Quantum phase-space representation for curved configuration spaces

Abstract

We extend the Wigner-Weyl-Moyal phase-space formulation of quantum mechanics to general curved configuration spaces. The underlying phase space is based on the chosen coordinates of the manifold and their canonically conjugate momenta. The resulting Wigner function displays the axioms of a quasiprobability distribution, and any Weyl-ordered operator gets associated with the corresponding phase-space function, even in the absence of continuous symmetries. The corresponding quantum Liouville equation reduces to the classical curved space Liouville equation in the semiclassical limit. We demonstrate the formalism for a point particle moving on two-dimensional manifolds, such as a paraboloid or the surface of a sphere. The latter clarifies the treatment of compact coordinate spaces as well as the relation of the presented phase-space representation to symmetry groups of the configuration space.

Typo in Eq. (82) corrected and missing terms in Eq. (85) added. These corrections are also described in the erratum Phys. Rev. A 106, 069904(E) (2022). 14 pages

References in corpus (7)

• Negativity and contextuality are equivalent notions of nonclassicality
• Particle self-bunching in the Schwinger effect in spacetime-dependent electric fields
• Directly estimating non-classicality
• Operational Dynamic Modeling Transcending Quantum and Classical Mechanics
• Semiclassical propagator of the Wigner function
• The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach II
• Wigner function for the orientation state

Cited by in corpus (17)

• Rotational friction and diffusion of quantum rotors
• Quasioptical modeling of wave beams with and without mode conversion: I. Basic theory
• Quantum Angular Momentum Diffusion of Rigid Bodies
• Spin-Controlled Quantum Interference of Levitated Nanorotors
• Theory of nanoparticle cooling by elliptic coherent scattering
• Semiclassical spectral function for matter waves in random potentials
• Disorder-induced dephasing in backscattering-free quantum transport
• Disorder-dressed quantum evolution
• Quantum evolution in disordered transport
• Orientational order parameters for arbitrary quantum systems
• A Wavefunction Description for a Localized Quantum Particle in Curved Spacetimes
• Envelope-function theory of inhomogeneous strain in semiconductor nanostructures
• Gauge-invariant gravitational waves in matter beyond linearized gravity
• Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility
• Sphere on a plane: Two-dimensional scattering from a finite curved region
• Quantum information in Riemannian spaces
• Thermalization of the Quantum Planar Rotor with external potential