However, in physical applications, the riemann zeta function appears much more often without any mention of rh. In fact, most people would say that this is some kind of mathematical trickery. We also give an extensive guide to the current literature on local zeta functions and its connections with other. There is a popular mathematical result which says that the sum of all natural numbers is 112. Lagarias and others published the lerch zeta function i. Noninvariant zetafunction regularization in quantum. Zetafunction method for regularization encyclopedia of.
For completeness, in appendices a and b we give some mathematical background on the analytic continuation of the polylogarithm, the function mainly involved in local zeta regularization for the case under investigation. In 6, the outcomes of 4 and 5 are combined to derive the casimir pressure on two parallel plates. Andrey todorov, the analogue of the dedekind eta function for cy threefolds, 2003 pdf. However, in some cases formal manipulations with zeta function regularization assuming linearity of sums lead to correct results.
The template sidebar with collapsible lists is being considered for merging. Everything is explained in detail, in particular the mathematical difficulties and tricky points, and several applications are given to show how the procedure works in practice, for example in the casimir effect, gravity and. Zeta function regularization is in general nonlinear. This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. In this paper, we would like to revisit the zeta function regularization approach for the evaluation of expectation values and their quantum. We apply zetafunction regularization to the kink and susy kink. A generic method in which you redistribute parts of the terms would violate this symmetry. The exponential of the derivative of the zeta function at n 0 n 0 also encodes the functional determinant of h h, a regularized version zeta function regularization of the naive and generally illdefined product of all eigenvalues.
Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. Renormalization of feynman amplitudes on manifolds by spectral. This looks very unintuitive to a first time observer. Heat kernels and zetafunction regularization for the mass of the. In the context of regularization in physics, zeta function regularization. More zeta functions for the riemann zeros 3 1 summary of previous results 1. What is the idea behind zeta function regularization. Polymeric quantum mechanics and the zeros of the riemann zeta function 7 conditions are directly related to the energy spectrum through the selfadjoint extension of the hamiltonian operat or. Here, the zetafunction regularization procedure consists in the following.
Analytic continuation leads to the finite renormalization of a quantum field theory. Spectral zeta functions of graphs and the riemann zeta function in the critical strip friedli, fabien and karlsson, anders, tohoku mathematical journal, 2017. It is known that not all summation methods are linear and stable. The selfadjointness condition provide a proper domain for the hamiltonian operator and the energy spectrum. In this paper, i will demonstrate an important fact about the zeros of the zeta function, and how it relates to the prime number theorem. This technique agrees with dimensional regularization where one generalises to n dimensions by adding extra flat dimensions. Basic misconceptions, misunderstandings and errors which keep appearing in important scientific journals. Abstract we have proposed a regularization technique and applied it to the euler product of the zeta functions in the part one. Zeta function regularization is a powerful method in perturbation theory, and this book is a comprehensive guide for the student of this subject. Moreover, in 1859 riemann gave a formula for a unique the socalled holo morphic extension of the function onto the entire complex plane c except s 1. On the estimation of the order of eulerzagier multiple zeta functions ishikawa, hideaki and matsumoto, kohji, illinois journal of mathematics, 2003. The riemann zeta function is defined for a complex variable s with res 1 as 8.
Submitted in partial fulllment of the requirements for the degree of master of science in quantum fields and undamenf tal orcesf of imperial college london. 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. It is thus no surprise that the preferred definition of the determinant for such operators is obtained through the corresponding zeta function. The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. The most popular of which for computing functional determinants is the zeta function regularization.
The riemann zeta function is defined as the analytic continuation of the function defined for. Naively, computation of the casimir pressure leads to infinite sums and therefore requires regularization. Pdf polymeric quantum mechanics and the zeros of the. Lerch zeta function, extending the complex variables viewpoint taken in part ii.
Like any other regularization, there are going to be possible ambiguities that unless treated consistently across a calculation will make the results of. The zetafunction regularization has a different feature than locality of counterterms that makes it special. An introduction to the theory of local zeta functions from. Before plunging into examples that involve more sophisticated zeta functions such will appear in later chapters we use this chapter to illustrate the regularization procedure in a few simple cases. By using the polymer representation, we obtain for both models, the associated polymeric quantum hamiltonians and the corresponding stationary wave functions. Leonhard euler considered the above series in 1740 for positive integer values of s, and later chebyshev extended the definition to re s 1. Zeta function regularization and physical applications of the quantum zeta functon are discussed in elizalde 1994. 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. Pdf zetafunction regularization is uniquely defined and. Multiloop zeta function regularization and spectral cutoff.
When one comes to specific calculations, the zeta function regularization method relies on the existence of simple formulas. It is our opinion that zeta function regularization is a very powerful tool at our disposal, as compared with other methods and that this issue deserves a careful investigation. For instance, this allows for the computation of the determinant of the laplace and dirac operators on a riemannian manifold, using the minakshisundarampleijel zeta function. The prime number theorem, hardys theorem on the zeros of. Zeta function and the theory of the distribution of primes, acta math 41, 119 1916 did much of the earlier work, by establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods g h hardy, divergent series clarendon press, oxford, 1949. Twovariable zeta functions and regularized products. Calculation of the riemann zetafunction from the physical.
The regularization for the zeta functions with physical. Several regulators can be used that yield the same answer. In mathematics and theoretical physics, zeta function regularization is a type of regularization or. What is the mathematical reason all regulators yield the same answer. Regularization of the casimir effect stack exchange. 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. Combining the two results one gets the formula 30 for the regularized. S 5 dual to the latitude 1 4bps wilson loop in n 4 superyangmills theory.
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 presented. He is actually doing zeta function regularization of the fibonacci series. If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide. To avoid certain ambiguities inherent to string theory on curved. As an example, perhaps not the best, we mention the problem of regularization of divergent expressions of field theorythe socalled zeta regularization of s. This is illustrated in a determination of the two loop renormalization group functions for in four dimensions. Zeta regularization method is to take for granted that for every s the identity 1 s n n s. Herein we concentrate on systems characterized by a nonrelativistic, realvalued one. Zeta function regularization can be thought of as analytic regularization with a special choice of the subtraction scheme. Local zeta regularization and the casimir effect pdf. In this paper we prove a regularized product expansion for the twovariable zeta functions of number.
Twovariable zeta functions and regularized products dedicated to kazuya kato christopher deninger received. Pdf zetafunction regularization of holographic wilson. 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. Combining both, we recover one of the relations which was proved. Iterated integrals in quantum field theory abstract. Here we consider the approach in which the determinant of a noncovariant operator is computed in the framework of the zeta function regularization and show that this procedure is equivalent to the noninvariant regularization of the green function at coincident points proposed by zz, and extensively used in. 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. In this paper that is the second part of the trilogy, we aim the nature of the nontrivial zero for the riemann zeta function. More generally, the zeta function approach can be used to regularize the whole energymomentum tensor in curved spacetime. As quantum vacuum fluctuations and zeta regularization the casimir effect and the cosmological constant qts5, valladolid, july 22, 2007 p.1403 570 1157 132 317 1341 145 718 868 591 653 495 222 1065 1222 474 32 1009 139 1207 605 68 62 1511 145 1500 1615 151 373 1167 1110 528 524 58 1049 345 638 944 485 638 1346