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For an arbitrary (possibly infinite-dimensional) pre-symplectic test function space \( (E, \sigma) \) the family of Weyl algebras \( \{\mathcal{W}(E, \hbar\sigma)\}_{\hbar\in\mathbb{R}} \) , 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 (Frechet-) *-algebras which are C*-norm-dense in \( \mathcal{W}(E, 0) \) , 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.

A closed, entirely quantum mechanical Josephson oscillator model is considered, consisting of two superconductors in tunneling contact, which weakly interact with the photon field. For each superconductor we use, for notational simplicity, the BCS model in the strong coupling approximation and restrict ourselves to Anderson’s quasi-spin formulation. The thermodynamic limit of the global (non-equilibrium) dynamics is formulated for a large variety of states. It arises a generalization of previously developed cocycle equations, connecting the collective behaviour of the two superconductors with the photon field dynamics. In the physically most interesting situations, where the two superconductors are in thermal equilibrium (below the critical temperature) at voltage difference V, we show, that for arbitrary initial states the outgoing multi-photon states are quantum optically all-order coherent and constitute an almost monochromatic radiation of frequency $ 2eV/\hbar $. Furthermore, we deduce in detail, how the collective behaviour of the superconductors influences the collective (that are the optical) features of the emitted microwave photons.

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.

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.

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.

For the case of infinitely many photon (Boson) modes we investigate the unitary implementability of a class of symplectic one-parameter groups (more exactly, of the associated groups of Bogoliubov automorphisms on the CCR algebra) in the Fock representation and in representations of the CCR algebra, which are symplectically related and inequivalent to Fock. Furthermore, the existence of the associated (squeezing) quadratic Hamiltonians is discussed. Finally, applications in the theory of quantization in QED and in the Luttinger model are pointed out.

Within the algebraic frame of smeared photon fields, the coherence properties of squeezed multi-photon states are elaborated. The squeezing Bogoliubov automorphisms are constructed by means of suitable one-mode symplectic transformations in the infinite-dimensional test-function space. Transposed into the state space, these transformations are shown to preserve first-order coherence when applied to coherent, classical, Fock-normal states in spite of rendering them nonclassical. The cases, where first-order coherence is destroyed, are also classified. Very special conditions on the initial state modes and on the operation parameters are worked out, under which even second-order coherence is still valid for the transformed classical coherent states, in spite of exhibiting squeezed fluctuations.

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).

A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre‐Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self‐contained fashion.

Bardeen–Cooper–Schrieffer (BCS)‐like models for permutation invariant electronic interactions are investigated in terms of recent rigorous mean‐field methods. Their limiting dynamics is constructed in the Heisenberg picture on (extensions of) the full canonical commutation relation (CAR) algebra and also restricted to (extensions of) the pair algebra. The classical part of the proper BCS dynamics on the CAR algebra is shown to exhibit 15 macroscopic collective variables, the time dependence of which is explicitly integrated and discussed.

As a microscopic description of the Josephson junction, two BCS models, are studied in the strict pair formulation with quite an arbitrary weak coupling potential. The modular formalism, the separate gauge transformations, and the limiting dynamics are analyzed for the interacting system in terms of the GNS representation of the uncoupled limiting Gibbs state. By means of the Connes theory the condensed Cooper pair and the quasiparticle spectrum is shown to be stable against weak perturbations. The modular formalism is used to construct a local approximation to the renormalized particle number operator and, by this, its time dependence, in spite of this observable not being affiliated with the von Neumann algebra of the temperature representation. The time derivation from this unbounded operator‐valued function coincides with the limit of the local currents and splits under a natural assumption into a sum of the Josephson and the quasiparticle current operator extending the two‐fluid picture also to the coupled model.

For a class of discrete mean field models the limiting dynamics is investigated in the representations of generalized grand canonical states. It is demonstrated that for a certain form of spontaneous symmetry breakdown the W*‐automorphism dynamics exhibits a uniquely determined nontrivial classical part, which is essential for the explanation of macroscopic quantum phenomena.