Time Gauge Fixing for 3d Loop Quantum Gravity
Abdulmajid Osumanu
10240100
THIS THESIS IS SUBMITTED TO THE UNIVERSITY OF GHANA,
LEGON IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE AWARD OF MPHIL MATHEMATICAL PHYSICS DEGREE
July, 2015
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DECLARATION
This thesis was written in the Department of Mathematics, University of Ghana,
Legon and Applied Mathematics Department, University of Waterloo, Canada
from September 2014 to July 2015 in partial fulfilment of the requirement for
the award of Master of Philosophy degree in Mathematical Physics under the
supervision of Dr. Prince Koree Osei of the University of Ghana, Legon and Dr.
Ma¨ıte´ Dupuis of the University of Waterloo, Canada.
I hereby declare that except where due acknowledgement is made, this work has
never been presented wholly or in part for the award of a degree at the University
of Ghana or any other University.
Student: Abdulmajid Osumanu
Signature:...... ................................
Supervisor: Dr. Prince Koree Osei
Signature:..... .................................
Supervisor: Dr. Ma¨ıte´ Dupuis
Signature:...... ..................................
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ACKNOWLEDGEMENTS
First and foremost I give thanks to Allah Almighty for giving me the knowledge
and strength to write this thesis.
The next goes to my supervisor, Dr. Prince Koree Osei, who directed me into this
interesting topic and guided me through it all. I also thank him for his patience,
constructive criticisms and careful reading of the manuscript.
Many thanks goes to Dr. Maite Dupuis, my co-supervisor who choose this inter-
esting topic. Another big thanks goes to her for hosting me for four months at
the University of Waterloo, Canada, where part of these work was carried out.
I acknowledge the African Institute for Mathematical sciences, Ghana for sup-
porting me financially through this program. Finally, I want to acknowledge the
support of my colleagues Leonard, Justice, Kinsley, Thomas, Christian, Micheal
and all staff of the Department of Mathematics, University of Ghana, Legon.
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DEDICATION
This work is dedicated to my father, Mr. Awudu Osumanu and my mother, Ms.
Emefa Agidi.
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Abstract
In this work, we review 3-dimensional gravity by canonically analyzing the Hilbert-
Palatini action. We apply a time gauge used in 4-dimensional loop quantum grav-
ity [14] to the simpler case of the 3-dimensional loop quantum gravity. By time
gauge fixing this Hilbert-Palatini action it leads to the Gauss constraints, the spa-
tial diffeomorphism constraints, the Hamiltonian constraint and a new constraint
C. The gauge symmetries generated by this new constraints are spacetime diffeo-
morphism and SO(2) gauge transformations. We solve the dynamics of the theory
by providing a regularization of the generalized projector operator in terms of the
Hamiltonian constraint. We provide the definition of the physical scalar product
which can be represented in terms of a sum over finite spinfoam amplitudes. Then
we establish a clear-cut link between the canonical quantization of the new theory
to the spinfoam model (Ponzano-Regge model) defined in terms of the SU(2) spin
networks.
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Contents
Declaration i
Acknowledgements ii
Dedication iii
Abstract iv
1 Introduction and Motivation 1
1.1 Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Loop Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Loops and Spin Networks . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Motivation and Goal of Thesis . . . . . . . . . . . . . . . . . . . . . 6
1.5 Organization of Work . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . . 8
2 Canonical Formulation of 3d Gravity 10
2.1 General Theory on Canonical Formalism . . . . . . . . . . . . . . . 11
2.2 Canonical Analysis of 3d Hilbert-Palatini Action . . . . . . . . . . . 13
2.3 Canonical Formalism and Analysis in Time Gauge . . . . . . . . . . 18
2.4 Time Evolution of Constraints . . . . . . . . . . . . . . . . . . . . . 22
2.5 Algebra of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Classical Representation of Phase Space Variables 34
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3.1 Holonomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Kinematical Hilbert Space 42
4.1 The quantization of edges(Quantum Operators) . . . . . . . . . . . 42
4.2 Cylindrical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Representation of Cylindrical Functions (Inner Product) . . . . . . 46
4.4 Orthonormal Basis of Hkin . . . . . . . . . . . . . . . . . . . . . . . 47
5 Dynamics 50
5.1 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 Conclusion and Outlook 55
Bibliography 59
A ADM Decomposition of 2 + 1 Gravity 60
B Representation Theory 64
B.1 Group Representation . . . . . . . . . . . . . . . . . . . . . . . . . 64
B.2 Peter-Weyl Decomposition Theorem . . . . . . . . . . . . . . . . . . 64
B.3 SU(2) Representation . . . . . . . . . . . . . . . . . . . . . . . . . 65
C Lie Derivative 67
C.1 Lie derivative of a function . . . . . . . . . . . . . . . . . . . . . . . 67
C.2 Lie derivative of a vector field . . . . . . . . . . . . . . . . . . . . . 68
D SO(2) Haar Measure and Useful Identities 70
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Chapter 1
Introduction and Motivation
1.1 Quantum Gravity
The task of quantizing gravity is one of the outstanding problems of modern
theoretical physics. Attempts to reconcile quantum theory and general relativity
(GR) date back to the early part of the twentieth century. Several decades of
hard work have yielded an abundance of insights into quantum field theory. But
despite this enormous effort, no one has yet succeeded in formulating a complete,
self-consistent quantum theory of gravity [5].
Einstein’s description of gravity as a curvature of spacetime (geometry of curved
spacetime) was seen as a game changer at the time, as this idea totally moved away
from the notion of gravity as a force. The consequence of this was the formulation
of the theory of GR which gave a deep insight into the geometric structure of the
universe at the macroscopic level.
Quantum mechanics (QM), which describes the microscopic world was formulated
based on the laws of probability. It allows for a discrete energy spectrum instead of
a continuous radiation. QM uses the time evolution of quantum states described by
the Schrodinger equation in which the Hamiltonian generates the time evolution.
This means that physical quantities can change only in discrete amounts and not
in a continuous way [27].
Our understanding of physics, and in general nature have been greatly influenced
by QM and GR. Both principles have revolutionized the pivotal structures of New-
tonian physics. The notion of absolute time and space ceases to exist, but rather
space and time and distances between points of the spacetime manifold, that is,
the metric themselves become dynamical objects [27, 28]. Further, the determin-
istic nature of Newton’s equations of motion evaporates at a fundamental level,
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rather dynamics is reigned by probabilities underlying the Heisenberg uncertainty
principle.
A usual phenomenon in fundamental physics is to reconcile two different concep-
tual theories to find a new synthesis, a satisfactory physical theory must therefore
combine both of these fundamental principles, QM and GR, in a consistent way
and which will be called quantum gravity (QG). However, despite the success sto-
ries of both QM and GR, the quantization of the gravitational field has turned
out to be one of the most challenging unsolved problems in theoretical and math-
ematical physics.
The reason for this incompatibility is that, GR requires spacetime to be a dy-
namical field, describing the notion of force as a geometry and also requires an
independent background structure. On the other hand, QM requires a dynamical
field to be quantized, and in addition requires a non-dynamical background space-
time (Hilbert space) for it to take place. In GR, spacetime is dynamical, while
in QM, any dynamical entity is made up of quanta and can be in probabilistic
superposition states.
Not only are there conceptual differences in quantizing gravity, there are techni-
cal obstacles also facing the new synthesis. Einstein’s equations which governs
the theory of gravity are highly non-linear, and hence very difficult to solve. In
addition, conventional field quantization methods which are based on the weak-
field perturbation expansion fail in their application to GR since they yield a
non-renormalizable theory. GR, hence must be replaced by a more fundamental
description that appropriately includes the quantum degrees of freedom of gravity
[24].
However, it is clear that while GR has given a very consistent framework for doing
classical physics, it has not yet been successful in bringing the quantum realm
in its wake [21]. Attempts have been made for decades to formulate a consistent
theory of quantum gravity, but with no success so far.
1.2 Loop Quantum Gravity
The past decades have seen an increase in active research in the field of quantum
theory of gravity, with some claiming to have found a quantum theory of gravity.
Notably among them is string theory, which is based on the physical hypothesis
that elementary objects are extended rather than particle-like [27]. A very rich
unified theory which includes Yang-Mills fields and gravitons, and free of ultravio-
let divergences are a few of string theory’s successful story. However, this comes at
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a cost of additional physics such as supersymmetry, extra dimensions, an infinite
number of fields with an arbitrary masses and spins.
In light of this another active area of research in QG is Loop Quantum Gravity
(LQG) which takes a different approach to string theory. LQG is a canonical
and background independent base approach to the quantization of gravity paying
special attention to the conceptual lessons of GR [9, 23, 27, 28]. It proposes that
spacetime is actually divided into small chucks, when you zoom out it appears
to be a smooth sheet, but when you zoom in, it is a bunch of dots connected by
lines or loops. These small fibres, which are woven together offer an explanation
of gravity.
LQG is based on a Hamiltonian formulation of GR at the classical level. In this
formulation, spacetime is foliated into a family of hypersurfaces of constant time
with coordinates on each slice. LQG is a straightforward quantization procedure
of GR with the physical inputs of the theory been the well-tested physical theories
of QM and GR, there are no major additional physical hypothesis or assumption
made in the construction of the theory [27]. All it tries to do is to merge the
conceptual ideas of GR into QM. In addition, LQG is based on assumption that
even if Einstein’s equations are modified at high energy, the general-relativistic
notion of space and time are assumed to be true.
The theory is formulated in the more physical four dimensional case. It was
initially formulated in terms of the metric variables which was due to Arnowitt,
Deser and Misner [28], hence the name ADM formulation. In this formulation, the
canonical variables are the metric tensor of three dimensional spatial slices qab(x, t)
(here a, b are 3d space indices) and it’s conjugate momentum piab [23, 28] following
an ADM decomposition of the action. The decomposition results in the spatial
diffeomorphism V (qab, piab) and Hamiltonian constraints S(qab, piab), however the
non-linear dependence on the canonical variables especially by the Hamiltonian
constraint makes it mathematically difficult to promote the constraints into quan-
tum operators.
To address the problem of promoting the constraints into operators, a tetrad
formulation was introduced. The tetrad is defined as eI = eIµdx
µ such that gµν =
eIµe
J
νηIJ (µ, ν are 4d spacetime tangent indices and I, J are 4d internal indices),
and ηIJ is the Minkowski metric. The extrinsic curvature
(
Kjb :=
1√
detE
KabEbjδ
ij
)
and the densitized triad
(
Eai :=
1
2ijk
abcejbe
k
c
)
become the new canonical variables
for general relativity after a 3+1 decomposition of spacetime.
In this new formulation, additional degrees of freedom are introduced due to the
redundancy exhibited by the new variables: this occurs as there are nine Eai
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to describe the six components of qab. As a result an additional constraints-
Gauss constraint- in terms of the new variables makes this redundancy manifest.
However, the problems similar to using the ADM formulation arise when one tries
to quantize the theory.
In the pursuit of solving the non-polynomial associated with the Hamiltonian
constraint, Ashtekar introduced a new configuration variable (Aia) that behaves
as a complex SU(2) connection. The three constraints now become polynomials
in the Ashtekar variables i.e the Ashtekar connection Aia and the densitized triad
Eai . This simplification open up ways to quantize the constraint.
However, despite this simplification there are still some issues in properly defining
the constraints and solving them. This comes from the fact that the Ashtekar
connection is defined as a complex variable, hence in the quantization of the theory
it is difficult to ensure one recovers real GR as opposed to complex GR, unless
the reality condition is imposed [27]. In addition, there are difficulties of a rather
technical nature related to the complete characterization of dynamics and the
quantization of the Hamiltonian constraint [23].
Despite these challenges, LQG has been able to predict the discreteness of geo-
metrical operators such as area and volume. This discreteness becomes important
at the Planck scale while the geometric operators crowds very rapidly at low en-
ergy scale. This property of the geometric operators is consistent with the smooth
spacetime picture of classical GR [23].
In carrying out canonical quantization, the guiding principle is mostly that of
Dirac. The first is to find a representation of the phase space variables as operators,
acting on some kinematical Hilbert space Hkin. This representation of the phase
space variables should map Poisson brackets into commutators. Next is to define
the constraints into self-adjoint operators in Hkin, and finally the solutions of
these constraints are characterize to define an inner product leading to a physical
Hilbert space Hphy.
The Poisson bracket between the Ashtekar-Barbero connection and the densitized
triad resulted in a delta function (distribution), this leads to a not well-defined
function. A regularization of the Poisson bracket is needed to address this problem,
and this is done by smearing the phase space variables to obtain the holonomy-flux
algebra. The regularization open up ways to quantize these constraints, and such
they (constraints) become operators on a kinematical Hilbert space. For more
on loop quantum gravity the reader is advice to read [23, 27, 28] and references
therein.
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1.3 Loops and Spin Networks
The choice of different algebra of basic field functions to promote to quantum
operators makes LQG stands out among other different quantization techniques.
That choice is a non-canonical algebra based on the holonomies of the gravitational
connection [23, 27].
The holonomy in LQG becomes a quantum operator creating ”loop states”1. In a
background independent theory like LQG, the position of a loop state is relevant
only with respect to other loops, and not with respect to the background [27].
The space spanned by these loop states are not too singular nor too many, thereby
providing a basis for the Hilbert space of LQG. A finite linear combination of some
of these loop states form a well defined orthonormal basis in the Hilbert space of
a lattice Yang-Mills theory. These orthonormal basis are the spin networks states
which eventually form the basis of the mathematical structure of LQG.
As stated earlier, one will like to know what the quanta of space (quantum prop-
erties of space) are, and how to describe them, in the pursuit of a quantum theory
of spacetime. Interactions of physical systems with other systems may occur in
”quanta” and these quanta are determined by the spectral properties of the op-
erators representing the quantities involved in the interaction with the systems.
The interactions with the gravitational field (spacetime) are through geometric
structures of the physical space resulting in measurements of length, area and
volume.
The spectrum of both volume and area turns out to be discrete. From the spectral
of the volume, the quanta of space can be thought of as grains of space. These
grains of space lie adjacent to each, where in this case adjacency means being
contiguous, or being in touch, or being nearby. If one consider a quantum state
of space formed by P grains of space, where some are in contact with one another
[27].
Then representing this state as an abstract graph Γ with P nodes: these nodes of
the graph represent the grains of the space; the links of the graph connect adjacent
grains and represent the surfaces separating two adjacent grains. The quantum
state is then characterized by the graph Γ, with the labels on the nodes and on
the links, and a graph with these labels is called a spin network.
LQG has been able to predict the discretness of the spectrum of geometrical op-
erators like area and volume. The formulation of LQG in terms of the Ashtekar-
1In the loop representation formulation of Maxwell theory, a loop state is a state in which
the electric field vanishes everywhere except along a single Faraday line
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Barbero variables result in the Gauss constraints, the spatial diffeomorphism con-
straints and the Hamiltonian constraint. Defining cylindrical functional as quan-
tum states of the Ashtekar connection, spin network states establish themselves
as solutions to both Gauss and spatial diffeomorphism constraints. Unfortunately,
this can’t be said about the Hamiltonian constraint.
1.4 Motivation and Goal of Thesis
General relativity in three dimensions is said to be ”simple” but not trivial mak-
ing it an interesting, and as such a large area of research providing an insight
into quantum gravity for the more physical four dimension. Physically three di-
mensional gravity is a topological field theory, meaning it has no degrees of [5]
freedom2 and its manifold is regarded as a two surface with time.
Conceptually both (2 + 1)-dimensional gravity and (3 + 1)-dimensional gravity
are equivalent and many of the fundamental issues carry over to quantum gravity
at the lower dimensional setting. Although (2+1)-dimensional model is much
simpler, mathematically and physically, and we can write down a quantum theory
for the model.
There are however few instances were (2+1)-dimensional solutions are quite dif-
ferent physically from those of (3+1)-dimensional, and in addition the (2+1)-
dimensional model does not help in understanding the dynamics of quantum grav-
ity. Based on the analysis of conceptual problems such as nature of time, the role
of topology and topology change, the relationship among different approaches to
quantization, the model has really proven highly instructive [5].
Canonical analysis of (2+1)-dimensional gravity [3] result in the flatness and Gauss
constraints, defining a normal to the hypersurface x0 =const, the Hamiltonian and
spatial diffeomorphism constraints are derived. These constraints generate equal
time hypersurface which can be deformed and pushed forward in various ways. A
geometric algebra reflected in the Poisson algebra of the constraints- known as the
Dirac’s hypersurface deformation algebra3 is satisfied by these deformations of the
hypersurface. However a quantum representation of this algebra (see [3] and other
references therein for more details) has not been reached yet even in the simple
case of three dimensional gravity.
2There are no gravitational waves in the classical theory, and no propagating gravitons in the
quantum theory.
3The Dirac algebra is universal, i.e. it is the same algebra for any theory of hypersurfaces
embedded in a higher dimensional manifold.
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In the 4d decomposition of spacetime, the Ashtekar connection transforms under
the local SU(2) or SO(3) transformation with the action being Lorentz invariant.
Since three dimensional gravity is simple, one will like to explore if it is possible
to time gauge just like in four dimensional gravity, and if successful what can we
learn from it with respect to the 4d case. Symmetry plays a crucial role in any
gauge theory, and as such performing a time gauge fixing in three dimensional
gravity, one expect a break in the original symmetry of the theory.
Description of a quantum space is an issue in quantum gravity, in (3+1)-dimensional
LQG and as well in (2 + 1)-dimensional LQG. The quantum space in both cases
are describe in terms of SU(2) spin networks, the question then is, if the 2+1 time
gauge fixing of the Palatini action is invariant under SO(2), then can we define
a quantum space of this theory in terms of spin networks invariant under SO(2)
transformation.
Addressing some of the questions raised above, and also understanding the physics
that can be studied as a result of the time gauge fixing in (2+1)-dimensional LQG.
In addition, how does the time gauge contribute meaningfully in understanding
loop quantum gravity, in general motivated this research.
1.5 Organization of Work
We have tried to make this work self-contained and to present all materials needed
for understanding the foundations and the theoretical framework of loop quantum
gravity and in particular the notion of time gauge fixing in (2 + 1)-dimensional
loop quantum gravity. Furthermore, we have tried to understand the dynamics of
the time gauge action in canonical quantization.
In chapter 2 we discuss the canonical formulation of 3d gravity. We begin with a
review on the main ideas of the general theory on canonical formalism. We then
discuss the canonical analysis of three dimensional Hilbert-Palatini action as a
example of the canonical formalism. A time gauge is introduced and a canonical
analysis of the Hilbert-Palatini action is carried out in terms of the time gauge.
Subsequent discussions on the time evolution and algebra of constraints are carried
out.
In chapter 3 we discuss the classical representation of the phase space variables.
To this the holonomies and the flux are introduced which are seen as the basic set
phase space function for quantization. Then the Poisson bracket between these
phase space variables are computed to give a much simpler expression (algebra)
thereby making quantization of these variables feasible.
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Chapter 4 looks at the kinematical Hilbert space needed for quantization of our
theory. Cylindrical functions as basic observables for quantization are discussed, in
addition an inner product of these cylindrical functions are defined leading to the
Cauchy completion of the space of the cylindrical functions. Then orthonormal
basis of the kinematical Hilbert space are introduced leading to an SO(2) spin
networks.
In chapter 5 we discuss the dynamics of the theory in terms of spin foam rep-
resentation. We begin by introducing a projector operator and then define a
regularization of the projector operator. Then the matrix element of this pro-
jector operator is constructed. This matrix element is seen as the start of the
quantization procedure. Finally, a spin foam representation of the matrix element
is carried out.
In chapter 6 a summary of the results of the work is given.
1.6 Conventions and Notations
The notations and conventions mentioned here will be used throughout this work
unless otherwise stated. Upper case Latin indices from the middle of the alphabet
I, J, .... = 0, 1, 2 are su(2) indices. Greek indices from the middle of the alphabet
µ, ν, ρ... = 0, 1, 2 are 3d spacetime indices. Lower case Latin indices from the
of the beginning alphabet a, b, .... = 1, 2 are 2d space indices. Lower case Latin
indices from the middle of the alphabet i, j, k, ... = 1, 2 are 2d internal indices. In
the case of a 2 + 1 decomposition of spacetime, timelike indices will be labeled
“t = 0” in the tangent space and “0” in the internal space. Throughout this work,
we use Einstein’s summation convention and, raise and lower indices with three
dimensional euclidean metric δIJ = dig(1, 1, 1).
The su(2) generators are denoted by TI , and in terms of these generators the Lie
bracket is
[TI , TJ ] = IJKT
K , (1.1)
where IJK denotes the fully antisymmetric tensor in three dimensions defined:
IJK =



1 for even permutation of I, J,K
−1 for odd permutation of I, J,K
0 otherwise.
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with the conventions 012 = 012 = 1.
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Chapter 2
Canonical Formulation of 3d
Gravity
In (3 + 1)-dimensional LQG with gauge group G = SL(2,C), working with a time
gauge normally breaks G into an SU(2) maximal compact subgroup. In this case
the canonical analysis simplifies and the phase space is parametrized by an su(2)-
valued connection [6, 14]. A time gauge describes the deviation of the normal to
the spacelike hypersurface t = 0 from the time direction. In [14], the time gauge
χ = (1, 0, 0, 0) was introduced in (3+1)-dimensional LQG for the parametrization
of the phase space.
We will carry out a similar construction for three dimensional gravity in this chap-
ter. Using this time gauge, we obtain at the end of the canonical analysis, a phase
space (for the Euclidean signature with Λ = 0, two copies of three-dimensional Eu-
clidean space as hyperplanes in four-dimensional Euclidean space [5]) parametrized
by a gauge connection and it’s conjugate momentum, as has been shown in the
(3 + 1)-dimensional situation [14].
This chapter is devoted to the canonical formulation of gravity, in particular,
Einstein’s theory of gravity in three dimensions (two space, one time). We shall
begin with a brief description of canonical formulation of gravity of an arbitrary
n-dimensional spacetime and later reduce our discussion to the 3-dimensional case.
A description of a three dimensional gravity’s phase space will be looked at.
Introducing a new time gauge for three dimensional gravity, a decomposition of the
triad field via the time gauge will then be carried out. Using this decomposition
of the triad, we will apply it to the (2 + 1)-dimensional Hilbert-Palatini action.
We look at the new gauge group resulting from the time gauge and the algebra
of constraints is computed. Some of the concepts (especially orthonormal frame)
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discuss in this chapter are extensively used in [16, 22].
2.1 General Theory on Canonical Formalism
A dynamical system in classical mechanics consists of a phase space1Γ which is
a Poisson (usually symplectic) manifold and a function Hcl : Γ −→ R called
the (classical) Hamiltonian of energy function. For example, for a single particle
moving along a real line, Γ is the cotangent bundle T (R)∗.
In quantum machanics, the phase space Γ is replaced by the set of rays in a
complex Hilbert space H, and the space F(Γ) of functions on Γ by the algebra
Op(H) of (not necessarily bounded) operators on H. For example, in the case of
a single particle moving along the real line, H is the space of square-integrable
functions of x ∈ R.
In section 1.4, three dimensional gravity was discussed as a topological theory
with no local degrees of freedom. This feature makes the description of spacetime
as a collection of patches (contractible open set of spacetime manifold), solutions
of the Einstein equations are isometric to a model spacetime which is determined
by the signature and the sign (or vanishing) of the cosmological constant.
For example, in Lorentzian signature with vanishing cosmological constant, the
model spacetime is Minkowski space while in the Euclidean signature and positive
cosmological constant it is the three-sphere, and so on [5]. This picture leads to
the description of (2 + 1)-dimensional gravity in terms of “geometric structures.”
Depending on the topology of the manifoldM, these local solutions may be glued
together to obtain non-trivial global solutions of the Einstein equations. This
is mathematically executed by the isometry group of the local model spacetime
(the 3d Poincare group ISO(2, 1) or the orthogonal group SO(4) in the examples
mentioned above). These groups are the local isometries of the model spacetime.
Matter in the form of point particles can be described geometrically in terms
of defects of the model spacetime. The space of all these geometric structures,
suitably defined, constitutes the phase space of 3d gravity. In addition, the phase
space provides us with a symplectic form (Poisson brackets) which is needed for
canonical quantization.
Canonical formulation plays a big role in the quantization of gravity. In chapter 1,
we discussed about quantum gravity precisely loop quantum gravity and the chal-
lenges in quantizing gravity. Some of the conceptual challenges like the meaning
1Where solutions of a system lie within a manifold
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of observables and what does unitary mean for a closed universe are more notice-
able in a canonical analysis. In carrying out a canonical analysis the meaning of
quantum gravity state becomes more meaningful.
We now describe in general a canonical formulation of an n-dimensional system
and subsequently it quantization. Consider an n-dimensional system where in a
canonical analysis the action is written as
S =
∫
L(qa, q˙b)dt =
∫
(paq˙b −H(qa, pb)) dt, (2.1)
where the momenta pa(t) are defined as
pa =
δS
δq˙b
=
∂L
∂q˙b
,
where the basic Poisson bracket is
{qa, pb} = δab. (2.2)
For general phase space functions f, g this gives
{f, g} =
∂f
∂qa
∂g
∂pb
−
∂g
∂qa
∂f
∂pb
. (2.3)
In a canonical analysis, writing a first order action in the form of the second term
in (2.1) one obtains the Hamiltonian. The Hamiltonian defines via its flow the
time evolution of the system. By varying the action (2.1) with respect to p and q
one obtains respectively the following equations of motions
q˙ =
∂H
∂p
= {q,H} , p˙ = −
∂H
∂q
= {p,H} . (2.4)
For an arbitrary phase space function, the time evolution is defined by the time
evolution of its basic variables thus
f˙ =
∂f
∂q
q˙ +
∂f
∂p
p˙, hence we have f˙ = {f,H}. (2.5)
The Hamiltonian H (or any other phase space function) generates a flow in a
parameter t in the sense that for any phase space function g we can find a one-
parameter solution g(t) which satisfies
d
dt
g(t) = {g,H}(t). (2.6)
Hence the Poisson bracket {g,H} describes the infinitesimal change under a trans-
12
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formation generated by H or any arbitrary phase space function. However the
notion of flow generalizes by replacing the Hamiltonian H by any other phase
space function F . The Poisson bracket {g, F} gives the infinitesimal change of a
phase space function g induced by the flow of F .
In the canonical analysis described above, the notion of phase space is provided
which is subsequently quantized [10]. This phase space parametrizes the space of
solutions with the help of initial data, usually the basic fields and its first time
derivatives (momenta). By picking an initial data, one would have to choose an
equal time hypersurface2. Solutions can then be reconstructed by solving the field
equations for the given initial data. One way then to solve this field equations is
to use the Hamiltonian to generate time evolution, i.e. the change of initial data
under the change of the equal time hypersurface. A particular hypersurface is
selected either by putting a restriction [25] on the coordinates Φ(xα), or by giving
parametrize equations of the form xα = xα(yi), where yi(i = 1, 2) are coordinates
intrinsic to the hypersurfaces.
2.2 Canonical Analysis of 3d Hilbert-Palatini Ac-
tion
Given the Einstein-Hilbert action in three dimension the basic variable is the
metric gµν which encodes the geometry of the three dimensional manifold M.
The metric could be define in terms of a field of n vectors, i.e. orthonormal frame
[22]. The orthonormal frame now contains the geometry of spacetime.
We say that a frame
{
eI
}
, {eI} is orthonormal if in this frame
g(eI , eJ) = eIµe
J
νg
µν = δIJ (2.7)
for all p ∈M.
Orthonormal frames always exist since at each point p ∈ M, we may choose for
example eI = dxI , where dxI are canonical orthonormal basis at the centre of a
geodesic coordinate system. For a given spacetime, (M, gµν), any orthonormal
frame yields a new orthonormal frame by transforming the basis through
e′I(x) = L(x)IJ e
J(x). (2.8)
2A hypersurface in an n-dimensional spacetime manifold is an (n−1)-dimensional submanifold
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If the linear maps L(x) preserve the orthonormality:
δIJ e
′I ⊗ e′J = δIJ e
I ⊗ eJ ,
i.e. if
LKIL
P
JδKP = δIJ ,
which is the defining equation for the Galilean transformations. Thus orthonormal
frames are unique up to Local transformations.
We now specify spacetime, through the data
(
M,
{
eI
})
which is equivalent to
(M, gµν). These are equivalent because for a differentiable manifold with atlas of
charts, if we specify eI , for example through functions LJI , so that
eI(x) = LIµ(x)dx
µ, (2.9)
then
g = δIJ e
I ⊗ eJ = δIJL
I
µL
J
νdx
µ ⊗ dxν , (2.10)
which implies that
{
eI(x)
}
indeed determines gµν
gµν = δIJL
I
µ(x)L
J
ν(x). (2.11)
The theory of interest to us is three dimensional Euclidean gravity in first order
formalism. The spacetime M is a three dimensional oriented smooth manifold.
The variable e is the triad, i.e a Lie algebra g valued 1-form overM transforming
through the adjoint representation, F [A] is the curvature of the three dimensional
spin connection A. In this case the triad and the spin connection are both treated
as dynamical variables and therefore varied independently.
For simplicity, we will consider euclidean gravity, hence the spin connection A
is defined on a principal SU(2)-bundle over M and the Lie algebra g = su(2).
Assume the spacetime topology to beM = Σ×R, where Σ is a Riemann surface
with arbitrary genus. In addition, we choose a slicing of the spacetime manifold
into equal time hypersurface. The materials used in this section can be viewed
from [4, 5, 19].
The triad and the spin connection are given as
eI = eIµdx
µ, AI = 12
IJKAµJKdxµ. (2.12)
The curvature of A is a 2-form given by FI = FIµνdxµ ∧ dxν . In component form
14
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the curvature is:
F Iµν = ∂µA
I
ν − ∂νA
I
µ + 
I
JKA
J
µA
K
ν . (2.13)
The torsion of the spin connection A with the triad e is defined by
T Iµν = ∂µe
I
ν − ∂νe
I
µ + 
IJKAµJeνK − 
IJKAνJeµK . (2.14)
Hence the Hilbert-Palatini action is
S[e, A] =
1
2
∫
M
eI ∧ FI [A]. (2.15)
By varying the action (2.15) with respect to the triad and the spin connection the
equations of motion obtained are
F = 0
dAe = de+ [A, e] = 0 (2.16)
the flatness condition and the torsion free condition respectively.
The canonical analysis reproduces the covariant picture [4], while preparing the
road to quantization. To see how this happens, we foliate the spacetime manifold
into a family of hypersurfaces at constant time t, denoted Σt. Without loss of
generality, we will denote coordinates now by t = x0, x1, x2 where x0 =const.
defines the equal time hypersurface.
Using the component form of the triad (2.12) and the curvature (2.13), the action
(2.15) takes the form:
S =
1
2
∫
eIµFIσρdx
µ ∧ dxσ ∧ dxρ,
=
1
2
∫
eIµFIσρ
µσρd3x. (2.17)
In the second line of (2.17), we made use of the fact that µσρd3x = dxµ∧dxσ∧dxρ,
where µσρ is the three dimensional anti-symmetric tensor.
Splitting only the spacetime indices in (2.17), we have
S =
1
2
∫
[
eI0FIab
0ab + eIaFI0b
a0b + eIaFIb0
ab0
]
dtd2x
=
1
2
∫
[
eI0FIab
ab − eIaFI0b
ab − eIaFI0b
ab
]
dtd2x, (2.18)
where we have used the fact that 0ab = ab0 = ab, a0b = −ab0 = −ab, and
15
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FIb0 = −FI0b. Here ab is the two dimensional anti-symmetric tensor. From (2.18)
we have
S =
1
2
∫
[
eI0FIab
ab − 2eIaFI0b
ab
]
dtd2x. (2.19)
Considering the second term in (2.19), we have
−eIaFI0b
ab = −eIa
[
∂0AIb − ∂bAI0 + 
JK
I A0JAbK
]
ab
= −ab
[
eIa∂0AIb − e
I
a∂bAI0 + e
J
a
I
JKAI0A
K
b
]
. (2.20)
Substituting (2.20) into (2.19), we obtain
S =
1
2
∫
[
abeI0FIab − 2
ab
[
eIa∂0AIb − e
I
a∂bAI0 + e
J
a
I
JKAI0A
K
b
]]
dtd2x
=
1
2
∫
[
2baeIa∂0AIb + 
abeI0FIab + 2AI0
ba
(
∂be
I
a + 
I
JKA
K
b e
J
a
)]
dtd2x
=
∫ [
baeIa∂0AIb +
1
2
abeI0FIab + AI0
baDbe
I
a
]
dtd2x, (2.21)
where DbeIa = ∂be
I
a + 
I
JKA
K
b e
J
a is the covariant derivative of the triad.
The canonical pairs of the spacetime decomposition are then formed by the com-
ponents of the connection AIb 3 and their momenta:
EbI :=
δS
δ(∂0AIb)
= baeIa. (2.22)
The the Poisson bracket between the pair is
{AJa (x), E
b
I} = δ
b
aδ
J
I δ
2(x− y), (2.23)
the other Poisson brackets involving the canonical variables vanishes
{AJa , A
K
b } = 0, {E
a
I , E
b
J} = 0.
The components eI0 and AI0 in (2.21) appear without time derivatives and hence
effectively act as Lagrange multipliers. Varying (2.21) with respect to these com-
ponents leads to the following associated equations of motion
GI := abDae
I
b = ∂aE
aI + IJKA
J
aE
aK = 0
F I :=
1
2
abF Iab = 
ab(∂aA
I
b +
1
2
IJKA
J
aA
K
b ) = 0. (2.24)
Here, F Iab is obtained from (2.13). Using the torsion (2.14), then the first term in
3This is the spacial part of the spin connection
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(2.24) is given by
GI :=
1
2
abT Iab = 
abDae
I
b . (2.25)
The associated equations of motions (2.24) constitute constraints on phase space
points as they involve only the spin connection AJb and the conjugate momenta E
a
I .
Hence only phase points in the constraint hypersurface described by (2.24) can
lead to a solution of the equations of motion. The appearance of these constraints
define a constraint hypersurface in the phase space and they represent some subset
of the equations of motion. In addition, a general feature of gauge theories: is
the constraints generate gauge symmetries. The constraints GI (Gauss constraint)
generates infinitesimal SU(2) gauge transformations, and FI (flatness constraint),
constrains the connection to be flat and generates translation symmetries.
Smearing the constraints in (2.24) with appropriate scalars, these constraints
which are first class form the algebra [3]:
{G[α],G[α′]} = G [[α, α′]]
{F [N ],G[α]} = F [[N,α]]
{F [N ],F [M ]} = 0, (2.26)
where in this case G[α] =
∫
d2x αIGI and F [N ] =
∫
d2x N IFI are the smeared
Gauss constraint and the smeared flatness constraint respectively. The algebra in
equation (2.26) is the semi direct product of rotation and translation i.e. ISU(2).
In a gauge theory, the momentum and connection variables of the action are
redundant: they describe both physical excitations and unobservable ’pure gauge’
degrees of freedom that merely represent coordinate changes [5]. Gauge equivalent
phase space points are defined by gauge equivalent solutions. Thus the set of
physically distinct solutions is given by the set of gauge orbits4 in the constraint
hypersurface. This reduced phase is then given by the constraint hypersurface
factored by the gauge transformation.
Alternatively, one can attempt to define this reduced phase space by finding a
gauge fixing conditions that would describe a hypersurface that intersect every
gauge orbit [10] in the constraint hypersurface in exactly one point. The reduced
phase space would then be given by the phase points satisfying the gauge condi-
tions and the set of constraint.
In the next section, we will apply a time gauge to the action (2.15), this time
gauge only affects the internal indices. The time gauge breaks the three dimen-
4These are sets whose points can be related to each other by gauge transformation.
17
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sional internal rotation to a two dimensional internal rotation, hence our three
dimensional phase space 5 will now be parametrized by a two dimensional spin
connection and a diad. In this circumstances, the gauge symmetries will be space-
time diffeomorphism and an SO(2) gauge transformation.
2.3 Canonical Formalism and Analysis in Time
Gauge
In general, canonical analysis in LQG leads to a Gauss, spatial diffeomorphism
and Hamiltonian constraints. In three dimensional LQG, the canonical analysis
yields the flatness and the Gauss constraints, the algebra between these constraints
is the semi-direct product of rotation and translation. An adhoc procedure is
introduced to get the spatial diffeomorphism and Hamiltonian constraints [4, 10]
in three dimensional LQG.
By time gauge fixing the Hilbert-Palatini action, one hope that the three con-
straints (Gauss, spatial diffeomorphism and Hamiltonian) of LQG emerge out of
the theory, without having to carry out an adhoc procedure to get the spatial
diffeomorphism and the Hamiltonian constraints.
In this section, we introduce a time gauge which we will use in the decomposition
of the triad into its various components. We will then use this decomposition
(components of the triad) to carry out a canonical analysis of the Hilbert-Palatini
action (2.15). It is necessary that we split both the spacetime and the internal
indices into their various components.
Let us choose a gauge (which we will refer to as the time gauge) such that
δI = (1, 0, 0) := nµeIµ, (2.27)
where nµ is the unit normal to the hypersurface. It is convenient to parametrize
it as [18]:
nµ =
(
1
N
,−
Na
N
)
, (2.28)
where N i is the shift vector and N (normalization constant) is the lapse function.
The lapse function describes the proper distance (proper time) by which the hy-
persurface is deformed in the normal direction, the shift describes the amount of
5Two copies of three-dimensional Euclidean space as hyperplanes in four-dimensional Eu-
clidean space.
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the tangential deformation (in the coordinates of the hypersurface).
From (2.27), making use of the definition of nµ, we have
δI := nµeIµ
= n0eI0 + n
aeIa
=
1
N
(
eI0 −N
aeIa
)
=
1
N
(
e00 −N
ae0a, e
i
0 −N
aeia
)
. (2.29)
Comparing (2.27) and (2.29) we have for the first entries
1
N
(
e00 −N
ae0a
)
= 1
e00 −N
ae0a = N. (2.30)
For (2.30) to hold implies that e0a = 0 and e
0
0 = N .
Similarly for the second entries of (2.27) and (2.29), we have ei0 − N
aeia = 0 ⇒
ei0 = N
aeia. Hence the decomposition of the triad via the time gauge gives:
e0µ = (N, 0) , e
I
0 = (N,N
aeia). (2.31)
The conditions in (2.31) derived are just like those used in [14] for the (3 + 1)-
dimensional decomposition of Hilbert-Palatini action. With the choice of the gauge
(2.27) and (2.31), our ultimate interest is to describe the phase space of three
dimensional gravity, on which the gauge symmetries are spacetime diffeomorphism
and SO(2) gauge transformations.
We are now in a position to split the spacetime and internal indices of the three
dimensional Palatini action, after which we will apply the result in (2.31) to per-
form the canonical analysis. Ignoring the integral in (2.15) and carry out the
summation with respect to the internal indices, we have
eI ∧ FI = e
0 ∧ F0 + e
i ∧ Fi. (2.32)
From (2.32), we consider the first term on the right hand side and carrying out
the summation as follows:
e0 ∧ F0 = e
0
µdx
µ ∧ F0νρdx
ν ∧ dxρ
= e0µ
µνρF0νρ d
3x
=
[
0bce00F0bc + 
a0ce0aF00c
]
dtd2x
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We now substitute the decomposition of the triad (2.31) into (2.33):
e0 ∧ F0 = 
bce00F0bc dtd
2x = bcNF0bc dtd
2x. (2.33)
Carrying out the summation on the second term of (2.32) (right hand side), we
have
ei ∧ Fi = e
i
µdx
µ ∧ Fiνρdx
ν ∧ dxρ
= eiµFiνρ
µνρ d3x
=
[
ei0
0νρFiνρ + e
i
a
aνρFiνρ
]
d3x
=
[
ei0
0bcFibc + e
i
a
a0cFi0c + e
i
a
ab0Fib0
]
dtd2x
=
[
ei0
0bcFibc − 2e
i
a
ac0Fi0c
]
dtd2x
=
[
Naeia
bcFibc − 2e
i
a
ac0Fi0c
]
dtd2x. (2.34)
Considering the second term in (2.34) and writing the curvature in component
form, we have
−ac0eiaFi0c = −
ac0eia
[
∂0Aci − ∂cA0i + 
IJ
i A0IAcJ
]
= −ac0eia
[
∂0Aci − ∂cA0i + 
k0
i A0kAc0 + 
0j
i A00Acj
]
= caeia
[
∂0Aci − ∂cA0i + 
k0
i A0kAc0 + 
0j
i A00Acj
]
. (2.35)
From (2.33), (2.34) and (2.35), the action in (2.13) becomes
S =
1
2
∫
[
2caeia∂0Aci + 2
caeia
(
−∂cA0i + 
k0
i A0kAc0
)]
dtd2x
+
1
2
∫
[
2caeia
0j
i A00Acj +N
abF0ab + 
bcNaeiaFibc
]
dtd2x. (2.36)
Carrying out an integration by part on the second term on the right hand side of
equation (2.36), we finally get
S =
1
2
∫
[
2caeia∂0Aci + 2
caA0i
(
∂ce
i
a + 
i0
ke
k
aAc0
)
+
]
dtd2x
+
1
2
∫
[
2A00
caeia
0j
i Acj +N
abF0ab + 
bcNaeiaFibc
]
dtd2x
=
1
2
∫ [
2caeia∂0Aci + 2
caA0iD˜ce
i
a + 2A00
caeia
0j
i Acj
]
dtd2x
+
1
2
∫
[
NabF0ab + 
bcNaeiaFibc
]
dtd2x, (2.37)
where D˜ceia = ∂ce
i
a + 
i0
ke
k
aAc0 is the covariant derivative of the diad. F0ab and Fibc
are the curvatures of the spin connections A0a and Aib respectively, and these are
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given by
F0ab = ∂aA0b − ∂bA0a + 
IJ
0 AIaAJb
= ∂aA0b − ∂bA0a + 
ij
0 (AiaAjb − AjaAib)
:= Fab
Fibc = ∂bAic − ∂cAib + 
JK
i AJbAKc
= ∂bAic − ∂cAib + 
0k
i (A0bAkc − AkbA0c) .
By splitting both the spacetime and internal indices of the action (2.15), the new
canonical variables from (2.37) are eai (diad) and Aci. The other variables which
do not appear with time derivatives act effectively as Lagrange multipliers i.e A00
A0i, N and Na, thereby making certain equations of motions act as constraints.
Thus we find the momenta (the densitized diad with vector densities of weight
one)
Eci :=
δS
δ(∂0Aci)
= caeia (2.38)
which is conjugate to Aci. Just like in section 2.2, the Poisson brackets of the
canonically conjugated variables can be read off from equation (2.23) as
{Aja(x), E
b
i (y)} = δ
b
aδ
j
i δ
(2)(x, y), (2.39)
the other Poisson brackets vanish:
{Ajb, A
i
a} = 0 and {E
b
i , E
a
j } = 0.
Varying the action in (2.37) with respect to the three Lagrange multipliers, we
obtain the following associated equations of motion:
C0 = 0ji 
caeiaAcj = 0, G
i = acD˜ae
i
c = 0
Va = 
bceiaFibc = 0, H = 
abFab = 0. (2.40)
As equations of motion these do have an unusual feature, that is they do not
include any time derivatives. These equations constitute constraints on phase
space points as they involve only the canonical variables, that is only phase space
points in the constraint hypersurface describe by (2.40) lead to a solution of all the
equations of motion. The multipliers A00, A0i, Na and N enforce the constraints
C0, Gi (Gauss), Va (spatial diffeomorphism) and H (Hamiltonian) respectively. In
subsequent discussions we will see that these constraints generate the symmetries
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of the theory.
Combining these four constraints in (2.40), the total Hamiltonian is given as
Htot = −
∫
d2x
[
A00C
0 + Ai0G
i +NaVa +NH
]
. (2.41)
Hence the system is described by the canonical pair (the diad and the spin con-
nection) and the totally constrained Hamiltonian (2.41).
2.4 Time Evolution of Constraints
How do the dynamical variables change (infinitesimal) under the influence of the
constraints (2.40)? This is our topic of discussion in this section. These infinites-
imal changes of the dynamical variables are describe by the time evolutions of
these constraints. In describing the time evolutions of constraints we follow the
literature in [10].
To compute the infinitesimal changes generated by these constraints in equation
(2.40), it will be useful to smear them with appropriate scalars in order to deal
with the delta functions in the Poisson brackets between the canonical variables:
C[µ] =
∫
d2x µ0C
0, G[λ] =
∫
d2x λiGi
V [
−→
N ] =
∫
d2x NaVa, H[N ] =
∫
d2x NH. (2.42)
Computing these infinitesimal changes will help in understanding the physics and
in making physical predictions of the theory. These infinitesimal changes will show
the Gauss constraints are generators of internal transformations, while the spatial
diffeomorphism constraints are generators of diffeomorphism transformations.
Starting with the spatial diffeomorphism constraints, it’s infinitesimal changes (i.e
acting the spatial diffeomorphism constraints on the densitized diad and the spin
connection) are computed below:
δ−→NA
j
b(x) = {A
j
b,V [
−→
N ]}
=
{
Ajb,
∫
d2y NaEciF
i
ac
}
=
{
Ajb,
∫
d2y NaEci
(
∂aA
i
c − ∂cA
i
a + 
i
JKA
J
aA
K
c
)
}
=
{
Ajb,
∫
d2y NaEci
[
∂aA
i
c − ∂cA
i
a + 
i
k0
(
AkaA
0
c − A
0
aA
k
c
)]
}
.
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Applying the distributive property of Poisson bracket to get
δ−→NA
j
b(x) =
{
Ajb,
∫
d2y (NaEci )∂aA
i
c
}
−
{
Ajb,
∫
d2y (NaEci )∂cA
i
a
}
+
{
Ajb,
∫
d2y (NaEci 
i
k0)
[
AkaA
0
c − A
0
aA
k
c
]
}
.
Integrating the second term by part and using the product rule of Poisson bracket
on the third term to get
δ−→NA
j
b(x) =
∫
d2y Na∂aA
i
c
{
Ajb, E
c
i
}
+
∫
d2y Aia∂cN
a
{
Ajb, E
c
i
}
+
∫
d2y (Naik0)
{
Ajb, E
c
i
} [
AkaA
0
c − A
0
aA
k
c
]
.
Using (2.39) to evaluate the Poisson bracket between the canonical variables above,
we have
δ−→NA
j
b(x) =
∫
d2y Na(∂aA
i
c)δ
c
bδ
j
i δ
2(x− y) +
∫
d2y Aia(∂cN
a)δcbδ
j
i δ
2(x− y)
+
∫
d2y (Naik0)
[
AkaA
0
c − A
0
aA
k
c
]
δcbδ
j
i δ
2(x− y). (2.43)
Considering the third term in (2.43), we have
(Naik0)
[
AkaA
0
c − A
0
aA
k
c
]
δcbδ
j
i = (N
aik0)A
k
aA
0
cδ
c
bδ
j
i − (N
aik0)A
0
aA
k
cδ
c
bδ
j
i
= (Naik0)
[
AkaA
0
c − A
0
aA
k
c
]
δcbδ
j
i
= 0. (2.44)
Integrating the dirac function in two dimensions in the first two terms in (2.43),
we get
δ−→NA
j
b(x) =
∫
d2y Na(∂aA
j
b)δ
2(x− y) +
∫
d2y Aja(∂bN
a)δ2(x− y)
= Na∂aA
j
b + A
j
a∂bN
a
:= L−→NA
j
b. (2.45)
Where L−→NA
j
b is the Lie derivative of A
j
b in the direction of the field N
a.
Similarly for the action of the spatial diffeomorphism constraints on the densitized
diad we have
δ−→NE
a
i = {E
a
i ,V [
−→
N ]}
=
{
Eai ,
∫
d2y N bEcjF
j
bc
}
.
23
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Expanding F jbc from the above equation, we have
δ−→NE
a
i =
{
Eai ,
∫
d2y N bEcj
(
∂bA
j
c − ∂cA
j
b + 
j
IKA
I
bA
K
c
)
}
=
{
Eai ,
∫
d2y N bEcj
[
∂bA
j
c − ∂cA
j
b + 
j
k0
(
AkbA
0
c − A
0
bA
k
c
)]
}
. (2.46)
By using the properties of Poisson brackets (2.46) becomes
δ−→NE
a
i =
{
Eai ,
∫
d2y (N bEcj )∂bA
j
c
}
−
{
Eai ,
∫
d2y (N bEcj )∂cA
j
b
}
+
{
Eai ,
∫
d2y (N bEcj 
j
k0)
[
AkbA
0
c − A
0
bA
k
c
]
}
.
Integrating the first and second term by part and using the product rule of Poisson
bracket on the third term to get
δ−→NE
a
i = −
∫
d2y N b∂bE
c
j
{
Eai , A
j
c
}
+
∫
d2y Ecj∂cN
b
{
Eai , A
j
b
}
+
∫
d2y (N bEcj 
j
k0)
({
Eai , A
k
b
}
A0c − A
0
b
{
Eai , A
k
c
})
.
By using equation (2.39) to the above, we obtain
δ−→NE
a
i =
∫
d2y N b(∂bE
c
j )δ
a
c δ
j
i δ
2(x− y)−
∫
d2y Ecj (∂cN
b)δab δ
j
i δ
2(x− y)
+
∫
d2y (N bEcj 
j
k0)
(
A0cδ
a
b δ
k
i − A
0
bδ
a
c δ
k
i
)
δ2(x− y). (2.47)
The third term in (2.47) is computed as
(N bEcj 
j
k0)
(
A0cδ
a
b δ
k
i − A
0
bδ
a
c δ
k
i
)
= (N bEcj 
j
k0)
(
A0cδ
a
b − A
0
bδ
a
c
)
δki
= 0. (2.48)
Hence integrating the first two terms in (2.47), we obtain
δ−→NE
a
i =
∫
d2y N b(∂bE
a
i )δ
2(x− y)−
∫
d2y Eci (∂cN
a)δ2(x− y)
= N b∂bE
a
i − E
c
i ∂cN
a
:= L−→NE
a
i . (2.49)
Where L−→NE
a
i is the Lie derivative of E
a
i in the direction of the field N
a.
The infinitesimal changes of the canonical variables under the influence of the
Gaussian constraints are computed as follows, beginning with the action of the
24
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Gauss constraints on the spin connection
{
Aib,G[λ]
}
= {Aib,
∫
d2y λjD˜aE
a
j }
=
{
Aib,
∫
d2y
[
D˜a(λ
jEaj )−
(
D˜aλ
j
)
Eaj
]}
.
Applying Stokes theorem to
∫
d2y D˜a(λjEaj ) at the boundary of Σ, the above
equation becomes
{
Aib,G[λ]
}
= −
{
Aib,
∫
d2y
(
D˜aλ
j
)
Eaj
}
.
On applying the Poisson product rule to the above equation, we have
{
Aib,G[λ]
}
= −D˜aλ
j
{
Aib,
∫
d2y Eaj
}
−
{
Aib,
∫
d2y D˜aλ
j
}
Eaj
= −D˜aλ
j
{
Aib,
∫
d2y Eaj
}
, since
{
Aib, D˜aλ
j
}
= 0
= −
∫
d2y D˜aλ
j(δab δ
i
j)δ
2(x− y)
= −D˜bλ
i = −
(
∂bλ
i + i0kλ
kAb0
)
. (2.50)
Carrying out similar procedure as in (2.50), the action of the Gauss constraints
on the momenta is computed as follows
{Eai ,G[λ]} = {E
a
i ,
∫
d2y λjD˜bE
b
j}
=
{
Eai ,
∫
d2y
[
D˜b(λ
jEbj )−
(
D˜bλ
j
)
Ebj
]}
.
Applying Stokes theorem to
∫
d2y D˜b(λjEbj ) at the boundary of Σ, the above
equation becomes
{Eai ,G[λ]} = −
{
Eai ,
∫
d2y
(
D˜bλ
j
)
Ebj
}
= −
{
Eai ,
∫
d2y D˜bλ
j
}
Ebj −
∫
d2y D˜bλ
j
{
Eai , E
b
j
}
.
But {Eai , E
b
j} = 0. Expanding D˜bλ
j, we have
{Eai ,G[λ]} = −
{
Eai ,
∫
d2y
(
∂bλ
j + j0kλ
kAb0
)
}
Ebj
= −
{
Eai ,
∫
d2y j0kλ
kAb0
}
Ebj .
25
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Simplifying the above equation, we obtain
{Eai ,G[λ]} = −
∫
d2y j0kλ
k {Eai , Ab0}E
b
j
= 0, since {Eai , Ab0} = 0. (2.51)
The Hamiltonian constraint is a function of only the spin connection and as such
it commute with the spin connection:
{Ajb,H[N ]} = 0. (2.52)
Next we look at the action of the Hamiltonian constraint on the densitized diad:
{Eai ,H[N ]} =
{
Eai ,
∫
d2y NbcFbc
}
=
{
Eai ,
∫
d2y Nbc
[
∂bA
0
c − ∂cA
0
b + 
0
kj
(
AkbA
j
c − A
j
bA
k
c
)]
}
.
Using the distributive properties of Poisson bracket, we obtain
{Eai ,H[N ]} =
{
Eai ,
∫
d2y Nbc∂bA
0
c
}
−
{
Eai ,
∫
d2y Nbc∂cA
0
b
}
+
{
Eai ,
∫
d2y Nbc0kj
(
AkbA
j
c − A
j
bA
k
c
)
}
.
Integrating the first two terms by part and applying the product properties of
Poisson bracket to the last term, to obtain
{Eai ,H[N ]} = −
∫
d2y bc(∂bN)
{
Eai , A
0
c
}
+
∫
d2y bc(∂cN)
{
Eai , A
0
b
}
+
∫
d2y Nbc0kjA
k
b
{
Eai , A
j
c
}
+
∫
d2y Nbc0kj
{
Eai , A
k
b
}
Ajc
−
∫
d2y Nbc0kjA
j
b
{
Eai , A
k
c
}
−
∫
d2y Nbc0kj
{
Eai , A
j
b
}
Akc .(2.53)
{Eai , A
0
b} = {E
a
i , A
0
c} = 0, since both A
0
b and A
0
c are not dynamical variables.
Hence equation (2.53) becomes
{Eai ,H[N ]} = 0. (2.54)
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The action of the last constraint on the connection is computed as follows
{
Akb , C[µ]
}
=
{
Akb ,
∫
d2x µ0
0i
jE
c
iA
j
c
}
=
∫
d2x µ0
0i
jA
j
c
{
Akb , E
c
i
}
=
∫
d2x µ0
0i
jA
j
cδ
k
i δ
c
bδ
2(x− y)
= µ0
0i
jA
j
cδ
k
i δ
c
b = µ0
0k
j A
j
b (2.55)
Similarly, the action of this constraint on the densitized diad is
{
Ebk, C[µ]
}
=
{
Ebk,
∫
d2x µ0
0i
jE
c
iA
j
c
}
=
∫
d2x µ0
0i
jE
c
i
{
Ebk, A
j
c
}
= −
∫
d2x µ0
0i
jE
c
i δ
b
cδ
j
kδ
2(x− y)
= −µ0
0i
jE
c
i δ
b
cδ
j
k = −µ0
0i
kE
b
i . (2.56)
In summary, the time evolution of the constraints (2.40) are
{Ajb,V [
−→
N ]} = Na∂aA
j
b + A
j
a∂bN
a = L−→NA
j
b
{Eai ,V [
−→
N ]} = N b∂bE
a
i − E
c
i ∂cN
a = L−→NE
a
i
{Aia,G[λ]} = −Daλ
i = −
(
∂aλ
i + i0kλ
kAa0
)
{Eai ,G[λ]} = 0
{
Akb , C[µ]
}
= µ0
0k
j A
j
b
{
Ebk, C[µ]
}
= −µ0
0i
kE
b
i
{Ajb,H[N ]} = 0
{Eai ,H[N ]} = 0. (2.57)
The first two result in (2.57) shows the spatial diffeomorphism constraints gen-
erate diffeomorphisms tangential to the hypersurface with generating vector field
Na. The Gauss constraints generate internal rotations in the indices i (two dimen-
sions) as seen from the third and fourth equations in (2.57). The time evolution
themselves are gauge transformation as seen from the various evolution of the
constraints. All these transformations exhibited by the constraints comes as no
surprise as this was the case when they were no time gauge imposed on the action
(2.15). However, the Hamiltonian constraint generate no gauge transformation,
since it is expected to describe the dynamics of the new theory.
27
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2.5 Algebra of Constraints
The appearance of constraints, which generate gauge symmetries, is, as one can
expert, a general feature of systems with gauge symmetries. The group structure
of gauge transformations is reflected in the algebra of constraints, in particular
by the fact that the Poisson brackets between constraints is again a combination
of constraints. If this is the case for a given set of constraints, it will be called
first class [10]. However not all constraints are related by gauge symmetries, in
particular there are second class constraints, for which the Poisson brackets do
not vanish on the constraint hypersurface. In what follows, we try to examine if
the constraints (2.40) are first class constraints, and of particular interest to us
of these Poisson bracket algebras, are whether if they completely determine the
theory.
We start with the spatial diffeomorphism constraints. A careful look at (2.45) and
(2.46), one can write down the action on the smeared spatial diffeomorphism con-
straints on any arbitrary phase space function. Hence, the action of the smeared
spatial diffeomorphism constraints on the spatial diffeomorphism constraints i.e.,
{Va,V [
−→
N ]} = L−→NVa
= ∂b
(
N bVa
)
+ Vb∂aN
b. (2.58)
Hence we have
{
∫
d2x MaVa,V [
−→
N ]} =
∫
d2x MaL−→NVa
=
∫
d2x Ma
[
∂b
(
N bVa
)
+ Vb∂aN
b
]
=
∫
d2x [M c∂cN
a −Na∂cM
c]Va
= V [L−→M
−→
N ], (2.59)
where we use integration by parts on the first term of the second line to get
the result in the third line. The Poisson bracket of the spatial diffeomorphism
constraints is therefore given by
{V [
−→
M ],V [
−→
N ]} = V [L−→M
−→
N ] = V [[
−→
M,
−→
N ]], (2.60)
where [
−→
M,
−→
N ] = L−→M
−→
N is the commutator of two vector fields, defining the algebra
of infinitesimal diffeomorphisms.
The action of the spatial diffeomorphism constraints on the Hamiltonian constraint
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is computed as
{
∫
d2x NH,V [
−→
M ]} =
∫
d2x NL−→MH
=
∫
d2x N (Ma∂aH +H∂aM
a)
= −
∫
d2x (Ma∂aN)H. (2.61)
Hence this gives
{H[N ],V [
−→
M ]} = −H[L−→MN ] (2.62)
for the Poisson bracket between a pseudo-flatness constraint and the spatial dif-
feomorphism constraints. This gives a Hamiltonian constraint with its smearing
field.
Similarly the action of the spatial diffeomorphism constraints on the Gauss con-
straints is computed as follows
{
∫
d2x λjGj,V [
−→
N ]} =
∫
d2x λjL−→NGj
=
∫
d2x λj
(
Na∂aGj + Gj∂bN
b
)
= −
∫
d2x
(
λj∂aN
a −N b∂bλ
j
)
Gj
= −G[L−→Nλ]. (2.63)
Hence this gives
{G[λ],V [
−→
N ]} = −G[L−→Nλ] (2.64)
for the Poisson bracket between the Gauss constraints and the spatial diffeomor-
phism constraints. Hence this gives Gauss constraint with a shift in its test field
λ.
Using the results in (2.50) and (2.51), the Poisson bracket between one Gauss
constraints and another is computed as follows:
{G[α],G[λ]} =
{
G[α],
∫
d2y λjD˜aE
a
j
}
= −
{
G[α],
∫
d2y
(
D˜aλ
j
)
Eaj
}
= −
∫
d2y D˜aλ
j
{
G[α], Eaj
}
−
∫
d2y
{
G[α], D˜aλ
j
}
Eaj .
29
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But
{
G[α], Eaj
}
= 0, hence the above equation becomes
{G[α],G[λ]} = −
∫
d2y
{
G[α], ∂aλ
j + j0kλ
kAa0
}
Eaj
=
∫
d2y j0kλ
k {G[α], Aa0}E
a
j
= 0, since {G[α], Aa0} = 0. (2.65)
The first term of the third line of (2.65) vanish by applying (2.51). The result in
(2.65) mirrors the so(2) algebra commutation relation as
[α, λ] = [(α)jτj, (λ)
kτk] = 0. (2.66)
Hence the Poisson algebra of the Gauss constraints is a representation of the so(2)
Lie algebra and we write
{G[α],G[λ]} = G [[α, λ]] = 0. (2.67)
The Hamiltonian constraints obviously commutes with each other since its only a
function of the spin connection
{H[M ],H[N ]} = 0. (2.68)
We now compute the Poisson algebra between the Gauss constraints and the
Hamiltonian constraint
{G[λ],H[N ]} =
{∫
d2x λj(D˜aE
a
j ),H[N ]
}
=
{∫
d2x
[
D˜a(λ
jEaj )−
(
D˜aλ
j
)
Eaj
]
,H[N ]
}
= −
{∫
d2x
(
D˜aλ
j
)
Eaj ,H[N ]
}
= −
∫
d2x
{
D˜aλ
j,H[N ]
}
Eaj −
∫
d2x D˜aλ
j
{
Eaj ,H[N ]
}
= 0, since
{
D˜aλj,H[N ]
}
=
{
Eaj ,H[N ]
}
= 0. (2.69)
We now compute the Poisson bracket between C[µ] and the Gauss constraint
{C[µ],G[λ]} =
{∫
d2x µ0C
0,G[λ]
}
=
{∫
d2x
(
µ0
0i
j
)
EciA
j
c,G[λ]
}
. (2.70)
30
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Using the properties of the Poisson brackets, equation (2.70) becomes
{C[µ],G[λ]} =
∫
d2x
(
µ0
0i
j
)
Eci
{
Ajc,G[λ]
}
+
∫
d2x
(
µ0
0i
j
)
{Eci ,G[λ]}A
j
c
= −
∫
d2x
(
µ0
0i
jE
c
i
)
D˜cλ
j, since {Eci ,G[λ]} = 0
= −
∫
d2x
(
µ0
0i
jE
c
i
) [
∂cλ
j + j0kλ
kAc0
]
(2.71)
Now carrying out an integration by part on the first term of equation (2.71), we
arrive at
{C[µ],G[λ]} = −
∫
d2x
(
µ0
0i
j
) [
−λj∂cE
c
i + E
c
i 
j0
kλ
kAc0
]
=
∫
d2x
(
µ0
0i
j
) [
λj∂cE
c
i + λ
jEck
k0
i Ac0
]
=
∫
d2x
(
λjµ0
0i
j
) [
∂cE
c
i + 
k0
i E
c
kAc0
]
. (2.72)
From the above equation, we have
{C[µ],G[λ]} =
∫
d2x
(
λjµ0
0i
j
)
D˜cE
c
i
=
∫
d2x βiD˜cE
c
i = G[β], (2.73)
where βi = λjµ00ij .
Poisson bracket between C[µ] and the diffeomorphism constraint is computed as
follows
{C[µ],V [
−→
N ]} =
{∫
d2x
(
µ0
0i
j
)
EciA
j
c,V [
−→
N ]
}
=
∫
d2x
(
µ0
0i
j
) [
Eci {A
j
c,V [
−→
N ]}+ {Eci ,V [
−→
N ]}Ajc
]
=
∫
d2x
(
µ0
0i
j
) [
EciL−→NA
j
c + A
j
cL−→NE
c
i
]
. (2.74)
Expanding the Lie derivatives of the dynamical variables, we obtain from equation
(2.74)
{C[µ],V [
−→
N ]} =
∫
d2x
(
µ0
0i
j
) [
Eci
(
N b∂bA
j
c + A
j
b∂cN
b
)
+ Ajc
(
N b∂bE
c
i − E
b
i ∂bN
c
)]
=
∫
d2x
(
µ0
0i
j
) [
Eci
(
N b∂bA
j
c
)
+ Ajc
(
N b∂bE
c
i
)]
=
∫
d2x
(
µ0
0i
j
) [
N b∂b
(
AjcE
c
i
)]
. (2.75)
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Integrating by parts the last line of equation (2.75), we obtain
{C[µ],V [
−→
N ]} = −
∫
d2x
(
µ0
0i
j
) [(
AjcE
c
i
)
∂bN
b
]
= −
∫
d2x
(
µ0
0i
j
) (
AjcE
c
i
)
∂bN
b = −
∫
d2x µ0C
0∂bN
b
= −
∫
d2x
(
µ0∂bN
b
)
C0
= C[L−→Nµ], (2.76)
where L−→Nµ = −µ0∂bN
b .
The Poisson bracket between C[β] and C[α] is computed as follows
{C[µ], C[α]} = {
∫
d2x
(
µ0
0i
j
)
EciA
j
c, C[α]}
=
∫
d2x
(
µ0
0i
j
) [
Eci {A
j
c, C[α]}+ {E
c
i , C[α]}A
j
c
]
=
∫
d2x
(
µ0
0i
j
) [
(α0
0j
k )E
c
iA
k
c − (α0
0k
i )A
j
cE
c
k
]
=
∫
d2x
(
µ0
0i
j
) [
α0C
0 − α0C
0
]
= 0. (2.77)
Poisson bracket between C[µ] and the Hamiltonian constraint is computed as fol-
lows
{C[µ],H[N ]} = {
∫
d2x
(
µ0
0i
j
)
EbiA
j
b,H[N ]}
=
∫
d2x
(
µ0
0i
j
) [
Ebi {A
j
b,H[N ]}+ {E
b
i ,H[N ]}A
j
b
]
= 0. (2.78)
In summary, the algebra of constraints associated with the Gauss constraints re-
sulting from the time gauge of the action (2.15) are
{G[λ],V [
−→
N ]} = −G[L−→Nλ], {G[α],G[λ]} = 0
{G[λ],H[N ]} = 0, {C[µ],G[λ]} = G[β]
{C[µ], C[α]} = 0, {C[µ],V [
−→
N ]} = C[L−→Nµ]
{C[µ],H[N ]} = 0. (2.79)
As stated earlier, the group structure of gauge transformations is reflected in
the algebra of constraints. The second equation of (2.79), shows the result of
32
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the Poisson bracket of a Gauss constraint with itself vanishes. This shows the
group structure of the theory is an SO(2) group structure. Physically, the time
gauge breaks the SU(2) symmetry of the initial theory to an SO(2) symmetry of
the new dynamical variables. This confirms our earlier assertion of SO(2) gauge
transformations as one of the gauge symmetries of the new theory.
The other algebra of constraints are
{V [
−→
M ],V [
−→
N ]} = V [L−→M
−→
N ] = V [[
−→
M,
−→
N ]]
{H[N ],V [
−→
M ]} = −H[L−→MN ]
{H[M ],H[N ]} = 0. (2.80)
From (2.79) and (2.80), one can see that the constraints are closed and consis-
tent, since the various Poisson bracket of the constraints computed vanish on the
constraint hypersurface. Acting any constraint on the spatial difffeomorphism con-
straint yields the other constraint with a shift in the smearing field. The reason
for this is that the smearing functions are not functions of the canonical variables
and so the spatial diffeomorphism does not generate diffeomorphisms on them.
They do however generate diffeomorphims on everything else. This is equivalent
to leaving everything else fixed while shifting the smearing field.
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Chapter 3
Classical Representation of Phase
Space Variables
The choice of a basic set of phase space function {f} is an integral part of canonical
quantization. This set is promoted into operators {fˆ} on some kinematical Hilbert
space. By promoting the set into operators, they should map Poisson brackets into
commutators, i.e
[fˆ , gˆ] = i~{f̂, g}. (3.1)
However, this basic set should be sufficiently large, so as to distinguish between
phase space points.
The result of the Poisson bracket between the diad and the spin connection at
the classical level gives a dirac delta function. At the quantum level, this delta
function has no equivalent proper operators. Hence one must smear the canonical
variables with an appropriate smooth and fast decaying function, in order to select
a proper set of the basic variables at the quantum level.
In selecting a proper set of the basic variable, the Poisson bracket between them
should result in a simple expression forming an algebra. By forming an algebra,
quantization of this basic set become feasible. Furthermore, one would like that the
phase space functions transform in a simple way under gauge transformations, so
that these can be implemented as gauge transformations in a kinematical Hilbert
space.
Subsequent discussions in this chapter will focus on the descretized form of the
spin connection and the diad, these form a basic set of the phase space function
to be quantized. Computation of the Poisson bracket between these basic set will
be carried out. The result will show a much simpler expression compared to the
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non-descretized version of the canonical variables.
3.1 Holonomies
Any kind of spin connection on a manifold gives rise, through its parallel transport
maps, to some notion of holonomy. The holonomy of a connection on a smooth
manifold is a general geometrical consequence of the curvature of the connection
measuring the extent to which parallel transport around closed loops fails to pre-
serve the geometrical data being transported. The holonomy of a connection is
closely related to the curvature of the connection, and represent finite parallel
transporters. The materials discussed in this section can be viewed from [3, 10].
Let the curve γ be a continuous, piecewise smooth map:
γ : [0, 1] 7−→M
s 7−→ xa(s)
such that γ˙a ≡ dx
a(s)
ds is the tangent vector to γ.
By parallel transporting an internal vector V = V jτj (τj is the generator of SO(2))
along γ, V transforms in the adjoint representation of the group SO(2) defined by
V (s) = h(s)V (0)h(s)−1. (3.2)
The holonomy h(s) along the curve γ is such that:
γ˙aDaV (s) = 0, V (0) = V , (3.3)
where DaV = ∂aV + [Aa, V ].
Computing for the holonomy, we have from (3.3):
γ˙aDaV (s) = 0
=
(
d
ds
h(s)
)
V h(s)−1 + h(s)V
(
d
ds
h(s)−1
)
+ γ˙aAah(s)V h(s)
−1 − h(s)V h(s)−1γ˙aA. (3.4)
The above equation is satisfied if:
d
ds
h(s) = −γ˙aAa(γ(s))h(s), h(0) = 1. (3.5)
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Integrating (3.5) along γ:
∫ s
0
dh(s)
ds
ds = −
∫ s
0
γ˙aAa(γ(s
′))h(s′)ds+ C
h(s) = −
∫ s
0
γ˙aAa(γ(s
′))h(s′)ds+ C. (3.6)
Applying the condition h(0) = 1 into (3.6), we get C = 1.
Hence the holonomy is given as
h(s) = 1−
∫ s
0
γ˙aAa(γ(s
′))h(s′)ds′. (3.7)
Carrying out an iteration of (3.7), the first two expressions are:
h(s) = 1−
∫ s
0
γ˙a(s1)Aa(γ(s1))
(
1−
∫ s
0
γ˙a(s2)Aa(γ(s2))h(s2)ds2
)
ds1
= 1−
∫ s
0
γ˙a(s1)Aa(γ(s1))ds1 −
∫ s
0
∫ s1
0
X − · · ·
=
∞∑
n=0
(−1)n
n!
∫ s
0
· · ·
∫ sn
0
γ˙a(s1)Aa(γ(s1)) · · · γ˙
a(sn+1)Aa(γ(sn+1))h(sn+1)
= P exp
(
−
∫
γ
γ˙a(s)Aa(γ(s))
)
= P exp
(
−
∫
γ
A
)
, (3.8)
where X = γ˙a(s1)Aa(γ(s1))γ˙a(s2)Aa(γ(s2))h(s2)ds1ds2, A = γ˙a(s)Aa(γ(s)) and
P is the path ordering, and it is from left to right i.e γ(s1) < γ(s2) < ......... <
γ(sn+1).
3.2 Fluxes
The holonomy (3.8) constituent half of the basic set phase space function to be
quantized, thus we need the other half to have a complete basic set of phase space
function. Smearing the diad over a one-dimensional submanifold we obtain the
other half of the basic set of phase space function. Given a piecewise analytic
curve γ∗, intersecting another curve γ transversely at one point, with the tangent
vectors of γ, γ∗ at this intersection point positively oriented, the geometric flux
variables is defined [10]:
Eγ =
∫
γ∗
hγ,γ∗(s)ea(γ
∗(s))γ˙∗
a
(s)(hγ,γ∗(s))
−1 ds, (3.9)
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where hγ,γ∗ is a parallel transport matrix from the point γ∗(s) on the curve γ∗ to
the beginning point of γ. The holonomies1 are inserted in (3.9) to ensure a local
transformation behaviour under internal rotations, i.e.
Eγ −→ g(γ(0))Eγg(γ(0))
−1 = E ′γ. (3.10)
The starting point of γ is used instead of γ∗, so that the transformation behaviour
of Eγ and hγ are related. The parallel transport hγ,γ∗(s) is defined by choosing a
parametrization of the curves γ, γ∗ such that the intersection point in both curves
corresponds to s = 12 . Then we have
hγ,γ∗ =
(
hγ
(
1
2
))−1
hγ∗
(
1
2
, s
)
, (3.11)
where hγ∗
(
1
2 , s
)
is the parallel transport from the parameter s to the parameter 12
on the curve γ∗.
Let us write the component form Eiγ of the flux (3.9). Define
Eγ = E
i
γτi
−2 tr(τiτj) = δij. (3.12)
Combing the two equations in (3.12), the component form of the flux:
δij = −2 tr (τiτj)
δijE
j = −2 tr
(
τiτjE
j
)
Ei = −2 tr (τiE) = −2 tr (Eτi) . (3.13)
Using (3.13) the component form of the flux (3.9) along γ is
(Eγ)i =
∫
γ∗
(−2) tr
[
τihγ,γ∗(s)e
j
bτj(hγ,γ∗(s))
−1
]
γ˙∗
b
ds
=
∫
γ∗
(−2) tr
[
τihγ,γ∗(s)˜baE
ajτj(hγ,γ∗(s))
−1
]
γ˙∗
b
ds
=
∫
γ∗
(−2) tr
[
τihγ,γ∗(s)τj(hγ,γ∗(s))
−1
]
γ˙∗
b
˜baE
aj(γ∗(s))ds
=
∫
γ∗
fij(s)γ˙∗
b
˜baE
aj(γ∗(s))ds, (3.14)
where fij(s) = (−2) tr [τihγ,γ∗(s)τj(hγ,γ∗(s))−1].
1This differs from standard LQG flux variables which are defined without the parallel trans-
port with holonomies.
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3.3 Poisson Brackets
The basic set of phase space function are now formed by the matrix components
of the holonomies (3.8) and by the (vector) components of the fluxes (3.9). We
now find the Poisson brackets between these phase space functions.
Consider the holonomy hγ, we make a split, as we require only the part near the
intersection of γ with γ∗. Hence we have
hγ = hγ
(
1,
1
2
+
ε
2
)
hγ
(
1
2
+
ε
2
,
1
2
−
ε
2
)
hγ
(
1
2
−
ε
2
, 0
)
(3.15)
for ε small. Approximate the middle term in (3.15) to get
hγ
(
1
2
+
ε
2
,
1
2
−
ε
2
)
= 1−
∫ 1
2 +
ε
2
1
2−
ε
2
γ˙b(s′)Ajb(γ(s
′))τj ds
′ +O(2). (3.16)
Substituting (3.16) into (3.15), we obtain
hγ = hγ
(
1,
1
2
+
ε
2
)
hγ
(
1
2
−
ε
2
, 0
)
+ hγ
(
1,
1
2
+
ε
2
)
O(2)hγ
(
1
2
−
ε
2
, 0
)
−hγ
(
1,
1
2
+
ε
2
)[∫ 1
2 +
ε
2
1
2−
ε
2
γ˙b(s′)Ajb(γ(s
′))τj ds
′
]
hγ
(
1
2
−
ε
2
, 0
)
. (3.17)
Using (3.14) and (3.17) we compute the Poisson bracket between the flux and the
holonomy as follows
{(Eγ)i, hγ} = −hγ
(
1,
1
2
+
ε
2
)
τihγ
(
1
2
−
ε
2
, 0
)∫ 1
2 +
ε
2
1
2−
ε
2
γ˙b(s′){(Eγ)i, A
i
b(γ(s
′))}ds′
= −hγ
(
1,
1
2
+
ε
2
)
τkhγ
(
1
2
−
ε
2
, 0
)∫
γ∗
∫ 1
2 +
ε
2
1
2−
ε
2
fij(s)˜baγ˙∗
b
γ˙c
× {Eaj(γ∗(s)), Akc (γ(s))}dsds
′. (3.18)
Making use of (2.39), we have (3.18) to be
{(Eγ)i, hγ} = hγ
(
1,
1
2
+
ε
2
)
τkhγ
(
1
2
−
ε
2
, 0
)
×
∫
γ∗
∫ 1
2 +
ε
2
1
2−
ε
2
fij(s)˜baγ˙∗
b
γ˙cδjkδac δ
2D(s− s′)dsds′
= hγ
(
1,
1
2
+
ε
2
)
τkhγ
(
1
2
−
ε
2
, 0
)
×
∫
γ∗
∫ 1
2 +
ε
2
1
2−
ε
2
fki (s)˜baγ˙∗
b
γ˙aδ2D(s− s′)dsds′. (3.19)
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On the assumption that γ, γ∗ are positively oriented at the intersection, we have
˜baγ˙∗
b
γ˙a > 0 near the intersection point. Solving the integral of the delta function
in (3.19), and taking the limit  −→ 0 then we have s −→ 12 and (3.19) becomes
{(Eγ)i, hγ} = hγ
(
1,
1
2
)
τkhγ
(
1
2
, 0
)
fki
(
1
2
)
. (3.20)
Now we solve for the last term on the right hand side of (3.20) to get
fki
(
1
2
)
= −(2) tr
[
τihγ,γ∗
(
1
2
)
τ k
(
hγ,γ∗
(
1
2
))−1
]
= (−2) tr
[
τ k
(
hγ,γ∗
(
1
2
))−1
τihγ,γ∗
(
1
2
)]
= (−2) tr
[
τ khγ
(
1
2
, 0
)
τi
(
hγ
(
1
2
, 0
))−1
]
. (3.21)
To get the second line of (3.21), we perform a cyclic permutation of the first line
of (3.21). Applying (3.11), the third line of (3.21) then follows.
Performing a cyclic permutation on the last line of fki
(
1
2
)
in (3.21), equation (3.20)
becomes
{(Eγ)i, hγ} = hγ
(
1,
1
2
)
τkhγ
(
1
2
, 0
)
(−2) tr
[
τi
(
hγ
(
1
2
, 0
))−1
τ khγ
(
1
2
, 0
)]
= hγ
(
1,
1
2
)
hγ
(
1
2
, 0
)
τk(−2) tr(τiτ
k)
(
hγ
(
1
2
, 0
))−1
hγ
(
1
2
, 0
)
.
Making use of (3.12), the above becomes
{(Eγ)i, hγ} = hγ
(
1,
1
2
)
hγ
(
1
2
, 0
)
τkδ
k
i
(
hγ
(
1
2
, 0
))−1
hγ
(
1
2
, 0
)
= hγ
(
1,
1
2
)
hγ
(
1
2
, 0
)
τi
(
hγ
(
1
2
, 0
))−1
hγ
(
1
2
, 0
)
= hγ
(
1,
1
2
)
hγ
(
1
2
, 0
)
τi
= hγτi. (3.22)
Finally, we obtain the Poisson brackets between a flux and a holonomy both
associated to the curve γ. The last line of (3.22) can be read in terms of matrix
component wise as
{
Eiγ, (hγ)mn
}
= (hγτ
i)mn. (3.23)
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The Poisson brackets between holonomies and holonomies will vanish since the
holonomies are functionals of the spin connection only:
{(hγ)mn, (hγ)m′n′} = 0. (3.24)
However the Poisson brackets between fluxes and fluxes initially appears not to
vanish due to the fact that the fluxes involve the diad variables and the holonomies,
but this is not the case. Here we demonstrate this, using the requirement that the
Jacobi identity has to hold for the following Poisson brackets:
{{
Eiγ, E
j
γ
}
, hγ
}
+
{{
Ejγ, hγ
}
, Eiγ
}
+
{{
hγ, E
i
γ
}
, Ejγ
}
= 0.
From which we get:
{{
Eiγ, E
j
γ
}
, hγ
}
= −
{{
Ejγ, hγ
}
, Eiγ
}
−
{{
hγ, E
i
γ
}
, Ejγ
}
=
{{
Eiγ, hγ
}
, Ejγ
}
−
{{
Ejγ, hγ
}
, Eiγ
}
. (3.25)
Applying the result of (??) to (3.25) we have
{{
Eiγ, E
j
γ
}
, hγ
}
=
{
hγτ
i, Ejγ
}
−
{
hγτ
j, Eiγ
}
= −hγτ
jτ i + hγτ
iτ j
= hγ[τ
j, τ i]
= hγ[0] = 0. (3.26)
This can be satisfy if
{
Eiγ, E
j
γ
}
= 0. (3.27)
The last line in (3.26) was possible because of the commutation relation of the
so(2) algebra.
In summary the Poisson brackets of the basic set of the phase space function are
{(Eγ)i, hγ} = hγτi
{(hγ)mn, (hγ)m′n′} = 0
{
Eiγ, E
j
γ
}
= 0. (3.28)
The first equation of (3.28), we have the holonomy appearing, hence the Poisson
bracket leads to a phase space function in our basic set selected for quantization.
In addition it involves only objects associated to one and the same curve. The
results in (3.28) form an algebra and this coincides with the Poisson algebra of
40
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functions on the co-tangent space T ∗SO(2) space of SO(2) which can be equipped
with a phase space structure.
41
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Chapter 4
Kinematical Hilbert Space
The choice of the canonical variables have been found. We will now aim at a
canonical (Dirac) quantization of the system, this will also provide the tools to
derive a (covariant) path integral quantization. In defining a kinematical Hilbert
space of the system, we first find a representation of the phase space variables
as operators on some space of functions, introduce the notion of a cylindrical
functions (space of functions of operators), after which we equip this cylindrical
functions with an inner product. This allows us to find an orthonormal basis for
the cylindrical functions.
4.1 The quantization of edges(Quantum Opera-
tors)
In this section we will follow the excellent literature in [10]. Consider a fixed pair of
curves γ, γ∗ and rename the label γ to an edge label e. Thus we can quantize this
edge as a subsystem from which we generalize to the whole system. The holonomies
as stated already are SO(2) group elements. The phase space functions which are
to be quantized are functions of these holonomies, i.e., functions on the group
SO(2). This means to quantize the functions of the holonomies as multiplication
operators on some space of functions
ψe : ge −→ fe(ge)
on the group SO(2). Thus for a state | ψe > in the holonomy representation
< ge | ψe >, we define the the action of a function of the holonomy fe(he) as
fˆ | ψe > = | feψe > . (4.1)
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By using (4.1), these multiplication operators obviously commute by using the
fact that the Poisson brackets should be mapped into commutators i.e.
i}{̂fe, ge} | ψe > = [fˆe, gˆe] | ψe >
=
(
fˆegˆe | ψe > −gˆefˆe | ψe
)
=
(
fˆe | geψe > −gˆe | feψe >
)
= (| fegeψe > − | gefeψe >)
= 0. (4.2)
This then satisfy the commutation relation as in the classical case, that is the
Poisson bracket between the holonomies in (3.24).
Next we compute the commutation relation between a flux and a holonomy, let
first consider the Poisson brackets between these two variables in terms of the
matrix elements of the holonomy i.e.,
{
Eie, (he)mn
}
=
(
heτ
i
)
mn
=
d
dt |t=0
(
he exp(tτ
i)
)
mn
=: Lie(he)mn, (4.3)
where Lie is the left invariant derivative (this is a Lie derivative on the group
manifold). The matrix elements can be understood as functions of the group and
this generalizes to functions of the holonomy
{
Eie, fe(he)
}
=
(
Liefe
)
(he)
= fe
(
heτ
i
)
=:
d
dt |t=0
f
(
he exp(tτ
i)
)
. (4.4)
Applying this result in (4.4) to the state | 1 > and mapping the Poisson brackets
into commutators, we arrive at
i}(̂Liefe) | 1 > = i} ̂{Ei, fe(he)} | 1 >
= [Eˆi, fˆe] | 1 >
= Eˆifˆe | 1 > −fˆeEˆ
i | 1 > . (4.5)
From the last line of (4.5) we get:
i}(̂Liefe) | 1 > = Eˆ
i | fe >, (4.6)
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the right handside of (4.6) was possible of since we take Eˆi to act as a derivative
operator.
Let us consider the state | ψe >= ψˆe|1 >, then (4.6) becomes
i}(̂Lieψe)|1 > = Eˆ
i
e|ψe >
i}|Lieψe > = Eˆ
i
e|ψe > . (4.7)
Hence the action of the fluxes Eie on a state |ψe > is given by
Eˆie|ψe > = i}|L
i
eψe > . (4.8)
Using (4.8) we compute the the commutation relation between two fluxes on the
state |ψe >:
[Eˆie, Eˆ
j
e ]|ψe > =
(
EˆieEˆ
j
e − Eˆ
j
eEˆ
i
e
)
|ψe >
= (i})2|(Lie ◦ L
j
e − L
j
e ◦ L
i
e)ψe > . (4.9)
Evaluating the commutator of the left invariant derivative of the left handside of
(4.9) on a matrix element (he)mn amounts to
(
Lie ◦ L
j
e − L
j
e ◦ L
i
e
)
(he)mn =
d
ds |s=0
d
dt |t=0
((
hee(sτ
i)e(tτ
j)
)
mn
−
(
hee(sτ
j)e(tτ
i)
)
mn
)
=
d
ds |s=0
d
dt |t=0
(
he
(
e(sτ
i)e(tτ
j) − e(sτ
j)e(tτ
i)
)
mn
)
=
(
he[τ
i, τ j]
)
mn
= (he × 0) = 0. (4.10)
Hence this generalizes to functions of matrix elements and we have
[Eˆie, Eˆ
j
e ] = i}
{
Êie, E
j
e
}
= 0. (4.11)
Confirming our result from the classical point of view thus mapping the Poisson
brackets into commutators. In addition, we have managed to find a representation
for the fluxes and holonomies as operators on the space of functions of the group
SO(2).
44
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4.2 Cylindrical Functions
In (2 + 1)-dimensional LQG, one usually works on a two dimensional space-like
hypersurface Σ and build states of the spin connection, which gives the geometry
of Σ. These states of spin connection are the cylindrical functions and they depend
on the spin connection through the holonomies of the edges of a graph. Hence
when defining a kinematical Hilbert space of the theory, the spin connection is
considered as the configuration variable.
Let G be a space of the smooth 2d real spin connection defined everywhere on Σ,
except, possibly, at isolated points. Fix the topology of Σ, say to a disk. In section
3.1, we noted that an oriented curve in Σ and a spin connection determine a group
element hγ[A], called the holonomy (fundamental variable) of the spin connection
along the path γ.
The holonomy transforms under gauge invariance of the gauge group G = SO(2):
hγ −→ ghγg
−1, where g ∈ G. (4.12)
For a given γ, the holonomy hγ[A] is a functional on G.
Let us consider an ordered collection Γ of smooth oriented paths γe with e =
1, 2, ....., N edges and a smooth function f(h1, ......, hN)
f : SO(2)Nγ −→ C
of N group elements.
The couple (Γ, f) defines a functional of the spin connection:
ϕ[A] := f (hγ1 [A], hγ2 [A], ......., hγN ) . (4.13)
Defining a space of all functionals ϕ[A], for all Γ and f , these functionals are called
”cylindrical functions”, and denote the space of these functionals as S. This is
the algebra of basic observables upon which the definition of Hkin will be based
upon. Similarly, the gauge symmetry of the cylindrical functions ϕ can be read
off from (4.12):
ϕ (h1, h2, ...., hN) = ϕ
(
gs(1)h1g
−1
t(1), ....., gs(N)hNg
−1
t(N)
)
, (4.14)
∀gi ∈ G. Where s(i) and t(i) are the source vertex and target vertex respectively.
Throughout this work we will be dealing with SO(2) gauge invariant function of
the spin connection. The simplest of these gauge invariant function of the spin
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connection is the Wilson loop. It is a function of the form:
Wγ[A] = tr (ρ(hγ[A])) (4.15)
for some loop in Σ and some representation ρ of the group SO(2). The invariance
of (4.15) is possible due to the invariance of the trace and of the holonomy from
(4.12). This makes the Wilson loop an element of ϕ according to the definition in
(4.13). The graph of the Wilson loop consists just of a single close edge (e = γ).
4.3 Representation of Cylindrical Functions (In-
ner Product)
These cylindrical functions defined in section 4.2 provide the possible states for
a Hilbert space structure and for quantization. In this section we will define a
kinematical Hilbert space Hkin for the space of cylindrical functions and give a
representation of these cylindrical functions. To this effect we will introduce a
positive normalized state µAL- the Ashtekar-Lewandowski measure-on the cylin-
drical functions and thus we obtain a definition of a kinematical inner product.
This inner product must be invariant under left and right translations and this is
given by the Haar measure of SO(2).
Given a functional ϕ[A] in the space S, µAL(ϕγ,f ) is defined as:
µAL(ϕγ,f ) =
∫
SO(2)
∏
e
dhe f (he1 , he2 , ......, heN ) , (4.16)
where he ∈ SO(2) and dh is the normalized Haar measure of SO(2).
Using the properties of µAL, we endow the space of cylindrical functions with an
inner product:
〈ϕγ,f |ϕγ′,g〉 := µAL (ϕγ,fϕγ′,g)
=
∫
SO(2)
∏
e
dhe f (he1 , ...., heN )g (he1 , ...., heN ) . (4.17)
Given the Haar measure of the group SO(2) to be dh = (2pi)−1dθ for θ ∈ [0, 2pi],
then µAL is obviously normalized as µAL(1) = 1, and positive i.e.,
µAL (ϕγ,fϕγ′,f ) =
∫
SO(2)
∏
e
dhe f (he1 , ...., heN )f (he1 , ...., heN ) ≥ 0. (4.18)
The state µAL is known as the Ashtekar-Lewandowski (AL) measure. Starting
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with the space of smooth functions on the group we can complete it to a Hilbert
space, thus Hkin = L2(S1) (space of complex-valued square-integrable function on
the circle) is the Cauchy completion of the space of cylindrical functions on the
AL measure. Next, we define spin networks as basis for the resulting L2 space.
4.4 Orthonormal Basis of Hkin
Introducing an orthonormal basis for Hkin requires that one uses the Peter-Weyl
decomposition theorem. The theorem states that given a compact Lie group, a
complete orthogonal basis is given by the set of all matrix elements of all the
unitary irreducible representation of the compact group. The Peter-Weyl decom-
position theorem can be viewed as a generalization of the Fourier theorem for
functions on the unit circle.
To introduce the orthonormal basis of Hkin, we will like to first redefine the cylin-
drical functions in equation (4.13). The holonomy hγ[A] been an element of SO(2),
can be written as
hγN = exp(inNθ), (4.19)
where N is the total number of edges within a particular graph.
Hence from (4.13) the cylindrical function is rewritten as
ϕf,γ[θ] := f (hγ1 [θ], ......., hγN [θ]) . (4.20)
From the Peter-Weyl decomposition, it states that given a function f ∈ L2[SO(2)],
the function can be expressed as a sum over unitary irreducible representation of
SO(2):
f(θ) =
∑
n
fnρn(h(θ)), (4.21)
where ρn(h(θ)) = exp(inθ) are the unitary one dimensional irreducible represen-
tation of SO(2) for n ∈ Z and
fn =
1
2pi
∫
SO(2)
dθ ρn(h(θ))f(θ) (4.22)
are the Fourier components.
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The orthogonality relation for unitary representation of SO(2) is
∫
SO(2)
dθ ρm(h(θ))ρn(h(θ)) = 2piδm,n (4.23)
where ρn(h(θ)) = exp(inθ) for n ∈ Z. These are orthonormal and hence provide
a basis for the space L2[SO(2)], a basis that corresponds to the decomposition
into irreducible of L2[SO(2)] as a representation of SO(2). One then has from the
Peter-Weyl theorem
(
ρ,L2[SO(2)]
)
=
⊕
n∈Z
(ρn,C) . (4.24)
A consequence of the Peter-Weyl decomposition theorem to functions f : SO(2)Nγ −→
C, is it allows us to write any cylindrical function (4.20) as
ϕγ,f [θ] =
∑
n1....nNl
fn1....nNlρn1 (hγ1 [θ]) .....ρnNl (hγN [θ]) , (4.25)
where fn1....nNl are the Fourier components.
Gauge transformations in equation (4.12) induces a gauge action on Fourier modes
which simply reads:
ρn(h) −→ ρn(gs)ρn(h)ρn(g
−1
t ). (4.26)
A basis of gauge invariant functions is then constructed by contracting the ten-
sor product of representation matrices in (4.25) with a so(2)-invariant tensors or
so(2)-intertwiners. By selecting an orthonormal basis of intertwiners iv, where v
labels the elements of the basis, one can write a basis of gauge invariant elements
of S called the spin network basis. Each spin network is labelled by a graph Γ ⊂ Σ,
set of irreducible representations ne labelling links (edges) of the graph Γ and a
set of intertwiners iv labelling the vertices (nodes) of the graph Γ.
The intertwiner iv lives in the tensor product of the SO(2) irreducible representa-
tion coming in and going out of the vertex v, precisely:
iv :
⊗
e|s(e)=v
V n
s
e −→
⊗
e|t(e)=v
V n
t
e , (4.27)
where V n is the representation space of SO(2).
Hence the spin network function is defined as
ψne,iv [θ] =
⊗
v
iv
⊗
e
ρne(he[θ]). (4.28)
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The inner product between any two of these SO(2) spin networks is computed as:
〈ψne,iv |ψn˜e ,˜iv〉 =
1
2pi
∫ 2pi
0
dθ ψne,ivψn˜e ,˜iv
=
∏
e
δne,n˜e
∏
v
〈iv |˜iv〉. (4.29)
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Chapter 5
Dynamics
The spinfoam approach of quantum gravity aims at constructing a rigorous well
defined mathematical notion of path integral for LQG as a tool for solving the
dynamics of the theory [19]. Characterization of the kernel of the quantum con-
straints of the theory given by quantization of the classical constraints of general
relativity defines the dynamics. The aim of these spinfoam models is to explicitly
construct a generalized projection operator from Hkin into the kernel of quantum
scalar constraint.
In [19], the authors establish a clear cut link between the spin foam approach and
the canonical quantization of three dimensional gravity. This they achieved by
constructing a physical scalar product (matrix element) of a regularized projector
operator which can be represented in terms of sum of finite spin foam amplitudes.
In this present work, we make a link between these approaches (i.e. spin foam and
canonical quantization), and to understand how the time gauge procedure from
SU(2) to SO(2) works at the quantum level. Essentially, we want to solve the
dynamics of our theory, by first constructing the matrix element of the projector
P on the physical Hilbert space.
In chapter 4, a kinematical Hilbert space of the theory in terms of the holonomies
was defined. And this can be seen as a first step towards understanding the
quantum geometry [17] of spin foams and its link to canonical formalism. In the
long run, the goal is to have a spin foam approach whose boundary states have an
SO(2) invariant spin networks.
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5.1 Canonical Quantization
Solving for the dynamics of the theory, we will provide a regularization of the
projector operator P -which is defined in terms of the Hamiltonian constraint.
This allows for the construction of a physical scalar operator (or matrix element)
of P . With the matrix element constructed, an illustration on how it admit a
spin foam state representation independent of the regularization [19, 26] is carried
out. The matrix elements of the projector P are called the transition amplitudes.
The projector constructed on the physical Hilbert space of the theory is formally
defined as
P =
∏
x∈Σ
δ(H(x)) =
∫
D[N ] exp
(
i
∫
Σ
NH(A)
)
. (5.1)
The expression (5.1) can be seen as a starting point for a spacetime representation
of quantum gravity. We will introduce a regularization of (5.1) which subsequently
will serve as a regularization of its matrix element < Pψ, ψ′ > for any pair of spin
network states ψ, ψ′ ∈ Hkin. We will follow the excellent work of [19] in carrying
out this task.
Let us define Γ and Γ′ to be graphs on which ψ and ψ′ are defined respectively.
An arbitrary cellular decomposition of Σ denoted by ΣΓΓ
′
 , where  ∈ R, is defined
such that:
1. The graphs of Γ and Γ′ are both contained in the graph defined by the union
of 0-cells and 1-cells in ΣΓΓ
′
 .
2. For each two-cells (plaquette) p there exists a ball B of radius  such that
p ⊂ B. Hence a consequence of this is that all 2-cells shrink to zero when  −→ 0.
Let us now consider a local patch U ⊂ Σ where we choose the cellular decompo-
sition to be a square with cells of coordinate length  (regulator). The integral in
the exponential in (5.1) can be written as a Riemann sum in the patch U as
H[N ] =
∫
U
tr[NH(A)]
= lim
−→0
∑
pi
2 tr[NpiHpi ] (5.2)
where pi labels ith plaquette in the path U , Npi ∈ so(2), and Hpi ∈ so(2) are
values at some interior point of the 2-cells.
Since the gauge group SO(2) is abelian, by using Stokes theorem the holonomy
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can be define as
hγ = P exp
(∫
S(γ)
H
)
, (5.3)
where S(γ) is the surface bounded by γ. Expanding (5.3), the holonomy hpi ∈
SO(2) around the 2-cells pi can be written as
hpi [A] = I+ 
2Hpi(A) +O(
4). (5.4)
Equation (5.2) can now be written as
H[N ] = lim
−→0
∑
pi
tr[Npihpi [A]]. (5.5)
Here we notice that the explicit dependence of  on the right hand side of (5.5)
has dropped from the sum, a sign that we can remove it upon quantization. Hence
the projector (5.1) takes the form
P =
∫ ∏
pi
dNpi exp

i lim
−→0
∑
pi
tr[Npihpi ]

 . (5.6)
We now define the physical scalar product of ψ and ψ′ as:
〈ψ, ψ′〉phy = 〈Pψ, ψ
′〉
:= 〈
∫ ∏
pi
dNpi exp

i lim
−→0
∑
pi
tr[Npihpi ]

ψ, ψ′〉
= lim
−→0
〈
∫ ∏
pi
dNpi exp

i
∑
pi
tr[Npihpi ]

ψ, ψ′〉. (5.7)
Since the exponential function is a continuous function, the limit can come out of
the scalar product in second line of (5.7).
Integrating (5.7) with respect to Npi we get
〈Pψ, ψ′〉 = lim
−→0
〈
∏
pi
δ(hpi)ψ, ψ
′〉, (5.8)
where δ(hpi) is the delta function on the SO(2) irreducible representation. These
irreducible representation of SO(2) are one-dimensional, are labelled by an integer
n, and their character is χn(g) = einφ, where g ∈ SO(2).
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Expanding the delta function, we have
δ(hpi) =
∑
n
χn(hpi). (5.9)
Hence (5.7) becomes
〈Pψ, ψ′〉 = lim
−→0
〈
∏
pi
δ(hpi)ψ, ψ
′〉 (5.10)
We now illustrate using the result in equation (5.10), on how the spin foam rep-
resentation arises. The spin foam representation of the generalized projection
operator (5.6) follows from inserting resolutions of the identity in Hkin [19] be-
tween consecutive delta distribution in (5.10). The resolutions of the identity in
Hkin is given by
1 =
∑
Γ⊂Σ,{n}Γ
|Γ, {n} >< Γ, {n}|. (5.11)
Inserting this resolution of identities between the delta distribution into the def-
inition of 〈Pψ, ψ′〉, from the large set of graphs in Σ, only a finite number of
intermediate graphs survive in the computation of 〈Pψ, ψ′〉.
Considering the ordering of the plaquettes such that the first delta distribution in
the regularization of P is evaluated on the lower central plaquette with respect to
the loop l. Then the first delta function acts on the initial state |n〉 as
δ(hp)|n〉 =
∑
m
χm(hp) n
,
(5.12)
where χm(hp) is the character or the trace of the one dimensional m-representation
matrix of U ∈ SO(2). Equation (5.12) becomes
δ(hp)|n〉 =
∑
m
tr(ρm(hp)) n
=
∑
m
(5.13)
where also tr(ρm(hp)) is known as the Wilson loop operator. The operator acts
on the initial state |n〉 in terms of spin network states as
δ(hp)|n〉 =
∑
p,m
Nn,k,pδp,m n
p
k .
(5.14)
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From (5.14), it implies that when inserting the resolution of the identity (5.11)
between the first and the second delta distributions, the corresponding term in
〈Pψ, ψ′〉 only the intermediate states on the right hand side of (5.14) will survive.
The delta function (5.14) acts on the initial state by attaching new infinitesimal
loop. Subsequent delta function have the same effect by creating new infinitesimal
loop associated to each corresponding plaquettes. Hence in this way each term in
〈Pψ, ψ′〉 explores the set of intermediate spin network states based only on those
graphs that are contained in the regulating cellular decomposition. Finally, when
contracted with the final state ψ′ only the sequence which represent consistent
histories remain.
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Chapter 6
Conclusion and Outlook
In this thesis, we time gauge fix the three dimensional Hilbert-Palatini action and
got some new physics resulting from the canonical analysis.
The triad is decomposed into its various components via the time gauge (2.27). In
addition, we applied the triad decomposition to carry out a canonical analysis on
the Hilbert-Palatini action. The diad and the spin connection (two dimensions)
emerge as the new dynamical variables of the new theory.
By time gauge fixing the Hilbert-Palatini action, the usual constraints associated
to an action upon a canonical analysis is carried are obtain, thus the Gauss,
the spatial diffeomorphism and the Hamiltonian constraints. However, a new
constraint, i.e., C also emerges. In computing the time evolution associated with
the Gauss constraint G, the new theory generates an internal rotation in two
dimensions. Additionally, diffeomorphism invariance is achieved from the time
evolution associated with the spatial diffeomorphism constraint V .
In computing the algebra of constraints for the constraints of the new theory,
we found out the new theory was consistent and closed, in the sense that all
the Poisson brackets between the different constraints vanish on the constraint
hypersurface. The gauge transformation of the new system is an SO(2) gauge
transformation.
In chapter 3, a descretized form of the new dynamical variables is defined. These
form a basic set of the phase space function to be quantized. The Poisson bracket
between these basic set result in a much simpler expression compared to the non-
descretized will be carried out. The result shows a much simpler expression com-
pared to the non-descretized version of the canonical variables.
A kinematical Hilbert space is constructed for the new theory with an SO(2) spin
network. In carry out quantization the new theory we follow a similar approach
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in [19]. We constructed a rigorous definition of a generalized projection operator
from the kinematical Hilbert space. In addition, we provided the definition of the
physical scalar product (matrix element) of the projection operator. This matrix
element represents a sum over finite spin-foam amplitudes. As a result a very
trivial spinfoam is obtained as compared to the Ponzano-Regge spinfoam model.
This trivial spinfoam model has SO(2) spin networks as it boundaries. However in
the Ponzano-Regge spinfoam model [27], there is a contraction of four 3j-symbols
(associated to a trivalent spin network) which results in a function of six spins
{j1, j2, j3, j4, j5, j6} with the faces bounded by the vertex of a tetrahedron. This
new function is the Wigner 6j-symbol.
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References
[1] Arvanitoyeorgos, A. (2003). An Introduction to Lie groups and Geometry of
Homogeneous Spaces. American Mathematical Society.
[2] Baez, J. C. (1999). An Introduction to Spin Foam Models of BF Theory and
Quantum Gravity. arXiv:gr-qc/9905087v1.
[3] Bonzom, V. and Dittrich, B. (2013). Dirac’s discrete hypersurface deformation
algebras. arXiv:1304.5983v1 [gr-qc].
[4] Bonzom, V. and Freidel, L. (2011). The Hamiltonian constraint in 3d Rieman-
nian loop quantum gravity. arXiv:1101.3524v3 [gr-qc].
[5] Carlip, S. (1998). Quantum Gravity in 2+1 Dimensions. Cambridge University
Press, Cambridge.
[6] Chao, Y. (2013). Towards a better understanding of the role Barbero-Immirzi
parameter in loop quantum gravity.
[7] Chazelle, B. (1991). Triangulating a Simple Polygon in Linear Time. Disc.
Comput. Geo., 6:485–524.
[8] Daniel Oriti, C. R. and Speziale, S. (2004). Spinfoam 2d quantum gravity and
discrete bundles. arXiv:gr-qc/0406063v1.
[9] Ditrrich, B. and Geiller, M. (2014). A new vacuum for loop quantum gravity.
arXiv: 140.664v1 [gr-qc].
[10] Dittrich, B. (2012). Lectures on Quantum Gravity for Perimeter Institute for
Theorectical Physics.
[11] Dona, P. and Speziale, S. (2010). Introductory lectures to loop quantum grav-
ity. arXiv:1007.0402v2[gr-qc].
[12] Dupuis, M. and Levine, E. R. (2010). Lifting SU(2) Spin Networks to Pro-
jected Spin Networks. arXiv:1008.4093v1 [gr-qc].
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[13] Freidel, L. and Louapre, D. (2003). Diffeomorphisms and spinfoam models.
arXiv:gr-qc/0212001v2.
[14] Geiller, M. and Noui, K. (2013). A note on the Holst action, the time gauge
and the Barbero-Immirzi parameter. arXiv:1212.5064v2 [gr-qc].
[15] Isham, C. J. (1999). Modern Differential Geometry for Physicists. World
Scientific Publishing Co. Pte. Ltd, London.
[16] Kempf, A. (2013). General relativity for cosmology. Available from
www.math.waterloo.ca/ akempf.
[17] Levine, E. R. (2002). Projected Spin Networks for Lorentz connection: Link-
ing Spin Foams and Loop Gravity. arXiv:gr-qc/0207084v1.
[18] Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation. W.H
Freeman and Company, San Francisco.
[19] Noui, K. and Perez, A. (2005). Three dimensional loop quantum gravity:
physical scalar product and spin foam models. arXiv:gr-qc/0402110v3.
[20] Oeckl, R. (2002). Generalized Lattice Gauge Theory, Spin Foams and State
Sum Invariants. arXiv:hep-th/0110259v2.
[21] Osei, P. K. (2007). General Relativity as a Gauge Theory of the Pioncare
Group. Master’s thesis, University of Ghana, Legon.
[22] Osei, P. K. (2012). Noncommutative Structure In Three Dimensional Quan-
tum Gravity. PhD thesis, University of Ghana, Legon.
[23] Perez, A. (2005). Introduction to Loop Quantum Gravity and Spin Foams.
arXiv:gr-qc/0409061v3.
[24] Perez, A. (2012). The Spin Foam Approach to Quantum Gravity.
arXiv:1250.2019v1[gr-qc].
[25] Poisson, E. (2002). An course in general relativity. University of Guelph,
Department of physics, Cambridge.
[26] Rovelli, C. (1999). Projector on physical states in loop quantum gravity. Phys-
ical Review D. Volume 59, 104015.
[27] Rovelli, C. (2004). Quantum Gravity. Cambridge University Press, Cam-
bridge.
[28] Thiemann, T. (2001). Introduction to modern quantum general relativity.
arXiv:gr-qc/0110034v1.
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[29] Witten, E. (1988). 2+1 Dimensional Gravity as an Exactly Soluble System.
pages 46–78. Nuclear Physics.
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Appendix A
ADM Decomposition of 2 + 1
Gravity
The unification of space and time play a crucial role in general relativity. Physi-
cally, general relativity is the discovery that spacetime and gravitational field are
the same entity. What we call “spacetime” is itself a physical object in many
respects similar to the electromagnetic field. This led to the discovery that there
is no spacetime at all, and so what Newton called “space” and Minkowski called
“spacetime” is nothing but a dynamical object called the gravitational field [27].
The Arnowitt-Deser-Misner (ADM) formalism however introduces an explicit- al-
though largely arbitrary division of spacetime into spatial and temporal divisions.
The decomposition of spacetime into space and time furnishes a natural setting
for the initial value problem, and it underlies Wheeler’s ’geometrodynamical’ pic-
ture of classical general relativity as the dynamics of evolving spatial geometries.
By providing a canonical description of the gravitational phase space, it leads
to a Hamiltonian version of classical gravity, and suggests a useful approach to
canonical quantization [5].
The ADM approach to 2 + 1-dimensional general relativity start with a slicing
of the spacetime manifold M into constant-time surfaces Σt, each provided with
a coordinate system {xi} and an induced metric gij(t, xi). To obtain the full
three-geometry, we must describe the way nearby time slice Σt and Σt+dt fit to-
gether. To do so, we start with a point on Σt with coordinates xi, and displace it
infinitesimally in the direction normal to Σt as illustrated in the diagram below.
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X
t+
dt
X
t
N
n
µδt
N
µδt
Σ
t+
dt
Σ
t
Figure A.1: The ADM decomposition of spactime
From figure A.1, define a time flow vector
τµ = Nnµ +Nµ
:=
∂xµ
∂t
= (1, 0, 0), (A.1)
where nµ is the unit normal to the hypersurface, Nµ is the unit tangential to the
hypersurface, and N is the lapse function serving as a normalization constant. It
is convenient to parametrize both nµ and Nµ as
nµ =
(
1
N
,−
N i
N
)
Nµ =
(
0, N i
)
. (A.2)
N i is the shift vector and describes the amount of the tangential deformation (in
the coordinates of the hypersurface). The spacetime metric is gµν which encodes
all the information of the geometry of the three dimensional manifold. It also
serves a lowering and raising tensor that is
gµνn
µnν = −1 =⇒ nµnµ = −1,
where the negative one shows nµ is timelike.
Now we compute the ADM form of the metric:
gµντ
µτ ν = g00τ
0τ 0
= g00. (A.3)
Here we apply Einstein’s summation convention in line one. Subsequently, we use
the second line of (A.1), thus τµ = (1, 0, 0) to get the result in the second line of
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(A.3). Use the first line of (A.1), we have
gµντ
µτ ν = gµν (Nn
µ +Nµ) (Nnν +Nν)
= gµνN
2nµnν + gµνN
µN ν + gµνNn
µN ν + gµνNn
νNµ
= −N2 + gijN
iN j. (A.4)
Here gµνNnµNν = gµνNnνNµ = 0, since nµ and N ν are perpendicular to each
other. For the second term in line two of (A.4) we made use of Nµ = (0, N i) to
get the result in the next line of (A.4). From (A.3) and (A.4) we have
g00 = −N
2 + gijN
iN j. (A.5)
Carrying out with the calculations we have:
gµντ
µN ν = gµν (Nn
µ +Nµ)Nν
= gµνNn
µN ν + gµνN
µN ν
= gijN
iN j. (A.6)
Repeating the same calculations as in (A.6) but using Einstein’s summation con-
vention to expand, and making use of τµ = (1, 0, 0) to get
gµντ
µNν = g00τ
0N0 + g01τ
0N1 + g02τ
0N2 + g10τ
1N0 + g11τ
1N1
+g12τ
1N2 + g20τ
2N0 + g21τ
2N1 + g22τ
2N2
= g00τ
0N0 + g01τ
0N1 + g02τ
0N2
= g0jN
j. (A.7)
From (A.6) and (A.7) we have
g0jN
j = gijN
iN j. (A.8)
Expanding the terms in (A.8)
g01N
1 + g02N
2 = g11N
1N1 + g12N
1N2 + g21N
2N1 + g22N
2N2
=
(
g11N
1 + g12N
2
)
N1 +
(
g21N
1 + g22N
2
)
N2. (A.9)
Equating corresponding terms we have: g01N1 = (g11N1 + g12N2)N1 =⇒ g01 =
g1jN j. Similarly, g02 = g2jN j, hence we have
g01 + g02 = g1jN
j + g2jN
j
g0i = gijN
j. (A.10)
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Since the spacetime metric is symmetric, from (A.10) we have gi0 = gijN j. The
line element for three-dimensional spacetime is given as
dS2 = gµνdx
µdxν
= g00dx
0dx0 + g01dx
0dx1 + g02dx
0dx2 + g10dx
1dx0 + g11dx
1dx1
+g12dx
1dx2 + g20dx
2dx0 + g21dx
2dx1 + g22dx
2dx2
=
(
−N2 + gijN
iN j
)
dt2 + gijN
jdtdxi + gijN
idxjdt+ gijdx
idxj
= −N2dt2 + gij
(
dxi +N idt
) (
dxj +N jdt
)
. (A.11)
Here to get to line three, we applied the results in (A.5) and (A.10) to the second
line. The result in (A.11) is the ADM form of the the metric. Expressing the
metric in a matrix form we have
gµν =
(
N iNi −N2 Ni
Nj gij
)
. (A.12)
The inverse of the matrix is
gµν =
(
− 1N2
Nj
N2
N i
N2
(
gij − N
jNj
N2
)
)
. (A.13)
The six components of the spacetime metric are now replaced by the three com-
ponents of the induced Riemannian metric gij of Σ plus the two components of
the shift vector N i and the lapse function.
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Appendix B
Representation Theory
B.1 Group Representation
Definition B.1.1. A finite dimensional representation of a Lie group G on a
vector space over a field K is a group homomorphism from G to GL(V ), the
general linear group on V . That is, a representation is a map [1]
φ : G −→ GL(V ) (B.1)
such that φ(g1g2) = φ(g1)φ(g2), for all g1, g2 ∈ G. The dimension of the represen-
tation is the dimension of the vector space V . Denote the representation of G in
V by (G, V ).
Definition B.1.2. Let (G, V ) be a representation. A subspace U of V is called
invariant or G-invariant if g.U ⊂ U for g ∈ G.
A representation (G, V ) has always at least two invariant subspaces, namely {0}
and V .
Definition B.1.3. A representation is called irreducible if the only invariant sub-
spaces are {0} and V .
B.2 Peter-Weyl Decomposition Theorem
Definition B.2.1. A matrix coefficient of a group G is a complex-valued function
ϕ on G given as the composition
ϕ = L ◦ pi
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where pi is a finite-dimensional group representation of G, and L is a linear func-
tional on the vector space of the endomorphisms of V which contains GL(V ) as
an open subset.
The Peter-Weyl decomposition theorem is a theory of harmonic analysis, applying
to topological groups that are compact but not necessarily abelian. The theorem
is a collection of results generalizing the significant facts about the decomposition
of the regular representation of any finite group.
The theorem has three parts: (i) The set of matrix coefficients of a group G is
dense in the space of continuous complex functions onG equipped with the uniform
norm. (ii) Let ρ be a unitary representation of a compact group G on a complex
Hilbert space H. Then H splits into an orthogonal direct sum of irreducible
finite dimensional unitary representation of G. (iii) The matrix coefficients for G,
suitably renormalized, are an orthonormal basis of the square integrable functions
of G (L2(G)). In particular, L2(G) decomposes into an orthogonal direct sum of
all the irreducible unitary representations.
B.3 SU(2) Representation
Commutation relation: The generators of the su(2) algebra are given as
Ti =
σi
2
, where i = {x, y, z} . (B.2)
Where σi’s are the Pauli matrices:
σx =
(
0 1
1 0
)
, σy =
(
0 −ı
ı 0
)
, σz =
(
1 0
0 −1
)
. (B.3)
The generators satisfy the following commutation relations:
[Ti, Tj] = ıijkTk (B.4)
Representation: Define the Casimir operator (commutes with all the Ti, i.e, [T 2, Ti] =
0)
T 2 = T 2x + T
2
y + T
2
z .
By convention choose Tz to be the observable to be diagonalized simultaneously
with T 2. Therefore we have
T 2|jm〉 = j(j + 1)|jm〉, and Tz|jm〉 = m|jm〉 (B.5)
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Define the non-Hermitian operators (raising and lowering operators):
T± = Tx ± ıTy (B.6)
The following relations exist between the non-Hermitian and the diagonalized
operators:
[
T 2, T±
]
= 0, [Tz, T±] = ±T±, [T+, T−] = 2Tz . (B.7)
The eigenvalues of the non-Hermitian operators are given as follows
T±|jm〉 =
√
(j ∓m) (j ±m+ 1)|jm± 1〉, (B.8)
where j ranges from −m to m.
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Appendix C
Lie Derivative
Functions, tensor fields and one-forms can be differentiated with respect to a
vector field. Lie derivative evaluates the change of a tensor field (including scalar
function, vector field and one-form), along the flow of another vector field. This
change is coordinate invariant and therefore the Lie derivative is defined on any
differentiable manifold. Considering vector fields as infinitesimal generators on
a manifold M, then the Lie derivatives are infinitesimal representation of the
diffeomorphism group on tensor fields.
C.1 Lie derivative of a function
Given a vector field V defined onM, there exist a smooth map σ : R×M −→M
called the flow of V , such that σ0(p) = p, σt(σs(p)) = σt+s(p) and
d
dt
σt = V (σt(p)) (C.1)
are satisfied. The follow of V is denoted by σt(p), where t ∈ R and p ∈ M.
Writing the component of the flow of V with respect to the local coordinates xµ
we have
σµt (p) = x
µ(p) + tV µ(p) +O(t2) (C.2)
Given a function f : M −→ R, the Lie derivative acting on this function with
respect to a vector field V changes along the flow of V . Therefore defining it using
the difference f(σt(p))− f(p), i.e. the difference between f in the point p and the
translated point σt(p). In that way we can get a measure of the change of f in the
direction of the flow of V . Hence, we define the Lie derivative LV f of the function
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f along the vector field V as
LV f)(p) = lim
t−→0
f(σt(p))− f(p)
t
. (C.3)
Using the expression in (C.2), we have for small t
f(σt(p)) = f(p) + tV
µ∂f(p)
∂xµ
+O(t2). (C.4)
In local coordinate system xµ, equation (C.3) is given as
LV f = V
µ ∂f
∂xµ
. (C.5)
C.2 Lie derivative of a vector field
We now want to define the Lie derivative of a vector field W with respect to a
vector field V , however let us first define an induced map. Let Φ :M−→M be a
map. For a given point p ∈ M we can define the induced map DpΦ : Tp(M) −→
TΦ(p)(M) as follows. Writing
V ′ = DpΦ(V ) (C.6)
we define the induced map as
V ′[f ](Φ(p)) = V [f ◦ Φ](p). (C.7)
In terms of coordinates we have that
V ′[f ] = V µ
∂Φν
∂xµ
∂f
∂xν
. (C.8)
The Lie derivative acting on W quantifies how much W changes along the flow of
V . Taking the difference W (σt(p))−W (p) i.e. the difference between W at p and
at σt(p), and then look at the limit t −→ 0. This however is not well defined since
W (σt(p)) ∈ Tσt(p)(M) and W (p) ∈ Tp(M) are vectors in two different tangent
spaces, hence one cannot subtract them from each other. Thus we need to find a
way to take W (σt(p)) and map it into a vector in the tangent space Tp(M). This
can be done with the induced map Dσt(p)σ−t : Tσt(p)(M) −→ Tp(M).
We now define the Lie derivative LVW of the vector field W along the vector field
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V as
LVW = lim
t−→0
Dσt(p)σ−t(W )−W (p)
t
, (C.9)
for any point p ∈M. The induced map in coordinates is given as
Dσt(p)σ−t(W )[f ] = W
ν(σt(p))
∂σµ−t
∂xν
(σt(p))
∂f
∂xµ
(p). (C.10)
Using the expression in (C.2), we have for small t
W ν(σt(p)) = W
ν(p) + tV ρ
∂W ν
∂xρ
(p) +O(t2)
∂σµ−t
∂xν
(σt(p)) = δ
µ
ν − t
∂V µ
∂xν
(p) +O(t2). (C.11)
Computing from this the components of the Lie derivative LVW we have
(LVW )
µ = V ν
∂W µ
∂xν
−W ν
∂V µ
∂xν
. (C.12)
We can see that this is nothing but the components of the commutator [V,W ] of
V and W , thus we have
LVW = [V,W ]. (C.13)
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Appendix D
SO(2) Haar Measure and Useful
Identities
A Haar measure is a way to assign an ”invariant volume” to subsets of locally
compact topological groups and subsequently define an integral functions on those
groups.
Definition D.0.1. A Haar on a group G is a measure µ : Σ −→ [0,∞), with Σ a
σ-algebra containing all Borel subsets of G, such that
• µ(G) = 1
• µ(γS) = µ(S) ∀γ ∈ G,S ∈ Σ.
Here γS = {γα|α ∈ S}
We may associate to any measure µ on G a bounded linear functional
E : L(G,Σ, µ) −→ C
by
E(f) =
∫
G
f(γ)dµ(γ). (D.1)
Hence in this case for the group SO(2) we have
∫
SO(2)
f(γ)dµ(γ) =
1
2pi
∫ 2pi
0
f(eiθ)dθ. (D.2)
We now present here the various identities used in section 5.1. The identities are
for the gauge group SO(2), for SU(2) and general treatment of these identities
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see [19, 20]. We start by recalling the graphical notation that is commonly used
in dealing with group representation theory. An n unitary irreducible SO(2)
representation matrix ρn is represented by an oriented line pointed downwards.
The group element g ∈ SO(2) on which the representation matrix is evaluated is
depicted by a dark box in the middle. Namely,
ρn(g) =
b
n
.
(D.3)
The tensor product of representation matrices is simply represented by a set of
parallel lines carrying the corresponding representation labels and orientation. An
important object is the integral of the tensor product of unitary irreducible rep-
resentations. We denote the Haar measure integration by a dark box overlapping
the different representation lines as follows:
I =
∫
dg ρn1(g)ρn2(g).......ρnN (g) =
b b b b
n1 n2 n3 nN .
(D.4)
The invariance of the Haar measure implies I2 = I and the invariance of I under
right and left action of the group; therefore, I defines the projection operator
In1n2....nN : n1 ⊗ n2 ⊗ .... ⊗ nN −→ Inv[n1 ⊗ n2 ⊗ .... ⊗ nN ]. With this, it is easy
to write the basic identities that follow from the properties of the Haar measure,
namely
n
= δn,0 (D.5)
and
n1 n2
= δn1,n2
⋃
n1
⋂
n2
(D.6)
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