No abstract

No abstract

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 multisymplectic setting for classical field theories subjected to non-holonomic constraints is presented. The infinite dimensional setting in the space of Cauchy data is also given.

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.