Mathematical Physics, Analysis and Geometry最新文献

In this paper, studying the inverse problem, we establish a curvature compatibility condition on a spherically symmetric Finsler metric. As an application, we characterize the spherically symmetric metrics of scalar curvature. We construct a Berwald frame for a spherically symmetric Finsler surface and calculate some associated geometric objects. Several examples are provided and discussed. Finally, we give a note on a certain general ((alpha ,beta ))-metric which appears in the literature.

In this paper we study the foundations of the algebraic treatment of classical and quantum field theories for Dirac fermions under external backgrounds following the initial contributions already present in various places in the literature. The treatment is restricted to contractible spacetimes of globally hyperbolic nature in dimensions (dge 4) and to external fields modelled with trivial principal bundles. In particular, we construct the classical Møller maps intertwining the configuration spaces of charged and uncharged fermions, and we show some of its properties in the case of a U(1) gauge charge. In the last part, as a first step towards a quantization of the theory, we explore the combination of the classical Møller maps with Hadamard bidistributions and prove that they are involutive isomorphisms (algebraically and topologically) between suitable (formal) algebras of functionals (observables) over the configuration spaces of charged and uncharged Dirac fields.

For a family of self-adjoint Dirac operators (-i c (alpha cdot nabla ) + frac{c^2}{2}) subject to generalized MIT bag boundary conditions on domains in (mathbb {R}^3), it is shown that the nonrelativistic limit in the norm resolvent sense is the Dirichlet Laplacian. This allows to transfer spectral geometry results for Dirichlet Laplacians to Dirac operators for large c.

We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of (H+A^{*}+A). Math. Phys. Anal. Geom. 23 (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind (H+A^{*}+A), where H and A play the role of the free field Hamiltonian and of the annihilation operator respectively. We give explicit representations of the resolvent and of the self-adjointness domain; the consequent Kreĭn-type resolvent formula leads to a characterization of these self-adjoint realizations as limit (with respect to convergence in norm resolvent sense) of cutoff Hamiltonians of the kind (H+A^{*}_{n}+A_{n}-E_{n}), the bounded operator (E_{n}) playing the role of a renormalizing counter term. These abstract results apply to various concrete models in Quantum Field Theory.

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of Male et al. (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419, 2023). and thus allow identifying the fluctuation around the thermal value in certain thermalization problems.

Ohno and Zagier (Indag Math 12:483–487, 2001) found that a generating function of sums of multiple polylogarithms can be written in terms of the Gauss hypergeometric function ({}_2F_1). As a generalization of the Ohno and Zagier formula, Ihara et al. (Can J Math 76:1–17, 2022) showed that a generating function of sums of interpolated multiple polylogarithms of Hurwitz type can be expressed in terms of the generalized hypergeometric function ({}_{r+1}F_r). In this paper, we establish q- and elliptic analogues of this result. We introduce elliptic q-multiple polylogarithms of Hurwitz type and study a generating function of sums of them. By taking the trigonometric and classical limits in the main theorem, we can obtain q- and elliptic generalizations of the Ihara, Kusunoki, Nakamura and Saeki formula.

Using Araki–Yamagami’s characterization of quasi-equivalence for quasi-free representations of the CCRs, we provide an abstract criterion for the existence of isomorphisms of second quantization local von Neumann algebras induced by Bogolubov transformations in terms of the respective one particle modular operators. We discuss possible applications to the problem of local normality of vacua of Klein-Gordon fields with different masses.

We consider the macroscopic limit for the space-time density fluctuations in the open symmetric simple exclusion in the quasi-static scaling limit. We prove that the distribution of these fluctuations converge to a gaussian space-time field that is delta correlated in time but with long-range correlations in space.

We study cover times of subsets of ({mathbb {Z}}^2) by a two-dimensional massive random walk loop soup. We consider a sequence of subsets (A_n subset {mathbb {Z}}^2) such that (|A_n| rightarrow infty ) and determine the distributional limit of their cover times ({mathcal {T}}(A_n)). We allow the killing rate (kappa _n) (or equivalently the “mass”) of the loop soup to depend on the size of the set (A_n) to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to (kappa _n^{-1}=|A_n|^{1-8/(log log |A_n|)},) showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order (kappa _n^{-1/2}=|A_n|^{1/2},) if (kappa _n^{-1}) exceeded (|A_n|,) the cover times of all points in a tightly packed set (A_n) (i.e., a square or close to a ball) would presumably be heavily correlated, complicating the analysis. Our result comes close to this extreme case.

This paper focuses on the characteristics of the Hankel determinant generated by a modified singularly Gaussian weight. The weight function is defined as:

where (alpha >1) and (tin (0,1)) are parameters. Using ladder operator techniques, we derive a series of difference formulas for the auxiliary quantities and recurrence coefficients. We present the difference equations for the recurrence coefficients and the logarithmic derivative of the Hankel determinant. We then use the “t-dependence" to obtain the differential identities satisfied by the auxiliary quantities and the logarithmic derivative of the Hankel determinant. To obtain the large n asymptotic expressions of the recurrence coefficients, we use the Coulomb fluid method together with the related difference equations, which depend on n either being odd or even. We also obtain the reduction forms of the second-order differential equations satisfied by the orthogonal polynomials generated by this weight. Two special cases coincide with the bi-confluent Heun equation and the double confluent Heun equation, respectively. Finally, we calculate the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by this weight. Our result not only covers the classical result of Szegö (Trans Am Math Soc 40:450–461, 1936) but also determines our next research direction.