Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization

Abstract

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.

Some Continuous Field Quantizations, Equivalent to the C*-Weyl Quantization

“…The more difficult part is, to demonstrate the characteristic mathematical properties for the modified Weyl quantizations. Accommodating the line of reasoning of [14] to the altered quantization maps, we prove, in fact, that we again have strict and continuous deformation quantizations, provided we use the concept of a continuous field of C*-algebras in the unrestricted sense of Dixmier to cope with the unbounded quantization factors. Also the Poisson algebra and the range I of the Planck parameter have to be carefully adjusted, if the quantization is to display the desired features.…”

“…We follow [14] for the construction of a continuous field of C*-Weyl algebras, where here I is set equal to R.…”

“…In Section 3 we present the conceptual frame for our intended class of quantizations, starting with a general, but preferentially infinite dimensional, pre-symplectic space (E, σ) and making use of the uniquely associated C*-Weyl algebras W(E, σ), as introduced in [14]. The Planck parameter varies in a subset I of R and accumulates at zero.…”

“…In Section 4 a generalization of the C*-algebraic Weyl quantization of [14] is put forward by associating the classical Weyl elements W 0 (f ) with the quantum mechanical ones W (f ) times a 'quantization factor'. There are various reasons for doing so.…”

“…In [14] it is shown that ∆ WF (E, σ) 1 is a . sup -dense sub-*-algebra of C * WF (E, σ), which consists of those sections K ∈ C * WF (E, σ) which have the unique decomposition…”