Quantization of time using symmetric and Born-Jordan orderings

Abstract

Quantization of a classical function, specifically time of arrival functions, may have different representations by using different orderings. In this paper we will be using the two common orderings, namely, symmetric and Born-Jordan orderings for distributions on the phase space  ℝ × S¹. These orderings are characterized by a real-valued function K such that K(0)=1. The function K is an additional integral factor of the standard Weyl quantization. These integral factors nultiplied by the Fourier transform of Weyl ordering will give the Fourier transforms of symmetric and Born-Jordan, respectively. Via the Wigner function, the method points to the determination of quasiprobability distributions corresponding to different orderings. This places the quantization scheme adopted here squarely within phase space quantum mechanics.