Existence Theorems for Ordered Variants of Weyl Quantization

Abstract

We consider some mathematical properties of Weyl-like quantizations based on two families of orderings of $e^{t(\alpha P+\delta Q)}$: the first family, $W_{(\lambda ,0)}$ interpolates between Wick ($\lambda =1$) and antiWick ($\lambda =–1$) ordering, while the second family, $W_{(0,\mu )}$, interpolates between the $Q^{-}$ ($\mu =1$) and $P^{-}$ ($\mu =–1$) orderings. The ordering $W_{(0,0)}$ common to both families is the unordered Weyl system.

The most important property is that of the existence of quantizations. For all orderings $W_{(0,\mu )}$ and for $W_{(\lambda ,0)}$ with $–1\le \lambda \le 0$ quantization is a well-defined map from the tempered distributions on phase space into the continuous linear operators from $\mathcal{S}(R)$ into $\mathcal{S}(R{)}^{\prime }$. For the orderings $W_{(\lambda ,0)}$ with $–1\le \lambda \le 0$ we have to restrict the class of wave functions from $\mathcal{S}(R)$ to a certain dense subset of it, and the resulting quantization procedure sends tempered distributions on phase space into sesquilinear forms on this subspace. For Wick ordering itself we have not been able to find any useable quantization scheme, and we doubt whether any one exists that is based on tempered distributions.

We also consider questions of boundedness, and determine the matrix coefficients for the quantizations of phase space functions of radius or of angle. In particular, we consider various quantizations of the angle function in phase space.

A correction to this paper is available.