Complete determination of the singularity structure of zeta functions

Elizalde, E.

doi:10.1088/0305-4470/30/8/019

Cited by 27 publications

(19 citation statements)

References 34 publications

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“…Then, the last term in the right hand side of (94) is nothing but the Wodzicki residue of A −2 [Wo84,Co88]. In other words, in the considered case, the last term in the right hand side of (94) is four times the Dixmier trace of A −2 [Di66] because of a known theorem by Connes [Co88,El97]. We conclude this section noticing that, differently to that argued in [Ev98], not only the local ζ function approach is consistent, but it also agrees with the point-splitting procedure and it is able to regularize and give a mathematically sensible meaning to formal identities handled by physicists 6 .…”

Section: 7mentioning

confidence: 99%

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

Moretti¹

1999

Communications in Mathematical Physics

108

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Some general properties of local ζ-function procedures to renormalize some quantities in D-dimensional (Euclidean) Quantum Field Theory in curved background are rigorously discussed for positive scalar operators −∆ + V (x) in general closed D-manifolds, and a few comments are given for nonclosed manifolds too. A general comparison is carried out with respect to the more known point-splitting procedure concerning the effective Lagrangian and the field fluctuations. It is proven that, for D > 1, the local ζ-function and point-splitting approaches lead essentially to the same results apart from some differences in the subtraction procedure of the Hadamard divergences. It is found that the ζ function procedure picks out a particular term w 0 (x, y) in the Hadamard expansion. The presence of an untrivial kernel of the operator −∆ + V (x) may produce some differences between the two analyzed approaches. Finally, a formal identity concerning the field fluctuations, used by physicists, is discussed and proven within the local ζ-function approach. This is done also to reply to recent criticism against ζ function techniques.Introduction.

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Section: 7mentioning

confidence: 99%

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

Moretti¹

1999

Communications in Mathematical Physics

108

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“…(14) is again the recurrent formula (17). More precisely, what is obtained in the limit is the reflected formula, which one gets after using Epstein zeta function's reflection Γ(s)Z(s; A)…”

Section: Extended Epstein Zeta Function In P Dimensionsmentioning

confidence: 97%

“…Summing up, we have thus checked that Eq. (14) is valid for any q ≥ 0, since it contains in a hidden way, for q = 0, the recurrent expression (17).…”

Section: Extended Epstein Zeta Function In P Dimensionsmentioning

confidence: 99%

“…The massless case is also obtained, with the same specifications, from the corresponding formula Eq. (17). In both cases no extra calculation needs to be done, and the physical results follow from a mere identification of the components of the matrix A with those of the metric tensor of the manifold in question 32 .…”

Section: Truncated Epstein Zeta Function In D =mentioning

confidence: 99%

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Zeta Function Regularization in Casimir Effect Calculations and J. S. Dowker's Contribution

Elizalde

2012

Int. J. Mod. Phys. Conf. Ser.

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A summary of relevant contributions, ordered in time, to the subject of operator zeta functions and their application to physical issues is provided. The description ends with the seminal contributions of Stephen Hawking and Stuart Dowker and collaborators, considered by many authors as the actual starting point of the introduction of zeta function regularization methods in theoretical physics, in particular, for quantum vacuum fluctuation and Casimir effect calculations. After recalling a number of the strengths of this powerful and elegant method, some of its limitations are discussed. Finally, recent results of the so called operator regularization procedure are presented.

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“…The Wodzicki residue continues to make sense for ΨDOs of arbitrary order and, even if the symbols a j (x, ξ ), j < m, are not invariant under coordinate choice, their integral is, and defines a trace. All residua at poles of the zeta function of a ΨDO can be easily obtained from the Wodzciki residue [13].…”

Section: The Wodzicki Residuementioning

confidence: 99%

On Zeta Regularization and Some of its Uses in Cosmology

Elizalde¹

2007

Proceedings of Fifth International Conference on Mathematical Methods in Physics — PoS(IC2006)

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Zeta regularization has been proven to be a fine, powerful and very reliable tool for the regularization of the vacuum energy density in ideal situations. With the additional help of the Hadamard calculus, we have shown it to yield also finite and physically meaningful answers in more involved cases, as when imposing physical boundary conditions in two-and higher-dimensional surfaces, being then able to mimic in a convenient way other ad hoc cut-offs, as non-zero depths. These recent developments are described in the first part of this presentation. Recently, those techniques have also been used in calculations of the contribution of the vacuum energy of the quantum fields which are presumably pervading the universe, to the cosmological constant. Naive calculations of the absolute contributions of all known fields lead to a value which is off by roughly 120 orders of magnitude, as compared with the results obtained from observational fits, what is known as the new cosmological constant problem. This is very difficult to solve and we address here such issue only indirectly, by means of some specific examples.

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Complete determination of the singularity structure of zeta functions

Cited by 27 publications

References 34 publications

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

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

Zeta Function Regularization in Casimir Effect Calculations and J. S. Dowker's Contribution

On Zeta Regularization and Some of its Uses in Cosmology

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