## Thesis title: Field theoretic methods in gravity

In this thesis, we study some phenomenological models which incorporate quantum gravitational effects into the standard field theory in a consistent manner and to look for their plausible observational signatures. In addition, the possibility of interpreting Hawking radiation as a tunneling phenomenon is investigated.

### Abstract

The fundamental interactions at present
known to physics can be divided
into three classes: strong, electro-weak and gravity. Quantum field
theory
has been successful as a theory describing the behaviour of the first
two
interactions up to energy scales of the order of *100 GeV*. The
electro-weak
interactions is successfully described by the Weinberg-Salam theory,
while
the strong interaction is described by quantum chromodynamics. Attempts
in the direction of grand unified theory to incorporate strong
interaction
into a wider gauge theory have also led to some success. However, we
still
lack a quantum description of the gravitational interaction -- __quantum
gravity__.

The past century has seen a large number of theoretical attacks on the problem of quantum gravity. Even though these approaches have had some successes, none of them have given a complete theory that works at Planck energy scales. The difficulties on the way to quantum gravity are of different kinds. First of all, the detection of quantum gravitational effects is by itself extremely difficult due to the weakness of the gravitational interaction. In addition one encounters technical problems in quantizing gravity resulting from basic, and peculiar, properties of general relativity such as the non-linearity of the Einstein equations and the invariance of the theory under the group of diffeomorphisms. Further, the fact that gravity couples via a dimensional coupling constant makes the theory intrinsically non-renormalisable. For sometime, it was believed that the supergravity theories might overcome the non-renormalisabilty of general relativity, but detailed calculations has led to the conclusion that they also suffer from the same problem. The modern viewpoint is that the non-renormalisability is a natural feature of a theory for which the action is not fundamental but arises as an effective action in some energy limit.

In spite of the fact that we are yet to
have a quantum theory of gravity,
there exist compelling reasons to believe that quantum gravitational
effects
will be important only at energy scales of the order of Planck energy .
There exist a domain of *17* orders of magnitude between the
Planck
energy and an energy scale of the order of *100 GeV*, where the
gravitational
field can be assumed to behave classically and the matter fields can be
assumed to have a quantum nature. Though, there exist other contesting
theories to describe the classical gravitational field, observations
have
pointed towards Einstein's general theory of relativity as a theory
describing
classical gravity. Thus, adopting general relativity as a theory
describing
classical gravity, one is led to the subject of quantum field theory in
curved space-times which has been an area of active research during the
past couple of decades.

Quantum field theory in curved spacetime is a semiclassical theory, in which the gravitational field is retained as a classical background while the matter fields are quantised according to the conventional quantum field theory. This theory describes the system propagating in a background curved spacetime based on the covariant version of the flat spacetime Lagrangian for the field. The formalism of quantum field theory in flat spacetime can be generalised to a curved spacetime in a straight forward manner. The quantisation of the field proceeds by defining a set of canonical commutation relations for the field operators. The evolution of the quantum field is governed by the behaviour of the normal modes of the field equation in the spacetime of interest.

The vacuum state of a quantum field
develops a non-trivial structure
in a classical gravitational background. As a result, three different
types
of phenomena occur in a classical background: (i) polarization of the
vacuum,
(ii) production of particles corresponding to the quantum field and
(iii)
the concept of a particle turns out to be coordinate dependent. In
particular,
quantum field theory predicts particle production in the gravitational
fields of black-hole and various cosmological models. In the case of a
body collapsing to a black hole, of mass *M,* quantum field
theory
predicts radiation of particles, at late times, in all modes of the
quantum
field, with characteristic thermal spectrum at a temperature equal to *(1/8
pi M)*.

One of the aims of this thesis, is to
understand Hawking radiation in
the Schwarzschild space-time as a tunneling phenomena and show that
Hawking
radiation is covariant. The second aim of the thesis is to incorporate
quantum gravitational effects in standard field theory and to quantify
the low energy effects of quantum gravity. As we have mentioned, there
is no viable complete theory of gravity as yet. The conceptual
difficulties
of integrating gravity into a quantum mechanical framework have proved
formidable so far. With no experimental evidence to guide the
construction
of such a theory, only selfconsistent frame works can be made. As
mentioned
in an earlier paragraph, the quantum gravitational effects are
significant
only at energies of the order of the Planck length. Such energies were
available in the universe at times and may not be obtainable at
the
present epoch. It is possible that there are ``low energy'' physical
effect
that could be experimentally tested. As a step in the direction to
quantify
low energy effects of quantum gravity, we use three basic __field
theoretic__
methods: (i) Hypothesis of path-integral duality, (ii) Dispersive field
theory models and (iii) Large extra dimension models.

A chapter wise summary of the thesis is given below:

In *chapter (1)*,
we introduce
the basic terminology and the mathematical
framework that is used to study the evolution of quantum fields in
curved
space-time. Some of the essential results that serve as a background
for
the chapters to follow are reviewed. The chapter begins by illustrating
the crucial differences in quantisation of fields between flat and
general
curved space-time. We also introduce effective action and Euclidean
path-integral
approach. Subsequently, Hawking radiation in a Schwarzschild space-time
is considered and three different well known methods of obtaining
Hawking
radiation are described. These different approaches illustrate the
semi-classical
approach usually taken when dealing with scalar fields in curved
space-time.
Finally, we discuss the three * field theoretic *approaches
-- path-integral duality, dispersive field theory model and large extra
dimension models -- which we use to quantify the low energy effects of
quantum gravity.

*Chapter (2)*
is concerned with
the covariance of Hawking radiation
in three different coordinate systems of the Schwarzschild space-time
using
the method of complex paths. The motivation for this approach is the
method
of complex paths used in standard non-relativistic semi-classical
quantum
mechanics to calculate transmission and reflection coefficients. We
show
that even though the two coordinate systems -- Lemaitre and Painleve--
do not posses a coordinate singularity, the action integral develops
poles
at the horizon. The method of complex paths, gives a suitable complex
contour
in order to regularise the singularity. This prescription takes into
account
the following: (i). the one-way nature of the horizon surface, (ii) the
multiple mapping of the coordinates. Using Novikov's R and T regions
analysis
of the space-time manifold, the standard Hawking temperature is
recovered.

In *chapter (3)*,
we use the
modified propagator for quantum field
based on a * principle of path integral duality *to
investigate
several results in quantum electrodynamics. This procedure modifies the
Feynman propagator by the introduction of a fundamental length (Planck
length) scale. We use this modified propagator for the Dirac particles
to evaluate the first order radiative corrections in quantum
electrodynamics.
We find that the extra factor of the modified propagator acts like a
regulator
at the Planck scales thereby removing the divergences that otherwise
appear
in the conventional radiative correction calculations of quantum
electrodynamics.
We also find that: (i) all the three renormalisation factors

*Z_1, Z_2, and Z_3*pick up finite corrections and (ii) the modified propagator breaks the gauge invariance at a very small level of

*O(10^{-45})*. The implications of this result to the generation of primordial seed magnetic fields are discussed.

*Chapter (4)*
is concerned with
the effects of the trans-Planckian
dispersion relation on the spectrum of primordial density perturbations
during inflation. In contrast to the earlier analyses in the
literature,
we do not assume any specific form of the dispersion relation and allow
the initial state of the field to be arbitrary. We obtain the spectrum
of vacuum fluctuations of the quantum field by considering a scalar
field
satisfying the linear wave equation with higher spatial derivative
terms
propagating in the de Sitter space-time. We show that the power
spectrum
* does
not *depend on the dispersion relation strongly and that the
form
of the dispersion relation does not play a significant role in
obtaining
the corrections to the scale invariant spectrum. We also show that the
signatures of the deviations from the flat scale-invariant spectrum
from
the cosmic microwave background radiation observations due to quantum
gravitational
effects

*be distinguished from the standard inflationary scenario with an*

__cannot__*initial state.*

__arbitrary__In *chapter (5)*,
we generalize
the results of Randall and Sundrum
to a wider class of four-dimensional space-times including the
four-dimensional
Schwarzschild background and de Sitter universe. We solve the equation
for graviton propagation in a general four dimensional background and
find
an explicit solution for a zero mass bound state of the graviton. We
find
that this zero mass bound state is normalisable only if the
cosmological
constant is * strictly* zero, thereby providing a dynamical
reason for the vanishing of cosmological constant within the context of
this model. We also show that the results of Randall and Sundrum can be
generalized without any modification to the Schwarzschild background.

The details of the calculations of the
work presented in the thesis
are given in the *Appendices A -- D*.
Some of the appendices also
provide some background material that may be useful in appreciating the
issues involved in the thesis.

Finally, in *chapter
(6)*, we present our conclusions and the future outlook.