One speaks of analytic regularization or zeta function regularization e. 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. Understanding zeta function regularization mathoverflow. Zeta function regularization of path integrals in curved spacetime. 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. Zeta function regularization quantum field theory boson. Zeta function regularization imperial college london. Dec 28, 2014 we can interpret this result by looking at the zeta function. Zetafunction method for regularization encyclopedia of.
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. Zeta function regularization, anomaly and complex mass term. The great advantage of the zeta function regularization is to provide an invariant regularization scheme as the eigenvalues. We consider a variant expression to regularize the euler product representation of the zeta functions, where we mainly apply to that of the riemann zeta function in this paper. However, in some cases formal manipulations with zeta function regularization assuming linearity of sums lead to correct results.
M 99, bcemz 03, section 2 if a feynman propagatorgreens function for a bosonic field, which is naively given by the expression tr 1 h tr\left\frac1h\right for h h the given wave operatorlaplace operator is made well defined by interpreting it as. The definition section of this article doesnt seem to give a definition of zeta function regularization. Zeta function regularization of path integrals in curved. 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. Twoparametric zeta function regularization in superstring theory lubo smotl october 5th, 1995 faculty of mathematics and physics1, charles university, prague, czech republic abstract in this paper some quite simple examples of applications of the zeta function regularization to superstring theories are presented. This technique agrees with dimensional regularization where one generalises to n dimensions by adding extra flat dimensions. Spectral zeta functions of graphs and the riemann zeta function in the critical strip friedli, fabien and karlsson, anders, tohoku mathematical journal, 2017. Searching on wikipedia, we have the page zeta function regularization which says. What is the idea behind zeta function regularization. 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. The euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are setwise coprime. Using complex analysis, we deduced that zeta function is consistently defined for every complex number. Hawkings zetafunction regularization procedure 1977 is shown to be rigorously and uniquely defined, thus putting an end to the spreading lore about different.
We consider several examples and show why this happens. 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. On the zeta function regularization as a simple illustration of the zeta function method, let us consider a massless scalar eld on a two dimensional spacetime of the form r s1, the length of the circle being a. It has evident analogs for all other zeta functions for some of which it has been proven. This technique agrees with dimensional regularization where one generalises to ndimensions by adding extra. Ten physical applications of spectral zeta functions emilio. Among the different methods, zetafunction regularization which is obtained by analytic continuation in the complex plane of the zetafunction of the relevant physical operator in each case might well be the most beautiful of all. The idea behind the zeta regularization method is to take for granted that for every s the identity 1 s n n s. The riemann zeta function david jekel june 6, 20 in 1859, bernhard riemann published an eightpage paper, in which he estimated \the number of prime numbers less than a given magnitude using a certain meromorphic function on c. Home journals canadian journal of physics list of issues volume 68, number 78, july 1990 zetafunction regularization of quantum field theory article previous toc next. Andrey todorov, the analogue of the dedekind eta function for cy threefolds, 2003 pdf.
On the estimation of the order of eulerzagier multiple zeta functions ishikawa, hideaki and matsumoto, kohji, illinois journal of mathematics, 2003. Menotti physics letters b 650 2007 432439 such a value is usually computed by the heat kernel technique,14, which we shall also employ in the following. Generalized zeta function will be 0 log n a s n n s. 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. I have never opened a math textbook and seen definition of derivative. As a simple illustration of the zeta function method, let us consider a massless scalar field on a twodimensional spacetime of the form r. Pdf zetafunction regularization of quantum field theory. The regularization itself is identical to that of the zeta function of the summation expression, but the nonuse of the moebius function enable us to confirm a finite behavior of residual terms. Zeta function regularization is a powerful method in perturbation theory, and this book is a comprehensive guide for the student of this subject. The regularization for the zeta functions with physical applications ii 2 31 using the nth order relation of eq.
Zeta function regularization in casimir effect calculations. Physics letters b 291 1992 2631 northholland physics letters b zeta function regularization for kaluzaklein finite temperature theories with chemical potentials andrei a. Nov 25, 20 a main class of sequences that admit a zeta regularization is given by the spectrum of a class of linear operators. The riemann hypothesis conjectures a characterization of the roots of this zeta function and is regarded as one of the outstanding problems in mathematics. May 31, 2012 if zeta function regularization is used and a complex mass term considered for fermions, the phase does not appear in the fermion determinant. 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. We have proposed a regularization technique and apply it to the euler product of zeta functions in the part one. Zetafunction regularization of quantum field theory. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. Noninvariant zetafunction regularization in quantum. Bytsenko, luciano vanzo a,b,2 and sergio zerbini a. In the context of regularization in physics, zeta function regularization. Zeta function regularization can be thought of as analytic regularization with a special choice of the subtraction scheme.
Multiloop zeta function regularization and spectral cutoff. Zeros of partial summs of the riemann zeta function borwein, peter, fee, greg, ferguson. The riemann zeta function, which i dont understand conceptually almost as well, but which is heavily studied and so is clearly important and natural. The riemann zeta function university of washington.
This is illustrated in a determination of the two loop renormalization group functions for in four dimensions. Pdf hawkings zeta function regularization procedure is shown to be rigorously and uniquely defined, thus putting. Pdf zetafunction regularization is uniquely defined and. So if that were true, then we just extrapolate the curve to get the value of the function at s1. Analytic continuation leads to the finite renormalization of a quantum field theory. 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. As a consequence, the zeta function does not provide a finite regularization of the effective action. Everything is explained in detail, in particular the mathematical difficulties and tricky points, and several applications are given to show how the. Zeta function regularization free download as pdf file. Pdf zetafunction regularization is uniquely defined and well. A generic method in which you redistribute parts of the terms would violate this symmetry.
One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. Operator regularization of quantum field theory 1, 2. Zetafunction regularization for kaluzaklein finite. Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. More generally, the zeta function approach can be used to regularize the whole energymomentum tensor in curved spacetime. For the case of the eigenvalues of a simple quantum harmonic oscillator in one dimension 10. The regularization for the zeta functions with physical. This is not a drawback of the regularization, which can recognize the phase through source terms, as demonstrated by the anomaly equation which is explicitly derived here for a complex mass term. This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime.
Use of this method yields, for instance, the vacuum energy corresponding to a quantum physical system with. Zeta function regularization is in general nonlinear. In mathematics and theoretical physics, zeta function regularization is a type of regularization or. The values obtained are the same as those obtained using ramanujans summation. When we do that, this is the value we get when s is 1. A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the \zetafunction approach is. Apr 08, 2020 we find that the spectral functions of this operator present an interesting behavior. Frontiers zeta regularization of infinite products physics. Sep 02, 2007 the zeta function regularization has a different feature than locality of counterterms that makes it special. Multiloop zeta function regularization and spectral cuto.
This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. Zeta function regularization equates two definitions of the zeta function at domain values where they contradict where the dirichlet series definition is divergent and riemanns definition is convergent. In the singular case, new results for the multiplicative anomaly are presented. Twoparametric zeta function regularization in superstring theory.
1161 1353 1147 379 1404 1093 1242 1330 1289 1516 1426 1255 195 1341 984 859 494 1460 402 1470 1132 308 1481 763 625 417 1246 316 711 303 1093 783 867 310 1224 1470 813 147 727 37 43 286 319 781 211