Quantum Mechanics as an Approximation to Classical Mechanics in Hilbert Space

Classical mechanics is formulated in Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket, and a quasidensity operator. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Classical mechanics can now be viewed as a deformation of quantum mechanics. The forms of semiquantum approximations to classical mechanics are indicated. While our understanding of the relation between quantum mechanics and classical mechanics has steadily increased over the past 75 years, as a result of many studies from various points of view (see [1]-[7] and references therein), few would claim that it is complete. Meanwhile, increasing attention has focussed on the interface between the quantum and classical domains, because of advances in experimental science and engineering, and the associated development of ‘nanotechnology.’ Classical mechanics is usually formulated in real, finite-dimensional phase space; quantum mechanics in complex, infinite-dimensional Hilbert space. However, a completely equivalent reformulation of quantum mechanics in phase space is known [8]-[15], which shows that quantum mechanics is a ∗Email: [email protected] On leave from Centre for Mathematical Physics, Department of Mathematics, University of Queensland, Brisbane 4072, Australia. It is a pleasure to thank G. Cassinelli for his generous hospitality, and to thank him and J.G. Wood for helpful discussions.