Summability methods sFrom Wikipedia, the free encyclopediaIn mathematics and theoretical physics, zetafunctionregularization is a type of regularization or summabilitymethod that assigns nite values to divergent sums or products, and in particular can be used to dene determinantsand traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but hasits origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.1 DenitionThere are several dierent summation methods called zeta function regularization for dening the sum of a possiblydivergent series a1 + a2 + ....One method is to dene its zeta regularized sum to be A(1) if this is dened, where the zeta function is dened forRe(s) large byA(s) =1as1+1as2+ if this sum converges, and by analytic continuation elsewhere.In the case when an = n, the zeta function is the ordinary Riemann zeta function, and this method was used by Eulerto sum the series 1 + 2 + 3 + 4 + ... to (1) = 1/12.Other values of s can also be used to assign values for the divergent sums (0)=1 + 1 + 1 + 1 + ... = 1/2, (2)=1 +4 + 9 + ... = 0 and in general (s) =n=1ns= 1s+2s+3s+. . . = Bs+1s+1, where B is a Bernoulli number.[1]Hawking (1977) showed that in at space, in which the eigenvalues of Laplacians are known, the zeta functioncorresponding to the partition function can be computed explicitly. Consider a scalar eld contained in a large boxof volume V in at spacetime at the temperature T=1. The partition function is dened by a path integral overall elds on the Euclidean space obtained by putting =it which are zero on the walls of the box and which areperiodic in with period .In this situation from the partition function he computes energy, entropy and pressureof the radiation of the eld . In case of at spaces the eigenvalues appearing in the physical quantities are generallyknown, while in case of curved space they are not known: in this case asymptotic methods are needed.Another method denes the possibly divergent innite product a1a2.... to be exp(A(0)). Ray & Singer (1971)used this to dene the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifoldin their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace of As.Minakshisundaram & Pleijel (1949) showed that if A is the Laplacian of a compact Riemannian manifold then theMinakshisundaramPleijel zeta function converges and has an analytic continuation as a meromorphic function to allcomplex numbers, and Seeley (1967) extended this to elliptic pseudo-dierential operators Aon compact Riemannianmanifolds. So for such operators one can dene the determinant using zeta function regularization. See "analytictorsion.Hawking (1977) suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta functionregularization in order to calculate the partition functions for thermal graviton and matters quanta in curved back-ground such as on the horizon of black holes and on de Sitter background using the relation by the inverse Mellintransformation to the trace of the kernel of heat equations.12 4 RELATION TO DIRICHLET SERIES2 ExampleThe rst example in which zeta function regularization is available appears in the Casimir eect, which is in a atspace with the bulk contributions of the quantum eld in three space dimensions. In this case we must calculate thevalue of Riemann zeta function at 3, which diverges explicitly. However, it can be analytically continued to s=3where hopefully there is no pole, thus giving a nite value to the expression. A detailed example of this regularizationat work is given in the article on the detail example of the Casimir eect, where the resulting sum is very explicitlythe Riemann zeta-function (and where the seemingly legerdemain analytic continuation removes an additive innity,leaving a physically signicant nite number).An example of zeta-function regularization is the calculation of the vacuum expectation value of the energy of aparticle eld in quantum eld theory. More generally, the zeta-function approach can be used to regularize the wholeenergy-momentum tensor in curved spacetime.The unregulated value of the energy is given by a summation over the zero-point energy of all of the excitation modesof the vacuum:0|T00|0 =n|n|2Here, T00 is the zeroth component of the energy-momentum tensor and the sum (which may be an integral) isunderstood to extend over all (positive and negative) energy modesn ; the absolute value reminding us that theenergy is taken to be positive. This sum, as written, is usually innite ( n is typically linear in n). The sum may beregularized by writing it as0|T00(s)|0 =n|n|2|n|swhere s is some parameter, taken to be a complex number. For large, real s greater than 4 (for three-dimensionalspace), the sum is manifestly nite, and thus may often be evaluated theoretically.The zeta-regularization is useful as it can often be used in a way such that the various symmetries of the physicalsystem are preserved. Zeta-function regularization is used in conformal eld theory, renormalization and in xingthe critical spacetime dimension of string theory.3 Relation to other regularizationsWe can ask if are there any relation to the dimensional regularization originated by the Feynman diagram. But nowwe may say they are equivalent each other. ( see .) However the main advantage of the zeta regularization is thatit can be used whenever the dimensional regularization fails, for example if there are matrices or tensors inside thecalculations i,j,k4 Relation to Dirichlet seriesZeta-function regularization gives a nice analytic structure to any sums over an arithmetic function f(n). Such sumsare known as Dirichlet series. The regularized formf(s) =n=1f(n)nsconverts divergences of the suminto simple poles on the complex s-plane. In numerical calculations, the zeta-functionregularization is inappropriate, as it is extremely slowto converge. For numerical purposes, a more rapidly convergingsum is the exponential regularization, given by3F(t) =n=1f(n)etn.This is sometimes called the Z-transform of f, where z = exp(t). The analytic structure of the exponential andzeta-regularizations are related. By expanding the exponential sum as a Laurent seriesF(t) =aNtN+aN1tN1+ one nds that the zeta-series has the structuref(s) =aNs N+ .The structure of the exponential and zeta-regulators are related by means of the Mellin transform. The one may beconverted to the other by making use of the integral representation of the Gamma function:(s + 1) =0xsexdxwhich lead to the identity(s + 1) f(s + 1) =0tsF(t) dtrelating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurentseries.5 Heat kernel regularizationThe sumf(s) =nanes|n|is sometimes called a heat kernel or a heat-kernel regularized sum; this name stems from the idea that the n cansometimes be understood as eigenvalues of the heat kernel.In mathematics, such a sum is known as a generalizedDirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the LaplaceStieltjestransform, in thatf(s) =0estd(t)where (t) is a step function, with steps of an at t= |n| .A number of theorems for the convergence of such aseries exist. For example, by the Hardy-Littlewood Tauberian theorem, ifL = limsupnlog |nk=1ak||n|then the series for f(s) converges in the half-plane (s) > L and is uniformly convergent on every compact subsetof the half-plane (s) > L . In almost all applications to physics, one has L = 04 8 REFERENCES6 HistoryMuch of the early work establishing the convergence and equivalence of series regularized with the heat kernel and zetafunction regularization methods was done by G.H. Hardy and J. E. Littlewood in 1916 and is based on the applicationof the CahenMellin integral. The eort was made in order to obtain values for various ill-dened, conditionallyconvergent sums appearing in number theory.In terms of application as the regulator in physical problems, before Hawking (1977), J. Stuart Dowker and RaymondCritchley in 1976 proposed a zeta-function regularization method for quantum physical problems. Emilio Elizaldeand others have also proposed a method based on the zeta regularization for the integrals axmsdx , here xsisa regulator and the divergent integral depends on the numbers (s m) in the limit s 0 see renormalization. Alsounlike other regularizations such as dimensional regularization and analytic regularization, zeta regularization has nocounterterms and gives only nite results.7 See alsoGenerating functionPerrons formulaRenormalization1 + 1 + 1 + 1 + 1 + 2 + 3 + 4 + Analytic torsionRamanujan summationMinakshisundaramPleijel zeta functionZeta function (operator)8 References^ TomM. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag New York.(See Chapter 8.)"^ A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti and S. Zerbini, Analytic Aspects of Quantum Fields,World Scientic Publishing, 2003, ISBN 981-238-364-6^ G.H. Hardy and J.E. Littlewood, Contributions to the Theory of the Riemann Zeta-Function and the Theoryof the Distribution of Primes, Acta Mathematica, 41(1916) pp. 119196. (See, for example, theorem 2.12)Hawking, S. W. (1977), Zeta function regularization of path integrals in curved spacetime, Communicationsin Mathematical Physics 55 (2): 133148, Bibcode:1977CMaPh..55..133H, doi:10.1007/BF01626516, ISSN0010-3616, MR 0524257^ V. Moretti, Direct z-function approach and renormalization of one-loop stress tensor in curved spacetimes,Phys. Rev.D 56, 7797 (1997).Minakshisundaram, S.; Pleijel, . (1949), Some properties of the eigenfunctions of the Laplace-operator onRiemannian manifolds, Canadian Journal of Mathematics 1: 242256, doi:10.4153/CJM-1949-021-5, ISSN0008-414X, MR 0031145Ray, D. B.; Singer, I. M. (1971), "R-torsion and the Laplacian on Riemannian manifolds., Advances in Math7: 145210, doi:10.1016/0001-8708(71)90045-4, MR 02953815Garca Moreta, Jos Javier http://prespacetime.com/index.php/pst/article/view/498 The Application of ZetaRegularization Method to the Calculation of Certain Divergent Series and Integrals Rened Higgs, CMB fromPlanck, Departures in Logic, and GR Issues & Solutions vol 4 N 3 prespacetime journal http://prespacetime.com/index.php/pst/issue/view/41/showTocHazewinkel, Michiel, ed. (2001), Zeta-function method for regularization, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4Seeley, R. T. (1967), Complex powers of an elliptic operator, in Caldern, Alberto P., Singular Integrals(Proc. Sympos. Pure Math., Chicago, Ill., 1966), Proceedings of Symposia in Pure Mathematics 10, Providence,R.I.: Amer. Math. Soc., pp. 288307, ISBN 978-0-8218-1410-9, MR 0237943^ J.S. Dowker and R. Critchley, Eective Lagrangian and energy-momentum tensor in de Sitter space, Phys.Rev.D 13, 3224 (1976).[1] Tao, Terence (10 April 2010). The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variableanalytic continuation.6 9 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES9 Text and image sources, contributors, and licenses9.1 Text Zeta functionregularization Source: https://en.wikipedia.org/wiki/Zeta_function_regularization?oldid=670941509 Contributors: MichaelHardy, TakuyaMurata, Charles Matthews, GreatWhiteNortherner, Giftlite, Tom harrison, Dratman, Lumidek, Linas, Salix alba, R.e.b.,Incnis Mrsi, Silly rabbit, Nberger, QFT, CRGreathouse, Karl-H, Headbomb, David Eppstein, Leyo, Policron, Lartoven, Brews ohare,Versus22, Addbot, Luckas-bot, Ptbotgourou, Niout, Intractable, Xqbot, Tom.Reding, Dewritech, ZroBot, Chris81w, Crown Prince,Bibcode Bot, Trevayne08, 123957a, BattyBot, Enyokoyama, Reak spoughly, Mark viking, Ardehali and Anonymous: 319.2 Images9.3 Content license Creative Commons Attribution-Share Alike 3.0