A systematic approach to the C*-Weyl algebra W(E,σ) over a possibly degenerate pre-symplectic form σ on a real vector space E of possibly infinite dimension is elaborated in an almost self-contained manner. The construction is based on the theory of Kolmogorov decompositions for σ-positive-definite functions on involutive semigroups and their associated projective unitary group representations. The σ-positive-definite functions provide also the C*-norm of W(E,σ), the latter being shown to be *-isomorphic to the twisted group C*-algebra of the discrete vector group E. The connections to related constructions are indicated. The treatment of the fundamental symmetries is outlined for arbitrary pre-symplectic σ. The construction method is especially applied to the trivial symplectic form σ=0, leading to the commutative Weyl algebra over E, which is shown to be isomorphic to the C*-algebra of the almost periodic continuous function on the topological dual Eτ′ of E with respect to an arbitrary locally convex Hausdorff topology τ on E. It is demonstrated that the almost periodic compactification aEτ′ of Eτ′ is independent of the chosen locally convex τ on E, and that aEτ′ is continuously group isomorphic to the character group Ê of E. Applications of the results to the procedures of strict and continuous deformation quantizations are mentioned in the outlook.

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.

For an arbitrary (possibly infinite-dimensional) pre-symplectic test function space (E, σ) the family of Weyl algebras {W(E, σ)} ∈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 (Fréchet-) *-algebras which are C*-norm-dense in 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.

Specifying their (normally ordered) characteristic functions we determine all states of the boson C*-Weyl algebra which satisfy Glauber's coherence condition and are not realizable as density operators in Fock space. The pure ones are shown to be just the eigenstates of the annihilation operators in their GNS-representations (in contrast to the Fock case) and are characterized in many equivalent manners. The central decomposition of an arbitrary coherent state has the macroscopic phase variable as parameter and is supported by the pure coherent states, which is in fact the only way for a maximal decomposition. The set of all coherent states with the same absolute factorizing function is proven to be a Bauer simplex. The appearence of a classical coherent field part is studied in detail in the GNS-representations and shown to correspond to an enlargement of the set of one boson states by just one additional mode. §1. Introduction and Preliminary ResultsIn one of his first papers on quantum optics [6] Glauber emphasizes the importance of states with a large number of photons for the description of light beams (and contrasts this with the few photon excitations in perturbational quantum electrodynamics). The relevant states, which have some degree of coherence, are investigated by him and his colleagues, however, only in the Fock representation. Since in laser beams there are in fact macroscopically many photons one may doubt if this formal limitation is justified. In [12] examples of fully coherent states are given, which cannot be represented by density operators in Fock space. A systematic study, where both Fock and non-Fock coherent states are investigated in the smeared field formalism, has been started recently in [13]. The aim of this paper is to carry on this analysis with the emphasis on non-Fock states.We start here from Glauber's factorization condition for a (fully) coherent