Field-theoretic Weyl quantization as a strict and continuous deformation quantization. (English) Zbl 1088.81066

Summary: 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, 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 \(\mathcal{W}(E, 0)\), 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.


81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products
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