Renormalized Stress Tensor Research Articles

Hadamard Regularization and Conformal Transformation

We investigate the conformal transformation of the renormalized stress tensor of conformally invariant scalar fields in the framework of the Hadamard regularization, We find that the renorma.l· ized stress tensor consists of three parts. The first is the simple scale transformation of the renormal· ized stress tensor, the second comes from the scale transformation of the logarithmic term in the Green's function and the third consists of curvature tensors, the conformal factor and their covariant derivatives. The method is applied to Friedmann universes and Schwarzschild black hole. § 1. Introduction Conformally invariant scalar field is one of the most interesting models to study the properties of quantized fields in curved spacetime. Conformal invariance of field equations helps us to construct mode solutions and the Green's functions. The conformal behaviour of the renormalized stress tensor is, however, not so trivial because we execute some regularization procedures to obtain a finite renormalized stress tensor and they break the conformal invariance and give the trace anomaly. Nevertheless in evaluating the renormalized stress tensor in concrete spacetimes, a' conformal transformation technique seems to be important. The de Sitter invariant vacuum state corresponds to the conformal vacuum with closed and fiat coordinatiza­ tion of de Sitter space.!) For the two-dimensional Schwarzschild black hole the static thermal state (Israel-Hartle-Hawking state)2) is equivalent to the conformal vacuum. Moreover Page has succeeded in obtaining an approximate renormalized stress tensor, in the Schwarzschild geometry with a conformal transformation technique. 3 ) He defined the optical metric which is conformally related to the static Einstein metric and where the trace anomaly of the renorma:lized stress tensor vanishes. In the optical metric the Gaussian approximation of the thermal propagator is very good. 4 ) He chose the renormalized stress tensor in the optical metric to have a thermal form and obtained the physical renormalized stress tensor by conformal transformation. His result remarkably agreed with numerical work by Howard and Candelas. 5 ) His method was improved by Zannias, Brown and Ottewill (Brown-Ottewill-Page-Zannias (BOPZ) Ansatz).6),7) These results suggest that more explicit investigation of the behaviour of the renormalized stress tensor under conformal tninsformation i~ impor­ tant to understand the behaviour of quantized fields in curved spacetime. There are two approaches to study the behaviour of the renormalized stress­ tensor under conformal transformation. The first one i~ to examine the conformal behaviour of the effective action. This method was first applied to the conformally fiat cases by Brown and Cassidy.S) For general cases the conformal transformation of the effective action was studied with dimensional regularization by Brown and OttewilF) and with zeta function regularization by Dowker.

Stress Tensor in De Sitter Space

The Hadamard regularization calculation is carried out to obtain the renormalized stress tensor of scalar fields with respect to the de Sitter invariant vacuum. We discuss how the condition of de Sitter invariance fixes the form of the Hadamard series expansion of the Hadamard elementary function. We obtain the value of the mass scale appearing in the logarithmic term in the Hadamard series expansion. We also compare the renormalized stress tensor of conformally invariant field in the de Sitter invariant and conformal vacua. For the purpose of investigating the role of quantized fields in cosmology, quantum field theory in de Sitter space is an interesting problem because the high symmetry of de Sitter space enables us to obtain the simple exact solution of the field equation. It is also expected to give fruitful information on the inflationary model of the early universe. It is one of the most important subjects to obtain the renormal­ ized expectation value of stress tensor of quantized fields with respect to certain vacuum states,. because the particle concept is very obscure in curved space and the renormalized stress tensor acts as the source term in the Einstein equation by con­ sidering the semi-classical back-reaction of quantized fields on the background geome­ try. The problem of how to select a physically meaningful vacuum state, however, remains unsolved. Vacuum states are sometimes characterized by symmetric prop­ erties or coordinatization of the background manifold. The most symmetric vacuum in de Sitter space is the de Sitter invariant one, which was first described by Chernikov and Tagirov 1 ) and have been studied by Schomblond, Spindel,2) Mottola 3 ) and Allen. 4 ) Several authors have evaluated the renormalized stress tensor with respect to the de Sitter invariant vacuum. Bunch and Davies investigated the behaviour of quantized scalar field and evaluated the renormalized stress tensor in de Sitter space with point-splitting regularization. 5 ) Their regularization method was formulated in the Friedmann universe not in the general curved spacetime. 6 ) Thus the subtracted divergent terms can be ambiguous. Dowker and Critchley evaluated the renormal­ ized stress tensor with the zeta function regularization,7) and Birrell and Davies did with the dimensional regularization. 8 ) Since they evaluated only the trace of the stress tensor with the effective Lagrangian, they could not obtain the state-dependent contribution. Allen, Bernard and Folacci applied their Hadamard regularization scheme to the de Sitter invariant vacuum. 9 ),IO) They used the Lagrangian with an additional term to give the limiting value in the flat space limit. There have been many works on this problem, but a few works with straightforward calculation. And the results for massive scalar fields are obscure. It is also still not clear how the

Photon propagators and the definition and approximation of renormalized stress tensors in curved space-time.

We present the symmetric Hadamard representation for scalar and photon Feynman Green's functions. We use these representations to give a simple definition for their associated renormalized stress tensors. We investigate the connection between the accuracy of the WKB approximation and the vanishing of the trace anomaly for these fields. We show that, although for scalars there is a direct connection, this is not true for photons, and we discuss the relevance of these results to the approximation of renormalized stress tensors in static Einstein space-times.

Conformally invariant quantum field theory in static Einstein space-times.

The conformal transformation law for the renormalized one-loop effective action is used to give approximate renormalized stress tensors for conformally invariant scalar, spinor, and vector field theories on static Einstein space-times, paying particular attention to the ambiguous coefficient of the term \ensuremath{\square}R in the vector trace anomaly. In black-hole space-times these tensors for the Hartle-Hawking thermal state are shown to be regular on the horizon. Geometrical expressions are given for the approximate energy densities there; these can all be expressed in terms of the scalar invariant ${C}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}\ensuremath{\tau}}$${C}^{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}\ensuremath{\tau}}$. A number of explicit examples are presented, including Schwarzschild space-time. In flat and conformally flat space-times these approximate stress tensors agree with the exact results wherever the latter are known. In more complicated space-times, where we do not always obtain the exact results, a brief discussion is given of how to obtain a more accurate approximation.

Effective actions and conformal transformations.

We give the conformal transformation law for the effective action for conformally invariant field theories on general curved space-times. It is shown that this law can be used to obtain almost all known renormalized stress tensors for such theories. It can also be used to obtain approximate renormalized stress tensors; for example, we derive as a special case Page's approximation for static Einstein space-times.

Comments on conformal Killing vector fields and quantum field theory

We give a comprehensive analysis of those vacuums for flat and conformally flat space-times which can be defined by timelike, hypersurface-orthogonal, conformal Killing vector fields. We obtain formulas for the difference in stress-energy density between any two such states and display the correspondence with the renormalized stress tensors. A brief discussion is given of the relevance of these results to quantum-mechanical measurements made by noninertial observers moving through flat space.

Adiabatic regularisation for scalar fields with arbitrary coupling to the scalar curvature

Adiabatic regularisation is applied to a scalar field propagating in a Robertson-Walker universe with arbitrary coupling to the scalar curvature. Explicit expressions for the expectation value of the quantum stress tensor in an adiabatic vacuum are obtained. This calculation yields the terms which are to be subtracted from the divergent mode-sum expressions for expectation values of the stress tensor to give a finite, renormalised stress tensor. It is shown that the removal of the infinite terms in this subtraction procedure corresponds to the renormalisation of coupling constants in Einstein's equation. A short description is given of the way in which adiabatic regularisation produces a trace anomaly.

On renormalisation of the quantum stress tensor in curved space-time by dimensional regularisation

Using dimensional regularisation, a prescription is given for obtaining a finite renormalised stress tensor in curved space-time. Renormalisation is carried out by renormalising coupling constants in the n-dimensional Einstein equation generalised to include tensors which are fourth order in derivatives of the metric. Except for the special case of a massless conformal field in a conformally flat space-time, this procedure is not unique. There exists an infinite one-parameter family of renormalisation ansatze differing from each other in the finite renormalisation that takes place. Nevertheless, the renormalised stress tensor for a conformally invariant field theory acquires a nonzero trace which is independent of the renormalisation ansatz used and which has a value in agreement with that obtained by other methods. A comparison is made with some earlier work using dimensional regularisation which is shown to be in error.

Stress tensor of massless conformal quantum fields in hyperbolic universes

The vacuum expectation value of the renormalized stress tensor for a conformally coupled massless scalar field propagating in an arbitrary hyperbolic Robertson-Walker spacetime is calculated. The vacuum state used is that obtained by conformal transformation to the static hyperbolic universe: The result differs from calculations performed using the vacuum defined by conformal transformation to Minkowski spacetime, and is also different from earlier results obtained using the conformally static vacuum which are shown to be incorrect.

Energy-momentum tensor of a massless scalar quantum field in a Robertson-Walker universe

The renormalized stress tensor of a free quantum field in a curved space-time should be obtainable by some covariant procedure from the short-distance behavior of the corresponding operator products evaluated at separated points. We perform the covariant expansion of the vacuum expectation value of the point-split stress tensor of a conformally invariant scalar field in a spatially flat Robertson-Walker geometry. When attempting to identify the finite direction-independent part of the coincidence limit of this quantity as the renormalized stress tensor, we encounter ambiguities beyond those corresponding to inevitable renormalizations of coupling constants in the gravitational field equations, but nevertheless limited to local geometrical tensors of fourth order or lower in derivatives of the metric. The ambiguities are resolved by requiring the tensor to be covariantly conserved and to have a trace of the form indicated by recent research on conformal anomalies. Various techniques useful in calculations of this kind are described.

Stress-tensor trace anomaly in a gravitational metric: Scalar fields

We compute the stress-tensor vacuum expectation value of a massive, scalar quantum field that is coupled to the metric of an arbitrary classical gravitational field. The renormalized tensor is defined by a dimensionally continued, proper-time representation. The stress tensor is calculated for arbitrary dimension in a potentially conformal-invariant manner so that its trace is formally proportional to the square of the scalar-field mass with this trace vanishing as the scalar field becomes massless. However, the renormalized stress tensor violates this formal identity with its trace containing additional, anomalous terms. These finite-trace anomalies are intimately related to the infinite counterterms that must be put into the action to make the stress tensor finite.