Aspects of extra dimensions and membranes Licentiate Thesis Martin Sundin

Aspects of extra dimensions and membranes Licentiate Thesis Martin Sundin
Licentiate Thesis
Aspects of extra dimensions and
membranes
Martin Sundin
Mathematical Physics, Department of Theoretical Physics,
School of Engineering Sciences
Royal Institute of Technology, SE-106 91 Stockholm, Sweden
Stockholm, Sweden 2011
Typeset in LATEX
Akademisk avhandling för avläggande av teknologie licentiatexamen (TeknL) inom
ämnesområdet teoretisk fysik.
Scientific thesis for the degree of Licentiate of Engineering (Lic Eng) in the subject
area of Theoretical physics.
ISBN 978-91-7415-934-9
TRITA-FYS-2011:14
ISSN 0280-316X
ISRN KTH/FYS/--11:14--SE
c Martin Sundin, April 2011
Printed in Sweden by Universitetsservice US AB, Stockholm April 2011
Abstract
This thesis is about thwo papers related to extra dimensions. Paper A discusses
extrinsic curvature effects, and paper B treats symmetries of supersymmetric membranes.
In the part of this thesis related to paper A, we extend the theory of nonrelativistic quantum particles confined to submanifolds to relativistic boson fields.
We show that a Klein-Gordon field constrained to a submanifold of a Lorentzian
manifold experiences an induced potential similar to the one for the Schrödinger
equation. We embedd the Schwarzschild solution and the Robertson-Walker spacetime and derive the induced potentials. Possible physical consequences of these
induced potentials are also discussed.
The second part is related to paper B, we study the dynamics of supersymmetric membranes, which are higher dimensional generalizations of supersymmetric
strings. We derive a supersymmetric analogue of a dynamical symmetry for bosonic
membranes.
Key words: Extra dimensions, brane world scenarios, supermembranes
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Preface
This thesis is the result of my research at the Department of Theoretical Physics
during the time period April 2009 to April 2011. The first part of the thesis contains
background material and results on the subjects constrained quantum mechanics,
constrained relativistic fields and membrane dynamics. The second part consists of
the scientific papers listed below.
List of papers
[A] Edwin Langmann and Martin Sundin
Extrinsic curvature effects in brane-world scenarios
arxiv:1103.3230, submitted for publication.
[B] Jonas de Woul, Jens Hoppe, Douglas Lundholm and Martin Sundin
A dynamical symmetry of supermembranes
arXiv:1004.0266, accepted for publication in the Journal of High Energy
Physics (JHEP).
The thesis author’s contribution to the papers
[A] I contributed ideas to most parts of the paper, and I wrote a first draft. The
results and final version of the paper weres obtained in collaboration of both
authors.
[B] This paper was a development of earlier results in [1]. The calculations and
writing of the paper was done in cooperation with the co-authors.
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Acknowledgments
I want to thank my supervisor Edwin Langmann for giving me the opportunity to
do research in theoretical physics and my assistant supervisor Teresia Månsson for
her help and support during my work. I am also very grateful to the collaborators of
paper B, Jens Hoppe, Jonas de Woul and Douglas Lundholm. Further I would like
to thank the other members of the Department of Theoretical physics for making
my stay enjoyable.
I am especially grateful to Erik Duse for many interesting, inspiring and encouraging discussions. Many thanks also to André, Joel and Sebastian at the Department of Mathematics.
Most of all I want to thank my family for their encouragement and support
during the work of this thesis.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preface
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Acknowledgments
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Contents
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I
1
Introduction and background material
1 Introduction
1.1 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .
2 Constrained quantum mechanics
2.1 Riemannian geometry . . . . . .
2.1.1 Manifolds . . . . . . . . .
2.1.2 The metric tensor . . . .
2.2 Submaifolds . . . . . . . . . . . .
2.3 The effective Hamiltonian . . . .
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3 Constrained relativistic fields
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Constrained Klein-Gordon field . . . . . . . . . . . . . . . . . . . .
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4 Extrinsic curvature effects in brane-world
4.1 Embedded cosmological models . . . . . .
4.2 Embedded Schwarzschild solution . . . . .
4.3 Embedded Robertson-Walker metric . . .
4.4 A model of the early universe . . . . . . .
4.4.1 The standard case . . . . . . . . .
4.4.2 The extended case . . . . . . . . .
4.5 Discussion . . . . . . . . . . . . . . . . . .
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scenarios
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x
Contents
5 Membrane dynamics
5.1 Point particle in the light-cone gauge . . . . . .
5.2 A dynamical symmetry . . . . . . . . . . . . .
5.3 The bosonic membrane . . . . . . . . . . . . . .
5.3.1 Light-Cone Gauge . . . . . . . . . . . .
5.4 Hamiltonian formalism . . . . . . . . . . . . . .
5.5 Mode expansion . . . . . . . . . . . . . . . . .
5.6 A dynamical symmetry for bosonic membranes
5.7 Supermembranes . . . . . . . . . . . . . . . . .
5.8 Poisson brackets . . . . . . . . . . . . . . . . .
5.9 A dynamical symmetry for supermembranes . .
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6 Summary and conclusions
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A The tubular neighbourhood theorem
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B Calculation of the induced potential
B.1 Calculation of γ̃ . . . . . . . . . . . .
B.2 Derivation of the induced potential .
B.3 The Schwarzschild solution . . . . .
B.3.1 Embedding for r > rs . . . .
B.3.2 Embedding for 0 < r < rs . .
B.4 Robertson-Walker metric . . . . . .
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Bibliography
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II
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Scientific papers
Part I
Introduction and background
material
1
2
Chapter 1
Introduction
Nature is full of geometry. The yellow disc flowers of the Ox-Eye Daisy are organised
in spirals and the planets move in elliptic orbits. With the idea of Einstein, that
space itself is geometrical, geometry has become an important part of all aspects
of nature. Einstein himself expressed this as [2]
Geometry [...] is evidently a natural science; we may in fact regard it
as the most ancient branch of physics.
One way to solve certain problems in physics is to consider scenarios where
the four dimensional world we experience is embedded in a higher dimensional
space. The first theory with extra dimensions was probably the theory proposed by
Gunnar Nordström [3, 4] in 1914. Nordström tried to unify Newtonian gravity with
electromagnetism by introducing an additional dimension. The theory, however,
was unable to explain certain phenomena such as light deflection. The theory
was therefore considered as non-physical by many. Three years later, Hermann
Weyl [3, 4] proposed a theory which is a classical predecessor of both Kaluza-Klein
and Yang-Mills theory. Weyl proposed that there existed an additional degree of
freedom in space. He also introduced a field which ”measured” the change of this
degree of freedom and called it a ”gauge field”1 . Weyl’s theory unified gravity with
electromagnetism, but received critique from Einstein and others who objected that
although the theory was mathematically beautiful it implied that measurements
dependes on the history of the measurement device. Despite the shortcomings of
the theory, it was eventually published in a journal (with Einsteins objections as an
appendix). It was later hypothesised that the additional degree of freedom could
actually be an extra dimension, as in Kaluza-Klein theory [5, 6], or a quantum
mechanical phase, as in Yang-Mills theory.
Today, there is a renewed interest in extra dimensions and many theories have
been proposed that use extra dimensions to solve certain physical problems (see e.g.
1 After
the verb ”gauge”, meaning ”to measure” or ”to estimate”.
3
4
Chapter 1. Introduction
[7] for references). One problem in physics is why the gravitational force is much
weaker then the other forces in nature. Extra dimensions might solve this problem
by allowing gravity to spread freely in the extra dimensions. This makes gravity
appear weaker to someone on a lower dimensional submanifold of the ambient
space. This idea underlies the ADD model (after Arkani-Hamed-Dimopoulos-Dvali
[8]) considered in particle physics. Another model is the Randall-Sundrum (RS)
model [9]. The model assumes that the universe is five dimensional with two fourdimensional membranes (branes), the ”Planck” and ”TeV” brane2 . In the theory,
the branes are separated in space. We live on the TeV brane while gravity is
localized on the Planck brane. According to the RS model, we experience gravity
as weaker than the other forces since it is localized on the other brane. There are
hopes that, in the future, experiments will either support or falsify the ADD and
Randall-Sundrum model.
Since extra dimensions have not been discovered to this day, there should be
some mechanism preventing us from detecting them (if they exists). One explanation is that the extra dimensions are so small that the energies needed to detect
them are very high. Another explanation is that some particles are confined to our
four-dimensional universe by some mechanism, e.g. a strong potential.
Classically, extra dimensions usually do not affect the dynamics of a particle
constrained to a lower dimensional space. It is therefore natural to assume that the
hamiltonian for a quantum particle on a manifold is proportional to the LaplaceBeltrami operator [10]. For a long time this was believed also to be true for quantum
mechanical particles. In the 1970’s Jensen and Koppe [11], and independently da
Costa [12, 13] in the 1980’s, derived the effective Hamiltonian for a non-relativistic
quantum particle confined by a strong potential to a surface embedded in three
dimensional Euclidean space. They found that, unlike classical particles, quantum
particles are affected by both the intrinsic and extrinsic geometry of a surface. This
is because the curvatures induce an additional potential.
One theory with extra dimensions is string theory (see e.g. the textbook by
Polchinski [14]). In string theory, point particles are replaced by vibrating strings.
The theory was first proposed in the context of the strong interaction, where it
was believed that quarks were bounded together by strings making them hard to
separate. This idea was later abandoned in favour of QCD, but string theory lived
on despite of this. The quantum theory of strings predicts an infinite number
of particles of different masses, spin and interactions. String theory is based on
the simple assumption that, in the same way as a relativistic particle moves in
a way that minimizes the length of its world-line, a string moves in a way that
minimizes the area of its world-sheet. Unlike other theories with extra dimensions,
string theory requires a specific number of space-time dimensions to be consistent.
Bosonic string theory requires D = 26 dimensions and supersymmetric string theory
requires D = 10.
2 This
model is often known as the RS1 model.
1.1. Overview of the thesis
5
One way to generalize string theory is to consider higher dimensional extended
objects, membranes, which (like strings) moves in a way that minimizes their worldvolume. The extra degrees of freedom and non-linearities of the theory makes the
equations of motion hard to solve explicitly. Because of this, it is not clear how
the quantum theory of membranes works. By understanding the symmetries of the
theory, one hopes to reduce the degrees of freedom. This might eventually lead to
an explicit solution of the classical theory and thereby also the quantum theory.
1.1
Overview of the thesis
Chapters 2,3 and 4 discusses material related to paper A. In chapter 2 we give a
short introduction to differential geometry and introduce the concept of constrained
quantum mechanics. This is further developed in chapter 3 where we derive the
induced potential for constrained Klein-Gordon field. In chapter 4 we discuss extrinsic curvature effects in cosmological brane-world scenarios, and we investigate
the effects of induced potentials the embedded Schwarzschild solution and the embedded Robertson-Walker universe.
Material related to paper B is discussed in chapter 5. We give a short introduction to membrane dynamics and derive a dynamical symmetry for bosonic
membranes. The dynamical symmetry is then generalized to supersymmetric membranes.
We summarize part I of the thesis in chapter 6 where we state and discuss
conclusions of the thesis. The scientific papers are in part II.
6
Chapter 2
Constrained quantum
mechanics
In classical mechanics, the dynamics of a particle is usually unaffected by whether
or not there exists an ambient space in which the particle cannot move. This
is because the extra dimensions are no degrees of freedom for the particle. How
does constraints of this type affect quantum particles? It was first proposed by
Schrödinger [10] in 1926 that the Hamiltonian of a quantum particle on a manifold
is proportional to the Laplace-Beltrami operator of the manifold. Almost 50 years
later that Jensen and Koppe [11] and independently da Costa [12, 13] examined
if this was also true for constrained particles. They found that the assumption
made by Schrödinger does not apply in this case. By considering quantum particles confined to a thin layer around a surface they found that constrained quantum
particles are affected by the intrinsic and extrinsic geometry of the surface. The
geometry of the surface induces a potential which affects the dynamics of the particle (this was found earlier by Marcus [15] in the context of quantum chemistry). In
short, this implies that constrained quantum particles is not the same as quantized
constrained particles.
The existence of an induced potential in the thin layer limit lead Exner and Seba
[16] to investigate if bound states exists also for layers of finite width. They found
that a strip of finite width does support bound states under certain conditions on
the curvature and thickness of the strip. It was later shown by Goldstone and
Jaffe [17] that any non-straight strip supports bound states. This has later been
confirmed experimentally (see e.g. [18]).
We begin this chapter by giving a short introduction to Riemannian geometry.
We then proceed to use Riemannian geometry to describe the confinement of a
quantum particle to a manifold embedded in Euclidean space. Thereafter we derive
the effective low-energy Hamiltonian for a non-relativistic constrained quantum
particle.
7
8
Chapter 2. Constrained quantum mechanics
2.1
Riemannian geometry
Classical geometry (see e.g. [19]) was studied in order to understand relations
between e.g. length, area, volume and angles of geometrical shapes in the plane
or in space. It was found that by assuming a set of postulates, many geometrical
relations could be proven rigorously. One postulate is the parallel postulate which
states that1 the angles α, β, γ of a triangle always satisfy
α+β+γ =π
(2.1)
where the angles are measured in radians. Many attempts were made to prove the
postulate, but none were successful. It was later realized that the postulate could
be omitted and that one can study spaces where the postulate does not hold. One
such example is the sphere (of radius 1) where the angles of a triangle2 does not
add up to π as in (2.1) but instead
α+β+γ =π+A
(2.2)
where A is the area enclosed by the triangle. By considering spaces where the
parallel postulate is omitted, the notion of geometry was generalized to what is
now known as differential geometry.
To measure the deviation from ”flatness”, one can use the notion of curvature.
By using curvature, the formulas in (2.1) and (2.2) can be generalized to
Z
α+β+γ =π+
KdA
(2.3)
C
for triangles on arbitrary surfaces. Here K is the (Gaussian) curvature of the surface
and C is the region enclosed by the triangle. The formula (2.3) is a generalization
of (2.1) and (2.2) since the curvature of the plane is K = 0 and the curvature of
the unit sphere is K = 1. The formula (2.3) is actually a special case of a more
general theorem, the Gauss-Bonnet theorem [19].
2.1.1
Manifolds
A manifold is the mathematical notion of a curved space. By a space we mean
something that locally is like Rn and by curved we mean something which in general
is not like Rn globally [20]. To make the definition precise we need some definitions.
Definition A function f : U → V between two sets is a homeomorphism if f
is a continuous and invertible function such that f −1 is also continuous. Two sets
are said to be homeomorphic if there exists a homeomorphism between the sets.
The homeomorphism f is said to be a diffeomorphism if both f and f −1 are
differentiable. Two sets are diffeomorphic if there exists a diffeomorphism between
1 Many
2 For
points.
different but equivalent formulations of this postulate exists.
a general space we can define a (geodesic) triangle as the shortest lines connecting three
2.1. Riemannian geometry
9
the sets. Sets being homeomorphic is more a statement about the topologies of the
sets, rather than of the sets themselves. In this sense, diffeomorphism is a stronger
condition then homeomorphism since, for differentiable functions we often use the
metric topology. With these definitions, we are ready to define what a manifold is.
Definition An n-dimensional topological manifold is a second countable set M
with a Hausdorff topology3 and such that every point p ∈ M has an open neighbourhood homeomorphic to an open subset of Rn . If every point of a manifold has
an open neighbourhood diffeomorphic to an open subset of Rn , then the manifold
is a differentiable manifold.
Somewhat simplified, one can say that the definition of a manifold states that
a manifold is not ”too big” (second countable), finite dimensional (locally homeomorphic to Rn ) and that the limit of a converging sequence is unique (Hausdorff
topology). By the theorem of Invariance of Domain [20] the dimension of a manifold is constant, hence a manifold cannot intersect itself. Intersecting ”manifold
like” spaces are sometimes called pseudomanifolds.
A homeomorphism ϕ : U → V ⊂ Rn is often called a chart. From the definition
of a manifold we find that, if Uα ∩ Uβ 6= ∅ and ϕα : Uα → Vα , ϕβ : Uβ → Vβ are
two homeomorphisms (diffeomorphisms), then
ϕβ ◦ ϕ−1
α : ϕα (Uα ∩ Uβ ) → ϕβ (Uα ∩ Uβ )
is also a homomorphism (diffeomorphism). One often does not need to work with
the underlying manifold explicitly, but can define the manifold in terms the transition functions ϕβ ◦ ϕ−1
α . A manifold can be given more structure by considering
transition functions belonging to different function classes. All manifolds considered
in this thesis are smooth manifolds, i.e. manifolds with C ∞ transitions functions.
2.1.2
The metric tensor
There are several ways to define vectors on differentiable manifolds (see e.g. [19]
for three different definitions). One way is to define a vector at a point p as the
evaluation (at the point p) of the derivative of a curve running through the point.
The tangent space Tp M is the set of all vectors at the point p and is diffeomorphic
to Rn when the manifold is n-dimensional. We can define an inner product of
vectors in the tangent space in the same way as for vectors in Rn by defining an
inner product as a bilinear function g : Rn × Rn → R such that
g(u, v) = g(v, u)
where u, v are vectors in Tp M. We define the norm ||u|| of a vector u ∈ Rn by the
relation g(u, u) = ||u||2 . Given a basis {eµ }nµ=1 of Tp M, we can express the vectors
3 Many of the properties in this section are stated for completness and will not be needed in
the following.
10
Chapter 2. Constrained quantum mechanics
as linear combinations of the basis vectors4 , u = uµ eµ and v = v µ eµ . Using this,
we can write the inner product in component form
g(u, v) = g(eµ , eν )uµ v ν = gµν uµ v ν
where gµν = g(eµ , eν ) is the metric tensor. The basis vectors and the metric tensor
on a manifold are in general position dependent.
The metric tensor can be used to calculate e.g. the length of a curve x(τ ) on a
manifold as
Z τ1 Z τ1 r
Z q
dx dxµ dxν
l=
dτ
=
dτ
=
g
(x(τ
))
gµν (x(τ ))dxµ dxν
µν
dτ dτ dτ
τ0
C
τ0
For this reason, the infinitesimal line element is often written as
ds2 = gµν dxµ dxν
so that we can write
Z
l=
ds
C
Similarly, the volume of a manifold can be computed as
Z
p
V ol(M) =
dn x |g|
M
where |g| is the absolute value of the determinant of (gµν ).
A manifold is said to be Riemannian if the inner product is positive definite. If
the inner product is indefinite and there exists a coordinate system and basis such
that5
gµν (x)uµ uν = ||u||2 = (u0 )2 − (u1 )2 − . . . (un−1 )2 + O(x2 )
then the manifold is said to be Lorentzian 6 .
2.2
Submaifolds
A submanifold is a subset of a manifold which itself is also a manifold. Submanifolds
can be considered as embedded in their ambient manifold. An embedding of a
manifold M in a manifold N is a map
f :M→N
such that f restricted to its image is a homeomorphism.
4 Throughout this thesis we use the Einstein summation convention, i.e. repeted indices are
summed over.
n−1
5 One often enumerates basis vectors as {e }n
µ µ=1 for Riemannian manifolds and as {eµ }µ=0
for Lorentzian manifolds.
6 This is the metric convention used throughout the thesis.
2.2. Submaifolds
11
In this chapter we use capital latin letters (A, B, C, . . . ) to denote indices running over 1, 2, . . . , n + p, greek letters (µ, ν, λ, σ, . . . ) to denote indices running over
1, 2, . . . n and lower case latin letters (i, j, k, l, . . . ) to denote indices running over
n + 1, n + 2, . . . , n + p.
If η is the metric tensor on N , then the metric on M is given by the pullback
of η by f , g = f ∗ η, i.e. in components
gµν = ηAB ∂µ f A ∂ν f B
If M is a n-dimensional Riemannian manifold embedded in Rn+p , then we can
(locally) parametrize the submanifold as
Z 1 , Z 2 , . . . , Z n+p = r(x)
where x = (x1 , x2 , . . . , xn ) and Z 1 , Z 2 , . . . , Z n+p are coordinates in a local chart
of M and the inertial coordinates in Rn+p respectivly. The components of the
metric tensor gµν are then given by
gµν =
∂r
∂Z A ∂Z B
∂r
·
= δAB
µ
ν
∂x ∂x
∂xµ ∂xν
In constrained quantum mechanics we want to study the dynamics of a quantum
particle under the influence of a confining potential which localizes the particle
to M ⊂ Rn+p . We therefore need to extend the embedding function r(x) to a
map R(x, y) covering an open neighbourhood of the submanifold. We can define
R(x, y) by introducing linearly independent normal vectors {ni (x)}n+p
i=n+1 to the
submanifold M at the point r(x). We choose the normal vectors to be orthogonal
ni (x) · nj (x) = δij
for all x and construct R(x, y) by setting
R(x, y) = r(x) + y i ni (x)
where (y n+1 , y n+2 , . . . , y n+p ) ∈ Rp . By the tubular neighbourhood theorem (see
Appendix A), there exists a constant > 0 such that F is one-to-one for all ||y|| < .
n+p
The tangent vectors {tµ = ∂µ r}nµ=1 and the normal vectors {ni }n+p
i=n+1 span R
at each point. Because of this, any vector can be written as a linear combination
of them. Especially we can write
∂µ ni = −αiµν tν − Aiµj nj
with αiµν and Aiµj defined by this equation.
(2.4)
12
Chapter 2. Constrained quantum mechanics
Using (2.4) we can compute the metric tensor in the coordinates (x, y) explicitly.
Using that GAB = ∂A R · ∂B R, we find that
Gµν Giµ
γµν + y i y j Aiµk Ajν l hkl −y k Akµl hlj
(GAB ) =
(2.5)
=
Gµj Gij
−y k Akν l hli
δij
where
γµν = gµν − 2y k αkµλ gλν + y k y l αkµσ αlν λ gλσ
We can also explicitly compute the quantity (see Appendix B)
γ̃ =
|G|
| det(GAB )|
=
= 1 − 2y i αiλλ + y i y j 2αiλλ αjσσ − αiλσ αjσλ + O(y 3 )
|g|
| det(gµν )|
which will be useful in later calculations.
2.3
The effective Hamiltonian
To derive the effective Hamiltonian of a particle confined to the submanifold M ⊂
Rn+p by a strong potential we start with the Hamiltonian of a free particle in
Rn+p . Thereafter we apply a confining potential which localizes the particle to the
submanifold. We then study the limit in which the extension of the wavefunction
in the normal directions tend to zero.
Consider the Schrödinger equation of a free particle in Rn+p
H0 Ψ = −
dΨ
1 2
∇ Ψ=i
2m
dt
where m is the mass of the particle and the wavefunction is normalized in Rn+p
Z
dn+p Z|Ψ|2 = 1
Rn+p
Changing to the coordinate system (x, y) induced by R, the Hamiltonian becomes
√
1
H0 = − √ ∂ A
GGAB ∂B
2m G
We now apply the confining potential Vconf (y) to confine the particle to the
submanifold M. Let Vconf be such that there exists a set {χα (y)}∞
α=0 of real
eigenfunctions satisfying
−
1 ij
δ ∂i ∂j χα (y) + Vconf (y)χα (y) = λα χα (y)
2m
2.3. The effective Hamiltonian
13
and
Z
dp yχα (y)χβ (y) = δαβ
B
where we for convenience have chosen the confining potential such that for all α,
χα (y) = 0 for ||y|| > . It is therefore sufficient to consider the eigenfunctions in
the ball B = {y ∈ Rp : ||y|| < }. We further choose the potential such that the
ground state is non-degenerate and · · · ≥ λn+1 ≥ λn ≥ · · · ≥ λ1 > λ0 = 0. We can
allow for more general potentials, provided that the solutions have proper decay
properties7 .
When studying quantum mechanics on M (rather then on Rn+p ) we want the
wavefunction to be normalized on M. We cannot localize the particle to the embedded manifold completely because of the uncertainty relation, but we can study
it for small but finite > 0.
To ensure that the wavefunction remains normalized after the particle has been
confined we expand it as
Ψ(x, y) = γ̃ −1/4
X
ψα (x)χα (y)
α
We then find that
XZ
√
dn xdp y G|Ψ|2 =
Z
1=
M×B
α
√
dn x g|ψα |2
M
To find the effective Hamiltonian acting on the wavefunctions ψα we need
to
compensate
for the scaling factor. We define the Hamiltonian H1 acting on
P
α ψα (x)χα (y) to be
H = H0 + Vconf = γ̃ −1/4 H1 γ̃ 1/4
We find that H1 is given by (for details see Appendix B)
√
γ̃ 1/4
√ ∂A
GGAB ∂B γ̃ −1/4 + Vconf (y)
2m G
√
1 k
= − √ ∂µ + y Akµi ∂i ( gg µν ) ∂ν + y l Alν j ∂j + V (x, y)
2m g
1 ij
−
δ ∂i ∂j + Vconf (y) + O(y)
2m
H1 = γ̃ 1/4 H γ̃ −1/4 = −
7 The precise conditions is that the expectation value of any polynomial in y j is finite and
decays as at least a for some a > 0 in the limit → 0.
14
Chapter 2. Constrained quantum mechanics
where V (x, y) is the potential
V (x, y) =
1 ij
3
δ ∂i ∂j γ̃ − (∂i γ̃)(∂j γ̃)
8m
4
= Vind (x) + O(||y||)
with
Vind (x) =
1 X λ σ
αiλ αiσ − 2αiλσ αiσλ
8m i
We find that the ”matrix elements” of the Hamiltonian H1 are
Z
p
(H1 )αβ =
dp y |h|χα (y)∗ H1 χβ (y)
B
=
X
σ
1
√
− √ (Dµ )ασ ( gg µν ) (Dν )σβ + Vind δαβ + λα δαβ + O()
2m g
where we defined the ”covariant derivative”
Z
p
(Dµ )αβ = δαβ ∂µ + (Aµ )αβ =
dp y |h|χα (y) ∂µ + y k Akµi ∂i χβ (y)
B
In the limit → 0, the terms of order can be neglected. The eigenvalues λα
scale as λα ∼ 12 and therefore become widely separated in the limit of small .
Since we have chosen the smallest eigenvalue to be zero, it is unaffected in this
limit. On physical grounds we can therefore assume that only the ground state
of the confining potential contributes to the effective dynamics of the constrained
particle in the limit → 0.
We find that
Z
Z
p
p
1
dp y |h|y k Akµi ∂i χ2α
(Aµ )αα =
dp y |h|χα y k Akµi ∂i χα =
2 B
B
1
1
1 k
= − δi Akµi = − δ ij ni · ∂µ nj = − δ ij ∂µ (ni · nj ) = 0
2
2
4
This implies that, for a non-degenerate ground state, the ”gauge field” Aµ does not
affect the dynamics of the constrained particle. If the ground state is N -degenerate,
then one can write the ground state wavefunction as a column vector and interpret
Aµ as a static gauge field acting on the components of the vector.
We have thus shown that, for a quantum mechanical particle confined to a
submanifold of Euclidean space by a confining potential with non-degenerate ground
state and small ”width” > 0, the effective Hamiltonian is [11–13, 21–24]
1
1 X λ σ
√
Hef f = − √ ∂µ ( gg µν ∂ν ) +
αiλ αiσ − 2αiλσ αiσλ
2m g
8m i
where we ignored terms of order . If we would have quantized a classical particle
confined to the embedded manifold, then we would only obtain the kinetic term
2.3. The effective Hamiltonian
15
of the Hamiltonian and not the induced potential. This shows that confining and
quantizing in general do not commute.
We end this chapter by considering two examples of constrained quantum mechanics. For a curve c(t) in R3 we can choose the normal vectors to be vectors of
the Frenet frame [19] of the curve. We then get that the only non-zero coefficient
is
α2tt = κ
where κ is the curvature of the curve. So the induced potential becomes
Vind = −
1 2
κ
8m
For a two dimensional surface embedded in R3 , there is only one linearly independent normal vector. We can diagonalize α3µν as (α3 is proportional to the
Weingarten map of the surface)
κ1 0
(α3µν ) =
0 κ2
where κ1 and κ2 are the principal curvatures of the surface (at the point x). We
then find that [11]
Vind =
i
1 h
1
2
2
(κ1 + κ2 ) − 2 κ21 + κ22 = −
(κ1 − κ2 )
8m
8m
So Vind ≤ 0 for a 2-dimensional surface embedded in three dimensional space. For
higher dimensional manifolds, this statement is not true in general.
16
Chapter 3
Constrained relativistic fields
3.1
Introduction
In this chapter we study the dynamics of constrained relativistic bosons (KleinGordon fields). In the derivation of the induced potential for quantum mechanical
particles, an important step was the rescaling of the wavefunction, which allowed
us to expand the wavefunction in modes. The wavefunction was rescaled in order to
preserve the total probability in the thin layer limit. For relativistic quantum particles, there is no conservation of probability because the relativistic mass-energy
equivalence makes any one-particle theory inconsistent. Relativistic quantum mechanics needs to be a theory with a variable number of particles, a quantum field
theory. We take a first step towards understanding constrained quantum field theories by considering classical relativistic constrained fields. In particular, we derive
the effective action for a classical Klein-Gordon field confined to an embedded manifold of a general manifold.
We find that the induced potential, which was originally discovered in the context of quantum mechanics, also appears for semi-classical constrained fields. The
derivation resembles the one in chapter 2 but with some subtle differences.
3.2
Constrained Klein-Gordon field
Let Φ be a free real-valued scalar field on a (n + p + q)-dimensional Lorentzian
manifold N where q is the number of time-like directions on M (N ). The action
of Φ is
Z
p
1 AB
1 2 2
n+p+q
S=
d
Z |η|
η ∂A Φ∂B Φ − m Φ
2
2
Rq,n+p
where ηAB is the metric of N .
17
18
Chapter 3. Constrained relativistic fields
As in the previous chapter, we change to the coordinate system (x, y) given
by the function F which extend the embedding function f . The metric in the
coordinates (x, y) is
0
GAB = ηA0 B 0 ∂A F A ∂B F B
0
We choose the map F such that
(GAB )|y=0 =
gµν
0
0
−hij
where gµν is the metric on M and hij is constant, non-degenerate and positive definite. To confine the field to the submanifold M ⊂ N we apply a confining potential
which localizes the field to the submanifold. Applying the confining potential, the
action becomes
Z
p
1
1
S=
dn+q xdp y |G|
∂A Φ∂B Φ − (m2 + Vconf (y))Φ2
2
2
M×B
We choose the potential to be such that any solution Φ of the equations of motion
fulfils Φ = 0 for ||y|| > 01 . It is therefore sufficient to consider the action in the
ball B = {y ∈ Rp : ||y|| < }.
We assume that Vconf is such that there exists a complete set of eigenfunctions
{χα (y)}∞
α=0 satisfying
−hij ∂i ∂j χα + Vconf χα = λα χα
with corresponding eigenvalue λα and
Z
dp y
p
|h|χα χβ = δαβ
B
We also choose the potential to be such that the ground state χ0 is non-degenerate
with λ0 = 0. We see here why it is important that all normal directions are spacelike. If one direction had been time-like, then the eigenvalues {λα } would not
have been bounded from below, and we would not be able to derive the low-energy
dynamics of the theory.
Naively expanding the field in terms of the eigenfunctions is not appropriate
since the volume element in general depends on the normal coordinate y. This is
1 More precisly we choose the potential to be such that, given a Cauchy surface (see e.g. [25]
for a review of the subject) of initial conditons for Φ, all solutions satisfy Φ = 0 for ||y|| > in the
future domain of dependence.
3.2. Constrained Klein-Gordon field
19
because the y-dependence of the volume element can give rise to an infinite sum of
mixed modes ψα ψβ (α 6= β). To solve this problem we instead expand the field as
Φ(x, y) = γ̃
−1/4
∞
X
φα (x)χα (y) = γ̃ −1/4 φ
α=0
where γ̃ −1/4 =
|G|
|g||h| .
Z
We then get that
dn+q xdp y
p
|G|Φ2 =
XZ
dn+q x
p
|g|φ2α
α
The kinetic term in the Lagrangian density becomes
Z
p
1
dn xdp y |G|GAB ∂A Φ∂B Φ
2
Z
p
1
dn xdp y |g||h|γ̃ 1/2 GAB ∂A (γ̃ −1/4 φ)∂B (γ̃ −1/4 φ)
=
2
Z
p
γ̃ −1 AB
γ̃ −2
1
dn xdp y |g||h| GAB ∂A φ∂B φ −
G (∂A γ̃)φ(∂B φ) +
∂A γ̃∂B γ̃φ2
=
2
2
16
Z
p
p
1
= dn xdp y |g| |h| GAB ∂A φ∂B φ − V (x, y)φ2
2
where
p
γ̃ −1 AB
3γ̃ −1
γ̃ −1
V (x, y) = −
|g|GAB (∂B γ̃)
G
∂A ∂B γ̃ −
∂A γ̃∂B γ̃ − p ∂A
8
4
8 |g|
To find the effective action we expand the potential as a series in y and use that
Z
p
p
i1 i2
iN
d y |h|y y . . . y χα χβ ≤ N
B
We thus find that, for all α and β,
Z
p
dp y |h|V (x, y)χα (y)χβ (y) = Vind (x)δαβ + O()
where Vind (x) = V (x, 0) is the induced potential
1
3
1
Vind = hij ∂i ∂j γ̃ − ∂i γ̃∂j γ̃ −
8
4
8
y=0
∂i Gij (∂j γ̃) (3.1)
y=0
The formula (3.1) expresses the induced potential in terms the reduced volume element γ̃. We belive that (3.1) is equivalent to the obvious generalization of
Mitchell’s result [26]. Mitchell’s result expresses the potential in terms the coefficients tµ · ∂ν ni and contractions of the Riemann tensor of N . The formula (3.1)
is not as explicit as Mitchells result, but we believe that it is more convenient for
practical computations. It is especially useful when the coordinate system is such
that the submanifold can be parametrized by letting some coordinates be constant.
20
Chapter 4
Extrinsic curvature effects in
brane-world scenarios
In this chapter we discuss possible physical implications of the induced potential
in different cosmological models. We consider embeddings of the Schwarzschild
solution and the Robertson-Walker metric, and we investigate possible physical
implications. We choose embeddings which we believe are the simplest possible, but
the extension to more complicated embeddings (e.g. motivated by some underlying
theory) is straightforward.
4.1
Embedded cosmological models
As was derived in chapter 2, the effective action for a Klein-Gordon field confined
to an (n + 1)-dimensional embedded submanifold f (M) ⊂ R1,n+p is
Z
p
1
1 µν
n+1
2
S=
d
x |g|
g ∂µ φ∂ν φ − Vind (x)φ
2
2
M
where Vind (x) is the induced potential
h
i
1
Vind (x) = hij αiλλ αjσσ − 2αiλσ αjσλ
8
4.2
Embedded Schwarzschild solution
The Schwarzschild metric is a solution to Einstein’s equation’s describing spacetime outside a spherically symmetric mass distribution. The line element of the
Schwarzschild solution is
rs 2 rs −1 2
ds2 = 1 −
dt − 1 −
dr − r2 (dθ2 + sin2 (θ)dϕ2 )
r
r
21
22
Chapter 4. Extrinsic curvature effects in brane-world scenarios
where rs = 2GM/c2 is the Schwarzschild radius with M the total mass and G
the gravitational constant. The angular variables θ and ϕ are the usual angular
variables used for spherical coordinates with 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π. The
variables t and r are the coordinate time and radius. Note that the coordinate time
and radius are different from the proper (physical) time and radius, e.g. the proper
distance between r0 and r1 is not r1 − r0 (for r1 > r0 > rs ), but
Z
r1
r
r0
1−
rs −1
dr
r
In 1921, Kasner [27] showed that the Schwarzschild metric cannot be embedded
in five dimensional Minkowski space. In 1959 Fronsdal [28] found an embedding in
six dimensional Minkowski space given by1
Z 0 = 2t0 (1 − rs /r)1/2 sinh(t/2t0 )
Z 1 = 2t0 (1 − rs /r)1/2 cosh(t/2t0 )
Z 2 = g(r)
Z 3 = r sin(θ) cos(ϕ)
Z 4 = r sin(θ) sin(ϕ)
Z 5 = r cos(θ)
where
g 0 (r)2 =
rs (r3 − t20 rs )
r3 (r − rs )
and t0 is an auxiliary parameter. It can be shown that, under certain assumptions,
this embedding, for t0 = rs , is a unique embedding of the Schwarzschild solution
[29]. For t0 6= rs , this embedding does not cover the entire Schwarzschild solution, but can be thought of as an embedding of space-time corresponding to some
spherically symmetric mass distribution.
The embedding can be modified to also cover the interior region 0 < r < rs [28]
by interchanging the hyperbolic functions, setting t0 = rs and replacing
(1 − rs /r)1/2 → (rs /r − 1)1/2
the embedding then becomes well defined for 0 < r < rs .
1 We here introduced the parameters r and t . Fronsdal’s original embedding is obtained by
s
0
setting t0 = rs = 1.
4.2. Embedded Schwarzschild solution
23
For t0 = rs the embedding can be written in the simple form (for derivation of
the induced potential in the case t0 6= rs see appendix B)
Z = 2rs |1 − rs /r|1/2 ev + g(r)e2 + rer
(4.1)
where we introduced the basis vectors
ev =
(sinh(t/(2rs )), cosh(t/(2rs )), 0, 0, 0, 0) , rs < r
(cosh(t/(2rs )), sinh(t/(2rs )), 0, 0, 0, 0) , 0 < r < rs
e2 = (0, 0, 1, 0, 0, 0)
er = (0, 0, 0, sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ))
One possible set of normal vectors to the embedding is
n4 = g 0 (r)|r/rs − 1|1/2 ev − (rs /r)3/2 e2
n5 = (rs /r)3/2 |1 − rs /r|1/2 ev + g 0 (r)(1 − rs /r)(r/rs )1/2 e2 − (rs /r)1/2 er
From this embedding we obtain the non-zero coefficients
r(r2 + rs r + rs2 )1/2
2(rs r)3/2
3rs2
=− 2 2
2r (r + rs r + rs2 )1/2
1 rs 3/2
= α511 = −
2rs r
r 3/2
1
s
= α544 =
rs r
α400 = −
α411
α500
α533
which gives us the induced potential
Vind (r) = −
(r3 + rs r2 + rs2 r + 9rs3 )(r2 + rs2 )(r + rs )
16rs2 r4 (r2 + rs r + rs2 )
We see that the induced potential is analytic in the whole region 0 < r < ∞.
We believe that this is because Fronsdal’s embedding covers the entire region 0 <
r < ∞.
24
Chapter 4. Extrinsic curvature effects in brane-world scenarios
The induced potential for t0 = rs = 1
0
−0.1
−0.2
−0.3
Vind
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9
−1
1
2
3
4
5
6
7
8
9
r
The potential has the asymptotic forms
(
Vind (r) =
9r 2
− 16rs4 1 +
1
− 16r
1+
2
s
r
2
9rs + O(r ) rs
−2
)
r + O(r
, for r ≈ 0
, for r rs
10
4.3. Embedded Robertson-Walker metric
25
1
For r rs , the potential approaches the constant value Vind (r) ≈ − 16r
2.
s
Translating this into a mass magnitude by the formula
~2 |Vind | = m2 c2
we find that the asymptotic value of the potential corresponds to a mass of
~p
~c
3.321 × 1010 kg GeV
m=
|Vind | ≈
≈
c
8GM
M
c2
A peculiar feature is that the mass is inverse proportional to M , the mass of the
gravitating object. This means that, for heavy bodies, the effect of the induced
potential is negligible, while for light bodies, the effect could be large. For e.g. the
sun (M ≈ 1.988 × 1030 kg) this gives a mass of about m ≈ 1.67 × 10−11 eV /c2 .
A super-heavy black hole, which are believed to be in the center of our galaxy
[30], with a mass of about 4.1 million solar masses the corresponding mass is m ≈
4 × 10−18 eV /c2 . We believe that these masses are negligible small. This suggests
that the induced potential can only produce measurable physical effects for small
black holes. One example are the hypothetical primordial black holes [31], which are
believed to have formed in the early universe. If primordial black holes exists, they
would offer good conditions for studying Hawking radiation and quantum effects in
gravity. Such small black holes are, however, believed to have evaporated because
of their Hawking radiation on a time scale of
3
M
64
τ = 10
y
M
This implies that a primordial black hole with mass M = 1010 kg would have
evaporated about 104 years after the Big-Bang.
4.3
Embedded Robertson-Walker metric
The Robertson-Walker metric describes a isotropic homogeneous universe. The line
element of the Robertson-Walker metric is
ds2 = dt2 − a(t)2 dr2 + S(r)2 dθ2 + sin2 θdϕ2
(4.2)
where a(t) is a scale factor corresponding to the ”radius” of the universe and S(r)
is a function describing the
R structure of the universe, i.e. the volume of the universe
is proportional to a(t)3 S(r)2 dr where the integral is finite if S is bounded. S(r)
depends on a parameter K and is given by
√

sin( Kr)
√

, for K > 0

K

r
, for K = 0
S(r) =
√

sinh( |K|r)


√
, for K < 0
|K|
For K > 0, S(r) is bounded so the universe has a finite volume and describes a
closed universe. For K ≤ 0, S(r) is unbounded, and the universe is therefore open.
26
Chapter 4. Extrinsic curvature effects in brane-world scenarios
Robertson-Walker space-time can be embedded in five dimensional Minkowski
space [32] as follows

√

 b(t)/
2 K R
1 r
Z0 =
+ r0 a +
2
r
0


a(t)C(r)


 a(t)C(r)
R
1 r2
1
Z =
2 r0 − r0 a +

p

b(t)/ |K|
, for K > 0
dt0
2r0 ȧ
, for K = 0
, for K < 0
, for K > 0
dt0
2r0 ȧ
, for K = 0
, for K > 0
2
Z = a(t)S(r) sin(θ) cos(ϕ)
Z 3 = a(t)S(r) sin(θ) sin(ϕ)
Z 4 = a(t)S(r) cos(θ)
where
C(r) =
√
√
cos( p
Kr)/ Kp
, for K > 0
cosh( |K|r)/ |K| , for K < 0
A normal vector to the embedding is
n4 =







ȧ(t)
√ e0 + ḃ(t) (C(r)e1 + S(r)er )
Kh
2
i
1
1
r
+
r
−
ȧ(t)
e0
0
2
r0
r0 ȧ(t) i
h
, for K > 0
, for K = 0
2
1

+ 12 ȧ(t) rr0 − r0 − r0 ȧ(t)
e1 + ȧ(t)rer




ȧ(t)
 √
e + ḃ(t) (C(r)e0 + S(r)er )
K 1
, for K < 0
Using this we find the induced potential (for details see Appendix B)
Vind =
1
4
ä
K + ȧ2
ä2
6 +3
−
2
a
a
K + ȧ2
We note that, even though the expressions for the embedding and the normal
vectors are very different for different values of K, the induced potential is described
by a single expression for all values of K.
For K = 0 we can write the induced potential as
Vind = Ḣ + 2H 2 −
H2 2
Ḣ 2
=
−
q + 6q − 3
4H 2
4
where H = ȧa is the Hubble parameter and q =
parameter [33–35].
aä
ȧ2
= −Ḣ − H 2 is the deceleration
4.4. A model of the early universe
27
In the a model where a(t) go to zero in the limit t → 0, a(t) can be modelled
as decaying as a power law a(t) = (t/t0 )x for some constant t0 and x > 0. The
induced potential is then
Vind =
8x2 − 4x − 1
4t2
(4.3)
This means that in the limit t → 0, the potential behaves as

 −∞ 0 < x < x∗
0
x = x∗
lim Vind =
t→0

+∞ x∗ < x
√
with x∗ = 1+4 3 ≈ 0.683. Thus under the condition that the potential is bounded,
it predicts a specific value of the exponent x. As we will see later, there is a solution
for a scalar field in the RW universe where, if the induced potential is included, the
scale factor grows as a(t) ∼ tx∗ short after t = 0.
4.4
A model of the early universe
The induced potential depends on the scale factor a(t) and its derivatives and
becomes large in some limits when a(t) becomes small. This suggests that the
induced potential could be important in the early universe. One model of the early
universe is the theory of cosmological inflation (see e.g. [33–35]).
In the usual Big Bang scenario [34], the expansion of the universe is determined
by the Friedmann equations
ρ̇ = −3H(ρ + P )
ρ
K
H2 =
− 2
3MP2 l
a
1
ä
= Ḣ + H 2 = −
(ρ + 3P )
a
6MP2 l
where ρ is the (energy) density and P is the pressure. In the following we will set
K = 0. It is natural to assume that these equations together with an equation of
state P = wρ gives the expanding universe we observe today. There are, however,
some problems which this model does not solve:
• The Flatness problem. If the curvature of the universe was non-zero in the
early universe, then the universe becomes more curved with time. The universe we observe today is measured to be flat with small uncertainty. This
means that there should have been some mechanism in the early universe,
making the universe more flat.
28
Chapter 4. Extrinsic curvature effects in brane-world scenarios
• The Horizon problem. As the universe expands, different regions separate
faster then they can exchange information. This implies that different regions
should have different temperature since they cannot come into thermal equilibrium fast enough. This contradicts observations, so this tendency should
have been suppressed in the early universe.
• Unwanted relics (or monopole problem). In the Big Bang, different exotic
particles are believed to have been produced. But since we do not observe
them today, they should have been diluted by some mechanism in the early
universe.
Cosmological inflation solves the above problems by postulating that a rapid
expansion occurred in the early universe [34]. This solves the problem of unwanted
relics since they then became strongly diluted in the expansion, the horizon problem
since all wavelength become red-shifted and the flatness problem since the rapid
expansion suppresses the curvature of the universe.
4.4.1
The standard case
In the standard case, where the induced potential is not present, the Friedmann
equations of the expanding universe can be obtained from Einstein’s equations
for the Robertson-Walker metric (4.2) with the Energy-Momentum tensor (Tµν ) =
diag(−ρ, p, p, p).
If matter consists of an isotropic Klein-Gordon field (φ = φ(t)), then
1 2
φ̇ + V(φ)
2
1
P = φ̇2 − V(φ)
2
ρ=
where V(φ) contains all interactions of the Klein-Gordon field. In this case the
equations of motion can also be derived from the action
Z
p
1 µν
2
4
S = d x |g| MP l (R − 2Λ) + g (∂µ φ)(∂ν φ) − V(φ)
2
where MP2 l =
~c
8πG
is the reduced Planck mass and
R = −6
ä ȧ2
+
a a2
is the Ricci (scalar) curvature of the Robertson-Walker universe and Λ is the cosmological constant. For concreteness we set
V(φ) =
1 2 2 λ 4
m φ + φ
2
4
4.4. A model of the early universe
29
which is a potential often studied in the literature [34]. Since both R and φ only
depends on the time variable, we can integrating over the variables r, θ and ϕ and
write the action as
Z
S = const
dtL
where
L = MP2 l (−3aȧ2 − Λa3 ) +
a3
2
φ̇2 − m2 φ2 −
λ 4
φ
2
It follows that the equations of motion for a and φ are
φ̈ + 3H φ̇ + m2 φ + λφ3 = 0
1
λ 4
2
2
2Ḣ + 3H 2 − Λ +
φ̇
−
m
φ
−
φ
=0
2MP2 l
2
(4.4)
(4.5)
Since L does not depend explicitly on time we have the conservation law
d ∂L
∂L
∂L
∂L
−
ä +
ȧ +
φ̇ − L
∂ ȧ
dt ∂ä
∂ä
∂ φ̇
1
λ 4
Λ
2
2 2
φ̇
+
m
φ
+
φ
= 3a3 MP2 l H 2 − −
3
6MP2 l
2
I=
(4.6)
from which we obtain the Friedmann equations by setting I = 0. To solve the
equations, one often uses the ”slow-roll” approximation where one approximates
(4.4) and (4.6) with
3H φ̇ + m2 φ + λφ3 = 0
1
λ 4
2
2
H −
m φ+ φ =0
6MP2 l
2
The approximation is valid when
φ̈ 3H φ̇
φ̇2 m2 φ2
Under this approximation, one can calculate a(t2 )/a(t1 ) exactly (for more details
see e.g. [34]). The third equation (4.5) is then automatically solved since it follows
from the other two equations.
30
Chapter 4. Extrinsic curvature effects in brane-world scenarios
We here use an alternative method to solve the equations. We expand the
solutions in series
φ(t) = ty φ0 + φ1 t + φ2 t2 + φ3 t3 + O(t4 )
x
H(t) = + H0 + H1 t + H2 t2 + H3 t3 + O(t4 )
t
and solve for each coefficient. We find the solution
φ21
φ1
t+
3H0 φ1 + m2 φ0 + λφ30 t2 + O(t3 )
2
2
2MP l
2MP l
1
3H0 φ1 + m2 φ0 + λφ30 t2 + O(t3 )
φ(t) = φ0 + φ1 t −
2
s
1
Λ
2 + m 2 φ2 + λ φ3
H0 =
+
φ
1
0
3
6MP2 l
2 0
H(t) = H0 −
The solution has two free parameters, φ0 = φ(0) and φ1 = φ̇(0), which describe
the initial amplitude and speed of the scalar
Rfield φ. We
see that the solution has
t
x = y = 0, implying that a(t) = a(0) exp 0 H(t0 )dt0 and φ(t) are analytic at
t = 0.
4.4.2
The extended case
In the extended case where the induced potential is present, we do not know how to
derive the Einstein equations since it is unclear how the induced potential changes
when the metric tensor is varied. We therefore, in analogy with the standard case,
postulate that the equations of motion for a(t) and φ(t) are the equations of motion
following from the action
p
1
1
λ
d4 x |g| m2P l (R − 2Λ) + g µν (∂µ φ)(∂ν φ) − (m2 + Vind )φ2 − φ4
2
2
4
Z
= const dtL
Z
S=
with the condition that the conservation law for this action satisfies I = 0. The
Lagrangian is
L = MP2 l (−3aȧ2 − Λa3 ) +
a3
2
1 ä
ȧ2
ä2
λ
φ̇2 − m2 +
6 +3 2 − 2
φ2 − φ4
4 a
a
ȧ
2
4.4. A model of the early universe
31
The equations of motion for a(t), φ(t) and the condition that I = 0 gives the
extended Friedmann equations
!
Ḣ
2
2
φ̈ + 3H φ̇ + m + Ḣ + 2H −
φ + λφ3 = 0
4H 2
1
1 2 1 2 2 λ 4
2
2Ḣ + 3H − Λ + 2
φ̇ − m φ − φ
MP l 2
2
4
"
... #
3
2
1
Ḧ
Ḣ
H
Ḣ Ḧ
5Ḣ
3Ḣ 2
H
+ 2
+
−
−
+
−
+
φ2
MP l
2H
4H 4
2
3H 3
12
8H 2
12H 2
!
!
#
2
Ḧ
Ḣ 2
Ḣ
1 Ḣ
2
− H+
−
φφ̇ +
−
φφ̈ + φ̇
=0
+
H
3
3H 2
2H 3
6H 2
3
Λ
1
λ 4
2
2
2 2
H − −
φ̇ + m φ + φ
3
6MP2 l
2
! #
"
!
1
Ḣ
1
Ḧ
1 2
Ḣ 2
1
φ2 = 0
+ 2
− H φφ̇ +
Ḣ +
− H −
MP l
6H
3
4
12H
6
8H 2
where H =
ȧ
a
as usual. For this case the conservation law is
I = − 3a
+
3
1
λ 4
Λ
2
2 2
2
φ̇ + m φ + φ
H − −
3
6MP2 l
2
!
! #
Ḧ
Ḣ
H2
Ḣ 2
1
Ḣ
H
+
−
−
φ2 + 2
−
φφ̇ = 0
12H
4
6
8H 2
MP l 6H
3
MP2 l
1
MP2 l
Using MAPLE we find that the equations have one solution with x = 43 , y = 1
and no free parameters
r 2
2m
126λMP2 l
2 1
3
φ(t) =
−
−
t + O(t )
λ 4t
15
65
2
2
m
594λMP2 l
H(t) =
+
−
t + O(t3 )
4t
5
65
Another solution with x = (1 +
√
3)/4 ≈ 0.683, y = 0 and no free parameters is
φ(t) = MP l 1.5 − 0.228m2 + 0.402λMP2 l + 0.0883Λ t2 + O(t4 )
x
H(t) = − 0.0142m2 + 0.120λMP2 l − 0.0690Λ t + O(t3 )
t
The exact values of the coefficients can easily be calculated, but are not very illuminating.
32
Chapter 4. Extrinsic curvature effects in brane-world scenarios
There is also a solution with x = y = 0
1
H12
2
2
2
φ(t) = φ0 + φ1 t −
φ0 t2 + O(t3 )
3H0 φ1 + m + λφ0 + H1 + 2H0 −
2
4H02
3H12
λ 2
3
2
2
H(t) = H0 + H1 t + m + φ0 + H0 − H1 H0 +
2
2
4H0
φ
H
1
0
+(2H02 − H1 )
+ (φ21 + 2ΛMP2 l − 6H02 MP2 l ) 2 t2 + O(t3 )
φ0
φ0
which has four free parameters, φ0 6= 0, φ1 , H0 and H1 . In contrast with the other
two solutions, the solution is analytical at t = 0.
We believe that each of these solutions has a finite radius of convergence, i.e. the
exact solution can be well approximated by these expansions (truncated at some
finite order) in some finite interval. We can therefore expand the solutions around a
initial point given initial values of φ and H. The solution can thereafter be extended
by expanding around another point in the interval to obtain an approximative
solution in a larger interval. For small t, the second solution has a power-law
behaviour
a(t) ∼ tx∗
with x∗ ≈ 0.683 precisely the exponent required for the potential (4.3) to have a
finite value in the limit t → 0.
4.5
Discussion
Since the extrinsic and intrinsic curvature of embedded submanifolds affects the
particles of our world through the induced potential, it can provide a means to
indirectly explore extra dimensions. If Schwarzschild space-time is embedded in
the way discussed in this chapter, then the induced potential could be present and
give measurable physical effects. Since the contribution is inverse proportional to
the mass of the gravitating object, the effect will probably be negligible for massive
objects like the sun and the earth. Only for small black holes, such as hypothetical
primordial black holes, could the induced potential give measurable effects.
The embedded Robertson-Walker universe produces an induced potential which
depends on the scale factor a(t) and its derivatives. This suggests that the presence
of the induced potential interacting with a scalar field could influence the expansion
of the universe. Especially, the potential affects the expansion of the early universe
where a(t) is small.
The embeddings considered in this chapter where chosen because they are (we
believe) the simplest possible. It is possible that other theories with extra dimensions, like e.g. string theory, suggest more complicated embeddings of the considered space-times. For such cases, the induced potential can be calculated in a
similar fashion.
4.5. Discussion
33
Rosen lists in his paper [32] embeddings of many solutions of general relativity.
An interesting future project would be to calculate the induced potentials for these
embeddings and to derive possible physical consequences. To calculate the induced
potential for more complicated embeddings motivated by an underlying physical
theory is another interesting project which could improve our understanding of
extra dimensions.
34
Chapter 5
Membrane dynamics
One often considers the fundamental particles of nature to be point particles. One
possible generalization of point particles can be obtained by considering higher
dimensional extended objects. One motivation for this is that one-dimensional
extended objects, so called strings, are believed to describe gravity and other fundamental forces (see e.g. the textbook by Polchinski [14]).
Two kinds of string theories are usually considered. Bosonic string theory, which
contains bosons, and supersymmetric string theory, which contains bosons and
fermions. String theory has the peculiar feature that the quantum theory requires
a certain number of space-time dimensions to be consistent1 . Bosonic string theory
requires D = 26 space-time dimensions, and supersymmetric string theory requires
D = 10. Five different kinds of supersymmetric string theory are known, and there
are hopes that the different supersymmetric string theories actually are limits of
a single underlying theory called M -theory [36]. It is not known what M -theory
exactly is, but it is known that it requires D = 11 space-time dimensions to be
consistent, and it is believed to contain quantum membranes.
Since M -theory and string theory contains quantum membranes, a better understanding of the dynamics of membranes is desired. Today, more is known about
quantum strings than about the quantum theory of higher dimensional extended
objects. It is, for example, not known if membranes can be quantized or what their
mass spectrum is. By examining symmetries of the theory, we hope to gain greater
insight and understanding of the theory [37–40]. Dynamical symmetries are certain
symmetries related to the dynamics of a system and are sometimes called ”hidden”
symmetries.
In this chapter we first review the light-cone gauge in a simple example and
then describe a well known dynamical symmetry. We then introduce the bosonic
membrane and study its Hamiltonian formulation in the light-cone gauge. The
dynamical symmetry of the bosonic membrane is described. In the final sections
1 This
is because of an anomaly related to the Lorentz invariance of the theory.
35
36
Chapter 5. Membrane dynamics
we introduce the supersymmetric membrane and study a dynamical symmetry for
the superembrane.
5.1
Point particle in the light-cone gauge
It is resonable to assume that the motion of a relativistic point particle in a Ddimensional Lorentzian manifold is such that the length of the particles world-line
is minimized (we here follow [41] and [14]). The action can be written as2
Z
τ1
S = −m
dτ
q
gαβ (x)ẋα ẋβ
(5.1)
τ0
where x = x(τ ), ẋ = ∂τ x, gαβ is the metric tensor of the ambient space and we
let greek letters denote the indices 0, 1, 2, . . . , D. By varying the action we find the
geodesic equation
ẍµ + Γµνσ ẋν ẋσ = 0
We can naively try to calculate the Hamiltonian of the relativistic point particle
in flat Minkowski space, where gαβ = ηαβ . The conjugate momenta is
pµ =
∂L
mẋµ
= −√
µ
∂ ẋ
ẋα ẋα
which gives us the Hamiltonian
q
mẋµ ẋµ
H = pµ ẋµ − L = − √
+
m
ẋβ ẋβ = 0
ẋα ẋα
Thus the Hamiltonian vanishes. This means that the variables of the theory are
not dynamical, but are determined by the conservation law
pµ pµ − m2 = 0
(5.2)
which follows from the definition of the canonical momenta or from Noether’s theorem [42].
The Hamiltonian vanishes because the action is reparametrization invariant [43],
i.e. the time variable, which is treated as a special variable in the Hamiltonian
formalism, can be substituted by a function of other variables without changing
the action. To obtain dynamical variables and a non-zero Hamiltonian we make
2 We
use the metric convention (+, −, −, . . . , −).
5.1. Point particle in the light-cone gauge
37
a gauge choice, i.e. we choose a specific time variable. One possible choice is the
light-cone gauge where we set
x+ =
x0 + xD−1
=τ
2
and x− = ζ = x0 − xD−1 . The action (5.1) then becomes
Z q
2
S = −m
2ζ̇ − ẋ dτ
where x = (x1 , x2 , . . . , xD−2 ). In this gauge, the metric is

 

η++ η+− η+j
0 1
0
0 
(ηµν ) =  η−+ η−− η−j  =  1 0
ηi+ ηi− ηij
0 0 −δij
where we use latin letters to denote indices 1, 2, . . . , D − 2. We find that
(pi ) = p =
π=
∂L
mẋ
=q
2
∂ ẋ
2ζ̇ − ẋ
−m
∂L
=q
2
∂ ζ̇
2ζ̇ − ẋ
mζ̇
p2 + m2
H = p · ẋ + π ζ̇ − L = q
=
−2π
2
2ζ̇ − ẋ
An important tool in the Hamiltonian formalism is the Poisson bracket. For
two function F (xµ , pν ) and G(xµ , pν ), the Poisson bracket is defined as
{F, G} =
D−1
X
µ=0
∂F ∂G
∂G ∂F
−
∂xµ ∂pµ
∂xµ ∂pµ
=
D−2
X ∂F ∂G
∂F ∂G ∂G ∂F
∂G ∂F
−
+
−
∂ζ ∂π
∂ζ ∂π
∂xi ∂pi
∂xi ∂pi
i=1
We have the Poisson brackets
{xi , pj } = δij
{ζ, π} = 1
Since the Hamiltonian is canonically conjugate to x− = x+ = τ (by that 1 =
τ̇ = {τ, H}) we set
p− = p+ = H
One notes that π is constant since
π̇ = {π, H} = −
∂H
=0
∂ζ
38
Chapter 5. Membrane dynamics
The other variables have the time dependence
pi
π
H
ζ̇ = {ζ, H} = −
π
ẋi = {xi , H} = −
with all other brackets zero. The generators of Lorentz transformations in the
light-cone gauge are
Mij = xi pj − xj pi
Mi− = xi H − ζpi
We find that the generators fulfil the Poisson bracket relations
{Mij , Mkl } = δik Mjl − δil Mjk − δjk Mil + δjl Mik
{Mi− , Mj− } = 0
{Mi− , Mkl } = −δik Ml− + δil Mk−
In the quantum theory of strings, the critical dimension is determined from the
requirement that the second relation holds also at the quantum level.
5.2
A dynamical symmetry
In classical mechanics, symmetries are closely related to conserved quantities by
Noethers theorem. For example, translation invariance is related to momentum
conservation, time independence to energy conservation and rotational symmetry
to conservation of angular momentum. In many cases, the existence of symmetries
allows us to reduce the number of degrees of freedom of a problem and possibly
obtain an exact solution. A symmetry which depends on the dynamics of the system
is called a dynamical symmetry.
An important example of a dynamical symmetry is the Kepler problem for which
the Hamiltonian is3
H=
p2
GM m
−
2m
r
where G is the gravitational constant, m is the mass of the particle and M is
the mass of a heavy particle often considered to be stationary. The generators of
angular momentum, Lij = xi pj − xj pi , generate the symmetry group SO(3) of
3 A quantum analogue is the Hydrogen atom model which can be obtained by substituting
GM m → e2 and replacing variables with operators.
5.2. A dynamical symmetry
39
spatial rotations and are conserved, i.e. {Lij , H} = 0. Another conserved quantity
in the problem is the Laplace-Runge-Lenz vector [44] which has the components
Ci =
1
GM m
pj Lij −
xi
m
r
The Lenz-vector and the generators of angular momentum have the Poisson brackets
{Lij , Lkl } = δik Ljl − δil Ljk − δjk Lil + δjl Lik
2H
{Ci , Cj } = −
Lij
m
{Ci , Lkl } = −δik Cl + δil Ck
The fact that the algebra is closed under Poisson brackets shows that Lij and Ci
are generators of a symmetry group of the Hydrogen atom. The symmetry group
is larger since the group of spatial rotations and depends on the Hamiltonian. This
shows that the problem has a dynamical symmetry.
To simplify the expressions we restrict ourselves to solution where H < 0
(bounded orbits) and set
r
m
Ck
Ak = −
2H
This gives the relations
{Lij , Lkl } = δik Ljl − δil Ljk − δjk Lil + δjl Lik
{Ai , Aj } = Lij
{Ai , Lkl } = −δik Al + δil Ak
By defining generators Iij and Jij
1
(Lij + ijk Ak )
2
1
Jij = (Lij − ijk Ak )
2
Iij =
we find that Iij and Jij have the Poisson brackets
{Iij , Ikl } = δik Ijl − δil Ijk − δjk Iil + δjl Iik
{Jij , Jkl } = δik Jjl − δil Jjk − δjk Jil + δjl Jik
{Iij , Jkl } = 0
These relations show that Iij and Jij together generate the symmetry group
SU (2) × SU (2) ∼
= SO(4). This shows that the Kepler problem atom has a larger
”hidden” SO(4) symmetry. For the quantum mechanical Hydrogen atom, this
dynamical symmetry can be used to calculate the spectrum [45].
40
Chapter 5. Membrane dynamics
5.3
The bosonic membrane
As a generalization of point particles, one can consider higher dimensional extended
objects moving in a Lorentzian manifold (for details see e.g. [38]). In analogy with
the point particle, we consider membranes which moves in a way that minimizes
the world volume of the membrane M as it moves in a Lorentzian manifold N .
The corresponding action is the Dirac-Nambu-Goto action4
Z
√
(5.3)
S = −V ol(M) = − dϕ0 dM ϕ G
where G is the absolute value of the determinant of the induced metric
Gαβ = ηµν ∂α xµ ∂β xν
of the membrane, xµ = xµ (ϕ) for ϕ = (ϕ0 , ϕ1 , . . . , ϕM ) and ηµν is the metric tensor
of N . The equations of motion for the membrane, which follows from (5.3), are
√
1
√ ∂α
GGαβ ∂β xµ + Gαβ ∂α xν ∂β xλ Γµνλ (x) = 0
G
In the following we will only consider the case where the embedding space is
Minkowski space, i.e. N = R1,D−1 and Γµνλ (x) = 0.
5.3.1
Light-Cone Gauge
Just as for the point particle, we consider the membrane in the light-cone gauge
where we set
x0 + xD−1
=τ
2
ζ = x0 − xD−1
ϕ0 =
The matrix Gαβ is then



(Gαβ ) = 

2ζ̇ − ẋ
u1
..
.
2
u1 . . . uM

−grs




uM
where
x = (x1 , x2 , . . . , xD−1 )
grs = ∂r x · ∂s x
ur = ∂r ζ − ẋ · ∂r x
4 The
action is often called the Nambu-Goto action in the string theory literature. In the
context of membranes one often also credits Dirac since he formulated a model where the electron
was modelled as a membrane [46].
5.4. Hamiltonian formalism
41
We thus find that
2
1
up
2ζ̇ − ẋ + ur g rs us
G = | det(G)| = det
0 −grp
−g pq uq
2
= g 2ζ̇ − ẋ + ur g rs us = gΓ
0
δsp
with Γ defined by this relation.
5.4
Hamiltonian formalism
As discussed earlier, membranes are an essential part of M -theory which is believed
to unify the different supersymmetric string theories in a single theory. M -theory
is a quantum mechanical theory, and the understanding of quantum mechanical
membranes is therefore important. One way to formulate a quantum theory is in
the operator formalism based on the Hamiltonian formulation of classical mechanics. We here work towards a quantum theory of membranes by investigating the
Hamiltonian structure of classical membranes [41].
Let us first (as before) naively try to find the Hamiltonian of the theory. The
canonical momenta are
pµ =
√
∂L
= − GG0α ηµν ∂α xν
µ
∂ ẋ
and the Hamiltonian (density) is found to be
√
√
√
√
H = pµ ẋµ − L = − GG0α ηµν ∂α xν ẋµ + G = − GG0α Gα0 + G = 0
i.e. the variables of the theory are not dynamical, but they are instead determined
by the constraints [47]
Ja := pµ ∂a xµ = 0
1
H̃ := (η µν pµ pν + det(µab )) = 0
2
i = 1, 2, . . .
where
µab = ηµν
∂xµ ∂xν
∂ϕa ∂ϕb
for a, b, = 1, 2, . . . , M . The constraints follow from the definition of the canonical
momenta.
As for the point particle, we choose a specific time variable to obtain a theory
with dynamical variables. We then lose the full Lorentz symmetry of the theory,
42
Chapter 5. Membrane dynamics
but obtain a (possibly) non-zero Hamiltonian and thereby dynamical variables. In
the light-cone gauge, the canonical momenta are
r
g
∂L
=
p=
ẋ
Γ
∂ ẋ
r
g
∂L
=−
π=
∂ζ
Γ
where we made the gauge choice ua = ∂r ζ − ẋ · ∂a x = 0. One finds that the
Hamiltonian density is
r
g
H = p · ẋ + π ζ̇ − L =
ζ̇
Γ
Like for the point particle, we can write the Hamiltonian density as
H=
p2 + g
−2π
We see that π is time-independent since
π̇ = −
∂H
=0
∂ζ
We therefore write π = π(ϕ) = −ηρ(ϕ), where ρ(ϕ) is a non-dynamical density
which fulfils
Z
ρ(ϕ)dM ϕ = 1
M
Just as for the point particle, we have that
p− = H
p+ = −π
which gives us that the Lorentz invariant mass squared is [41]
2
Z
M = 2P+ P− − P · P =
p2 + g M
d ϕ−
ρ
where
Z
Pµ =
is the zero-mode of pµ .
pµ dM ϕ
Z
2
pi d ϕ
M
5.5. Mode expansion
5.5
43
Mode expansion
We introduce a complete orthonormal set of (non-constant) basis functions {Yα (ϕ)}α∈I
defined on the membrane satisfying
Z
Yα (ϕ)Yβ (ϕ)ρ(ϕ)dM ϕ = δαβ
M
∆Yα (ϕ) = −µα Yα (ϕ)
and expand the dynamical variables in terms of the basis functions
X
x=X+
xα Yα (ϕ)
(5.4)
α6=0
X
p
pα Yα (ϕ)
=P +
ρ
(5.5)
α6=0
where X and P are the zero-modes
Z
X=
Z
P =
x ρdM ϕ
p dM ϕ
of the position and momentum variables. In the following we will use the convention
that α denotes indices different from zero and we sum over repeated indices.
For functions F (xµ (ϕ), pν (ϕ)) and G(xµ (ϕ0 ), pν (ϕ0 )) we define the Poisson bracket
as
Z
δF
δG
δG
δF
M
{F, G} = d ϕ̃
−
δxµ (ϕ̃) δpµ (ϕ̃) δxµ (ϕ̃) δpµ (ϕ̃)
We find the Poisson bracket
{xi (ϕ), pj (ϕ̃)} = δij δ(ϕ, ϕ̃)
which gives us that
{xiα , pjβ } = δij δαβ
(5.6)
{Xi , Pj } = δij
(5.7)
and all other brackets zero. Using the basis functions, we can construct the ”Green’s
function”
G(ϕ, ϕ̃) =
X −1
α
µα
Yα (ϕ)Yα (ϕ̃)
44
Chapter 5. Membrane dynamics
satisfying
δ(ϕ, ϕ̃)
∆ϕ̃ G(ϕ, ϕ̃) =
− 1,
ρ(ϕ)
Z
G(ϕ, ϕ̃)ρ(ϕ)dM ϕ = 0
From the equations for ζ
p
· ∂a x
ρ
p2 + g
2η 2 ζ̇ =
ρ2
η∂a ζ =
we can reconstruct ζ using the Green’s function. We get that [48]
Z
1
p ˜
a
˜
ζ(ϕ) = ζ0 +
G(ϕ, ϕ̃)∇
· ∇a x (ϕ̃)ρ(ϕ̃)dM ϕ̃
η
ρ
5.6
A dynamical symmetry for bosonic
membranes
As for the Kepler problem, there is a dynamical symmetry for the membranes
[1, 39]. For the bosonic membrane, the generators of Lorentz transformations in
the light-cone gauge are
Z
Mij = (xi pj − xj pi ) dM ϕ
Z
Mi− = (xi H − ζpi ) dM ϕ
The generators fulfil the same commutation relations as the Lorentz generators for
the point particle [49]
{Mij , Mkl } = δik Mjl − δil Mjk − δjk Mil + δjl Mik
(5.8)
{Mi− , Mj− } = 0
(5.9)
{Mi− , Mkl } = δik Ml− − δil Mk−
(5.10)
Using the expansions (5.4) and (5.5) we expand the angular momentum generators as
Mij = Xi Pj − Xj Pi + Mij
where Mij = xiα pjα − xjα piα only contains internal degrees of freedom. By using
the relation (5.6) we get that
{Mij , Mkl } = δik Mjl − δil Mjk − δjk Mil + δjl Mik
Thus the purely internal angular momentum generate a symmetry group.
(5.11)
5.7. Supermembranes
45
We can expand the Lorentz generators Mi− in the same way using that
2ηζ(ϕ) = 2ηζ0 + 2P · xα Yα (ϕ) + (internal modes)
2
2ηH(ϕ) = ρP + 2ρP · pα Yα (ϕ) + (internal modes)
which gives us that
2
2ηMi− = Xi P − 2ηζ0 Pi + 2Pj Mij + 2ηMi−
(5.12)
where Mi− only contains internal modes. We find that ηMi− satisfy
{ηMi− , ηMj− } = M2 Mij
and also that
{ηMi− , Mkl } = −ηδki Mj− + ηδkj Mi−
(5.13)
From (5.11), (5.12) and (5.13) we see that the purly internal Lorentz generators
Mij and ηMi− generate a dynamical symmetry similar to the one of the Kepler
problem in section 5.2. This shows that the bosonic membrane has a dynamical
symmetry.
5.7
Supermembranes
One objection to bosonic string theory as a physical theory is that it only contains bosons. A string theory with fermions is supersymmetric string theory (or
superstring theory) which is a string theory with bosonic and fermionic variables related by supersymmetry. Because of the different statistics of bosons and fermions,
bosonic variables commute, xµ xν = xν xµ , while fermionic variables anti-commute
θα θβ = −θβ θα . The action for supersymmetric strings is called the Green-Schwarz
action. The generalization of the Green-Schwarz action to p-dimensional supermembranes, p-branes, is [50]
Z q
1
1 µ
αβγ
µ ν
ν
|det(Gαβ )| + ∂α x Eβ + θΓ ∂α θθΓ ∂β θ θΓµν ∂γ θ dp ϕdϕ0
S=−
2
6
where
Gαβ = Eα · Eβ = ηµν Eαµ Eβν
Eαµ = ∂α xµ + θΓµ ∂α θ
and the matrices Γµ are the D-dimensional gamma matrices which satisfy
Γµ Γν + Γν Γµ = 2η µν
Since the theory of classical bosonic membranes can be formulated in spacetime of arbitrary dimension, it is only at the quantum level that restrictions on
46
Chapter 5. Membrane dynamics
the number of space-time dimensions (like for the quantum bosonic string) might
occur. The supersymmetric membrane, however, must be formulated in such a way
that the bosonic and fermionic degrees of freedom match [51]. This means that,
even at the classical level, a p-dimensional supermembrane can only be formulated
in D space-time dimensions, where D depends on p. The possible supermembrane
theories are
p=1 2
3
4
5
D=3
X
4
X
X
5
X
6
X
X
7
X
8
X
9
X
10
X
X
11
X
where we have put a X to indicate that the dynamics is consistent. Note that
p = 1 corresponds to the string. The classical superstring can be formulated in
D = 3, 4, 6 or 10 dimensions, while the quantum superstring requires D = 10
dimensions. All different superstring theories are believed to be unified in M -theory
which requires D = 11 space-time dimensions. Since this is also the dimension
required for the 2-dimensional supermembrane, 2-dimensional supermembranes is
the most probable candidate for M -theory [52].
In the light-cone gauge, the generalized Green-Schwarz Lagrangian is
q
√
2
L = − g 2ζ̇S + ẋ + 2θΓ− θ̇ + rs ∂r xa θΓ− Γa ∂s θ
g = det(grs )
grs = ∂r x · ∂s x
We find that
r
∂L
g
p=
=
ẋ − ur g rs ∂s x
Γ
∂ ẋ
r
∂L
g
π=
=
Γ
∂ ζ̇
r
∂L
g −
S=
=−
Γ θ
Γ
∂ θ̇
p2 + g
H = p · ẋ + π ζ̇ + S θ̇ − L =
− rs ∂r xi θΓ− Γi ∂s θ
−2π
By super symmetry, we can reduce the number of fermionis variables by imposing
the constraint
(Γ0 + ΓD )θ = 0
5.9. A dynamical symmetry for supermembranes
47
It is convenient to factorize the gamma matrices as
√ 0
2
√0 0 ⊗ I
Γ+ =
⊗ I Γ− =
2 0
0 0
Γi =
0
1
1
0
⊗ γi
where I is the identity matrix and γ a are the 9-dimensional (SO(9)) gamma matrices (for details see e.g. [52]).
5.8
Poisson brackets
In order for the Poisson bracket to be formulated consistently5 we need the special
property that for fermionic variables [53]
{θα , θβ } = {θβ , θα }
{θα θβ , θγ } = θα {θβ , θγ } − {θα , θγ }θβ
The bosonic and fermionic variables have the non-zero Poisson brackets
{xi (ϕ), pj (ϕ)} = δij δ(ϕ, ϕ̃)
iδαβ
δ(ϕ, ϕ̃)
{θα (ϕ), θβ (ϕ̃)} = −
ρ
Expanding the variables in terms of the basis functions
xi (ϕ) = Xi + xiA YA (ϕ)
pi (ϕ)/ρ = Pi + piA YA (ϕ)
θα (ϕ) = θα0 + θαA YA (ϕ)
we get the Poisson brackets
{Xi , Pj } = δij
{θα0 , θβ0 } = −iδαβ
5.9
{xiA , pjB } = δij δAB
{θαA , θβB } = −iδαβ δAB
A dynamical symmetry for supermembranes
In analogy with the bosonic membrane, there is a corresponding dynamical symmetry for the supermembrane [40]. We find the symmetry by calculating the Poisson
brackets of the generators of Lorentz transformations.
5 We want the Poisson bracket to be such that e.g.
F (xµ , pν , θα ).
Ḟ = {F, H} for any function F =
48
Chapter 5. Membrane dynamics
The generators of Lorentz transformations for the supermembrane are
Z i
Jij =
xi pj − xj pi − θγ ij θρ d2 ϕ
4
Z i
i
Ji− =
xi HS − ζS pi − θγ ik θpk − rs ∂r xj ∂s xk θγ ijk θ d2 ϕ
2η
8η
with
Z
i
p ˜
1
Gr (ϕ, ϕ̃)
· ∂r x + θ∂˜r θ ρ(ϕ̃)d2 ϕ̃
η
ρ
2
2
p +g
i
HS =
− θγ i rs ∂r xi ∂s θ
2ηρ
2η
1
γ ij = γ [i γ j] = [γ i , γ j ]
2
1 i j k
γ ijk = γ [i γ j γ k] =
γ γ γ + γj γk γi + γk γiγj − γj γiγk − γk γj γi − γiγk γj
3!
ζS = ζ0 −
It can be shown that (see [54, 55]) the generators fulfil the Poisson bracket
relations
{Jij , Jkl } = δik Jjl − δil Jjk − δjk Jil + δjl Jik
{Ji− , Jk− } = 0
We separate the zero modes by defining
(0)
Jij = Jij + J ij
(0)
Ji− = Ji− + J˜i− + J i−
where
i
(0)
Jij = Xi Pj − Xj Pi − θ0 γ ij θ0
4
i
(0)
Ji− = (Xi HS − ζ0 Pi ) − θ0 γ ik θ0 Pk
4η
1
i
J˜i− = J ij Pk − θ0 γ i Q
η
2η
Z 1 rs
ij
i
i
Qβ =
pi γβα θα + ∂r xi ∂s xj γβα θα d2 ϕ − Pi γβα
θ0α
2
We find that
{ηJ i− , ηJ k− } = M2 J ik − Qik
where
i
ik
Qik = − Qα γαβ
Qβ
4
5.9. A dynamical symmetry for supermembranes
49
Calculating the Poisson brackets we find that
{Qij , Qkl } = (−δjk Qil + δik Qjl − δil Qjk + δjl Qik ) M2
{J ik , ηJ l− } = −δkl ηJ i− + δil J k−
{Qik , ηJ l− } = 0
{J ij , Qkl } = −δjk Qil + δik Qjl − δil Qjk + δjl Qik
This gives us that J ik , ηJ i− and Mik = M2 J ik − Qik generate the dynamical
symmetry
{Mij , Mkl } = −δjk Mil + δik Mjl − δil Mjk + δjl Mik M2
{η J¯i− , η J¯k− } = Mik
{Mik , η J¯l− } = −δkl η J¯i− + δil η J¯k− M2
This shows that the supermembrane has a dynamical symmetry similar to the
dynamical symmetry of the bosonic membrane. We hope that this symmetry, in
the future, will help us to better understand the dynamics of both classical and
quantum supermembranes.
50
Chapter 6
Summary and conclusions
Extra dimensions have been proposed in many theories. Since we do not observe
extra dimensions in our daily life, it is plausible to assume that some mechanism
prevents particles from moving in the extra dimensions (if they exists). For such
scenarios, extrinsic curvature effects could result in physical effects, which might
be accessible to experiments.
In paper A, we derived the induced potential for a relativistic scalar field
(Klein-Gordon field). This generalizes the notion of constrained quantum mechanics to relativistic fields. We also investigated embeddings of certain cosmological models, which gave indications about how extrinsic curvature effects can
produce measurable physical effects. We first considered the Fronsdal embedding
of the Schwarzschild solution and found that the induced potential is continuous
for all values 0 < r < ∞. We believe that this is a consequence of that Fronsdal’s
embedding covers the entire Schwarzschild space-time. Thereafter we calculated
the induced potential for a simple embedding of the Robertson-Walker space-time,
which describes the evolution of a homogeneous isotropic universe. The induced
potential appears to be negligible for measured values of the Hubble constant and
deceleration parameter, but it could have been important in the early universe. We
formulated a model from which we deduced equations extending the usual Friedman equations. It was found that the extended model allows a scaling solution not
possible in the original model. This implies that the induced potential can give rise
to physical effects in this scenario.
If, as suggested by many theories, we live on a membrane embedded in some
higher dimensional ambient space, then the membrane should be dynamical, i.e.
it should be affected by particles on the membrane. A simple model is to assume
that the dynamics of a membrane is analogous to that of a point-particle, i.e. a
membrane moves in a way that minimizes its world volume. By understanding
the symmetries of the theory we hope to gain insight into the dynamics of such
membranes.
51
52
Chapter 6. Summary and conclusions
In paper B, we derived a dynamical symmetry of a supermembrane. We found
that both the bosonic and fermionic variables satisfy Poisson relations like the ones
for the bosonic membrane. This shows that the supermembrane has a dynamical
symmetry with richer structure then that of the bosonic membrane.
Appendix A
The tubular neighbourhood
theorem
In this appendix we prove the tubular neighbourhood theorem which proves that
the embedding function can always be extended in the way done in chapter 2 and
3. All maps and manifolds considered will be assumed to be smooth.
Theorem [Tubular neighbourhood theorem] Let f : M → N be a smooth
embedding of a smooth n-dimensional manifold M in a smooth (n + p)-dimensional
manifold N and let Ba = {y ∈ Rp : ||y|| < a}. Then there exists an extension
F : M × Rp → N of f in the sense that F (x, 0) = f (x) for all x ∈ M and such that
for every compact set U ⊂ M there exists a constant > 0 such that F restricted
to U × B is one-to-one.
Proof At each point q ∈ M, the tangent space Tf (q) N can be decomposed as
Tf (q) N = Tq M ⊕ Nq M
Nq M is called the normal bundle of M at f (q) and is isomorphic to Rp . Let
n+1 n+2
{ni }n+p
,y
, . . . , y n+p ) ∈ Nq M. Denote
i=n+1 be a basis of Nq M and set y = (y
n+p
by ϕq : Wq → R
a local chart of N , where Wq is an open neighbourhood of
f (q) ∈ N . Define the map F to be
F : M × Rp → N
−1 i
F (x, y) = ϕ−1
q (ϕq (f (x)) + dϕq (y ni ))
where q is such that x ∈ f −1 (Wq ∩ f (M)). This shows the existence of F .
It remains to show that there exists some > 0 such that F |U ×B is one-to-one.
For each x ∈ M, F has full rank at y = 0. By the inverse function theorem there
exists an open neighbourhood Wx ⊂ N of F (x, 0) = f (x) such that F restricted
53
54
Appendix A. The tubular neighbourhood theorem
to F −1 (Wx ) is one-to-one. Since Wx is open, there is a constant x > 0 such that
F (Ux × Bx ) ⊂ Wx , where Ux = f −1 (Wx ∩ f (M)).
Let U ⊂ M be compact. For every x ∈ U there is an x > 0 such that F |Ux ×Bεx
is one-to-one. Since U is compact, there is a finite set of points x1 , x2 , . . . , xm in U
such that U ⊂ ∪m
i=1 Uxi . Setting
= min{x1 , x2 , . . . , xm }
we get that F |U ×B is one-to-one. This proves the theorem. If M is compact, then there is a constant > 0 such that F is one-to-one for
all (x, y) ∈ M × B . If M is non-compact, there is in general no constant > 0
such that the above statement holds for all x ∈ M. A simple counterexample
is the manifold {(x, y) ∈ R2 : x2 y 2 = 1}. For non-compact manifolds one can
use Urysohn’s lemma to construct a contionous function (x) > 0 such that F is
one-to-one for all (x, y) ∈ M × Rp such that ||y|| < (x).
Appendix B
Calculation of the induced
potential
We here present details of the calculations of the induced potential.
B.1
Calculation of γ̃
The expression (2.5) is an explicit expression for the metric tensor GAB . In order
to calculate γ̃ we use the block-matrix relation
γµν + y a y b Aaµk Abν l hkl −y a Aaµk hkj
Gµν Gµj
(GM N ) =
=
Giν Gij
−y b Abν l hli
−hij
λ
σ
a
k
0
δν
−y Aaµ
δµ
γλσ
0
=
0
−hkl
0
δi k
−y b Abν l δj l
which gives us that
G = | det(GM N )| = | det(γµν )|| det(hij )| = |γ||h|
where |γ| = | det(γµν )| and |h| = det(hij ). This gives us that
γ̃ =
|G|
= | det(γµν )|/|g|
|g||h|
= | det(gµν − 2y i αiµλ gλν + y i y j αiµλ αjν σ gλσ )|/|g|
= det(δν β − 2y i αiν β + y i y j αiγ λ αjν σ gλσ g γβ )
(B.1)
A useful relation for determinants is
det(A) = etr(log(A))
55
(B.2)
56
Appendix B. Calculation of the induced potential
To simplify calculation when expanding (B.1) we introduce the matrices I, Ai and
Bij with components
I = (δν β )
Ai = (−2αiν β )
Bij = (αiγ λ αjν σ gλσ g γβ )
= (αiσλ αjν σ gλγ g γβ )
= (αiσβ αjν σ )
Using these expressions and (B.2) we find that
det(I + y i Ai + y i y j Bij ) = exp tr(log(I + y i Ai + y i y j Bij ))
1 i j
i
i j
3
= exp tr(y Ai + y y Bij − y y Ai Aj + O(y ))
2
1
1
i
i j
= 1 + y tr(Ai ) + y y tr(Bij ) − tr(Ai Aj ) + tr(Ai )tr(Aj ) + O(y 3 )
2
2
1
1
λ
λ
i
λ
i j
σ
2
σ
2
λ
σ
= 1 − 2y αiλ + y y αiλ αjσ − (−2) αiλ αjσ + (−2) αiλ αjσ + O(y 3 )
2
2
= 1 − 2y i αiλλ + y i y j 2αiλλ αjσσ − αiλσ αjσλ + O(y 3 )
B.2
Derivation of the induced potential
In the derivation of the induced potential, we expanded the scalar field as Φ =
γ̃ −1/4 φ, where
φ(x, y) =
X
α
φα (x)χα (y)
B.2. Derivation of the induced potential
We get that the Hamiltonian H1 is
√
γ̃ 1/4
H1 = γ̃ 1/4 H γ̃ −1/4 = − √ ∂A
GGAB ∂B γ̃ −1/4 + Vconf (y)
2 G
γ̃ −1/4 √ 1/2 µν −1/4 γ̃ −1/4 √ 1/2 µσ k
gγ̃ γ ∂µ γ̃
− √ ∂µ
gγ̃ γ y Akσj ∂j γ̃ −1/4
= − √ ∂µ
2 g
2 g
−1/4
γ̃
γ̃ −1/4 √ 1/2 ij −1/4
√ 1/2 νσ k
− √ ∂i
gγ̃ γ y Akσi ∂ν γ̃ −1/4 − √ ∂i
gγ̃ b ∂j γ̃
2 g
2 g
γ̃ −1/4 √ 1/2 k l
gγ̃ y y Akσi Alρj γ σρ ∂j γ̃ −1/4 + Vconf (y)
− √ ∂i
2 g
√
1
1
√
gg µσ y k Akσj ∂j
= − √ ∂µ ( gg µν ∂µ ) − √ ∂µ
2 g
2 g
1
1
√
√
− √ y k Akσi ∂i ( gg νσ ∂ν ) − √ y k Akσi ∂i ( gg σρ )y l Alρj ∂j
2 g
2 g
3
1
1 ij
+ δ ∂i ∂j γ̃ − (∂i γ̃)(∂j γ̃) − δ ij ∂i ∂j + Vconf (y) + O(y)
8
4
2
1
√
= − √ ∂µ + y k Akµi ∂i ( gg µν ) ∂ν + y l Alν j ∂j + V (x, y)
2 g
1 ij
− δ ∂i ∂j + Vconf (y) + O(y)
2
where V (x, y) is the potential
1 ij
3
V (x, y) = δ ∂i ∂j γ̃ − (∂i γ̃)(∂j γ̃)
8
4
3
1X
λ
σ
σ
λ
λ
σ
=
2 2αiλ αiσ − αiλ αiσ − (−2αiλ )(−2αiσ ) + O(y)
8 i
4
X
1
=
αiλλ αiσσ − 2αiλσ αiσλ + O(y) = Vind (x) + O(y)
8 i
using that
Z
Z
p
p
dp y |h|y i1 y i2 . . . y iN χ∗α χβ ≤ dp y |h||y i1 ||y i2 | . . . |y iN ||χ∗α ||χβ |
Z
p
≤ N dp y |h||χ∗α ||χβ | ≤ N
gives us that
XZ
α,β
dp y
p
|h|V (x, y)χα (y)χβ (y) = Vind (x)δαβ + O()
57
58
Appendix B. Calculation of the induced potential
B.3
The Schwarzschild solution
We here describe the details of the derivation of the induced potential for the
embedded Schwarzschild solution for general t0 (not necessarily t0 = rs ). We first
discuss the case r > rs and then the case 0 < r < rs .
B.3.1
Embedding for r > rs
The embedding for r > rs can be written as
Z 0 = 2t0 (1 − rs /r)1/2 sinh(t/2t0 )
Z 1 = 2t0 (1 − rs /r)1/2 cosh(t/2t0 )
Z 2 = g(r)
Z 3 = r sin(θ) cos(ϕ)
Z 4 = r sin(θ) sin(ϕ)
Z 5 = r cos(θ)
where
g 0 (r)2 =
rs (r3 − t20 rs )
r3 (r − rs )
To simplify calculations we introduce the basis vectors
et = (cosh(t/2t0 ), sinh(t/2t0 ), 0, 0, 0, 0)
eτ = (sinh(t/2t0 ), cosh(t/2t0 ), 0, 0, 0, 0)
eg = (0, 0, 1, 0, 0, 0)
er = (0, 0, 0, sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ))
So that the embedding can be written as
Z = f = 2t0 (1 − rs /r)1/2 eτ + g(r)eg + rer
A set of normal vectors is
s
r3 − t20 rs (1 − rs /r)1/2 eτ + f (r)eg
2
r (r − rs )
r
t20 rs
r
1/2
n5 =
(1
−
r
/r)
e
+
h(r)e
−
e
s
τ
g
r
r3
t0
n4 =
B.3. The Schwarzschild solution
59
where
s
f (r) = −
s
h(r) =
t20 rs (r − rs )
r(r3 − t20 rs )
(r3 − t20 rs )(r − rs )
t20 rs r
From the embedding we can calculate the second derivatives of the embedding
function, tµν = ∂µ ∂ν Z, and use them to find the components
αiµν = αiµλ gλν = ni · tµν
which in turn allows us to calculate αiµν = g νλ αiµλ . A useful way to organize the
coefficients αiµν is to define matrices αi = (αiµν ). With these conventions we find
that


−1/2t0
0
0 0
s
3t0 rs (r−rs )
r3 − t20 rs 
0
− 2r(r
0 0 
3 −t2 r )


0 s
α4 =
r2 (r − rs ) 
0
0
0 0 
0
0
0 0
and
− 2t10
2
t0 rs 
 0
3
r  0
0
0

r
α5 =
− 2t10
0
0
0
0
1
t0
0

0
0 

0 
1
t0
This finally gives us the induced potential
Vind =
B.3.2
2
1X
(r4 − 4t2 rs r + 3t20 rs2 )2
rs
− 3
tr(αi )2 − 2tr(αi2 ) = − 2 4 0
2
3
4 i=1
16t0 r (r − rs )(r − t0 rs ) r
Embedding for 0 < r < rs
For 0 < r < rs the embedding can be written as
Z 0 = 2t0 (rs /r − 1)1/2 cosh(t/2t0 )
Z 1 = 2t0 (rs /r − 1)1/2 sinh(t/2t0 )
Z 2 = g(r)
Z 3 = r sin(θ) cos(ϕ)
Z 4 = r sin(θ) sin(ϕ)
Z 5 = r cos(θ)
60
Appendix B. Calculation of the induced potential
As before we can simplify calculations by introducing the basis vectors so that
Z = 2t0 (rs /r − 1)1/2 et + g(r)eg + rer
A set of normal vectors to the embedding is
1/2 −1/2 r
t0 rs rs
0
n4 = 1 −
−1
g (r)et − 2
eg
rs
r
r
−1/2
rs − r t0 rs rs
rs
0
−1
er
n5 = √
et − g (r)eg +
rs r
r2
r
r − rs
Using this we find that



(α4µν ) = 


− 2t10
q
t20 rs −r 3
r 2 (rs −r)
0
− 3t2r0 r2s
0
0
0
q
0
rs −r
t20 rs −r 3
0
0
0



0 

0 0 
0 0
0
and
p rs
1
− 2r
r

0
(α5µν ) = 

0
0

0p
rs
1
− 2r
0
0
r
1
r
0
0
p
rs
r
0
1
r
0
0
0
p

rs
r



So we find that the induced potential for 0 < r < rs is
Vind = −
(r4 − 4t20 rs r + 3t20 rs2 )2
rs
− 3
2
2
4
3
4(2t0 ) r (t0 rs − r )(rs − r) r
Note that the expression for the induced potential is is the same as for r > rs .
The parameter t0 needs to be chosen such that g 0 (r)2 is non-negative. When t0 6= rs
one can study the regions 0 < r < t0 < rs and rs < t0 < r separately, but for t0 = rs
the embedding covers the entire region 0 < r < ∞.
Setting t0 = rs we can simplify the expression for the potential as
Vind = −
(r + rs )(r2 + rs2 )(r3 + rs r2 + rs2 r + 9rs3 )
16rs2 r4 (r2 + rs r + rs2 )
Which is the expression given in chapter 4.
B.4. Robertson-Walker metric
B.4
61
Robertson-Walker metric
The Robertson-Walker metric can be embedded in five dimensional Minkowski
space by setting [32]

√

 b(t)/
2 K R
1 r
Z0 =
2 r0 + r0 a +


a(t)C(r)


 a(t)C(r)
R
1 r2
Z1 =
2 r0 − r0 a +

p

b(t)/ |K|
, for K > 0
dt0
2r0 ȧ
, for K = 0
, for K < 0
, for K > 0
dt0
2r0 ȧ
, for K = 0
, for K > 0
2
Z = a(t)S(r) sin(θ) cos(ϕ)
Z 3 = a(t)S(r) sin(θ) sin(ϕ)
Z 4 = a(t)S(r) cos(θ)
where
ḃ(t)2 = K + ȧ(t)2
√
√
cos( p
Kr)/ Kp
C(r) =
cosh( |K|r)/ |K|
√

sin( Kr)
√

, for K

K

r √
, for K
S(r) =

|K|r)
 sinh(

√
, for K
|K|
, for K > 0
, for K < 0
>0
=0
<0
For the different values of K, a normal vector is
n4 =







ȧ(t)
√ e0 + ḃ(t) (C(r)e1 + S(r)er )
Kh
2
i
1
r
1
ȧ(t)
+
r
−
e0
0
2
r0
h
r0 ȧ(t) i
2
1

+ 12 ȧ(t) rr0 − r0 − r0 ȧ(t)
e1 + ȧ(t)rer




ȧ(t)
 √
e + ḃ(t) (C(r)e0 + S(r)er )
K 1
, for K > 0
, for K = 0
, for K < 0
where
e0 = (1, 0, 0, 0, 0)
e1 = (0, 1, 0, 0, 0)
er = (0, 0, sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ))
62
Appendix B. Calculation of the induced potential
From this we find that
α400 = −
ä
ḃ
α411 = α422 = α433 = −
ḃ
a
with ḃ = ȧ for K = 0. This gives us that the induced potential is
1
ä
K + ȧ2
ä2
Vind =
6 +3
−
4
a
a2
K + ȧ2
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Part II
Scientific papers
67
68
Paper A
Edwin Langmann and Martin Sundin
Extrinsic curvature effects in brane-world scenario
arXiv: 1103.3230
Paper B
Jonas de Woul, Douglas Lundholm, Jens Hoppe and Martin Sundin
A dynamical symmetry for supermembranes
arXiv:1004.0266, accepted for publication in the Journal of High
Energy Physics
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