Phase Space Descriptions of Quantum Phenomena

In this paper we show the deep connection between the WignerMoyal approach and the Bohm approach to quantum mechanics. We point out that the key equations used in the Bohm approach were already contained in Moyal’s classic 1949 paper. Furthermore we argue that these two approaches can be seen as different but related aspects of standard quantum formalism when the algebraic approach, rather than the Hilbert space approach is taken as primary. This leads us to consider the non-commutative Moyal *algebra in which two key equations obtained are shown to be the analogue of algebraic equations derived by Brown and Hiley using the algebraic approach. The different phase spaces exploited in each case appear as necessary consequences of the non-commutative algebra since they are but shadow manifolds already implicit in the algebra defining the quantum formalism.