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, coincides with the set of all cylindrical measures acting on the algebraic dual of the space of test functions for the field, and this reveals a very rich semiclassical structure compared to the finite-dimensional case. We characterize the cylindrical Wigner measures and the \emph{a priori} properties they inherit from the corresponding quantum states.Comment: 59 page

Wigner measures approach to the classical limit of the Nelson model: Convergence of dynamics and ground state energy

We consider the classical limit of the Nelson model, a system of stable nucleons interacting with a meson field. We prove convergence of the quantum dynamics towards the evolution of the coupled Klein-Gordon-Schr\"odinger equation. Also, we show that the ground state energy level of $N$ nucleons, when $N$ is large and the meson field approaches its classical value, is given by the infimum of the classical energy functional at a fixed density of particles. Our study relies on a recently elaborated approach for mean field theory and uses Wigner measures.Comment: 37 page

Self-Adjointness criterion for operators in Fock spaces

In this paper we provide a criterion of essential self-adjointness for operators in the tensor product of a separable Hilbert space and a Fock space. The class of operators we consider may contain a self-adjoint part, a part that preserves the number of Fock space particles and a non-diagonal part that is at most quadratic with respect to the creation and annihilation operators. The hypotheses of the criterion are satisfied in several interesting applications.Comment: 20 page

Classical limit of the Nelson model with cut off

In this paper we analyze the classical limit of the Nelson model with cut off, when both non-relativistic and relativistic particles number goes to infinity. We prove convergence of quantum observables to the solutions of classical equations, and find the evolution of quantum fluctuations around the classical solution. Furthermore we analyze the convergence of transition amplitudes of normal ordered products of creation and annihilation operators between different types of initial states. In particular the limit of normal ordered products between states with a fixed number of both relativistic and non-relativistic particles yields an unexpected quantum residue: instead of the product of classical solutions we obtain an average of the product of solutions corresponding to varying initial conditions.Comment: 42 page

Concentration of cylindrical Wigner measures

In this note we aim to characterize the cylindrical Wigner measures associated to regular quantum states in the Weyl C*-algebra of canonical commutation relations. In particular, we provide conditions at the quantum level sufficient to prove the concentration of all the corresponding cylindrical Wigner measures as Radon measures on suitable topological vector spaces. The analysis is motivated by variational and dynamical problems in the semiclassical study of bosonic quantum field theories.Comment: 23 page

<i>De novo</i> synthesis of budding yeast DNA polymerase alpha and <i>POL1</i> transcription at the G₁/S boundary are not required for entrance into S phase

The POL1 gene, encoding DNA polymerase α(pol α) in Saccharomyces cerevisiae, is transiently transcribed during the cell cycle at the G1/S phase boundary. Here we show that yeast pol α is present at every stage of the cell cycle, and its level only slightly increases following the peak of POL1 transcription. POL1 mRNA synthesis driven by a GAL1 promoter can be completely abolished without affecting the growth rate of logarithmically growing yeast cultures for several cell divisions, although the amount of the pol α polypeptide drops below the physiological level. Moreover, α-factor-arrested cells can enter S phase and divide synchronously even if POL1 transcription is abolished. These results indicate that the level of yeast pol α is not rate limiting and de novo synthesis of the enzyme is not required for entrance into S phase

Scattering theory for Lindblad master equations

International audienceWe study scattering theory for a quantum-mechanical system consisting of a particle scattered off a dynamical target that occupies a compact region in position space. After taking a trace over the degrees of freedom of the target, the dynamics of the particle is generated by a Lindbladian acting on the space of trace-class operators. We study scattering theory for a general class of Lindbladians with bounded interaction terms. First, we consider models where a particle approaching the target is always re-emitted by the target. Then we study models where the particle may be captured by the target. An important ingredient of our analysis is a scattering theory for dissipative operators on Hilbert space