LIE ALGEBRAS IN CLASSICAL AND QUANTUM MECHANICS

Yair Shapira

Yingfei Gu

This article presents a mathematical framework for group-theoretical formalism of quantum mechanics and discusses the geometric quantization of the classical systems associated with a Lie group. The classical-quantum correspondence is realized by identifying quantum observable algebra and its classical analogue with the set of distributions with compact supports on a Lie group and on the associated Lie algebra respectively, both having a convolution-type associative algebraic structure. The general mathematical constructs are illustrated by studying systems associated with the Heisenberg-Weyl group. It is shown that the exponential mapping from Heisenberg-Weyl algebra to the corresponding Lie group gives Weyl quantization.

Yair Shapira

Andrei Yu. Khrennikov

Jonathan Kauffman

In this paper, the connections between classical propositional logic and Boolean algebras are explored. Then, an introduction to quantum logic explains that the distributive property does not hold, which gives rise to the need for a different structure for quantum logic. One of these systems, algebraic test spaces, is described, along with their connection to orthoalgebras, which are also analyzed.

Maria Clara Nucci, P. G. L. Leach