Analog of Lie Algebra and Lie Group for Quantum Non-Hamiltonian Systems

TL;DR

This paper introduces a new algebraic and group-theoretic framework for quantum non-Hamiltonian systems, extending Lie algebra and Lie group concepts to accommodate dissipative quantum dynamics.

Contribution

It defines the commutant Lie algebra and Valya loop as analogs of Lie algebra and Lie group for dissipative quantum systems, providing a new mathematical foundation.

Findings

The commutant Lie algebra generalizes the Heisenberg-Weyl algebra for dissipative systems.

The Valya loop is a non-associative loop with an associative commutant subloop.

The tangent algebra of the Valya loop is a commutant Lie algebra.

Abstract

Quantum mechanics of Hamiltonian (non-dissipative) systems uses Lie algebra and analytic group (Lie group). In order to describe non-Hamiltonian (dissipative) systems in quantum theory we need to use non-Lie algebra and analytic quasigroup (loop). The author derives that analog of Lie algebra for quantum non-Hamiltonian systems is commutant Lie algebra and analog of Lie group for these systems is analytic commutant associative loop (Valya loop). A commutant Lie algebra is an algebra such that commutant (a subspace which is generated by all commutators) is a Lie subalgebra. Valya loop is a non-associative loop such that the commutant of this loop is associative subloop (group). We prove that a tangent algebra of Valya loop is a commutant Lie algebra. It is shown that generalized Heisenberg-Weyl algebra, suggested by the author to describe quantum non-Hamiltonian (dissipative) systems, is a commutant Lie algebra. As the other example of commutant Lie algebra, it is considered a generalized Poisson algebra for differential 1-forms. Note that non-Hamiltonian (dissipative) quantum theory has a broad range of application for non-critical strings in "coupling constant" phase space and bosonic string in non-Riemannian (for example, affine-metric) curved space which are non-Hamiltonian (dissipative) systems.