Relevant bibliographies by topics / Zeta regularization

Academic literature on the topic 'Zeta regularization'

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Published: 9 March 2023
Last updated: 11 March 2023

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Journal articles on the topic "Zeta regularization"

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Lavrov, P. M. "Generalized zeta-function regularization." Soviet Physics Journal 30, no. 5 (May 1987): 359–62. http://dx.doi.org/10.1007/bf00900080.

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Allouche, Jean-Paul. "Zeta-regularization of arithmetic sequences." EPJ Web of Conferences 244 (2020): 01008. http://dx.doi.org/10.1051/epjconf/202024401008.

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Is it possible to give a reasonable value to the infinite product 1 × 2 × 3 × · · · × n × · · · ? In other words, can we define some sort of convergence of the finite product 1 × 2 × 3 × · · · × n when n goes to infinity? One way is to relate this product to the R√iemann zeta function and to its analytic continuation. This approach leads to: 1 × 2 × 3 × · · · × n × · · · = 2π. More generally the “zeta-regularization” of an infinite product consists of introducing a related Dirichlet series and its analytic continuation at 0 (if it exists). We will survey some properties of this generalized product and allude to applications. Then we will give two families of possibly new examples: one unifies and generalizes known results for the zeta-regularization of the products of Fibonacci, balanced and Lucas-balanced numbers; the other studies the zeta-regularized products of values of classical arithmetic functions. Finally we ask for a possible zeta-regularity notion of complexity.
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AKIRA, ASADA. "REGULARIZED CALCULUS: AN APPLICATION OF ZETA REGULARIZATION TO INFINITE DIMENSIONAL GEOMETRY AND ANALYSIS." International Journal of Geometric Methods in Modern Physics 01, no. 01n02 (April 2004): 107–57. http://dx.doi.org/10.1142/s0219887804000071.

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A method of regularization in infinite dimensional calculus, based on spectral zeta function and zeta regularization is proposed. As applications, a mathematical justification of appearance of Ray–Singer determinant in Gaussian Path integral, regularized volume form of the sphere of a Hilbert space with the determinant bundle, eigenvalue problems of regularized Laplacian, are investigated. Geometric counterparts of regularization procedure are also discussed applying arguments from noncommutative geometry.
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Reuter, M. "Chiral anomalies and zeta-function regularization." Physical Review D 31, no. 6 (March 15, 1985): 1374–85. http://dx.doi.org/10.1103/physrevd.31.1374.

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ELIZALDE, E., and A. ROMEO. "REGULARIZATION OF GENERAL MULTIDIMENSIONAL EPSTEIN ZETA-FUNCTIONS." Reviews in Mathematical Physics 01, no. 01 (January 1989): 113–28. http://dx.doi.org/10.1142/s0129055x89000055.

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We study expressions for the regularization of general multidimensional Epstein zeta-functions of the type [Formula: see text] After reviewing some classical results in the light of the extended proof of zeta-function regularization recently obtained by the authors, approximate but very quickly convergent expressions for these functions are derived. This type of analysis has many interesting applications, e.g. in any quantum field theory defined in a partially compactified Euclidean spacetime or at finite temperature. As an example, we obtain the partition function for the Casimir effect at finite temperature.
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Fermi, Davide, and Livio Pizzocchero. "Local Zeta Regularization and the Casimir Effect." Progress of Theoretical Physics 126, no. 3 (September 2011): 419–34. http://dx.doi.org/10.1143/ptp.126.419.

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Shiekh, A. Y. "Zeta-function regularization of quantum field theory." Canadian Journal of Physics 68, no. 7-8 (July 1, 1990): 620–29. http://dx.doi.org/10.1139/p90-093.

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Analytic continuation leads to the finite renormalization of a quantum field theory. This is illustrated in a determination of the two loop renormalization group functions for [Formula: see text] in four dimensions.
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Shiekh, A. Y. "Operator regularization of Feynman diagrams at multiloop order." Canadian Journal of Physics 89, no. 11 (November 2011): 1149–54. http://dx.doi.org/10.1139/p11-110.

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It has been previously believed not possible to use operator regularization with Feynman diagrams, but such an option would greatly simplify matters, as operator regularization is otherwise limited to the more complicated Schwinger approach. Further, operator regularization, unlike zeta function regularization, is not limited to one-loop order, and preserves supersymmetry, unlike dimensional regularization. In this work, we investigate operator regularization at the more probing two-loop order, and find not only compatibility but that the simplification associated with Feynman diagrams is retained.
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ZHAI, XIANG-HUA, YANG YANG, and JIE LAI. "FINITE TEMPERATURE CASIMIR EFFECT FOR PERFECTLY CONDUCTING PARALLEL PLATES IN (D + 1)-DIMENSIONAL SPACETIME." International Journal of Modern Physics: Conference Series 07 (January 2012): 202–8. http://dx.doi.org/10.1142/s2010194512004278.

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We study the finite temperature Casimir effect between perfectly conducting parallel plates in (D + 1)-dimensional spacetime by using zeta-function regularization technique. We get the analytical results for Casimir energy, Casimir free energy, Casimir entropy and Casimir pressure expressed by Riemann zeta function and Bessel function and give the asymptotic expressions for low and high temperature limits. In the case of D = 3, through mathematic transformation, we reproduce the standard results in the literature which is in most times obtained by using Green's function regularization technique.
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Lin, Rui-Hui, and Xiang-Hua Zhai. "Equivalence of zeta function technique and Abel–Plana formula in regularizing the Casimir energy of hyper-rectangular cavities." Modern Physics Letters A 29, no. 35 (November 17, 2014): 1450181. http://dx.doi.org/10.1142/s0217732314501818.

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Zeta function regularization is an effective method to extract physical significant quantities from infinite ones. It is regarded as mathematically simple and elegant but the isolation of the physical divergency is hidden in its analytic continuation. By contrast, Abel–Plana formula method permits explicit separation of divergent terms. In regularizing the Casimir energy for a massless scalar field in a D-dimensional rectangular box, we give the rigorous proof of the equivalence of the two methods by deriving the reflection formula of Epstein zeta function from repeatedly application of Abel–Plana formula and giving the physical interpretation of the infinite integrals. Our study may help with the confidence of choosing any regularization method at convenience among the frequently used ones, especially the zeta function method, without the doubts of physical meanings or mathematical consistency.
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Dissertations / Theses on the topic "Zeta regularization"

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Wang, Stephen. "Zeta Function Regularization and its Relationship to Number Theory." Digital Commons @ East Tennessee State University, 2021. https://dc.etsu.edu/etd/3895.

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While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to zeta function regularization and explore more fully the relationship between operators in physics and classical zeta functions of mathematics. In so doing, we highlight intriguing connections to number theory that arise.
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Filippi, Antonio. "The multiplicative anomaly and zeta-function regularization in quantum field theory." Thesis, Imperial College London, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.408158.

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Fermi, D. "A FUNCTIONAL ANALYTIC FRAMEWORK FOR LOCAL ZETA REGULARIZATION AND THE SCALAR CASIMIR EFFECT." Doctoral thesis, Università degli Studi di Milano, 2016. http://hdl.handle.net/2434/360961.

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It is developed a functional analytic framework allowing to formulate a rigorous implementation of zeta regularization for a canonically quantized scalar field, living on an arbitrary spatial domain and interacting with a classical background potential. This framework relies on the construction of an infinite scale of graded Hilbert spaces associated to the real powers of some given, positive self-adjoint operator. When the latter is a Schr\"odinger-type differential operator, this formulation provides a natural language to study the integral kernels related to a large class of operators, fulfilling minimal regularity requirements; particular attention is devoted to the regularity of these kernels and to the construction of their analytic continuations with respect to some parameters. Within this framework, complex powers of the elliptic operator giving rise to the Klein-Gordon equation are used to define a zeta-regularized version of the Wightman field whose pointwise evaluation is well-posed. This regularized field determines regularized local observables (such as the stress-energy tensor), whose vacuum expectation values can be expressed in terms of the above mentioned integral kernels. This allows to make contact with the theory of the Casimir effect. Renormalization is achieved by analytic continuation, which is proved to give finite results for the previously mentioned expectation values in most cases of interest. Finally, to exhibit the computational efficiency of the above methods, some explicit examples are discussed.
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Rondelli, Andrea. "Functional methods in quantum field theory." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/15839/.

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Iniziamo introducendo l'integrazione su manifold di Hilbert, tramite l'approssimazione dello spazio tangente alla varietà. Passiamo poi a descrivere due tecniche per regolarizzare integrali funzionali o di cammino quadratici (che presentano un laplaciano nell'azione): la regolarizzazione e rinormalizzazione tramite zeta function e il cutoff nel tempo proprio. Cerchiamo di confrontare i due diversi risultati (finiti) così ottenuti. Sussessivamente applichiamo l'integrazione funzionale agli integrali di cammino usando il formalismo della quantizzazione in qp-simboli ottenendo così un'ampiezza di probabilità. Infine iniziamo a sviluppare questi argomenti per le teorie di gauge. In particolare ci soffermeremo su vari aspetti geometrici dei campi di gauge, quali la connessione e la curvatura (usando il formalismo dei fibrati). In ultimo introduciamo l'integrazione funzionale per le teorie di gauge.
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Lu, Rongmin. "Regularized equivariant Euler classes and gamma functions." 2008. http://hdl.handle.net/2440/50479.

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We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus.
Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008
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Books on the topic "Zeta regularization"

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E, Elizalde, ed. Zeta regularization techniques with applications. Singapore: World Scientific, 1994.

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Fermi, Davide, and Livio Pizzocchero. Local Zeta Regularization and the Scalar Casimir Effect: A General Approach Based on Integral Kernels. World Scientific Publishing Co Pte Ltd, 2017.

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Book chapters on the topic "Zeta regularization"

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Williams, Floyd. "Zeta Regularization." In Topics in Quantum Mechanics, 317–20. Boston, MA: Birkhäuser Boston, 2003. http://dx.doi.org/10.1007/978-1-4612-0009-3_16.

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Dittrich, Walter. "Riemann’s Zeta Function Regularization." In SpringerBriefs in History of Science and Technology, 39–43. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91482-4_8.

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de Bernard, Wit, Proeyen Van Antoine, Majid Shahn, Reinhard Oehme, Duplij Steven, Marcinek Władysław, Duplij Steven, et al. "Regularization, zeta function method." In Concise Encyclopedia of Supersymmetry, 345. Dordrecht: Springer Netherlands, 2004. http://dx.doi.org/10.1007/1-4020-4522-0_456.

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Dittrich, Walter. "Riemann’s Zeta Function Regularization." In SpringerBriefs in History of Science and Technology, 39–43. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-61049-4_8.

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Yoshimoto, Masami. "Two Examples of Zeta-Regularization." In Analytic Number Theory, 379–93. Boston, MA: Springer US, 2002. http://dx.doi.org/10.1007/978-1-4757-3621-2_22.

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Voros, André. "Infinite Products and Zeta-Regularization." In Lecture Notes of the Unione Matematica Italiana, 9–22. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-05203-3_2.

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Elizalde, Emilio. "Applications of Zeta Function Regularization in QFT." In Quantum Field Theory Under the Influence of External Conditions, 122–30. Wiesbaden: Vieweg+Teubner Verlag, 1996. http://dx.doi.org/10.1007/978-3-663-01204-7_24.

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Cognola, Guido, and Sergio Zerbini. "Generalized Zeta Function Regularization and the Multiplicative Anomaly." In Springer Proceedings in Physics, 355–64. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-19760-4_33.

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Elizalde, Emilio. "Miscellaneous Applications Combining Zeta with Other Regularization Procedures." In Ten Physical Applications of Spectral Zeta Functions, 147–74. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-29405-1_7.

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Lohiya, D. "Zeta Function Regularization and Effective Action in Curved Spacetime." In Gravitation, Gauge Theories and the Early Universe, 315–41. Dordrecht: Springer Netherlands, 1989. http://dx.doi.org/10.1007/978-94-009-2577-9_16.

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Conference papers on the topic "Zeta regularization"

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Asada, Akira. "Zeta-regularization and calculus on infinite dimensional spaces." In GLOBAL ANALYSIS AND APPLIED MATHEMATICS: International Workshop on Global Analysis. AIP, 2004. http://dx.doi.org/10.1063/1.1814716.

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COGNOLA, GUIDO, and SERGIO ZERBINI. "GENERALISED ZETA-FUNCTION REGULARIZATION FOR SCALAR ONE-LOOP EFFECTIVE ACTION." In Proceedings of the MG10 Meeting held at Brazilian Center for Research in Physics (CBPF). World Scientific Publishing Company, 2006. http://dx.doi.org/10.1142/9789812704030_0289.

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Elizalde, Emilio. "On Zeta Regularization and Some of its Uses in Cosmology." In Fifth International Conference on Mathematical Methods in Physics. Trieste, Italy: Sissa Medialab, 2007. http://dx.doi.org/10.22323/1.031.0008.

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FILIPPI, ANTONIO. "NEW PHYSICS IN THE CHARGED RELATIVISTIC BOSE GAS USING ZETA-FUNCTION REGULARIZATION?" In Proceedings of the SEWM2000 Meeting. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812799913_0045.

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