A generalized Schrödinger formalism as a Hilbert space representation of a generalized Liouville equation

Ram K. Varma

Diógenes Campos, Juan Diego Urbina, Carlos Viviescas

We develop a general `classicalization' procedure that links Hilbert-space and phase-space operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in an ℏ-Taylor series and fix parameters with the condition that coefficients of ℏ 0 , -iℏ 1 ∂/∂ Q and iℏ 1 ∂/∂ P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact . In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S ( Q , P , t ) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.

Stephen Mark Barnett, Bryan J. Dalton

The thermofield representation of a thermal state by a pure-state wavefunction in a doubled Hilbert space is generalised to arbitrary and pure states. The authors employ a Liouville space formalism to investigate the connection between these generalised thermofield wavefunctions and a generalised thermofield state vector in Liouville space which is valid for all cases of the quantum density operator. The system dynamics in the Schrodinger and Heisenberg pictures are discussed.

Е. А. Соловьев, E. Yankovsky

John S. Nodvik

Partha Guha

The Koopman–von Neumann (KvN) formulation recasts deterministic classical mechanics in a Hilbert-space operator framework analogous to quantum mechanics. In this approach, the Liouville equation is expressed as a Schrödinger-type evolution equation generated by a Hermitian operator, yielding unitary propagation for phase-space-volume-preserving dynamics. We first clarify the relationship among the Koopman, Perron–Frobenius, and Liouville operators through an explicit example. Wilczek's construction shows that the KvN formalism is not only restricted to canonical Hamiltonian mechanics but this can be generalized any first-order classical dynamical system. Wilczek shows how a general autonomous first-order system can be generated by a Hermitian temporal-evolution operator after the introduction of auxiliary conjugate variables. We develop the KvN description of systems driven by Hamiltonian curl forces–position-dependent, non-conservative forces that are non-dissipative and preserve phase-space volume. Finally, we extend the KvN formalism to non-Hamiltonian systems using the Jacobi Last Multiplier and examine its connection to the Boltzmann H-theorem.