André Verbeure

The smallest

We consider theC*-algebras which contain the Weyl operators when the symplectic form which defines the C.C.R. is possibly degenerate. We prove that the C.C.R. are all obtained as a quotient of a universalC*-algebra by some of its ideals, and we characterize all these ideals.

C^*

We rigorously characterize the KMS and the limiting Gibbs states for mean field models. As an application we prove the convergence of the Gibbs states for the Dicke Maser model in the infinite volume limit.

-algebra for canonical commutations relations

SummaryNon-commutative central limit theorems are derived. The CCR-C*-algebra of fluctuations is analyzed in detail. The stability of the central limit is studied by means of the notion of relative entropy.

Equilibrium states for mean field models

For short range interactions and forL1-space clustering states it is proved that there exists a bonafide time evolution on the set of normal fluctuations. This dynamics is applied to derive the notion of equilibrium state of the algebra of fluctuations.

Non-commutative central limits

Ground and temperature quantum Gibbs states are constructed for a ferroelectric anharmonic quantum oscillator model with small masses. It is shown that they possess mixing properties. The construction relies on the Feynman–Kac–Nelson representation of the conditional reduced density matrices and on the cluster expansions for the corresponding Gibbs field of trajectories.

Dynamics of fluctuations for quantum lattice systems

For a large class of quantum models of mean-field type the thermodynamic limit of the free energy density is proved to be given by the Gibbs variational principle. The latter is shown to be equivalent to a non-commutative version of Varadhans asymptotic formula.

A QUANTUM CRYSTAL MODEL IN THE LIGHT-MASS LIMIT: GIBBS STATES

Given any operator on the testfunction space, the general form of the induced completely positive map of the C*-algebra of the canonical commutation relations is characterized.

Asymptotics of varadhan-type and the gibbs variational principle

Correlation inequalities are used to prove rigorously the following for the imperfect Bose gas: (i) existence of condensation in and only in the ground state, (ii) unicity and existence of the limiting Gibbs states.

Completely positive maps on the CCR-algebra

The temperature states of the spin-boson model consisting of a two-level atom in a Bose field are studied. It is proved that for all temperatures there exists a unique solution, hence there is no spontaneous reflection symmetry breaking.

The condensed phase of the imperfect Bose gas

A complete description of the fluctuation operator algebra is given for a quantum crystal showing displacement structural phase transitions. In the one-phase region, the fluctuations are normal and its algebra is non-Abelian. In the two-phase region and on the critical line (Tc>0) the momentum fluctuation is normal, the displacement is critical, and the algebra is Abelian; atTc=0 (quantum phase transition) this algebra is non-Abelian with abnormal displacement and supernormal (squeezed) momentum fluctuation operators, both being dimension dependent.