Field-theoretic Weyl Quantization as aStrict and Continuous Deformation Quantization

For an arbitrary (possibly infinite-dimensional) pre-symplectic test functionspace 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 kindof a quantization map is demonstrated to realize a continuous strict deformationquantization in the sense of Rieffel and Landsman. The quantization is proved tobe equivariant under the automorphic actions of the full affine symplectic group.The relationship to formal field quantization in theoretical physics is discussed bysuggesting a representation dependent direct field quantization in mathematicallyconcise terms.Communicated by Joel FeldmanSubmitted 07/10/03, accepted 07/11/03