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

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.