Is there a conceptual handwavy is fine explanation for zeta function regularization, and for how this expression in the zeta function is capturing the idea of a determinant. Zeta function regularization might be the most mathematically convenient way of computing the casimir force, but it is probably also the most confusing and least physical. Properties 1 and 2 below follow from the definition of zeta regularized products. Zetafunction regularization of sums over known spectrum.
Abstract aps in this paper, we develop a systematic technique to regularize double summations of landau levels and analytically evaluate the photon vacuum polarization at an external magnetic field. Justification of the zeta function renormalization in rigid string model. This looks very unintuitive to a first time observer. Pdf local zeta regularization and the casimir effect. It is regarded as mathematically simple and elegant but the isolation of the physical divergency is hidden in its analytic continuation. Like any other regularization, there are going to be possible ambiguities that unless treated consistently across a calculation will make the results of. The generalized zeta function can be expressed as a mellin transform of the kernel of the heat. There is a lot you can tell about a complex mathematical function just by looking at its behavior in some limited area. On the zetafunction regularization of a twodimensional series of epstenhurwitz type. Hawkings zeta function regularization procedure is shown to be rigorously and uniquely defined, thus putting and end to the spreading lore about different difficulties associated with it. When one comes to specific calculations, the zetafunction regularization method relies on the existence of simple formulas. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zetafunction regularization analytic continuation, asymptotic expansions, many of which appear here, in book format, for the first time are.
An example of zetafunction regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. Zeta regularizations and qanalogue of ring sine functions 3 zeta regularizations. Zetafunction regularization can be thought of as analytic regularization with a special choice of the subtraction scheme. In this paper that is the second part of the trilogy, we aim the nature of the nontrivial zero for the riemann zeta function which gives us another evidence to demonstrate the riemann hypotheses by way of the approximate functional equation. The idea behind the zeta regularization method is to take for granted that for every s the identity 1 s n n s. Analytic continuation of the hurwitz zeta function with. Noninvariant zetafunction regularization in quantum.
We study a generalization of the zeta regularization method applied to the. In this paper we deal with the case of qlinear factors fnx an. Thus, a direct comparison between their result and the one obtained by schwinger was missing. Zeta function regularization nicolas m robles department of theoretical physics imperial college london september 25th 2009 supervior. Mar 03, 2014 in this paper, we have developed a systematic technique to regularize double summations of landau levels and analytically evaluated the photon vacuum polarization at an external magnetic field.
Zeta regularization techniques with applications e. Zeta function regularization of the photon polarization. I have recently read about zeta function regularization, a way of ascribing values to functions having simple poles in a point and to divergent series. An example of zeta function regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. Frontiers zeta regularization of infinite products physics. The regularization for the zeta functions with physical. Zetafunction method for regularization encyclopedia of. This technique agrees with dimensional regularization where one generalises to ndimensions by adding extra. Equivalence of zeta function technique and abelplana.
The schwinger propertime formalism is used to define the zeta function. After introducing the zeta function rigorously we compute the partition function of bosonic and fermionic harmonic oscillators in quantum mechanics and study the generating func. The divergences in the series representation arise from the. The values obtained are the same as those obtained using ramanujans summation. We have proposed a regularization technique and applied it to the euler product of the zeta functions in the part one. Zeta function regularization can be thought of as analytic regularization with a special choice of the subtraction scheme. In the case of the zeta function, we have a nice definition for values of s greater than 1. The regularization for the zeta functions with physical applications ii 2 31 using the nth order relation of eq. In fact, most people would say that this is some kind of mathematical trickery. A main class of sequences that admit a zeta regularization is given by the spectrum of a class of linear operators. A brief survey of the zeta function regularization and multiplicative anomaly issues when the associated zeta function of fluctuation operator is the regular at the origin regular case as well as when it is singular at the origin singular case is presented. More generally, the zetafunction approach can be used to regularize the whole energymomentum tensor in curved spacetime. In theorem 2, we establish that the relationship given by voros.
These occur in a lot of one loop quantum corrections. Pdf zetafunction regularization is uniquely defined and well. In addition to penalizing large values of the solution vector x, for su ciently large values of the scalar this yields solutions that are sparse in terms of x having many values set to exactly 0. We propose regularization with the generalized zeta function in order to calculate oneloop corrections to the original action in quantum field theory. The riemann zeta function, which i dont understand conceptually almost as well, but which is heavily studied and so is clearly important and natural. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web. In theorem 1, we show that if j2xks is zeta regularizable, then so is yhk xk. Zeta regularization, spectral zeta functions, eisenstein series, and casimir energy 57 8. When one comes to specific calculations, the zeta function regularization method relies on the existence of simple formulas for obtaining the analytic continuation above. In theoretical physics, dimensional regularization is a method introduced by giambiagi and bollini as well as independently and more comprehensively by t hooft and veltman for regularizing integrals in the evaluation of feynman diagrams. The heat kernel can likewise be used to analyze the functional determinant in alternative regularization procedures, such as zetafunction.
How exactly does riemann zeta function regularization apply. Nov 25, 20 a main class of sequences that admit a zeta regularization is given by the spectrum of a class of linear operators. More generally, the zeta function approach can be used to regularize the whole energymomentum tensor in curved spacetime. Zeta regularization of infinite products physics frontiers.
What is the idea behind zeta function regularization. There is a popular mathematical result which says that the sum of all natural numbers is 112. Like any other regularization, there are going to be possible ambiguities that unless treated consistently across a calculation will make the results of a naiveminimal subtraction result incorrect. Related to ii, a multiplicative anomaly of zeta regularized products is studied in kv. How exactly does riemann zeta function regularization. By contrast, abelplana formula method permits explicit separation of. Sep 02, 2007 the zeta function regularization has a different feature than locality of counterterms that makes it special.
We compare the zeta spectral action with the usual cuto based spectral action and discuss its origin, predictive power, stressing the importance of the issue of. We study a generalization of the zeta regularization method applied to the case of the regularization of divergent integrals 0 xdxs. By contrast, abelplana formula method permits explicit separation of divergent terms. Regularization using the generalized zeta function in the. The theory also provides an under standing of the functional equations for simple and multiple gamma functions, the dedekind eta function, and the jacobi theta. Spectral zeta functions of graphs and the riemann zeta function in the critical strip friedli, fabien and karlsson, anders, tohoku mathematical journal, 2017. In this paper, whose aims are mainly pedagogical, we illustrate how to use the local zeta regularization to compute the stressenergy tensor of the casimir effect. Zeta regularizations and analogue of ring sine functions. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zetafunction regularization analytic continuation, asymptotic expansions, many of which appear here, in book format, for the first time are presented.
The zetafunction regularization has a different feature than locality of counterterms that makes it special. Zeta function regularization in casimir effect calculations. In mathematics and theoretical physics, zeta function regularization is a type of regularization or. Zeta function regularization free download as pdf file.
Zeta function regularization of path integrals in curved. A generic method in which you redistribute parts of the terms would violate this symmetry. Ten physical applications of spectral zeta functions. Dec 28, 2014 as we can see from the figure here, this is what the zeta function looks like. Zeta function regularization quantum field theory boson. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta function regularization analytic continuation, asymptotic expansions, many of which appear here, in book format, for the first time are. Although the riemann zetafunction is an analytic function with a deceptively simple definition, it keeps bouncing around almost randomly without settling down to some regular asymptotic pattern. This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. Pdf zeta function regularization in casimir effect calculations and. Singer was led to propose that strict positivity of the corresponding ricci tensor computable from the rigorously defined curvature tensor through a suitable zeta function regularization procedure could play a fundamental role in establishing that the associated schrodinger operator admits a spectral gap. We find that the tensor of vacuum polarization is split into not only longitudinal and transverse parts but. Understanding zeta function regularization mathoverflow. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. It is thus no surprise that the preferred definition of the determinant for such operators is obtained through the corresponding zeta function.
This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. We compare the zeta spectral action with the usual cuto based spectral action and discuss its origin, predictive power, stressing the importance of the issue of the three dimensionful fundamental constants, namely the. There are alternative ways of computing the casimir force that are le. Pdf zetafunction regularization is uniquely defined and. What links here related changes upload file special pages permanent link page. Zeta function regularization of path integrals in curved spacetime.
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some selfadjoint operators. The method of local zeta regularization arose from some ideas of hawking 6 and wald 14. The method is illustrated by calculation of the oneloop corrections to the maxwell lagrangian in qed with anomalous moments. The riemann zetafunction displays the essence of chaos in quantum mechanics, analytically smooth, and yet seemingly unpredictable. The final results are described by lerch transcendent. Equivalence of zeta function technique and abelplana formula. The riemann zeta function regularization is useful in a class of quantum processes where the contribution to the effective action is a sum of eigenvalues. Does the fact that zeta function regularization works in. Zeta function regularization is an effective method to extract physical significant quantities from infinite ones. Lectures on zeta functions, lfunctions and modular forms. On the estimation of the order of eulerzagier multiple zeta functions ishikawa, hideaki and matsumoto, kohji, illinois journal of mathematics, 2003.