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C*-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-*-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-*-algebras instead of C*-algebras. Fourier transformation and representation theory of the measure Banach-*-algebras are combined with the theory of continuous projective group representations to arrive at the genuine C*-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general) infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter h. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation.

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 (Fréchet-) *-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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]