We discuss several aspects of quantum field theory of a scalar field in a Friedmann universe. (i) We begin by showing that it is possible to map the dynamics of a scalar field with a given mass, in a given Friedmann background to another scalar field of a different mass in another Friedmann universe. In particular one can map the dynamics of (1) a massless scalar field in a universe with power-law expansion to (2) a massive scalar field in the de Sitter spacetime. This allows us to understand several features of either system in a simple manner and clarifies several issues related to the massless limit. (ii) We relate the Euclidean Green's function for the de Sitter spacetime to the solution of a hypothetical electrostatic problem in D=5 and obtain, in a very simple manner, a useful integral representation for the Green's function. This integral representation is helpful in the study of several relevant limits, and in recovering some key results which are -though known earlier -not adequately appreciated. One of these results is the fact that, in any Friedmann universe, sourced by a negative pressure fluid, the Wightman function for a massless scalar field is divergent. This shows that the divergence of Wightman function for the massless field in the de Sitter spacetime is just a special, limiting, case of this general phenomenon. (iii) We provide a generally covariant procedure for defining the power spectrum of vacuum fluctuations in terms of the different Killing vectors present in the spacetime. This allows one to study the interplay of the choice of vacuum state and the nature of the power spectrum in different co-ordinate systems, in the de Sitter universe, in a unified manner. (iv) As a specific application of this formalism, we discuss the power spectra of vacuum fluctuations in the static (and Painlevé) vacuum states in the de Sitter spacetime and compare them with the corresponding power spectrum in the Bunch-Davies vacuum. We demonstrate how these power spectra are related to each other in a manner similar to the power spectra detected by the inertial and Rindler observers in flat spacetime. This also gives rise to a notion of an invariant vacuum noise in the corresponding spacetimes which is observer independent. (v) In addition, several conceptual and technical issues regarding quantum fields in general cosmological spacetimes are clarified as a part of this study. * kinjalk@iisermohali.ac.in † karthik@iucaa.in ‡
A harmonic oscillator with time-dependent mass m(t) and a time-dependent (squared) frequency ω 2 (t) occurs in the modelling of several physical systems. It is generally believed that systems, with m(t) > 0 and ω 2 (t) > 0 (normal oscillator) are stable while systems with m(t) > 0 and ω 2 (t) < 0 (inverted oscillator) are unstable. We show that it is possible to represent the same physical system either as a normal oscillator or as an inverted oscillator by redefinition of dynamical variables. While we expect the physics to be invariant under such redefinitions, it is not obvious how this invariance actually comes about. We study the relation between these two, normal and inverted, representations of an oscillator in detail both in Heisenberg and Schrödinger pictures to clarify several conceptual and technical issues. The situation becomes more involved when the oscillator is coupled to another (semi)classical degree of freedom C(t) and we want to study the back-reaction of the quantum system q(t) on C(t), in the semi-classical approximation. We provide a simple prescription for the backreaction based on energy conservation and study that the dynamics of the full system in both normal and the inverted oscillator representation. The physics again remains invariant but there are some extra subtleties which we clarify. The implications of these results for quantum field theory in cosmological backgrounds are discussed briefly in an appendix. * karthik@iucaa.in † sumantac.physics@gmail.com ‡ paddy@iucaa.in
A spatially homogeneous, time-dependent, electric field can produce charged particle pairs from the vacuum. When the electric field is constant, the mean number of pairs which are produced depends on the electric field and the coupling constant in a non-analytic manner, showing that this result cannot be obtained from the standard perturbation theory of quantum electrodynamics. When the electric field varies with time and vanishes asymptotically, the result may depend on the coupling constant either analytically or non-analytically. We investigate the nature of this dependence in several specific contexts. We show that the dependence of particle production on coupling constant is non-analytic for a class of time-dependent electric fields, with the leading order non-analytic behaviour being controlled by a specific parameter which can be identified. We also demonstrate that, for another class of electric fields, which vary rapidly, the dependence of particle production on coupling constant is analytic. Finally, we describe what happens to these results when we go beyond the leading order, using some specific examples. * karthik@iucaa.in † sumantac.physics@gmail.com ‡ paddy@iucaa.in
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