States on a Weyl algebra of field observables

Fabio Acerbi

Reinhard Honegger, Alfred Rieckers

Benjamin H. Feintzeig

This paper considers states on the Weyl algebra of the canonical commutation relations over the phase space R2n. We show that a state is regular if and only if its classical limit is a countably additive Borel probability measure on R2n. It follows that one can “reduce” the state space of the Weyl algebra by altering the collection of quantum mechanical observables so that all states are ones whose classical limit is physical.

Reinhard Honegger, Alfred Rieckers

Starting from a (possibly infinite dimensional) pre-symplectic space (E, σ) , we study a class of modified Weyl quantizations. For each value of the real Planck parameter ℏ , we have a C*-Weyl algebra \mathcal W(E,ℏ_σ) , which altogether constitute a continuous field of C*-algebras, as discussed in previous works. For ℏ = 0 , we construct a Fréchet–Poisson algebra, densely contained in \mathcal W(E, 0) , as the classical observables to be quantized. The quantized Weyl elements are decorated by so-called quantization factors, indicating the kind of normal ordering in specific cases. Under some assumptions on the quantization factors, the quantization map may be extended to the Fréchet–Poisson algebra. It is demonstrated to constitute a strict and continuous deformation quantization, equivalent to the Weyl quantization, in the sense of Rieffel and Landsman. Realizing the C*-algebraic quantization maps in regular and faithful Hilbert space representations leads to quantizations of the unbounded field expressions.

Volker Schomerus