Local Zeta Regularization and the Scalar Casimir Effect

Abstract

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.Printed in Singapore Local Zeta Regularization and the Scalar Casimir Effect Downloaded from www.worldscientific.com by 54.202.27.70 on 05/12/18. For personal use only. PrefaceZeta regularization is a method to treat the divergent quantities appearing in several areas of mathematics and physics; these are managed by introducing a complex parameter, with the role of a regulator, and defining their "renormalized versions" in terms of the analytic continuation with respect to the regulator. The standard textbook example of this procedure deals with the divergent series +∞ =1(1) which, in the zeta approach, is interpreted as the analytic continuation at s = −1 of the regularized series ζ(s) := +∞ =1 −s . The latter series only converges for s > 1 but the function s → ζ(s) defined in this way, the well-known Riemann zeta function, possesses a unique analytic extension to C \ {1} and ζ(−1) = −1/12; in this sense, the sum (1) "equals" −1/12. In a more pictorial language, one could say that −1/12 is the "renormalized" value of the series (1).The application of zeta regularization to the divergences of quantum field theory was first proposed by Dowker and Critchley [46], Hawking [84] and Wald [146] to renormalize local observables, especially the vacuum expectation value (VEV) of the stress-energy tensor. The ultimate purpose was the semiclassical treatment of quantum effects in general relativity, e.g. using the stress-energy VEV as a source term in Einstein's equations. Due to the attention to vacuum states, the zeta approach was connected from its very beginning to Casimir physics. However, the above mentioned pioneers focused their investigations on the conceptual validity of the method, rather than on its implementation for actual computations in specific configurations. For example, the elegant formula of Dowker-Critchley-Hawking, which gives the VEV of the stress-energy tensor as the functional derivative of the (renormalized) effective field action with respect to the space-time metric is not useful for actual computations in a given geometry, apart from very special cases. This is due to the fact that its use would require computing the effective action for all spacetime metrics close the one under consideration. For this reason, Birrel and Davies have described the application of this formula as "impossibly difficult" (see [18], page 190).Moreover, papers [46,84, 146] deal with a quantized field on the whole spacetime manifold and do not consider the possibility of confining the field to a given space ...

The Casimir energy anomaly for a point interaction

“…where λ is a real parameter related to the strength of the potential, G z := e i √ z |x| 4π|x| with Im √ z > 0 and H 2 (R 3 ) is the usual Sobolev space of order two. The non-negativity of A (necessary for a consistent formulation of the field theory; see [12]) is ensured assuming λ 0; in this case, A has purely absolutely continuous spectrum σ(A) = [0, ∞). Besides, note that the choice λ = 0 corresponds to the free theory where no delta potential is present (correspondingly, DomA| λ=0 = H 2 (R 3 )).…”

“…Continuing the analysis begun in Ref. [13], here the total Casimir energy for the above model is investigated within a general framework for zeta regularization developed in previous works [8,9,10,11,12]. In addition to a regular bulk contribution which is finite after renormalization, there also appears an anomalous boundary term which remains infinite even after implementing the standard renormalization procedure.…”

Local Casimir Effect for a Scalar Field in Presence of a Point Impurity

“…Quantum field theory and the fundamental operator. In the present section we briefly recall the general setting of [31] for the quantum theory of a scalar field on a space domain with boundary conditions, possibly in presence of a static external potential; this formulation will be methodically employed in the sequel.…”

“…This representation of the stress-energy VEV, depending on the regulating parameter u ∈ C and on the infrared cutoff ε > 0, is reformulated in Section 7 in terms of Bessel functions; this also allows to determine the analytic continuation of the map u → 0|T u,ε µν |0 to a meromorphic function on the whole complex plane, possessing a simple pole at u = 0. We compute the regular part of 0|T u,ε µν |0 at this point in Section 8 and subsequently evaluate the limit ε → 0 + of the resulting expression in Section 9; according to a general prescription of [31], these operations determine the renormalized stress-energy VEV 0|T µν |0 ren . The final expressions thus obtained for the non-vanishing components of 0|T µν |0 ren are reported in the conclusive Section 10; therein, we also analyze the asymptotic behavior of the renormalized stress-energy VEV in various regimes, discussing especially the expansions for small and large distances from the point impurity (see, respectively, subsections 10.1 and 10.2).…”

Integrating Gauge Fields in the ζ-Formulation of Feynman’s Path Integral

“…For example, if A is a positive operator whose spectrum σ(A) is discrete and free from accumulation points, then we could define A(z) := A z and ζ(A) is given by the meromorphic extension of z → λ∈σ(A)\{0} λ z (counting multiplicities); hence, giving rise to the name "operator ζ-function." This is precisely how Hawking [20] employed ζ-regularization, it has been used successfully in many physical settings (e.g., the Casimir effect, defining one-loop functional determinants, the stress-energy tensor, conformal field theory, and string theory [1,2,3,4,5,6,7,8,9,10,11,20,21,24,25,26,27,28,29,35,36,39]), and is related to Hadamard parametrix renormalization [14]. This approach has been fundamental for many subsequent developments as it allows for an effective Lagrangian to be defined [2] as well as heat kernel coefficients to easily be computed [3], and implies non-trivial extensions of the Chowla-Selberg formula [7].…”