A variational scheme for the derivation of generalized, symmetry-induced continuity equations for Hermitian and non-Hermitian quantum mechanical systems is developed. We introduce a Lagrangian which involves two complex wave fields and whose global invariance under dilation and phase variations leads to a mixed continuity equation for the two fields. In combination with discrete spatial symmetries of the underlying Hamiltonian, the mixed continuity equation is shown to produce bilocal conservation laws for a single field. This leads to generalized conserved charges for vanishing boundary currents and to divergenceless bilocal currents for stationary states. The formalism reproduces the bilocal continuity equation obtained in the special case of $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics and paraxial optics.A variational scheme for the derivation of generalized, symmetry-induced continuity equations for Hermitian and non-Hermitian quantum mechanical systems is developed. We introduce a Lagrangian which involves two complex wave fields and whose global invariance under dilation and phase variations leads to a mixed continuity equation for the two fields. In combination with discrete spatial symmetries of the underlying Hamiltonian, the mixed continuity equation is shown to produce bilocal conservation laws for a single field. This leads to generalized conserved charges for vanishing boundary currents and to divergenceless bilocal currents for stationary states. The formalism reproduces the bilocal continuity equation obtained in the special case of $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics and paraxial optics.

arXiv:1607.06730Physical Review ANov 21, 20166 citations

Formal solutions of stargenvalue equations

Nuno Costa Dias, João Nuno Prata

The two most important equations of Wigner quantum mechanics are the Moyal and the stargenvalue equations. The latter equation is the Weyl-Wigner transform of the eigenvalue equation of standard operator quantum mechanics and, in the context of the Wigner formulation, is of identical importance. The aim of this letter is to present the formal solution of a general stargenvalue equation in arbitrary dimension, both for continuous and discrete spectra. The properties of the formal solution will be studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. These results provide further insight into the mathematical structure of phase space quantum mechanics and are especially relevant for the construction of a complete formal solution of Wigner quantum mechanics in the Heisenberg picture. 1The two most important equations of Wigner quantum mechanics are the Moyal and the stargenvalue equations. The latter equation is the Weyl-Wigner transform of the eigenvalue equation of standard operator quantum mechanics and, in the context of the Wigner formulation, is of identical importance. The aim of this letter is to present the formal solution of a general stargenvalue equation in arbitrary dimension, both for continuous and discrete spectra. The properties of the formal solution will be studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. These results provide further insight into the mathematical structure of phase space quantum mechanics and are especially relevant for the construction of a complete formal solution of Wigner quantum mechanics in the Heisenberg picture. 1

Oct 5, 200118 citations

Hamiltonian Formulation of Quantum Mechanics

Domenico Giulini

Springer eBooksJan 1, 20030 citations

Continuity equation in quantum mechanics with spontaneous localization

Fabio Benatti, T. Weber

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BOct 1, 19914 citations

The Continuous Formalism of Discrete Physics: Why Unitary Time Evolution Is an Approximation, Not a Foundation

Erol Karazincir

This paper argues that unitary time evolution — the continuous, deterministic, reversible evolution of the wave function via the Schrodinger equation — is not a fundamental feature of quantum reality but a continuous approximation of an underlying discrete process, structurally parallel to how Newtonian mechanics approximates discrete molecular dynamics. The measurement problem is identified not as a feature of quantum reality but as an artifact of imposing continuous formalism on discrete physics. A continuous equation cannot produce discrete events; this is mathematical, not mysterious. Caldirola's finite-difference quantum mechanics is cited as the formal precedent. Companion paper to DOI: 10.5281/zenodo.19540088.This paper argues that unitary time evolution — the continuous, deterministic, reversible evolution of the wave function via the Schrodinger equation — is not a fundamental feature of quantum reality but a continuous approximation of an underlying discrete process, structurally parallel to how Newtonian mechanics approximates discrete molecular dynamics. The measurement problem is identified not as a feature of quantum reality but as an artifact of imposing continuous formalism on discrete physics. A continuous equation cannot produce discrete events; this is mathematical, not mysterious. Caldirola's finite-difference quantum mechanics is cited as the formal precedent. Companion paper to DOI: 10.5281/zenodo.19540088.

Apr 12, 20240 citations

\mathcal{P}\mathcal{T}

-symmetric quantum mechanics and paraxial optics.