT models, stating only the results
• Ponzano-Regge. The 3d SU (2) GFT model with trivial kinetic term and
tetrahedron-like interaction term, generates exactly the 3d BF path integral (8.25) for SU (2). The action is
Z
1
dg1 dg2 dg3 φ2 (g1 , g2 , g3 )
S[φ] =
2
Z Y
6
λ
+
dgi φ(g1 , g2 , g3 )φ(g3 , g4 , g6 )φ(g6 , g2 , g4 )φ(g5 , g4 , g1 ) . (8.91)
4! i=1
13
Think of n-point functions of QFT and you will get the idea. However, I will not go in much
more details on this, I refer the reader to [183] and references therein.
136
8.5. Spinfoams as a field theory: Group Field Theory
φ
g1 g2 g3
φ
φ
φ
g1
g4
g5
g3
g4
g6
φ
g5 g2 g6
φ
Figure 8.6.: Propagator and interaction term of the 3d GFT model (8.91). The
propagator is simply a delta over the group, while the interaction
term is defined by a tetrahedron-like contraction.
The interaction term is easily understood looking to figure 8.6. It is indeed a tetrahedron like graph by itself; moreover, if you take the dual in
this sense strand → edge (of a triangle), field (i.e. 3-strand) → triangle,
interaction → tetrahedron. In this sense the interaction term is precisely
telling us how to glue the 3 edges of 4 triangles to produce a tetrahedron.
In this precise sense, gluing interaction terms through propagators you
create Feynman diagrams as well as a 3d triangulation.
Taking the Fourier representation (i.e. the Peter-Weyl decomposition) of
the field you get for the vertex the Ponzano-Regge amplitude.
• 4d GFT model. Take the following action
1
S[φ] =
2
Z
λ
dg1 dg2 dg3 dg4 φ (g1 , g2 , g3 , g4 ) +
5!
2
Z Y
10
dgi φ(g1 , g2 , g3 , g4 )×
i=1
φ(g4 , g5 , g6 , g7 )φ(g7 , g3 , g8 , g9 )φ(g9 , g6 , g2 , g10 )φ(g10 , g8 , g5 , g1 ) . (8.92)
Here the interaction vertex is a 4-simplex (see figure 8.7), thus Feynman
diagrams are dual to a 4d triangulation, and the vertex value, once decomposed into irreducible representations of the underlying group, is exactly
the 4d BF vertex amplitude, as in equation (8.26).
• Barrett-Crane. As a final example I give a sketch of how it is possible to
deduce the Barrett-Crane spinfoam model 8.3.1 from a GFT model. The
gauge group is of course G = Spin(4). The trick is to introduce a (slightly)
137
8. The spinfoam approach to the dynamics
φ
1 2 3 4
φ
4
5
6
7
1
5
8
10
10 6
2 9
φ
9 3 8 7
φ
φ
Figure 8.7.: Interaction term of the 4d GFT model (8.92).
more complicated kinetic term in (8.92), namely
Z
1
T [φ] =
d4 g (P φ(g1 , . . . , g4 ))2 ,
2
(8.93)
with
Z
P φ(g1 , . . . , g4 ) =
Z
dg
SL(2,C)
d4 u φ(g1 gh1 , g2 gh2 , g3 gh3 , g4 gh4 ) .
SU (2)4
(8.94)
It is matter of calculations to prove that this GFT gives exactly the BC
spinfoam amplitude (8.39)
For a derivation of the EPRL spinfoam model from a GFT, I recommend the
reading of [198].
Bubble divergences A very delicate and important issue of the spinfoam approach to Quantum Gravity is the one regarding divergences. I want here only to
inform the reader of the importance of the issue, and give some useful references.
Firstly a remark: inside the LQG framework one always talks about infrared divergences, i.e. divergences arising when summing over the large-distance-scale
degrees of freedom, i.e. large spins. In LQG there are no ultraviolet divergences,
since there is an ultimate minimum length-scale, of the order of the Planck length
(see section 7.5).
138
8.5. Spinfoams as a field theory: Group Field Theory
Bubble divergences arise in the sum over spins. You can see it as a loop
integral of QFT. Simple power counting techniques reveal the presence of divergences depending on the nature of the manifold and of the kind of vertex one is
considering (how many edges converge to that vertex, how many faces, and so
on). Many studies have been done in this respect (See [208, 209, 210] and references therein). Moreover, it has been noted that the face amplitude is crucial in
determining whether a bubble is or is not divergent [210]. We shall talk about
this in chapter 9.
Here I want to stress that the GFT approach gives powerful tools to handle
and analyze these divergent bubbles and may provide a unified strategy to fix
them14 . I refer the reader to the recent work [211] and to [212, 213].
This concludes this brief detour into GFT framework. As I have said at the
beginning of this section, GFT is not the focus of my research papers, so this is
not the right place to discuss it in more details. What I wanted to give, is the
idea of the existence of a (in my opinion) promising unifying framework, inserting
Quantum Gravity into a QFT approach while preserving its non-perturbative
nature. Indeed recall that the perturbation of the Feynman diagram series in
GFT is a perturbation in the coupling λ, i.e. is a perturbation on the number
of vertices of the triangulations, that is on the “complexity” of the graph15 and
not a perturbation around a fixed background geometry. I really encourage
the interested reader to look up the bibliography I have referred to during this
section, in particular [200, 201, 202] for good reviews on the subject.
14
Incidentally, notice that the GFT approach provide with an intriguing duality between UV
and IR divergences: indeed the Quantum Gravity IR divergences (namely the large spin
divergences) can also be seen as UV divergences on the group.
15
Of course we have no clue on the coupling parameter, but this is another problem.
139
9. A proposal for fixing the face
amplitude in Quantum
Gravity spinfoam models
The present chapter is intended to present the content of the paper [5], in which
we propose a way for fixing the face amplitude of a general spinfoam model. The
proposal is motivated by the requiring of a sort of “unitarity” of time evolution of
space geometries, in a sense that will be made clear in what follows. We found
that this requirement imposes the face amplitude to be equal to the dimension of
the SU (2)-projected representation of the SO(4)(SL(2, C)) one attached to the
face.
9.1. Introduction and resume of the content of
the paper [5]
In this section I present the content of the paper [5], which collects the results
of the work I have done in Marseille, in collaboration with Eugenio Bianchi and
Carlo Rovelli.
I have repeatedly focused on the fact that, while for BF theory the face amplitude of the spinfoam sum is well determined by the path integral discretization
procedure and it is given by the dimension of the representation labeling the
face (8.24),(8.25),(8.26), for Quantum Gravity the situation is much more subtle. One starts with a certain BF theory, obtains a spinfoam sum formula, and
then imposes constraints in the vertex amplitude. What happens to the face
amplitude? The BC model, for instance, simply let it be the BF face amplitude, i.e. the square of the SU (2) irreducible dimension. In SO(4)-models with
the Immirzi parameter (such as SO(4)-EPRL) one has1
Aρf = (2j+ + 1)(2j− + 1) = (2γ+ jf + 1)(2γ− jf + 1).
1
(9.1)
In [5] we have worked in the euclidean (SO(4)) case. However, everything can be done in
the lorentzian (SL(2, C)) as well, mutatis mutandis. See section 8.3.2 for discussions about
this point.
141
9. A proposal for the face amplitude
However this is not well motivated, indeed doubts can be raised against this
argument. For instance, Alexandrov [221] has stressed the fact that the implementation of second class constraints into a Feynman path integral in general
requires a modification of the measure, and here the face amplitude plays precisely the role of such measure, since Av ∼ ei Action . Do we have an independent
way of fixing the face amplitude?
Let me recall that all the spinfoam models share the same form of partition
function, namely the sum
X Y
Y
Z∆ =
Av (ρf , ιe ) ,
(9.2)
Aρf
ρ,ι
f
v
where, as usual (recall section 8.2), that sum is intended to be a sum over all
the possible labellings of the faces (edges) of ∆∗ with irreducible representations
(intertwiners) of an appropriate group G.
In [5] we argued that the face amplitude is uniquely determined for any spinfoam sum of the form (9.2) by three inputs: 1. the choice of the boundary
Hilbert space, 2. the requirement that the composition law holds when gluing
two-complexes; and 3. a particular “locality" requirement, or, more precisely, a
requirement on the local composition of group elements.
We argued that these requirements are implemented if the partition function
Z is given by the expression
Z
Y
Y
(9.3)
δ(Ufv1 ...Ufvk ) ,
Z∆ = dUfv
Av (Ufv )
v
f
where Ufv ∈ G, v1 ...vk are the vertices surrounding the face f , and Av (Ufv ) is the
vertex amplitude Av (jf , ie ) expressed in the group element basis [222]. Then we
showed that this expression leads directly to (9.2), with arbitrary vertex amplitude, but a fixed choice of face amplitude, which turns out to be the dimension
of the representation j of the group G,
Aj = dim(j) .
(9.4)
In particular, for Quantum Gravity this implies that the BF face amplitude (9.1)
is ruled out, and should be replaced (both in the Euclidean and in the Lorentzian
case) by the SU (2) dimension
Aj = dim(j) = 2j + 1 .
(9.5)
Equation (9.3) is the key expression of the whole paper.
I organize this chapter as the paper from which is taken, i.e.: in section 9.2 I
show that SO(4) BF theory (the prototypical spinfoam model) can be expressed
142
9.2. BF theory
h−1
v 1 ek
v1
h v 1 e1
Ue1
h−1
v2 e1
vk
Ufv1
v2
f
...
v3
Figure 9.1.: Schematic definition of the group elements hve , Ufv and Ue associated
to a portion of a face f of the two-complex.
in the form (9.3). Then I discuss the three requirements above and I show that
(9.3) implements these requirements. (Section 9.3). Finally I show that (9.3)
gives (9.2) with the face amplitude (9.4) (Section IV).
The problem of fixing the face amplitude has been discussed also by Bojowald
and Perez in [220]. Bojowald and Perez demand that the amplitude be invariant
under suitable refinements of the two-complex. This request is strictly related
to the composition law that we considered in [5], and the results we obtained
are consistent with those of [220].
9.2. BF theory
Take the general expression of the BF partition function in terms of group
elements (see equation (8.16))
Z Y
Y
δ(Ue1 ...UeN ) ,
(9.6)
Z∆ =
dUe
e
f
where Ue are group elements associated to the oriented edges of σ, and (e1 , ..., eN )
are the edges that surround the face f . Let us introduce group elements hve ,
labeled by a vertex v and an adjacent edge e, such that
Ue = hve h−1
v0 e ,
(9.7)
where v and v 0 are the source and the target of the edge e (see figure 9.1). Then
we can trivially rewrite (9.6) as
Z
Y
−1
Z∆ = dhve
δ (hv1 e1 h−1
(9.8)
v2 e1 ) ... (hvN eN hv1 eN ) .
f
143
9. A proposal for the face amplitude
Now define the group elements
Ufv = h−1
ve hve0
(9.9)
associated to a single vertex v and two edges e and e0 that emerge from v and
bound the face f (see figure 9.1). Using these, we can rewrite (9.6) as
Z
Z
Y
Y
0
h
)
δ(Ufv1 ...UfvN ) ,
Z∆ = dhve dUfv
δ(Ufv , h−1
ve ve
v,f v
f
where the first product is over faces f v that belong to the vertex v, and then a
product over all the vertices of the 2-complex.
Notice that this expression has precisely the form (9.3), where the vertex
amplitude is
Z
Y
v
Av (Uf ) = dhve
δ(Ufv , hve h−1
(9.10)
ve0 ) ,
fv
which is the well-known expression of the 15j Wigner symbol (the vertex amplitude of BF in the spin network basis) in the basis of the group elements
(cfr.(8.29)).
We have shown that the BF theory spinfoam amplitude can be put in the
form (9.3). We shall now argue that (9.3) is the general form of a local spinfoam
model that obeys the composition law.
9.3. Three inputs
(a) Hilbert space structure. Equation (9.2) is a coded expression to define the
amplitudes
XY
Y
A∆ (jl , ιn ) =
Ajf
Av (jf , ιe ; jl , ιn ) ,
(9.11)
j,ι
f
v
defined for a triangulation ∆ with boundary, where the boundary graph Γ is
formed by links l and nodes n. The spins jl are associated to the links l, as well
as to the faces f that are bounded by l; the intertwiners ιn are associated to the
nodes n, as well as to the edges e that are bounded by n. The amplitude of the
vertices that are adjacent to these boundary faces and edges depend also on the
external (thus fixed) variables (jl , ιn ).
In a quantum theory, the amplitude A(jl , ιn ) must be interpreted as a (covariant) vector in a space HΓ of quantum states.2 We assume that this space has a
2
If Γ has two disconnected components interpreted as “in" and an ”out" spaces, then HΓ
can be identified as the tensor product of the “in" and an ”out" spaces of non-relativistic
quantum mechanics. In the general case, HΓ is the boundary quantum state in the sense
of the boundary formulation of quantum theory [151, 223].
144
9.3. Three inputs
Hilbert space structure, which we know. In particular, we assume that
HΓ = L2 [GL , dUl ] ,
(9.12)
where L is the number of links in Γ and dUl is the Haar measure. Thus we can
interpret (9.11) as
A∆ (jl , ιn ) = hjl , ιn | Ai ,
(9.13)
where |jl , ιn i is the spin network function (cfr. equation (7.28))
O
O
hUl | jl , ιn i = ψjl ,ιn (Ul ) =
ιn .
Djl (Ul ) ·
Using the scalar product defined by (9.12), we have
Z
0 0
hjl , ιn | jl , ιn i =
dUl ψjl ,ιn (Ul )ψjl0 ,ι0n (Ul )
Y
Y
dim(jl ) δjl jl0
διn ι0n .
=
l
(9.14)
n
l
(9.15)
n
where dim(j) is the dimension of the representation j. Therefore the spinnetwork functions ψjl ,ιn (Ul ) are not normalized. (These dim(j) normalization
factors are due to the convention chosen: they have nothing to do with the
dimension of the representation that appears in (9.4).) The resolution of the
identity in this basis is
X Y
11 =
dim(jl ) |jl , ιn ihjl , ιn | .
(9.16)
jl ,ιn
l
(b) Composition law. In non relativistic quantum mechanics, if U (t1 , t0 ) is the
evolution operator from time t0 to time t1 , the composition law reads
U (t2 , t0 ) = U (t2 , t1 )U (t1 , t0 ) .
(9.17)
That is, if |ni is an orthonormal basis,
X
hf |U (t2 , t0 )| ii =
hf |U (t2 , t1 )| nihn |U (t1 , t0 )| ii .
n
Let us write an analogous condition of the spinfoam sum. Consider for simplicity
a two-complex σ = σ1 ∪ σ2 without boundary, obtained by gluing two twocomplexes σ1 and σ2 along their common boundary Γ. Then we require that W
satisfies the composition law
Zσ1 ∪σ2 = hAσ2 | Aσ1 i ,
(9.18)
145
9. A proposal for the face amplitude
...
v2
vk
Ufv2
Ufvk
f
l
Ufv1
v1
Ul
l
Figure 9.2.: Cutting of a face of the 2-skeleton. The holonomy Ul is attached to
a link of the boundary spin network and satisfies equation (9.21).
– where now I label Z directly with the 2-skeleton σ ⊂ ∆∗ – as discussed by
Atiyah in [224]. Notice that to formulate this condition we need the Hilbert
space structure in the space of the boundary states.
(c) Locality. As a vector in HΓ , the amplitude A(jl , ιn ) can be expressed on
the group element basis
A(Ul ) = hUl | Ai =
X
Y
jl ,ιn
l
dim(jl ) ψjl ,ιn (Ul )A(jl , ιn ) .
(9.19)
Similarly, the vertex amplitude can be expanded in the group element basis
Av (Ufv ) = hUfv | Av i
X Y
dim(jfv ) ψjfv ,ιvn (Ufv )Av (jfv , ιvn ) .
=
jfv ,ivn
(9.20)
fv
Notice that here the group element Ufv and the spin jfv are associated to a vertex
v and a face f adjacent to v. Similarly, the intertwiner ιvn is associated to a
vertex v and a node n adjacent to v. Consider a boundary link l that bounds
a face f (see figure 9.2). Let v1 ...vk be the vertices that are adjacent to this
face. We say that the model is local if the relation between the boundary group
element Ul and the vertices group elements Ufv is given by
Ul = Ufv1 ... Ufvk .
(9.21)
In other words: if the boundary group element is simply the product of the group
elements around the face.
Notice that a spinfoam model defined by (9.3) is local and satisfies composition
146
9.4. Face amplitude
law in the sense above. In fact, (9.3) generalizes immediately to
Z
Aσ (Ul ) =
dUfv
Y
Av (Ufv )
v
×
Y
δ(Ufv1 ...Ufvk )
internal f
Y
δ(Ufv1 ...Ufvk Ul−1 )
.
(9.22)
external f
Here the first product over f is over the (“internal") faces that do not have an
external boundary; while the second is over the (“external") faces f that are
also bounded by the vertices v1 , ..., vk and by the the link l. It is immediate
to see that locality is implemented, since the second delta enforces the locality
condition (9.21).
Furthermore, when gluing two amplitudes along a common boundary we have
immediately that
Z
dUl Aσ1 (Ul ) Aσ2 (Ul ) = Zσ1 ∪σ2 ,
(9.23)
because the two delta functions containing Ul collapse into a single delta function
associated to the face l, which becomes internal.
Thus, (9.3) is a general form of the amplitude where these conditions hold.
In [220], Bojowald and Perez have considered the possibility of fixing the face
amplitude by requiring the amplitude of a given spin/intertwiner configuration
to be equal to the amplitude of the same spin/intertwiner configuration on a
finer two-simplex where additional faces carry the trivial representation. This
requirement imply essentially that the amplitude does not change by splitting a
face into two faces. It is easy to see that (9.3) satisfies this condition. Therefore
(9.3) satisfies also the Bojowald-Perez condition.
9.4. Face amplitude
Finally, let us show that (9.3) implies (9.2) and (9.4). To this purpose, it is
sufficient to insert (9.20) into (9.3). This gives
Zσ =
Z
dUfv
YX Y
v jfv ,ivn
×
Y
dim(jfv ) ψjfv ,ιvn (Ufv )Av (jfv , ιvn )
fv
δ(Ufv1 ...Ufvk ) .
(9.24)
f
147
9. A proposal for the face amplitude
Expand then the delta function in a sum over characters
Z
YX Y
Zσ = dUfv
dim(jfv ) ψjfv ,ιvn (Ufv )Av (jfv , ιvn )
v
×
jfv ,ιvn
Y X
f
jf
(9.25)
fv
dim(jf ) Tr(Djf (Ufv1 ) · · · Djf (Ufvk )) .
We can now perform the group integrals. Each Ufv appears precisely twice in
the integral: once in the sum over jfv and the other in the sum over jf . Each
integration over the group, gives a delta function (recall equation (8.21))
Z
1
v
(9.26)
δjfv ,jf =
dUfv Djf (Ufv )Djf (Ufv ) ,
dim(jf )
which can be used to kill the sum over jfv dropping the v subscript. Notice that
the dimensional factor involved here, exactly cancels – once done the product–
the first dimensional factor in (9.25), i.e. the one coming from the spin network
normalization. Following the contraction path of the indices, it is easy to see
that these contract the two intertwiners at the opposite side of each edge. Since
intertwiners are orthonormal, this gives a delta function διvn ,ιvn0 which reduces the
0
sums over intertwiners to a single sum over ιn := ιvn = ιvn . Bringing everything
together, we have
X Y
Y
Zσ =
dim(jf )
Av (jfv , ιvn ) .
(9.27)
jι
f
v
This is precisely equation (9.2), with the face amplitude given by (9.4).
Notice that the face amplitude is well defined, in the sense that it cannot be
absorbed into the vertex amplitude (as any edge amplitude can). The reason
is that any factor in the vertex amplitude depending on the spin of the face
contributes to the total amplitude at a power k, where k is the number of sides
of the face. The face amplitude, instead, is a contribution to the total amplitude
that does not depend on k. This is also the reason why the normalization
chosen for the spinfoam basis does not affect the present discussion: it affects
the expression for the vertex amplitude, not that for the face amplitude.
By an analogous calculation one can show that the same result holds for the
amplitudes W : equation (9.11) follows from (9.22) expanded on a spin network
basis.
In conclusion, we have shown that the general form (9.3) of the partition function, which implements locality and the composition law, implies that the face
amplitude of the spinfoam model is given by the dimension of the representation
of the group G which appears in the boundary scalar product (9.12).
148
9.4. Face amplitude
In general relativity, in both the Euclidean and the Lorentzian cases, the
boundary space is
HΓ = L2 [SU (2)L , dUl ] ,
(9.28)
therefore the face amplitude is dj = dimSU (2) (j) = 2j + 1, and not the SO(4)
dimension (9.1), as previously supposed.
Notice that such dj = 2j+1 amplitude defines a theory that is far less divergent
than the theory defined by (9.1). In fact, the potential divergence of a bubble
is suppressed by a power of j with respect to (9.1). In [210], it has been shown
that the dj = 2j + 1 face amplitude yields a finite main radiative correction to
a five-valent vertex if all external legs set to zero.
149
Conclusion
Let me now briefly review in a bird-flight manner the results which have been
achieved.
We have worked on phantom dark energy models in classical Cosmology. Let
me recall that phantom models (i.e. models with some kind of phantom energy)
are not ruled out by observations, even if the standard model of Cosmology,
ΛCDM, scientifically still “wins”, by virtue of Occam’s razor. Crossing of the
phantom divide line is not excluded as well. We have analyzed what seem to
us one of the best candidates to reproduce this crossing: two-field cosmological
models with one scalar and one phantom field.
We have discovered that, starting from an expression for the Hubble variable
h(t), there is an infinite number of models (i.e. of potentials) that have, among
their solutions, precisely that h(t). This freedom, namely to have infinite different models which are compatible with the same dynamics for the universe, is
the most important result presented in [1].
One possible approach to discriminate between such models, is to couple the
cosmological fields with other, observable, fields, such as cosmic magnetic fields.
The calculations and the numerical simulations have indeed shown that such a
coupling gives in principle observable results, capable of select between different
models [2].
However, we are aware that phantom fields have been heavily criticized, and
actually they do deserve the label of ‘exotic’. The negative sign of the kinetic
term is indeed something that disturbs even just to write it. Thus, we tried to
find out some mechanism so that the negative sign was only effective, while the
“fundamental” field is a standard one, with no stability problem. Accepting to
work in the P T symmetric Quantum Theory framework, we succeeded in this
respect, and find a possible way to have an effective phantom field, stable to
quantum fluctuations [3].
These were the contributions in the phantom/crossing-of-the-phantom-line
branch of Cosmology. Aside, we analyzed one-field models giving evolutions
for the universe with an interesting kind of singularities: sort of Big Bang/Big
Crunch with a finite non-zero radius [4]. Notice that this is in line with all
Quantum Gravity theories, namely to have a minimum length scale. Thus, we
thought that the study of non-zero radius singularities could be useful also in
perspective of future results from the Quantum Gravity community.
151
Conclusion
Quantum Gravity is essential to properly understand what is space and what
is time. Maybe the total absence of experimental results give to these kind of
studies an halo of philosophical vagueness. I could agree with this, but still I
firmly think that this kind of fundamental research must be carried on. Indeed,
many results of LQG and spinfoam theory are very suggestive and stimulating,
even if on purely theoretical basis. The attempt to find a ‘sum-over-histories’
formulation of Quantum Gravity, based on the LQG approach to quantization, is
a recent and promising field of research, where results crowd from many different
and far areas of Theoretical Physics. Specifically we have proposed a way of fixing
the face amplitude of a general spinfoam model. Up till now, the face amplitude
has been taken to be the one of the BF theory underlying the quantization
procedure. But we argued that this is not the case, if one want to assure the
proper “gluing” of spinfoam amplitudes [5].
152
Acknowledgements
Many are the people that deserve my thanks, for different reasons. Honestly, I
wouldn’t be able to thank properly everybody. So I use this occasion to give my
explicit thanks to my thesis advisor, Alexander Kamenshchik, who helped me as
a professor, as a research colleague and, most importantly, as a friend, especially
in some difficult moments. My only other explicit thank goes to Carlo Rovelli,
who guested me in his amazing group in Marseille, where I learned a lot, both
from the scientific and personal viewpoint.
I warmly hope that all the others will feel thanked as well, please. Thank You.
Grazie. Merci.
4lab
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