Local ζ-Function Techniques vs. Point-Splitting Procedure: A Few Rigorous Results

Moretti, Valter

doi:10.1007/s002200050558

Cited by 69 publications

(114 citation statements)

References 31 publications

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“…Recently it has been shown that this procedure leads to the same results as the point-splitting technique [32]. The odd-dimensional case is simpler, as the local ζ-function and its derivative are finite in s = 1: the field theory is super-ζ-regular and the field fluctuations are simply given by the value in s = 1 of the local ζ-function.…”

Section: Vacuum Expectation Value Of the Field Fluctuationsmentioning

confidence: 90%

“…In particular, the field fluctuations and the stress tensor will be computed with the recently proposed direct ζ-function approach [30,31]. The equivalence of this approach with the more standard (euclidean) point-splitting procedure has been shown for compact spaces [32,33], and holds only formally in the noncompact case. We shall deal with operators in H N with continuous spectrum, and our results are an important test of the generalization of this equivalence when the hypothesis of compactness misses.…”

Section: Introductionmentioning

confidence: 99%

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Quantum scalar fields on anti-de Sitter space-time

Caldarelli

1999

Nuclear Physics B

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We investigate the propagation of arbitrarily coupled scalar fields on the N -dimensional hyperbolic space H N . Using the ζ-function regularization we compute exactly the one loop effective action. The vacuum expectation value of quadratic field fluctuations and the one loop renormalized stress tensor are then computed using the recently proposed direct ζ-function technique. Our computation tests the validity of this approach in presence of a continuous spectrum. Our results apply as well to the N -dimensional anti-de Sitter spacetime, whose appropriate euclidean section is the hyperbolic space H N .

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Section: Vacuum Expectation Value Of the Field Fluctuationsmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Quantum scalar fields on anti-de Sitter space-time

Caldarelli

1999

Nuclear Physics B

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“…The substantial equivalence between the formula above and the result of point splitting procedure has been analyzed in [28]. In D = 3, the local ζ-function is regular at s = 1 and the dependence on the scale parameter µ 2 drops out; thus one has…”

Section: The Vacuum Expectation Value Of the Stress Tensormentioning

confidence: 98%

“…Making use of the Epstein functional equation with a 1 = (β/2) 2 , a 2 = π 2 , one has 28) so that, after a first order expansion,…”

Section: The Effective Action For a Scalar Field In The Btz Groumentioning

confidence: 99%

Quantum scalar field on the massless(2+1)-dimensional black hole background

Binosi¹,

Moretti

Vanzo

et al. 1999

Phys. Rev. D

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The behavior of a quantum scalar field is studied in the metric ground state of the (2+1)-dimensional black hole of Bañados, Teitelboim and Zanelli which contains a naked singularity. The one-loop BTZ partition function and the associate black hole effective entropy, the expectation value of the quantum fluctuation as well as the renormalized expectation value of the stress tensor are explicitly computed in the framework of the ζ-function procedure. This is done for all values of the coupling with the curvature, the mass of the field and the temperature of the quantum state. In the massless conformally coupled case, the found stress tensor is used for determining the quantum back reaction on the metric due to the scalar field in the quantum vacuum state, by solving the semiclassical Einstein equations. It is finally argued that, within the framework of the 1/N expansion, the Cosmic Censorship Hypothesis is implemented since the naked singularity of the ground state metric is shielded by an event horizon created by the back reaction.

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“…The expression (27) involves the divergent terms coming from the poles of Γ(z) at z = −k (k = 0, 1, 2, ...)…”

Section: Local Zeta Functionmentioning

confidence: 99%

Quantum Effects in a Rotating Spacetime

Radu

Astefanesei

2002

Int. J. Mod. Phys. D

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The behavior of a arbitrary coupled quantum scalar field is studied in the background of the Gödel spacetime. Closed forms are derived for the effective action and the vacuum expectation value of quadratic field fluctuations by using ζ-function regularization. Based on these results, we argue that causality violation presented in this spacetime can not be removed by quantum effects. Introduction and motivationWhether or not the laws of physics permit the existence of closed timelike curves (CTCs) is one the important problems in the research field of modern theoretical physics. A number of familiar spacetimes make it clear that general relativity, as it is normally formulated, does not exclude the violation of causality in large scale, despite its local Lorentzian character. 1 where Ω and m are constants; m 2 can be continued to negative values, for m 2 → 0 obtaining SomRaychaudhuri spacetime [3]. The solution originally proposed by Gödel corresponds to the case m 2 = 2Ωwhere Ω ≥ 0 is a constant parameter related to the vorticity of the matter, and −∞ < t, z < ∞, 0 ≤ r < ∞, 0 < ϕ ≤ 2π. The source of this geometry is a perfect fluid with constant density ρ and no pressure (p = 0). Einstein's equation's with a cosmological constant Λ are satisfied if between Ω, Λ and ρ the following relation holdsThe study of geodesics showed that this spacetime is geodesically complete (and so singularity free) [4,5].Furthermore, because the universe is spacetime homogeneous, there are CTCs through every event [6] (hence the causal violation is not localized to some small region).In the last decade many authors have studied features of quantum field theory on a spacetime background that contains CTCs but most of the papers are dealing with confined causality violating spacetime.In these, CTCs are confined within some regions and there exist at least one region free of them; the regions with CTCs are separated from the well behaved spacetime by Cauchy horizons. Being a highly symmetric homogeneous spacetime, Gödel spacetime is an excellent model to investigate questions of principle related to the quantization of fields propagating on curved background, the interaction with a global vorticity and the issues related to the lack of global hyperbolicity. Even if one's primary interest is in quantum field theory in a confined causality violating spacetime, we hope that, by widening the context to Gödel spacetime, one may achieve a deeper appreciation of the theory. In particular, one may hopes to attain more general features of a quantum field propagating in a nonglobally hyperbolic spacetime, whether or not containing a Cauchy horizon. Of particular interest is the question whether causality violation which occurs in Gödel spacetime implies divergence of vacuum polarization fluctuations and may be removed by quantum effects. Thus it seems important to study the vacuum polarization for different physical fields in this background. 2A different reason to study quantum field theory (QFT) in this background emerges from the fact ...

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Local ζ-Function Techniques vs. Point-Splitting Procedure: A Few Rigorous Results

Cited by 69 publications

References 31 publications

Quantum scalar fields on anti-de Sitter space-time

Quantum scalar fields on anti-de Sitter space-time

Quantum scalar field on the massless(2+1)-dimensional black hole background

Quantum Effects in a Rotating Spacetime

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