Annales Henri Poincaré最新文献

In this paper, we prove a polynomial central limit theorem for several integrable models and for the (beta )-ensembles at high temperature with polynomial potential. Furthermore, we connect the mean values, the variances and the correlations of the moments of the Lax matrices of these integrable systems with the ones of the (beta )-ensembles. Moreover, we show that the local functions’ space-correlations decay exponentially fast for the considered integrable systems. For these models, we also established a Berry–Esseen-type bound.

Given an Axiom A attractor for a (C^{1+alpha }) flow ((alpha >0)), we construct a countable Markov extension with exponential return times in such a way that the inducing set is a smoothly embedded unstable disk. This avoids technical issues concerning irregularity of boundaries of Markov partition elements and enables an elementary approach to certain questions involving exponential decay of correlations for SRB measures.

Analogue Hamiltonian simulation is a promising near-term application of quantum computing and has recently been put on a theoretical footing alongside experiencing wide-ranging experimental success. These ideas are closely related to the notion of duality in physics, whereby two superficially different theories are mathematically equivalent in some precise sense. However, existing characterisations of Hamiltonian simulations are not sufficiently general to extend to all dualities in physics. We give a generalised duality definition encompassing dualities transforming a strongly interacting system into a weak one and vice versa. We characterise the dual map on operators and states and prove equivalence ofduality formulated in terms of observables, partition functions and entropies. A building block is a strengthening of earlier results on entropy preserving maps—extensions of Wigner’s celebrated theorem- –to maps that are entropy preserving up to an additive constant. We show such maps decompose as a direct sum of unitary and antiunitary components conjugated by a further unitary, a result that may be of independent mathematical interest.

Buchholz and Grundling (Commun Math Phys 272:699–750, 2007) introduced a (hbox {C}^*)-algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle (T^*mathbb {T}^n) of an n-torus by first generalising the classical analogue of the resolvent algebra defined by the first author of this paper in earlier work (van Nuland in J Funct Anal 277:2815–2838, 2019) and subsequently applying Weyl quantisation. We prove that this quantisation is almost strict in the sense of Rieffel and show that our resolvent algebra shares many features with the original resolvent algebra. We demonstrate that both our classical and quantised algebras are closed under the time evolutions corresponding to large classes of potentials. Finally, we discuss their relevance to lattice gauge theory.

Let (H_0) be the free Dirac operator and (V geqslant 0) be a positive potential. We study the discrete spectrum of (H(alpha )=H_0-alpha V) in the interval ((-1,1)) for large values of the coupling constant (alpha >0). In particular, we obtain an asymptotic formula for the number of eigenvalues of (H(alpha )) situated in a bounded interval ([lambda ,mu )) as (alpha rightarrow infty ).

Soft symmetries for Yang–Mills theory are shown to correspond to the residual Hamiltonian action of the gauge group on the Ashtekar–Streubel phase space, which is the result of a partial symplectic reduction. The associated momentum map is the electromagnetic memory in the Abelian theory, or a nonlinear, gauge-equivariant, generalisation thereof in the non-Abelian case. This result follows from an application of Hamiltonian reduction by stages, enabled by the existence of a natural normal subgroup of the gauge group on a null codimension-1 submanifold with boundaries. The first stage is coisotropic reduction of the Gauss constraint, and it yields a symplectic extension of the Ashtekar–Streubel phase space (up to a covering). Hamiltonian reduction of the residual gauge action leads to the fully reduced phase space of the theory. This is a Poisson manifold, whose symplectic leaves, called superselection sectors, are labelled by the (gauge classes of the generalised) electric flux across the boundary. In this framework, the Ashtekar–Streubel phase space arises as an intermediate reduction stage that enforces the superselection of the electric flux at only one of the two boundary components. These results provide a natural, purely Hamiltonian, explanation of the existence of soft symmetries as a byproduct of partial symplectic reduction, as well as a motivation for the expected decomposition of the quantum Hilbert space of states into irreducible representations labelled by the Casimirs of the Poisson structure on the reduced phase space.