Mathematical Physics, Analysis and Geometry最新文献

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.

On a d-dimensional Riemannian, spin manifold (M, g) we consider non-linear, stochastic partial differential equations for spinor fields, driven by a Dirac operator and coupled to an additive Gaussian, vector-valued white noise. We extend to the case in hand a procedure, introduced in Dappiaggi et al (Commun Contemp Math 27(07):2150075, 2022), for the scalar counterpart, which allows to compute at a perturbative level the expectation value of the solutions as well as the associated correlation functions accounting intrinsically for the underlying renormalization freedoms. This framework relies strongly on tools proper of microlocal analysis and it is inspired by the algebraic approach to quantum field theory. As a concrete example we apply it to a stochastic version of the Thirring model proving in particular that it lies in the subcritical regime if (dle 2).

We study the limiting spectral distribution of quantum channels whose Kraus operators sampled as ( ntimes n) random Hermitian matrices satisfying certain assumptions. We show that when the Kraus rank goes to infinity with n, the limiting spectral distribution (suitably rescaled) of the corresponding quantum channel coincides with the semi-circle distribution. When the Kraus rank is fixed, the limiting spectral distribution is no longer the semi-circle distribution. It corresponds to an explicit law, which can also be described using tools from free probability.

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.

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.

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.

Abstract

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.