A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The $1$ -- D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The $1$ -- D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.

arXiv:2307.12046Jul 22, 20230 citations

The continuity equation and the Hamiltonian formalism in quantum mechanics

L. Ferrari

Foundations of PhysicsApr 1, 19874 citations

Discrete Phase Space: Quantum mechanics and non-singular potential functions

Anadijiban Das, Andrew DeBenedictis

The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space $\mathbb{E}_{3}$ where the potential is singular but invariant under the continuous inhomogeneous orthogonal group $\mathcal{I}O(3)$ . The invariance under the translation subgroup is compared to the corresponding unitary transformation in the Schrödinger representation of quantum mechanics. This scenario is well known but serves as a reference point for the other scenarios. (ii) Next, the discrete potential equation as a partial difference equation in a three-dimensional lattice space is studied. In this arena the potential is non-singular but invariance under $\mathcal{I}O(3)$ is broken. This is the usual picture of lattice theories and numerical approximations. (iii) Next we study the six-dimensional continuous phase space. Here a phase space representation of quantum mechanics is utilized. The resulting potential is singular but possesses invariance under $\mathcal{I}O(3)$ . (iv) Finally, the potential is derived from the discrete phase space representation of quantum mechanics, which is shown to be an \emph{exact} representation of quantum mechanics. The potential function here is both non-singular and possesses invariance under $\mathcal{I}O(3)$ , and this is proved via the unitary transformations of quantum mechanics in this representation.The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space $\mathbb{E}_{3}$ where the potential is singular but invariant under the continuous inhomogeneous orthogonal group $\mathcal{I}O(3)$. The invariance under the translation subgroup is compared to the corresponding unitary transformation in the Schrödinger representation of quantum mechanics. This scenario is well known but serves as a reference point for the other scenarios. (ii) Next, the discrete potential equation as a partial difference equation in a three-dimensional lattice space is studied. In this arena the potential is non-singular but invariance under $\mathcal{I}O(3)$ is broken. This is the usual picture of lattice theories and numerical approximations. (iii) Next we study the six-dimensional continuous phase space. Here a phase space representation of quantum mechanics is utilized. The resulting potential is singular but possesses invariance under $\mathcal{I}O(3)$. (iv) Finally, the potential is derived from the discrete phase space representation of quantum mechanics, which is shown to be an \emph{exact} representation of quantum mechanics. The potential function here is both non-singular and possesses invariance under $\mathcal{I}O(3)$, and this is proved via the unitary transformations of quantum mechanics in this representation.

arXiv:1504.01788Apr 8, 20150 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

Discrete Phase Space, String-Like Phase Cells, and Relativistic Quantum Mechanics

Anadijiban Das, Sourav Haldar

The discrete phase space representation of quantum mechanics involving a characteristic length is investigated. The continuous (1 + 1)-dimensional phase space is first discussed for the sake of simplicity. It is discretized into denumerable infinite number of concentric circles. These circles, endowed with “unit area”, are degenerate phase cells resembling closed strings.Next, Schrödinger wave equation for one particle in the three dimensional space under the influence of a static potential is studied in the discrete phase space representation of wave mechanics. The Schrödinger equation in the arena of discrete phase space is a partial difference equation. The energy eigenvalue problem for a three dimensional oscillator is exactly solved.Next, relativistic wave equations in the scenario of three dimensional discrete phase space and continuous time are explored. Specially, the partial finite difference-differential equation for a scalar field is investigated for the sake of simplicity. The exact relativistic invariance of the partial finite difference-differential version of the Klein-Gordon equation is rigorously proved. Moreover, it is shown that all nine important Green’s functions of the partial finite difference-differential wave equation for the scalar field are non-singular.In the appendix, a possible physical interpretation for the discrete orbits in the phase space as degenerate, string-like phase cells is provided in a mathematically rigorous way.The discrete phase space representation of quantum mechanics involving a characteristic length is investigated. The continuous (1 + 1)-dimensional phase space is first discussed for the sake of simplicity. It is discretized into denumerable infinite number of concentric circles. These circles, endowed with “unit area”, are degenerate phase cells resembling closed strings.Next, Schrödinger wave equation for one particle in the three dimensional space under the influence of a static potential is studied in the discrete phase space representation of wave mechanics. The Schrödinger equation in the arena of discrete phase space is a partial difference equation. The energy eigenvalue problem for a three dimensional oscillator is exactly solved.Next, relativistic wave equations in the scenario of three dimensional discrete phase space and continuous time are explored. Specially, the partial finite difference-differential equation for a scalar field is investigated for the sake of simplicity. The exact relativistic invariance of the partial finite difference-differential version of the Klein-Gordon equation is rigorously proved. Moreover, it is shown that all nine important Green’s functions of the partial finite difference-differential wave equation for the scalar field are non-singular.In the appendix, a possible physical interpretation for the discrete orbits in the phase space as degenerate, string-like phase cells is provided in a mathematically rigorous way.

Physical Science International JournalFeb 5, 20183 citations

Quantum mechanics on phase space

Nicolaas P. Landsman

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsJan 1, 19996 citations

A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The

1

-- D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.

The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space

\mathbb{E}_{3}

where the potential is singular but invariant under the continuous inhomogeneous orthogonal group

\mathcal{I}O(3)

. The invariance under the translation subgroup is compared to the corresponding unitary transformation in the Schrödinger representation of quantum mechanics. This scenario is well known but serves as a reference point for the other scenarios. (ii) Next, the discrete potential equation as a partial difference equation in a three-dimensional lattice space is studied. In this arena the potential is non-singular but invariance under

\mathcal{I}O(3)

is broken. This is the usual picture of lattice theories and numerical approximations. (iii) Next we study the six-dimensional continuous phase space. Here a phase space representation of quantum mechanics is utilized. The resulting potential is singular but possesses invariance under

\mathcal{I}O(3)

. (iv) Finally, the potential is derived from the discrete phase space representation of quantum mechanics, which is shown to be an \emph{exact} representation of quantum mechanics. The potential function here is both non-singular and possesses invariance under

\mathcal{I}O(3)

, and this is proved via the unitary transformations of quantum mechanics in this representation.