Related papers: Free quantitative fourth moment theorems on Wigner…

This paper proves a Berry--Esseen theorem for sample quantiles of strongly-mixing random variables under a polynomial mixing rate. The rate of normal approximation is shown to be $O(n^{-1/2})$ as $n\to\infty$, where $n$ denotes the sample…

We establish the conditions under which a conservation law associated with a non-invertible operator may be realized as a symmetry in quantum physics. As established by Wigner, all quantum symmetries must be represented by either unitary or…

Let the Ornstein-Uhlenbeck process $\{X_t,\,t\geq 0\}$ driven by a fractional Brownian motion $B^H$ described by $d X_t=-\theta X_t dt+ d B_t^H,\, X_0=0$ with known parameter $H\in (0,\frac34)$ be observed at discrete time instants $t_k=kh,…

We prove an analogue of the Baum-Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a $ \gamma $-element and that $ \gamma = 1 $. It follows that free orthogonal quantum groups are $…

This study shows how Aronszajn's theory of reproducing kernels can be of use for the construction the Hilbert spaces of quantum theory. We show that the Feynman propagator is an example of a reproducing kernel under a boundedness condition.…

We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the…

This work explores the intersection of quantum mechanics and curved spacetime by employing the Wigner formalism to investigate quantum systems in the vicinity of black holes. Specifically, we study the quantum dynamics of a probe particle…

The nonnegativity of the density operator of a state is faithfully coded in its Wigner distribution, and this places constraints on the moments of the Wigner distribution. These constraints are presented in a canonically invariant form…

In this paper, we present an oscillatory version of the celebrated Breuer-Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random…

In this note, we prove a multidimensional counterpart of the central limit theorem on the free Poisson chaos recently proved by Bourguin and Peccati (2014). A noteworthy property of convergence toward the semicircular distribution on the…

We prove two results on arithmetic quantum chaos for dihedral Maass forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level $1$…

We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano-Wichmann relations and a representation of the Poincare' group on the one-particle Hilbert space. The abstract real…

We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of "homogenization" for Dyson Brownian Motion, this yields the universality of quantities which depend on…

We study quantum field theories with boundary by utilizing non-invertible symmetries. We consider three kinds of boundary conditions of the four dimensional $\mathbb{Z}_2$ lattice gauge theory at the critical point as examples. The weights…

The Wentzel-Kramers-Brillouin semiclassical method is formulated for quasiparticles with quartic-in-momentum dispersion which presents the simplest case of a soft energy-momentum dispersion. It is shown that matching wave functions in the…

We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current~$\bm J$ as an alternative to the popular Moyal…

We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner's…

Wigner's celebrated theorem, which is particularly important in the mathematical foundations of quantum mechanics, states that every bijective transformation on the set of all rank-one projections of a complex Hilbert space which preserves…

We define the quantile set of order $\alpha \in \left[ 1/2,1\right) $ associated to a law $P$ on $\mathbb{R}^{d}$ to be the collection of its directional quantiles seen from an observer $O\in \mathbb{R}^{d}$. Under minimal assumptions these…

We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically…