Annales Henri Poincaré最新文献

In this work, we consider a suitable generalization of the Feynman path integral on a specific class of Riemannian manifolds consisting of compact Lie groups with bi-invariant Riemannian metrics. The main tools we use are the Cartan development map, the notion of oscillatory integral and the Chernoff approximation theorem. We prove that, for a class of functions of a dense subspace of the relevant Hilbert space, the Feynman map produces the solution of the Schrödinger equation, where the Laplace–Beltrami operator coincides with the second-order Casimir operator of the group.

We develop a framework for studying consistent interactions of local gauge theories, which is based on the presymplectic BV-AKSZ formulation. The advantage of the proposed approach is that it operates in terms of finite-dimensional spaces and avoids working with quotient spaces such as local functionals or functionals modulo on-shell trivial ones. The structure that is being deformed is that of a presymplectic gauge PDE, which consists of a graded presymplectic structure and a compatible odd vector field. These are known to encode the Batalin–Vilkovisky (BV) formulation of a local gauge theory in terms of the finite dimensional supergeometrical object. Although in its present version the method is limited to interactions that do not deform the presymplectic structure and relies on some natural assumptions, it gives a remarkably simple way to analyse consistent interactions. The approach can be considered as the BV-AKSZ extension of the frame-like approach to consistent interactions. We also describe the underlying homological deformation theory, which turns out to be slightly unusual compared to the standard deformations of differential graded Lie algebras. As an illustration, the Chern-Simons and YM theories are rederived starting from their linearized versions.

We calculate the transition rate of an Unruh–DeWitt detector coupled to a non-equilibrium steady state (NESS) of a free massless scalar field on four-dimensional Minkowski spacetime. Bringing two semi-infinite heat baths at different temperatures into thermal contact along a surface, the NESS arises at asymptotically late times as a stationary state that has modewise thermal properties and features a heat flow between the reservoirs. The detector couples linearly to the field by a monopole interaction, and it moves inertially along the axis of the NESS heat flow. We contrast the transition rate with the case of a detector that is coupled to an inertial thermal equilibrium state. The results illustrate that the monopole does not couple to the heat flow, causing the detector to only register kinematical effects. Hence, dynamical features of the NESS are hidden from this detector model.

The use of quantum stochastic models is widespread in dynamical reduction, simulation of open systems, feedback control and adaptive estimation. In many applications only part of the information contained in the filter’s state is actually needed to reconstruct the target observable quantities; thus, filters of smaller dimensions could be in principle implemented to perform the same task. In this work, we propose a systematic method to find, when possible, reduced-order quantum filters that are capable of exactly reproducing the evolution of expectation values of interest. In contrast with existing reduction techniques, the reduced model we obtain is exact and in the form of a Belavkin filtering equation, ensuring physical interpretability. This is attained by leveraging tools from the theory of both minimal realization and non-commutative conditional expectations. The proposed procedure is tested on prototypical examples, laying the groundwork for applications in quantum trajectory simulation and quantum feedback control.

We provide a class of examples of interacting particle systems on (mathbb {Z}^d), for (din {1,2}), that admit a unique translation-invariant stationary measure, which is not the long-time limit of all translation-invariant starting measures, due to the existence of time-periodic orbits in the associated measure-valued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measure-valued dynamics needs to be a time-stationary measure.

We consider mapping properties of resolvents and the orthonormal Strichartz estimates for the Schrödinger operator. First we prove the Kato–Yajima-type uniform resolvent estimates for the repulsive Schrödinger operator with logarithmic decaying weight functions. This is new even when without perturbations. Moreover, our class of perturbations includes logarithmic decaying potentials, which is also new. The proof depends on the microlocal analysis and the Mourre theory. Then we use the Kato–Yajima estimates to prove the orthonormal Strichartz estimates. In the proof, we also use an abstract Ginibre–Velo or Keel–Tao-type theorem for the orthonormal Strichartz estimates, which means that the dispersive estimates yield the orthonormal Strichartz estimates for strongly continuous unitary groups. This abstract theorem is a partial extension of the recent results by Nguyen (J Funct Anal 286(1):110196, 2024) and also applies to many other Schrödinger propagators which are difficult to treat by the smooth perturbation theory, for example, local-in-time estimates for the Schrödinger operator with unbounded electromagnetic potentials, for the (k, a)-generalized Laguerre operators and global-in-time estimates for the Schrödinger operator with scaling critical magnetic potentials including the Aharonov–Bohm potentials. Finally, we also discuss mapping properties of resolvents of the repulsive Schrödinger operator on the Schwartz class. Since the principle symbol of our Hamiltonian does not belong to good symbol classes, resolvents do not preserve the Schwartz class. However, we prove that functions in the range of resolvents are Schwartz functions microlocally outside the radial sink. These results are different from related works Taira (Commun Math Phys 388(1):625–655, 2021) and Kameoka (J Spectr Theory 11(2):677–708, 2021).

We investigate the possibility of performing full quantum tomography based on the homogeneous time evolution of a single expectation value. Remarkably, every non-trivial binary measurement evolved by any quantum channel, except for a null set, in principle enables full quantum state tomography. We show that this remains true when restricted to Lindblad semigroups, although unitary evolution—with or without added simply depolarizing noise—is insufficient beyond the qubit case, highlighting the necessity of non-trivial noise. We establish an analog of Takens’ embedding theorem for quantum channels, which incorporates prior information into the framework. We also provide estimation bounds for finite statistics and analyze the feasibility of recovering an infinite time series of expectation values from a finite one using only spectral properties of the evolution.

We construct a (P(phi )_2) Gibbs state on infinite volume periodic surfaces (namely, with discrete “time translations”) by analogy with 1-dimensional spin chains and establish the mass gap for our Gibbs state; there are no phase transitions. We also derive asymptotic properties of the (P(phi )_2) partition function on certain towers of cyclic covers of large degrees that converge to the periodic surface in some appropriate sense. This gives the first construction of an interacting quantum field theory on surfaces of infinite genus with a mass gap. The main ingredient in our approach is to reconcile the so-called (P(phi )_2) model from classical constructive quantum field theory (CQFT) with Riemannian version of the axioms proposed by Segal (in: Tillmann U (ed) Topology, geometry, and quantum field theory. Lecture notes, vol 308, London Mathematical Society, London, pp 421–577, 2004) in the 90 s. We show the (P(phi )_2) model satisfies these axioms, appropriately adjusted. One key, new ingredient in our proof is a simple observation on a symmetry between conditional probabilities out of some infinite-dimensional “Bayes formulae”. We also give a precise statement and full proof of the locality of the (P(phi )_2) interaction.

We construct and study the Kirkwood-Dirac (KD) representations naturally associated to the Fourier transform of finite abelian groups G. We identify all pure KD-positive states and all KD-real observables for these KD representations. We provide a necessary and sufficient condition ensuring that all KD-positive states are convex combinations of pure KD-positive states. We prove that for (G=mathbb {Z}_{d}), with d a prime power, this condition is satisfied. We provide examples of abelian groups where it is not. In those cases, the convex set of KD-positive states contains states outside the convex hull of the pure KD-positive states.

We study how to extend Sanov theorem to the quantum setting. Although a quantum version of the Sanov theorem was proposed in Bjelakovic et al. (Commun. Math. Phys., 260, p.659 (2005)), the classical case of their statement is not the same as Sanov theorem as follows. The original Sanov theorem discusses the behavior of the empirical distribution when the empirical distribution is different from the true distribution, but the above paper studied a problem related to quantum hypothesis testing, whose classical version can be shown by classical Sanov theorem. We introduce a quantum analog of the empirical distribution by combining the empirical distribution based on a specific basis and Schur sampling. Using the above type of quantum analog of the empirical distribution, we propose another quantum version of Sanov theorem.