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Quantum Gravity via Loop and String Theories (1)
Loop Quantum Gravity Theory vs. String Theory The
strength of the loop quantum gravity theory is
its capacity to describe the quantum spacetime in
a background independent non-perturbative manner
in combining quantum mechanics with general
relativity. The quantum gravity may also be
studied by string theory whose aim is to unify
all known fundamental physics into a single
theory. (2) Merits and Demerits of String
Theory The main merits of string theory are that
it provides elegant unification of known
fundamental physics through perturbation
expansion and finite order. Its main
incompletenesses are that its non-perturbative
regime is poorly understood and it lacks the
background-independent formulation of the theory,
thus it is difficult to obtain Planck scale
physics from string theory. Except for these
demerits, string theory allows computations of
some high energy scattering amplitudes and
derivations of the Bekenstein-Hawking black hole
entropy. (3) Merits and Demerits of Loop Quantum
Gravity The main merit of loop quantum gravity is
that it provides a well-defined and
mathematically rigorous formulation of a
background-independent, non-perturbative,
covariant quantum field theory at the Planck
scale. The main incompleteness of the theory is
regarding the dynamics, formulated in several
variants. (4) Towards Unification of Both Loop
and String Theories Strings and loop gravity may
not necessarily be competing theories. Perhaps
the two approaches might even, to some extent,
converge. There are similarities between the two
theories both theories start with the idea that
the relevant excitations at the Planck scale are
one-dimensional objects call them loops or
strings.
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• Loop Quantum Gravity
• 1. Review of Quantum Field Theory
• 1.1 Strong Interactions
• The force binding the nucleus together can be
  mediated by the exchange of p mesons (1940)
• The nonperturbative effects, extremely difficult
  to calculate, are dominant.
• (b) Goldberg (1950-60) worked with S matrix
  satisfying dispersion relations.
• (c) The SU(3) quark theory of hadron spectrums
  Gell-Mann, Néeman, Zweig
• (d) Lie group SU(3)
• 1.2 Weak interactions (strongly interacting
  particles decayed into lighter particles via a
  much weaker force
• These lighter particles are leptons
  etc.

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1.3 Gauge Revolution Weinberg-Salam
Model SU(2) ? U(1) Symmetric Group Vector
mesons 1.4 Quarks in QCD ? color ?
flavor 1.5 Standard Model The forces
between the leptons and quarks are mediated by
the massive vector mesons for the weak
interactions and the massless gluons for the
strong interactions. 1.6 Grand Unified Theory
(GUT) Big Bang?
Today?
0(10) ?SU(5)?SU(3)?SU(2)?U(1)?SU(3)?U(1) Can we
solve the ultraviolet divergences with
renormalization?
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1.7 Action Principle Lagrangian Hamiltonian
Poisson Bracket Commutators Hamilton
s Equation
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1.8 Quantization First Quantization Treat the
coordinates of qi and Pi as quantized
variables Second Quantization Fields are
quantized with an infinite number of degrees of
freedom Consider a Lagrangian
with such
that Euler Lagrange equation of
motion with which leads to the Klein-Gordon
equation where
. 1.9 Noethers
Theorem The variation of the fields together with
the equations of motion we obtain the
energy-momentum tensor such that with
or
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1.10 Group Theory Summary O(N) Orthogonal
complex or real matrices O(N,C) ? N(N-1)
independent parameters O(N,R) ? N(N-1)/2
independent parameters SO(N) Orthogonal complex
or real matrices with unit determinants SO(N,C)
? N(N-1) independent parameters SO(N,R) ?
N(N-1)/2 independent parameters U(N) Unitary
matrices U(N) ? N2 independent parameters SU(N)
Unitary matrices with unit determinants SU(N)
? (N2-1) independent parameters
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Group Dimension Rank
SO(N)(N even) (1/2)N(N-1) N/2
SO(N)(N odd) (1/2)N(N-1) (N-1)/2
SU(N) N2-1 N-1
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1.11 Quantum Gravity Feynman Diagram One-Loop
quantum gravity Feynman diagram
Fig. 1. Finite probability for loop diagrams in a
quantum theory of gravity is obtained by
including gravitinos in the interaction. The
diagrams shown here are those for the interaction
between two photons. The first diagram, labeled
Maxwell-Einstein theory, consolidates all the
one-loop diagrams that involve only gravitons and
photons the contribution from these diagrams is
equal to an infinite quantity multiplied by the
coefficient 137/60. Five one-loop diagrams
involving gravitinos can be constructed each of
them is proportional to the same infinite term
multiplied by the coefficients shown in brackets.
Only the sum of the diagrams is observable, and
adding the coefficients shows that the sum is
zero. Hence the infinite contributions of the
gravitinos cancel those of the gravitons and the
diagrams have a finite probability 249.
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Two-Loop Quantum Gravity Feynman Diagram
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1.12 SYMMETRIES AND GROUP THEORY (a) SO(2),
Orthogonal group in 2D For small
angles The
rotation group O(2) is called the orthogonal
group in two dimensions, defined as the set of
all real, two-dimensional orthogonal
matrices.
We conclude that all elements of O(2) are
parametrized by one angle ?. Thus, O(2) is a
one-parameter group.
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The resulting subgroup is called SO(2)
or special orthogonal matrices in 2D
Let us introduce an operator
L Such that Scalar
Vector
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(b) Representations of SO(2) and U(1) Consider
the process of symmetrizing and antisymmetrizing
all possible tensor indices to find the
irreducible representations
Let The set of all
one-dimensional unitary matrices U(?) ei?
defines a group called U(1)
There is a correspondence between the two, even
though they are defined in two different
spaces The correspondence between
O(2) and U(1)
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Or For small ?, Invariants
under O(2) or U(1) Here all elements
in O(2) commute with each other. This is known
as Abelian group. (c) Representations of SO(3)
and SU(2). In non-Abelian groups, elements do
not commute each other. We define O(3) as the
group that leaves distances in three dimensions
invariant. For O(3), we have
with or
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These antisymmetric matrices correspond to the
Lie algebra involved in the permutation symbol
(structure constants) ?ijk such that For
small angles Let us introduce the
operator Li Which satisfies
SO(3) Scalar field Vector
field For higher order tensor fields we
decompose a reducible tensor and take various
symmetric and antisymmetric combinations of the
indices. Irreducible representations can be
extracted by using dij and ?ijk
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Consider the set of all unitary, 2x2
matrices with unit determinent. These matrices
form a group, called SU(2), known as the special
unitary group in two dimensions. SU(2), special
unitary group has 8-4-13 independent
elements Hermitian matrix Hermitian
conjugate Hermitian Pauli spin
matrix

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3x3 Orthogonal 2x2 Unitary
matrix
matrix Let SU(2) transforms
as This is equivalent to the SO(3)
transformation (d) Representations of
SO(N) The number of independent elements in each
member of O(N) is Number of
constraints arising from the orthognality
conditions This is the same as the number of
independent antisymmetric NxN matrices.
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tiantisymmetric matrices with purely
imaginary elements ?irotation angle or the group
parameter Commuting all antisymmetric
matrices Define the generator of O(N) as The
matrix is antisymmetric in i and j. There are
N(N-1)/2 such matrices Let us define
the operator Construct SO(N) as Adjoint
representation Field Transformation
O(2) U(1) , O(3) SU(2) , SO(4) SU(2) ?
SU(2) , SO(6) SU(4)
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1.13 LORENTZ AND POINCARÉ GROUPS Lorentz
Group A Lorentz transformation can be
parametrized by which
characterizes the O(4) group if all signs are the
same. For Lorentz group (1 1 1 1), however, we
have O(3,1). Other properties for the
Lorentz group remain the same as O(4).
where pµi? µ. Similarly,
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Let us define with For a vector
field fµ, we have with the Lorentz
transformation where the velocity
components ux , vy , uz are the parameters of
Lorentz boosts. Using the standard
definitions, we obtain
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Let us define or
or Similarly
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Note that a boost in the x-direction, followed by
a boost in the y-direction, does not generate
another Lorentz boost. Thus we must introduce the
generators of the ordinary rotation group
O(3).

Thus we have
It can be shown that the Lorentz group can be
split up into two pieces.

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With each piece generating a separate SU(2) and
Ai , Bj 0. If we change the sign of the metric
so that we only have compact groups, then the
Lorentz group can be written as SU(2) ?
SU(2). This implies that irreducible
representations ( j) of SU(2) where j0, ½, 1,
3/2, etc. can be used to construct
representations of the Lorentz group. Poincaré
Group The Poincaré group is obtained by adding
translations to the Lorentz group. This
leads to With the
translation generator Pµ given by
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The rank and dimension of SO(N) and SU(N) are
given by where the rank of a Lie group is the
number of generators that simultaneously commute
among themselves. Casimir operators of the
Poincaré group are those operators that commute
with all generators of the algebra. Let
us define the Pauli-Lubanski tensor
Then the Casimir operators are We
find
Group Dimension Rank
SO(N) (N even) ½ N(N-1) N / 2
SO(N) (N odd) ½ N(N-1) N-1 / 2
SU(N) N2-1 N-1
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where S is the spin eigenstate of the
particle for (massless particles) we
have where h is the velocity
In nature, the physical spectrum
of states is known only for the first two
categories with
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1.14 3D and 2D (Third Order and Second Order)
Permutation Symbols and Their Products The
three-dimensional permutation symbol can be
written as Using the matrix algebra A B
ATB , we obtain Similarly,
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• 1.15 Classical Quantum Gravity
• Equivalence Principle
• The laws of physics in a gravitational field are
  identical to those in a local accelerating frame.
• Covariant Action
• The Einstein-Hilbert action in 4-dimensions
• Small variation in the metric
• Taking the variation of the action we find the
  equations of motion
• The scalar matter couples to gravity

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• Vierbeins and spinors in General Relativity
• The generally covariantation for scalar and
  Yang-Mills fields
• Vierbein
• Dirac matrices
• Spinor
• Coordinate transformation
• Lorentz transformation

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Covariant derivative for gauging the
Lorentz group The generally covariant
Dirac equation New version curvature
tensor Riemann tensor Connection
field (4) GUTs and Cosmology Redshift
Nuclear synthesis 3 Background
radiation
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cosmological constant
Kinetic energy Potential energy Conservation of
energy For , eliminate For
Radius of Universe (Expanding Friedman
Universe)

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• Baryon-Antibaryon asymmetry (
  Antibaryon)
• Parity P
• Charge C
• Time T
• For baryon asymmetry
• Breaking of C and CP symmetry and Baryon number
  at the origin of time
• A cosmological phase when these C and CP
  violating processes were out of equilibrium
• (5) Inflation
• Flatness
• Horizon - Isotropic

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(6) Cosmological constant problem (7)
Kaluza-Klein Theory Where is the 5th
dimension, possibly curled up in the radius of
Plank length?
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Einsteins action in 5-dimensional space, with
the 4-dimensional theory yielding the Maxwells
theory coupled to generate relativity Dirac
equation Coupling of Fermion Equating the
coefficients The electric charge gt
Plank length
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(8) Generalization to Yang-Mills Theory
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Let Then Yang-Mills
tensor Yang-Mills in 4N dimensions Kaluza-Klein
Yang-Mills theory (1) Can the standard model
gauge group be included? (2) Can complex,
representation of fermions be included? (3) Is
the theory renormalizable? (4) Why should
higher-dimensional space compactify? (5) What
about the cosmological constant? 7 dimensional
manifold Thus 4711 is the minimal number of
total dimensions that we must have in order to
have a standard model gauge group.
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Dirac operator on a (4N) dimensional
product manifold If the Dirac operator on
the B manifold has eigen value m, we have
mplank mass gt lepton and quark mass We
must achieve renormalizability, not losing
unitarity. (Paulli-Villars cutoff, ghost
states) (9) Quantizing gravity To see why
general relativity is not renormalizable we must
explain how to quantize the theory. Begin
the process of quantization by power expanding
the metric tensor around some classical
solution of the equations of motion
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The existence of a dimensional Newtons constant
is the origin of the problem of the
nonrenormalizability of gravity. Tto circumvent
this difficulty we may resort to the Feyman
rules. The Feyman rules for the graviton
propagator can be obtained by extracting the
lowest order term quadratic in the gravitation
field. The Lagrangian reduces to Add a term
here to break the gauge Thus we obtain
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Invert the matrix to obtain the final propogator
(one can check that the propagator is singular if
the gauge-breaking part is missing). (10)
Counter terms in Quantum Gravity For quantum
gravity we have no symmetries by which to cancel
the higher-loop graphs. For cancellations to
happen, higher-loop counterterms must be
forbidden by some unknown mechanism. If we can
show that these higher order-loop counterterms
cannot exist, then the theory might have a chance
at being finite. Let us first enumerate the
total number of one-loop counterterms that are
invariant. The total number of counterterms that
are invariant is just three, given by the set
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In the background field method, the counterterms
are gauge invariant, and we are allowed to
eliminate some of them via the equations of
motion. If we set then
, which implies .
Thus we are left with only one possible
counterterm .
The question is Can some unforseen identity or
symmetry prevent this invariant from appearing as
a counterterm? If so, then general relativity
would be one-loop finite even without computing a
single Feyman diagram. The answer is yes. There
is an identity. The Gauss-Bonnet identity, that
allows us to eliminate this last invariant as a
possible counterterm. To see this, we first note
that, as in Yang-Mills theory, there is a
topological invariant corresponding to the square
of curvature tensors Total derivative Reducing
out the product of two antisymmetric constant
tensors, we obtain
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Where e is the determinant of the Vierbein and we
sum over the permutations in the indices, which
preserves the antisymmetries of the antisymmetric
tensor. Thus Total derivative This means that
any counterterm that may appear at the one-loop
level can be eliminated. The second and third
tensors are eliminated by the equations of
motion, and the first tensor is eliminated by the
Gauss-Bonnet identity. For the two-loop level,
the following term cannot be cancelled by the
equations of motion or any known identity
the usual divergence Weyl
curvature tensor composed of Riemann tensor The
fact that this term does not cancel indicates
that perturbative quantum gravity is not a finite
theory. The quantum gravity becomes a divergent
theory when coupled to matter (spin 0, 1/2 , 1).
For spin 3/2 field the theory is less divergent
in supergravity. However, supergravity diverges
at the third-loop level.
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2. Hyperbolic Formulations of Einstein
Equation 2.1 Ashtekar Variables Consider a
three-dimensional manifold M, a smooth real SU(2)
connection , a vector density on
M, spin connection , and an extrinsic
curvature related as follows The
constraint equations are Hamiltonian Constraint
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Momentum Constraint Gauss Constraint Dynamical
Equations in Hyperbolic Forms Weakly
hyperbolic Strongly hyperbolic Symmetric
hyperbolic
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2.2 ADM Variables The ADM dynamical equations
are with the constraint equations 2.3
Transformation of Variables (a) Define the triad
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• (b) Obtain the inverse triad from
• (c) Calculate the density
• Obtain the densitized triad
• Calculate the connection form
• (f)
• Obtain ADM from Ashtekar______
  
• Calculate the density e as
• Obtain the three inverse metric
• Prepare
• Calculate the connection
• Calculate
• Obtain