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.


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 cannot be distinguished from the standard inflationary scenario with an arbitrary initial state.

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.

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  • S. Shankaranarayanan, Is there an imprint of Planck scale physics on inflationary cosmology? Class. Quant. Grav. 20, 75(2003); gr-qc/0203060
  • S. Shankaranarayanan, T. Padmanabhan and K. Srinivasan, Hawking radiation in different coordinate settings: Complex paths approach, Class. Quant. Grav. 19, 2671-2688 (2002); gr-qc/0010042.
  • S. Shankaranarayanan, K. Srinivasan and T. Padmanabhan, Method of complex paths and general covariance of Hawking radiation, Mod. Phys. Letts. A 16, 571-578 (2001); gr-qc/0007022.
  • S. Shankaranarayanan and T. Padmanabhan, Hypothesis of path integral duality: Applications to QED, Int. J. Mod. Phys. D 10, 351 - 365 (2001); gr-qc/0003058.
  • T. Padmanabhan and S. Shankaranarayanan, Vanishing of the cosmological constant in nonfactorizable geometry, Phys. Rev. D 63, 105021 (2001); hep-th/0011159.

Kruskal diagram

trans-Planckian modes

trans-Planckian modes