This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. Pdf hawkings zeta function regularization procedure is shown to be rigorously and uniquely defined, thus putting and end to the spreading. On the other hand, the dirichlet series diverges when the real part of s is less than or equal to 1, so, in particular, the. The latter series is an example of a dirichlet series.
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. Zeros of partial summs of the riemann zeta function borwein. When the real part of s is greater than 1, the dirichlet series converges, and its sum is the riemann zeta function. 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. Dr arttu rajantie submitted in partial fulfillment of the requirements for the degree of master of science in quantum fields and fundamental forces of imperial college london abstract the technique of zeta. Zeta function regularization imperial college london. On the estimation of the order of eulerzagier multiple zetafunctions ishikawa, hideaki and matsumoto, kohji, illinois journal of mathematics, 2003.
A generic method in which you redistribute parts of the terms would violate this symmetry. 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. This is precisely what we report in the present note. Zeta function regularization and physical applications of the quantum zeta functon are discussed in elizalde 1994.
Zetafunction regularization 2 is a powerful and increasingly liked method, but to our knowledge it has never been completely applied to electromagnetic e. 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. Local zeta regularization and the casimir effect arxiv. Previous results on the quantum zeta function include those of voros 1980 and berry 1986, who made use of various identities involving quantum theoretical constructs such as resolvent operators and green functions. Spectral zeta functions of graphs and the riemann zeta function in the critical strip friedli, fabien and karlsson, anders, tohoku mathematical journal, 2017. This is illustrated in a determination of the two loop renormalization group functions for in four dimensions. 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. Such a regularization is called the zeta function regularization since the vacuum energy is defined by 2s 1 e0. Zetafunction regularization of quantum field theory. The exposition is selfcontained, and required a preliminary knowledge of only the ele. Zetafunction regularization can be thought of as analytic regularization with a special choice of the subtraction scheme. Zeta function regularization quantum field theory boson. Zeta function regularization nicolas m robles department of theoretical physics imperial college london september 25th 2009 supervior.
Zeta function regularization, anomaly and complex mass term. In the context of regularization in physics, zeta function regularization is a methodprescription for extracing finite values for traces of powers of laplace operatorsdirac operators by considering s s powers for all values of s s in the complex plane. Zeta regularization techniques with applications e. Zetafunction regularization of quantum field theory article pdf available in canadian journal of physics 6878. Zeta regularization method is to take for granted that for every s the identity 1 s n n s.
Zetafunction method for regularization encyclopedia of. What is the idea behind zeta function regularization. Zeta function regularization is in general nonlinear. Zeta function regularization in casimir effect calculations. The casimir effect is the attraction of two neutral metallic plates and was initially predicted theoretically in 1948. Zeta function regularization is a technique that can be used to derive casimir energies associated with vacuum energies present between atoms. The zetafunction regularization has a different feature than locality of counterterms that makes it special. The zetaregularization is useful as it can often be used in a way such that the various symmetries of the physical system are preserved. Zeta function regularization, anomaly and complex mass. Basic misconceptions, misunderstandings and errors which keep appearing in important scientific journals. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. Yeah, im jealous the riemann hypothesis is named after the fact that it is a hypothesis, which, as we all know, is the largest of the three sides of a right triangle. We consider several examples and show why this happens. Zeta function of degenerate plane curve singularity quy thuong, le, osaka journal of mathematics, 2012.
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. As it so happens, it still works if s is a complex number as long as the real part of s is greater than 1. Zetafunction regularization of holographic wilson loops. The idea behind the zeta regularization method is to take for granted that for every s the identity 1 s n n s. Zeta function regularization quantum field theory complex. In the case of the zeta function, we have a nice definition for values of s greater than 1. 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. The zeta function is an important function in mathematics. On the estimation of the order of eulerzagier multiple zeta functions ishikawa, hideaki and matsumoto, kohji, illinois journal of mathematics, 2003. Complete zetafunction approach to the electromagnetic. Properties 1 and 2 below follow from the definition of zeta regularized products. Zeta function regularization can be thought of as analytic regularization with a special choice of the subtraction scheme. However, in some cases formal manipulations with zeta function regularization assuming linearity of sums lead to correct results.
Local casimir effect, renormalization, zeta regularization. Zeta function regularization free download as pdf file. Wodzicki residue, multiplicative or noncommutative. Hawkings zetafunction regularization procedure 1977 is shown to be rigorously and uniquely defined, thus putting an end to the spreading lore about different. Fluctuations of quantum fields via zeta function regularization. For the case of the eigenvalues of a simple quantum harmonic oscillator in one dimension 10. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. In mathematics and theoretical physics, zeta function regularization is a type of regularization or.
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. They are also essential in supersymmetry calculations. 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. P is expressed in terms of the generalized zeta function as x 1.
Noninvariant zetafunction regularization in quantum. 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. A main class of sequences that admit a zeta regularization is given by the spectrum of a class of linear operators. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. Critical dimension of bosonic string theory and zeta. During a description of zeta function regularization procedures in a book by elizalde 1995 that contains several detailed. If zeta function regularization is used and a complex mass term considered for fermions, the phase does not appear in the fermion determinant. 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. Analytic continuation leads to the finite renormalization of a quantum field theory. Zeta function regularization in casimir effect calculations and stuart.
The goal is to define the determinant of a positive self. In the context of regularization in physics, zeta function regularization is a. The riemann zeta function is defined as the analytic continuation of the function defined for. 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. Generalized zeta function will be 0 log n a s n n s. The great advantage of the zetafunction regularization is to provide an invariant regularization scheme as the eigenvalues. 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. Use of this method yields, for instance, the vacuum energy corresponding to a quantum physical system with. Pdf zeta function regularization and vacuum energy. Searching on wikipedia, we have the page zeta function regularization which says. 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. P is the zeta function corresponding to an elliptic boundary value. 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.
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. However, the formula 2 cannot be applied anymore if the real part. However, in some cases formal manipulations with zeta function regularization assuming. This technique agrees with dimensional regularization where one generalises to n dimensions by adding extra flat dimensions. In the context of regularization in physics, zeta function regularization is a methodprescription for extracing finite values for traces of powers of laplace operatorsdirac operators by considering s s powers for all values of s s in the complex plane where the naive trace does make sense and then. 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 regularization for the zeta functions with physical.
Zeta function regularization of path integrals in curved. Dec 28, 2014 in the case of the zeta function, we have a nice definition for values of s greater than 1. 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. Sep 02, 2007 the zeta function regularization has a different feature than locality of counterterms that makes it special. Zeta function regularization is used in conformal field theory, renormalization and in fixing the critical spacetime dimension of string theory. G h hardy, divergent series clarendon press, oxford, 1949.
Nicolas m robles department of theoretical physics imperial college london september 25th 2009 supervior. I think i found a flaw in riemann zeta function regularization. Zetafunction regularization is used in conformal field theory, renormalization and in fixing the critical spacetime dimension of string theory. Understanding zeta function regularization mathoverflow. The prime number theorem, hardys theorem on the zeros of. The zeta regularization is useful as it can often be used in a way such that the various symmetries of the physical system are preserved. The regularization for the zeta functions with physical applications ii 2 31 using the nth order relation of eq. This technique agrees with dimensional regularization where one generalises to ndimensions by adding extra. In theorem 1, we show that if j2xks is zeta regularizable, then so is yhk xk. Regularization and renormalization procedures are essential issues in contemporary physics without which it would simply not exist, at least in the form known today 2000.
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. In this paper, we would like to revisit the zeta function regularization approach for the evaluation of expectation values and their quantum. Nov 25, 20 a main class of sequences that admit a zeta regularization is given by the spectrum of a class of linear operators. Moreover, in 1859 riemann gave a formula for a unique the socalled holomorphic extension of the function onto the entire complex plane c except s 1. Now lets go ahead and extend this definition to complex numbers. 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. Pdf zetafunction regularization is uniquely defined and well.