We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry $s\to (1-s)$ in the critical strip. By modifying one integral representation of the zeta-function with a cut-off that does exhibit the symmetry $x\mapsto 1/x$, we obtain a generalized functional equation involving Bessel functions of second kind. Next, with another cut-off that does exhibit the same symmetry, we obtain a generalization for the functional equation involving only one Bessel function of second kind. Some connection between one regularized zeta-function and the Laplace transform of the heat kernel for the Euclidean and hyperbolic space is discussed.
The zeta-regularization allows to establish a connection between Feynman's path integral and Fourier integral operator zeta-functions. This fact can be utilized to perform the regularization of the vacuum expectation values in quantum field theories. In this proceeding, we will describe the concept of the zeta-regularization, give a simple example and demonstrate that quantum computing can be employed to numerically evaluate zeta-regulated vacuum expectation values on a quantum computer.
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $ζ$-functions to efficiently compute values of spectral $ζ$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $τ$. Depending on the underlying boundary conditions, we express the $ζ$-function values in terms of a fundamental system of solutions of $τy = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $ζ$-function through …
Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $χ$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $Δ$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $χ$. From the spectral theory of $Δ$, there are three distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$ eigenfunctions, the second coming from resonances associated to the continuous spectrum, and the third being the set of negative integers. Using these se…
It has recently been shown that vacuum expectation values and Feynman path integrals can be regularized using Fourier integral operator $ζ$-function, yet the physical meaning of these $ζ$-regularized objects was unknown. Here we show that $ζ$-regularized vacuum expectations appear as continuum limits using a certain discretization scheme. Furthermore, we study the rate of convergence for the discretization scheme using the example of a one-dimensional hydrogen atom in $(-π,π)$ which we evaluate classically, using the Rigetti Quantum Virtual Machine, and on the Rigetti 8Q quantum chip "Agave" d…
We use the asymptotic expansion of the heat trace to express all residues of spectral zeta functions as regularized sums over the spectrum. The method extends to those spectral zeta functions that are localized by a bounded operator.
Lattice field theory is a very powerful tool to study Feynman's path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined. On the other hand, a recently developed regularization scheme based on Fourier integral operator $ζ$-functions can treat Feynman's path integral non-pertubatively in Lorentzian background metrics. In this article, we formally $ζ$-regularize lattice theories with Lorentzian backgrounds and identify conditions for the Fourier integral operator $ζ$-function regularization to be applicable. Furthermore, we show that the clas…
We construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices. The regularized spectral determinant of the Dirac operator is also obtained as the derivative of the zeta function at a special value. In each case the zeta function is formulated using a contour integral method, which extends results obtained for Laplace and Schrodinger operators on graphs.
Epstein vector zeta functions related to the ADE Lie algebras
National Research Centre ”Kurchatov Institute”（库尔恰托夫国家研究中心）；Institute for Information Transmission Problems RAS (Kharkevich Institute)（俄罗斯科学院信息传输问题研究所（哈尔克维奇研究所））
We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also …
Given a number field $K$ with a Hecke character $χ$, for each place $ν$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $χ$. These theories appear in the study of $p$-adic string theory and $p$-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $χ$. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $ν$, the field theory t…
We establish rigorously the regularization of the p-adic open string amplitudes, with Chan-Paton rules and a constant B-field, introduced by Ghoshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematics parameters, the B-field and the Chan-Paton factors. We show that these integrals admit meromorphic continuations in t…
The exact Kohn-Sham iteration of generalized density-functional theory in finite dimensions witha Moreau-Yosida regularized universal Lieb functional and an adaptive damping step is shown toconverge to the correct ground-state density.
