Partially Classical States of a Boson Field

Employing positive-definiteness arguments we analyse Boson field states, which combine classical and quantum mechanical features (signal and noise), in a constructive manner. Mathematically, they constitute Bauer simplexes within the convex, weak-*-compact state space of the C*-Weyl algebra, defined by a presymplectic test function space (smooth one-Boson wave functions) and are affinely homeomorphic to a state space of a classical field. The regular elements are expressed in terms of weak distributions (probability premeasures) on the dual test function space. The Bauer simplex arising from the bare vacuum is shown to generalize the quantum optical photon field states with positive P-functions.     

Power series of the free field as operator-valued functional on spaces of typeS

Infinite series of Wick powers of the free, massive Bose field are analysed in terms of test function spaces of typeS for arbitrary space dimension. By direct estimates of the smeared phase space integrals sufficiency conditions for the existence of the vacuum expectation values are derived. These conditions are shown to be precise. The field-operators are defined on a dense invariant domain in Fock space, where they satisfy the Wightman axioms with the possible exception of locality. Localisable and nonlocalisable fields are dealt within the same frame. The behaviour of spectral functions and the strength of singularities are discussed.This work was supported by the Deutsche Forschungsgemeinschaft.This work is part of a thesis submitted to the Naturwissenschaftlichen Fakultät der Universität München.     

On higher-order coherent states of the Weyl algebra

It is shown that for classical, analytical states of the Weyl algebra, the quantum optical coherence condition of second order implies those of nth order for all n2.     

Coherence and incompatability inWu* -algebraic quantum theory

In the framework of generalized quantum theory using aW ^*-algebraic formalism, we introduce a completely symmetric coherence relation for states which is also applicable to nonpure states. Making use of lattice theoretic results the properties of this relation, especially its connection with incompatibility, are investigated. By means of algebraic decomposition theory the investigation is reduced to the case of factors where a complete classification of the coherence classes is given.     

Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization

For an arbitrary (possibly infinite-dimensional) pre-symplectic test function space the family of Weyl algebras , introduced in a previous work [1], is shown to constitute a continuous field of C^*-algebras in the sense of Dixmier. Various Poisson algebras, given as abstract (Fréchet-) ^*-algebras which are C^*-norm-dense in , are constructed as domains for a Weyl quantization, which maps the classical onto the quantum mechanical Weyl elements. This kind of a quantization map is demonstrated to realize a continuous strict deformation quantization in the sense of Rieffel and Landsman. The quantization is proved to be equivariant under the automorphic actions of the full affine symplectic group. The relationship to formal field quantization in theoretical physics is discussed by suggesting a representation dependent direct field quantization in mathematically concise terms. Communicated by Joel FeldmanSubmitted 07/10/03, accepted 07/11/03     

Canonical versus Grand-Canonical Free Energies and Phase Diagrams of a Bipolaronic Superconductor Model

We continue the discussion of a bipolaronic superconductor (resp. an anisotropic antiferromagnet in quasispin formulation) as formulated in a previous work, based on a quantum-statistical, microscopic mean-field model. The grand-canonical thermodynamic limit is compared with the canonical thermodynamic limit in terms of a net of perturbations, becoming singular in the infinite lattice limit. A generalized thermostatistical framework is elaborated which covers model potentials with infinite parts. The function of the limiting free energy density in selecting the (stable) phases with broken symmetry is graphically illustrated. The phase diagrams for the two types of ensembles are shown to differ in the region where both the gauge symmetry and the invariance under sublattice exchange are broken. In particular, the type of the phase transitions, the order of the critical points, and the shape of some phase boundaries are found to depend on the ensemble, which clarifies certain controversial topics for these models. The uniqueness of the limiting Gibbs states with free boundary conditions in all thermodynamic phase regions is proved, and their decomposition into pure phase states in terms of a symmetric measure is evaluated. The field operators of the condensed particles are determined in the representations over the limiting Gibbs states.     

Global C*-Dynamics and Its KMS States of Weakly Inhomogeneous Bipolaronic Superconductors

We establish the limiting dynamics of a class of inhomogeneous bipolaronic models for superconductivity which incorporate deviations from the homogeneous models strong enough to require disjoint representations. The models are of the Hubbard type and the thermodynamics of their homogeneous part has been already elaborated by the authors. Now the dynamics of the systems is evaluated in terms of a generalized perturbation theory and leads to a C*-dynamical system over a classically extended algebra of observables. The classical part of the dynamical system, expressed by a set of 15 nonlinear differential equations, is observed to be independent from the perturbations. The KMS states of the C*-dynamical system are determined on the state space of the extended algebra of observables. The subsimplices of KMS states with unbroken symmetries are investigated and used to define the type of a phase. The KMS phase diagrams are worked out explicitly and compared with the thermodynamic phase structures obtained in the preceding works.     

Limiting Gibbs States and Phase Transitions of a Bipartite Mean-Field Hubbard Model

In the frame of operator-algebraic quantum statistical mechanics we calculate the grand canonical equilibrium states of a bipartite, microscopic mean-field model for bipolaronic superconductors (or anisotropic antiferromagnetic materials in the quasispin formulation). Depending on temperature and chemical potential, the sets of statistical equilibrium states exhibit four qualitatively different regions, describing the normal, superconducting (spin-flopped), charge ordered (antiferromagnetic), and coexistence phases. Besides phase transitions of the second kind, the model also shows phase transitions of the first kind between the superconducting and the charge ordered phases. A unique limiting Gibbs state is found in its central decomposition for all temperatures, even in the coexistence region, if the thermodynamic limit is performed at fixed particle density (magnetization).     

Effective dynamics of the quantum mechanical Weiβ-Ising model

The quantum-mechanical Weiβ-Ising model is discussed anew in the light of recent results of algebraic quantum mechanics. The limiting Gibbs states are calculated by direct convergence estimations. Starting from the molecular field operator the dynamics is constructed in every temperature representation. The spectra of the effective Hamiltonians are determined by means of the Connes theory. The representations of the quasilocal algebra given by the pure phase states below the transition temperature are shown to be factors of type III_λ, λ ∈ (0, 1). An infinity of ground states together with their effective Hamiltonians are constructed and investigated.