The quantum-to-classical transition: contraction of associative products

TL;DR

This paper explores how the quantum-to-classical transition can be understood through contractions of associative algebras, illustrating the process with phase space functions and Lie algebra examples.

Contribution

It introduces methods for algebra contractions that unify the quantum and classical descriptions of physical observables.

Findings

Classical phase space algebra arises as a contraction of the quantum Moyal star-product.

Contractions of Lie algebra-associated associative algebras are analyzed, including Weyl-Heisenberg and SU(2).

The framework provides a mathematical bridge between quantum and classical algebraic structures.

Abstract

The quantum-to-classical transition is considered from the point of view of contractions of associative algebras. Various methods and ideas to deal with contractions of associative algebras are discussed that account for a large family of examples. As an instance of them, the commutative algebra of functions in phase space, corresponding to classical physical observables, is obtained as a contraction of the Moyal star-product which characterizes the quantum case. Contractions of associative algebras associated to Lie algebras are discussed, in particular the Weyl-Heisenberg and $SU(2)$ groups are considered.