Mean field limit for Bosons with compact kernels interactions by Wigner measures transportation

Mean field limit for bosons and infinite dimensional phase-space analysis

This article proposes the construction of Wigner measures in the infinite dimensional bosonic quantum field theory, with applications to the derivation of the mean field dynamics. Once these asymptotic objects are well defined, it is shown how they can be used to make connections between different kinds of results or to prove new ones.

Cylindrical Wigner measures

In this paper we study the semiclassical behavior of quantum states acting on the C*-algebra of canonical commutation relations, from a general perspective. The aim is to provide a unified and flexible approach to the semiclassical analysis of bosonic systems. We also give a detailed overview of possible applications of this approach to mathematical problems of both axiomatic relativistic quantum field theories and nonrelativistic many body systems. If the theory has infinitely many degrees of freedom, the set of Wigner measures, i.e. the classical counterpart of the set of quantum states, coi…

On the general principle of the mean-field approximation for many-boson dynamics

The mean-field approximations of many-boson dynamics are known to be effective in many physical relevant situations. The mathematical justifications of such approximations rely generally on specific considerations which depend too much on the model and on the initial states of the system which are required to be well-prepared. In this article, using the method of Wigner measures, we prove in a fairly complete generality the accuracy of the mean-field approximation. Roughly speaking, we show that the dynamics of a many-boson system are well approximated, in the limit of a large number of partic…

Quantum-field-theoretical approach to phase-space techniques: Symmetric Wick theorem and multitime Wigner representation

In this work we present the formal background used to develop the methods used in earlier works to extend the truncated Wigner representation of quantum and atom optics in order to address multi-time problems. The truncated Wigner representation has proven to be of great practical use, especially in the numerical study of the quantum dynamics of Bose condensed gases. In these cases, it allows for the simulation of effects which are missed entirely by other approximations, such as the Gross-Pitaevskii equation, but does not suffer from the severe instabilities of more exact methods. The numeric…

Jordan-Wigner Dualities for Translation-Invariant Hamiltonians in Any Dimension: Emergent Fermions in Fracton Topological Order

Inspired by recent developments generalizing Jordan-Wigner dualities to higher dimensions, we develop a framework of such dualities using an algebraic formalism for translation-invariant Hamiltonians proposed by Haah. We prove that given a translation-invariant fermionic system with general $q$-body interactions, where $q$ is even, a local mapping preserving global fermion parity to a dual Pauli spin model exists and is unique up to a choice of basis. Furthermore, the dual spin model is constructive, and we present various examples of these dualities. As an application, we bosonize fermionic s…

非平凡的自能和质量重整化旋 Boson 模型

Non-Trivial Renormalization of Spin-Boson Models with Supercritical Form Factors

本文通过非单位ary 的 dressing 变换，对具有超临界形式因子的旋 Boson 模型进行非平凡重整化，解决了单位ary 重整化超临界旋 Boson 模型的平凡性问题。

On The Weak-Coupling Limit for Bosons and Fermions

In this paper we consider a large system of Bosons or Fermions. We start with an initial datum which is compatible with the Bose-Einstein, respectively Fermi-Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time $t$, agrees with the analogous expansion for the solution to the Uehling-Uhlenbeck equation. This paper follows in spirit the companion work [\rcite{BCEP}], where the authors invest…

Exactly-solvable system of one-dimensional trapped bosons with short and long-range interactions

We consider trapped bosons with contact interactions as well as Coulomb repulsion or gravitational attraction in one spatial dimension. The exact ground state energy and wave function are identified in closed form together with a rich phase diagram, unveiled by Monte Carlo methods, with crossovers between different regimes. A trapped McGuire quantum soliton describes the attractive case. Weak repulsion results in an incompressible Laughlin-like fluid with flat density, well reproduced by a Gross-Pitaevskii equation with long-range interactions. Higher repulsion induces Friedel oscillation and …

Dirac equation for sphercially symmetric $AdS_5$ space-time and application to a boson star in EGB gravity

We discuss the Dirac equation in a curved 5-dimensional spherically symmetric space-time. The angular part of the solutions is thoroughly studied, in a formulation suited for extending to rotating space-times with equal angular momenta. It has a symmetry $SU(2)\times U(1)$ and is implemented by the Wigner functions. The radial part forms a Dirac-Schrödinger type equation, and existence of the analytical solutions of the massless and the massive modes is confirmed. The solutions are described by the Jacobi polynomials. Also, the spinor of the both large and small components is obtained numerica…