One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The great advantage of the zeta function regularization is to provide an invariant regularization scheme as the eigenvalues. However, in some cases formal manipulations with zeta function regularization assuming linearity of sums lead to correct results. Appendix a2 has been rewritten to make the calculations more transparent. Apr 30, 2019 one speaks of analytic regularization or zeta function regularization e. Feb 05, 2012 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 of the sum of all natural.
Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. Zeta function regularization of path integrals in curved. A main class of sequences that admit a zeta regularization is given by the spectrum of a class of linear operators. Less fundamental applications of zeta regularization. In theorem 1, we show that if j2xks is zeta regularizable, then so is yhk xk. Pdf zetafunction regularization is uniquely defined and well. Zetafunction regularization of sums over known spectrum. Zetafunction method for regularization encyclopedia of. Xin terms of the zeta function associated with the corresponding laplace operators on x. Zeta regularization techniques with applications world scientific.
Zetafunction regularization of holographic wilson loops. Field theory world scientific lecture notes in physics. Reassessing riemanns paper on the number of primes less. An example of zetafunction regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. Lectures on zeta functions, lfunctions and modular forms. I have seen zeta function regularization in string theory. We now set up the framework for calculating this anomaly. Euclidean partition function, regularised by zeta function techniques, reads 5 lnz. Zeta function regularization is a powerful method in perturbation theory, and this book is a comprehensive guide for the student of this subject. Motivated by the study of quantum fields in a friedmanrobertsonwalker frw spacetime, the oneloop effective action for a.
These sums with k1 or k3 appears in several calculations. The latter series is an example of a dirichlet series. Modular forms of nonpositive weight, the entropy of a zero weight form. Ten physical applications of spectral zeta functions emilio.
Zeta function regularization and physical applications of the. Zetafunction regularization of determinants and traces. Pdf fluctuations of quantum fields via zeta function. 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.
However, the formula 2 cannot be applied anymore if the real part. Effective action for scalar fields and generalized zeta function regularization. Twoparametric zeta function regularization in superstring theory. 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.
Stack overflow for teams is now free for up to 50 users, forever. Zeta function regularization of path integrals 7 note that the volume element which appears in the 2. Generalized zeta function will be 0 log n a s n n s. Noninvariant zetafunction regularization in quantum. Pdf effective action for scalar fields and generalized zeta. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. These latter methods are quite useful in the covariant study of quantum field theories on nontrivial backgrounds with or without boundaries as well as in the study of casimir energy. Zetafunction regularization, the multiplicative anomaly and the wodzicki residue. 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. Epstein zeta meets gravity in extra dimensions 66 9. 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. The definition section of this article doesnt seem to give a definition of zeta function regularization. Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and ramanujan summation, with its shortcut to the eulermaclaurin formula. By contrast, abelplana formula method permits explicit separation of.
Pdf zetafunction regularization of quantum field theory. Ice zeta function regularization of quantum vacuum. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. Zetafunction regularization of holographic wilson loops jeremas aguileradamia1, alberto faraggi2, leopoldo a.
This technique agrees with dimensional regularization where one generalises to n dimensions by. Zeta function regularization imperial college london. The method of regularization using a cutoff function can smooth the series to arrive at. Within the zeta function regularization procedure, one has to deal with traces of complex powers of elliptic operators. Zeta function regularization is an effective method to extract physical significant quantities from infinite ones. Pdf analytic continuation leads to the finite renormalization of a quantum. It is regarded as mathematically simple and elegant but the isolation of the. Ashfaque in zeta function regularization, one temporarily changes the power of the frequency. The functional integral can be solved exactly and by using zeta function regularization technique 34, 35, 36 we obtain a closed expression for the free energy of the system. Zeta function regularization is in general nonlinear. Zetafunction regularization, the multiplicative anomaly and the wodzicki residue emilio elizalde1. In chapter 16 we shall employ a zeta function, for example. Pdf zetafunction regularization and the thermodynamic. The riemann zeta function or eulerriemann zeta function.
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. Equivalence of zeta function technique and abelplana. Pdf zetafunction regularization is uniquely defined and. Frontiers zeta regularization of infinite products physics. Dos, zeta functions, determinants, and traces wodzicki residue, multiplicative or noncommutative anomaly, or defect singularities of. By contrast, abelplana formula method permits explicit separation of divergent terms. 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. In this paper that is the second part of the trilogy, we aim the nature of the nontrivial zero for the riemann zeta function. What has now come to be known as the riemann zeta function. On the other hand the volume element that appears in the action is g112 ig12. The details of the calculation are also much more simple than in paulivillars and zeta function regularization. As a matter of fact, it is not strictly necessary that. The zeta function is also useful for the analysis of dynamical systems 38. Zetafunction regularization of quantum field theory canadian.
Zeta functions play a very useful role in physics, especially as they provide for a natural means of regularizing certain quantities which a priori are infinite, or meaningless. Zetafunction regularization, the multiplicative anomaly. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to. This sum is cubically divergent, but it can be regularized in zeta function regularization, or by simply multiplying each term by an exponential damping factor e. In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums. It is regarded as mathematically simple and elegant but the isolation of the physical divergency is hidden in its analytic continuation. The zeta function regularization is a method to assign finite values to divergent integration or sums. In one notable example, the riemann zetafunction shows up explicitly in one method of calculating the casimir effect. Effective action for scalar fields and generalised zeta function regularisation guido cognola. Zeta function regularization of quantum field theory. Ten physical applications of spectral zeta functions.
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. The integral representation of the zeta function converges by the analytic continuation. Zeta functions have been deeply investigated in mathematics, mainly in. Use of this method yields, for instance, the vacuum energy corresponding to a quantum physical system with. Pdf zetafunction regularization, the multiplicative. The application of zeta regularization method to the. In chapter 14 we presented a zeta function representation of the free energy of a harmonic oscillator. When the real part of s is greater than 1, the dirichlet series converges, and its sum is the riemann zeta function. Zetafunction regularization, the multiplicative anomaly and the wodzicki residue sergio zerbini introductionwithin the oneloop or external field approximation, the importance of zetafunction regularization for functional determinants, as introduced in 1, is well known, as a powerful tool to deal with the ambiguities ultraviolet. This technique agrees with dimensional regularization where one generalises to n dimensions by adding extra flat dimensions. Among the different methods, zeta function regularization which is obtained by analytic continuation in the complex plane of the zeta function of the relevant physical operator in each case might well be the most beautiful of all.
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of selfsimilar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their selfinteractions. International centre for theoretical physics, miramare, 3400 trieste, italy and. Zeta regularization, spectral zeta functions, eisenstein series, and casimir energy 57 8. This is achieved by expressing the zeta function of the laplace operators on t1. Zeta function regularization of the determinant of the laplacian, for example on a torus, might qualify as a purely mathematical application. One method of the regularization is the analytic continuation. See, for example, on functional determinants of laplacians in polygons and simplicial complexes or zeta functions and regularized determinants on projective spaces.
The first variation of the effective action is well defined by the use of zeta function regularization, while the second variation is intrinsically ill defined, unless one makes use of suitable variations with disjoint supports. Zeta function regularization and the thermodynamic potential for quantum fields in symmetric spaces. What is the idea behind zeta function regularization. Pdf effective action for scalar fields and generalized. This revised and enhanced new edition contains three new chapters, two on the application of riemanns zeta function regularization to obtain the partition function of a bose fermi oscillator and one on the zeta function regularization in quantum electrodynamics. Zeta function regularization of sums over known spectrum. What has now come to be known as the riemann zeta function has its roots traced. Pdf rigorous extension of the proof of zetafunction.
I have never opened a math textbook and seen definition of derivative. The regularization for the zeta functions with physical. The proper time formalism was the precursor of the zeta function regularization and the heat kernel method which we will discuss in the next chapter. The zeta function regularization method relies, for its practical application 32,33, on the existence of explicit and particularly simple expressions, which provide the analytic continuation of.
Riemann zeta function, functional determinant, zeta regularization, divergent series. Pdf analytic continuation leads to the finite renormalization of a quantum field theory. Equivalence of zeta function technique and abelplana formula. For the case of the eigenvalues of a simple quantum harmonic oscillator in one dimension 10. 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. Pdf zeta function regularization and vacuum energy. Everything is explained in detail, in particular the mathematical difficulties and tricky points, and several applications are given to show how the. Properties 1 and 2 below follow from the definition of zeta regularized products.
More generally, the zeta function approach can be used to regularize the whole energymomentum tensor in curved spacetime. The riemann zeta function plays a pivotal role in analytic number theory and has applications. But even if no infinities arose in loop diagrams in quantum field theory, it. 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. On the other hand, the zeta function also has a series representation. Zetafunction regularization of quantum field theory.
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