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Item Hypothesis of path integral duality. II. Corrections to quantum field theoretic results(American Physical Society, 1998-07-13) Srinivasan, K.; Sriramkumar, L.; Padmanabhan, T.In the path integral expression for a Feynman propagator of a spinless particle of mass m, the path integral amplitude for a path of proper length R(x,x'\|gμν) connecting events x and x' in a spacetime described by the metric tensor gμν is exp \{-[m R(x,x'\|gμν)]\}. In a recent paper, assuming the path integral amplitude to be invariant under the duality transformation R-->(L2P/R), Padmanabhan has evaluated the modified Feynman propagator in an arbitrary curved spacetime. He finds that the essential feature of this ``principle of path integral duality'' is that the Euclidean proper distance (Δx)2 between two infinitesimally separated spacetime events is replaced by [(Δx)2+4L2P]. In other words, under the duality principle the spacetime behaves as though it has a ``zero-point length'' LP, a feature that is expected to arise in a quantum theory of gravity. In Schwinger's proper time description of the Feynman propagator, the weightage factor for a path with a proper time s is exp [-(m2s)]. Invoking Padmanabhan's ``principle of path integral duality'' corresponds to modifying the weightage factor exp [-(m2s)] to exp \{-[m2s+(L2P/s)]\}. In this paper, we use this modified weightage factor in Schwinger's proper time formalism to evaluate the quantum gravitational corrections to some of the standard quantum field theoretic results in flat and curved spacetimes. In flat spacetime, we evaluate the corrections to (1) the Casimir effect, (2) the effective potential for a self-interacting scalar field theory, (3) the effective Lagrangian for a constant electromagnetic background and (4) the thermal effects in Rindler coordinates. In arbitrary curved spacetime, we evaluate the corrections to (1) the effective Lagrangian for the gravitational field and (2) the trace anomaly. In all these cases, we first briefly present the conventional result and then go on to evaluate the corrections with the modified weightage factor. We find that the extra factor exp [-(L2P/s)] acts as a regulator at the Planck scale thereby ``removing'' the divergences that otherwise appear in the theory. Finally, we discuss the wider implications of our analysis.Item Hawking radiation in different coordinate settings: complex paths approach(IOP Publishing, 2002-04-30) Shankaranarayanan, S.; Padmanabhan, T.; Srinivasan, K.We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild spacetime. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semiclassical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation as far as these coordinates are concerned.Item Method of complex paths and general covariance of Hawking radiation(World Scientific Publishing Company, 2001-02-05) Shankaranarayanan, S.; Srinivasan, K.; Padmanabhan, T.We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space{time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semiclassical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation. This also shows that there is no correspondence between the particles detected by the model detector and the particle spectrum obtained by the quantum field theoretic analysis | a result known in other contexts as well.Item Formal Analysis of two Dimensional Gravity(American Astronomical Society, 1998-09-17) Engineer, Sunu; Srinivasan, K.; Padmanabhan, T.Several investigations in the study of cosmological structure formation use numerical simulations in both two and three dimensions. In this paper we address the subtle question of ambiguities in the nature of two-dimensional gravity in an expanding background. We take a detailed and formal approach by deriving the equations describing gravity in (D ] 1) dimensions using the action principle of Einstein. We then consider the Newtonian limit of these equations and Ðnally obtain the necessary Ñuid equations required to describe structure formation. These equations are solved for the density perturbation in both the linearized form and in the spherical top-hat model of nonlinear growth. We Ðnd that, when the special case of D \ 2 is considered, no structures can grow. We therefore conclude that, within the frame work of Einstein's theory of gravity in (2 ] 1) dimensions, formation of structures cannot take place. Finally, we indicate the di erent possible ways of getting around this difficulty, so that growing struc-tures can be obtained in two-dimensional cosmological gravitational simulations, and discuss their implications.Item Possible quantum interpretation of certain power spectra in classical field theory(World Scientific Publication Company, 1997-03-13) Srinivasan, K.; Sriramkumar, L.; Padmanabhan, T.In this paper we report an analogue for the vacuum state in classical field theory and its Planckian nature with respect to uniformly accelerated observers. When a real, monochromatic, mode of a scalar field is Fourier analyzed with respect to the proper time of a uniformly accelerating observer, the resulting power spectrum consists of three terms none of which have a simple classical meaning. Specifically, the three terms are (i) a factor (1/2) that is typical of the ground state energy of a quantum ascillator, (ii) a Planckian distribution N(Ω) and — most importantly — (iii) a term , which is the root mean aquare fluctuations about the Planckian distribution. It is the appearance of the root mean square fluctuations that motivates us to attribute a "thermal" nature to the power spectrum. Such a power spectrum also arises when we Fourier analyze a real, monochromatic, plane electromagnetic wave in the frame of a uniformly accelerating observer. We also present a model of a detector whose response is the Fourier spectrum of the field with respect to its proper time, which illustrates that it should, in principle, be possible to physically measure the power spectrum we have obtained. These results show that some of the "purely" quantum mechanical results might have a classical analogue.Item Plane waves viewed from an accelerated frame: Quantum physics in a classical setting(American Physical Society, 1997-04-14) Srinivasan, K.; Sriramkumar, L.; Padmanabhan, T.We report here an analogue for the vacuum state in classical field theory and its Planckian nature with respect to uniformly accelerated observers. We find that when a real, monochromatic mode of a classical field is Fourier transformed with respect to the proper time of a uniformly accelerating observer, the resulting power spectrum has three separate terms none of which have a simple classical meaning. But they bear a striking resemblance to the quantum mechanical description. Specifically, the three terms are (i) a factor (1/2) that is typical of the ground state energy of a quantum oscillator, (ii) a Planckian distribution N(Ω) and, most importantly, (iii) a term proportional to √N(N+1), which is the root mean square fluctuations about the Planckian distribution. The implications of this result are discussed.Item Particle production and complex path analysis(American Physical Society, 1999-06-14) Srinivasan, K.; Padmanabhan, T.This paper discusses particle production in Schwarzschild-like spacetimes and in a uniform electric field. Both problems are approached using the method of complex path analysis which is used to describe tunnelling processes in semiclassical quantum mechanics. Particle production in Schwarzschild-like spacetimes with a horizon is obtained here by a new and simple semiclassical method based on the method of complex paths. Hawking radiation is obtained in the (t,r) coordinate system of the standard Schwarzschild metric without requiring the Kruskal extension. The coordinate singularity present at the horizon manifests itself as a singu-larity in the expression for the semiclassical propagator for a scalar field. We give a prescription whereby this singularity is regularized with Hawking’s result being recovered. The equation satisfied by a scalar field is also reduced to solving a one-dimensional effective Schro¨ dinger equation with a potential (21/x2) near the hori-zon. Constructing the action for a fictitious nonrelativistic particle moving in this potential and applying the above mentioned prescription, one again recovers Hawking radiation. In the case of the electric field, standard quantum field theoretic methods can be used to obtain particle production in a purely time-dependent gauge. In a purely space-dependent gauge, however, the tunnelling interpretation has to be resorted to in order to recover the previous result. We attempt, in this paper, to provide a tunnelling description using the formal method of complex paths for both the time and space dependent gauges. The usefulness of such a common description becomes evident when ‘‘mixed’’ gauges, which are functions of both space and time variables, are analyzed. We report, in this paper, certain mixed gauges which have the interesting property that mode functions in these gauges are found to be a combination of elementary functions unlike the standard modes which are transcen-dental parabolic cylinder functions. Finally, we present an attempt to interpret particle production by the electric field as a tunnelling process between the two sectors of the Rindler spacetime.