Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Anyang Normal University

The U(1) Symmetry of the Schrodinger Equation and the Continuity Equation

HONG-JUN HAO

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Based on the field theory of quantum mechanics,this paper consider the U(1) transformation on the Schrodinger fields.It find that the conserved Neother's current is related to the continuity equation,and the conserved charge is the probability of all space.

Related papers

On charge conservation and the continuity equation for real wavefunctions in unified quantum mechanics

Pushpendra Singh

This paper presents a rigorous mathematical derivation of the continuity equation within a reformulated quantum mechanical framework-Unified Quantum Mechanics (UQM)-where the fundamental imaginary unit \(i\) is systematically replaced by the temporal Hilbert transform operator \(\mathcal{H}_t\). This substitution enables a complete description of quantum dynamics using exclusively real-valued wavefunctions while preserving the essential conservation laws of quantum theory. We demonstrate that this real-valued formalism naturally gives rise to a conserved quantity analogous to the probability density in conventional quantum mechanics. By constructing a coupled system of evolution equations for the primary wavefunction \(\psi\) and its Hilbert transform \(\phi = \mathcal{H}_t[\psi]\), we identify an underlying global SO(2) symmetry that is mathematically isomorphic to the U(1) gauge symmetry responsible for charge conservation in standard complex-valued quantum mechanics. Through direct computation, we derive the continuity equation \(\frac{\partial\rho}{\partial t} + \nabla \cdot \vec{j} = 0\) and obtain explicit expressions for the conserved density \(\rho\) and current density \(\vec{j}\). This result establishes that the fundamental principle of local charge conservation emerges naturally within the UQM dynamics, thereby validating its mathematical consistency and physical completeness. Moreover, we extend the result to demonstrate the local conservation of mass, deriving the corresponding continuity equation for mass density.

Next research.•Oct 7, 2026•0 citations

Symmetries of the Schrödinger bundle

V. Aldaya, J. A. de Azcárraga

We exhibit in this paper the invariance of the Schrödinger Lagrangian density under the eleven-parameter group 𝒢(m), central extension of the Galilei group 𝒢. As a result, the quantum mechanical probability ℱρ d3x turns out to be the conserved charge associated with the central generator of 𝒢(m), and the continuity equation is simply the expression of the conservation of the corresponding Noether current.

Journal of Mathematical Physics•Jul 1, 1981•8 citations

Generalized continuity equations from two-field Schrödinger Lagrangians

A. G. B. Spourdalakis, George Pappas, Christian V. Morfonios +3 more

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:1607.06730•Physical Review A•Nov 21, 2016•6 citations

Generalized Continuity Equations for Schrödinger and Dirac Equations

A. Katsaris, P. A. Kalozoumis, F. Κ. Diakonos

The concept of the generalized continuity equation (GCE) was recently introduced in [J. Phys. A: Math. and Theor. {\bf 52}, 1552034 (2019)], and was derived in the context of $N$ independent Schrödinger systems. The GCE is induced by a symmetry transformation which mixes the states of these systems, even though the $N$-system Lagrangian does not. As the $N$-system Schrödinger Lagrangian is not invariant under such a transformation, the GCE will involve source terms which, under certain conditions vanish and lead to conserved currents. These conditions may hold globally or locally in a finite domain, leading to globally or locally conserved currents, respectively. In this work, we extend this idea to the case of arbitrary $SU(N)$-transformations and we show that a similar GCE emerges for $N$ systems in the Dirac dynamics framework. The emerging GCEs and the conditions which lead to the attendant conservation laws provide a rich phenomenology and potential use for the preparation and control of fermionic states.

arXiv:2103.00052•Feb 26, 2021•0 citations

A New Formulation of Quantum Mechanics

Arbab I. Arbab, Faisal A. Yassein

A new formulation of quantum mechanics based on differential commutator brackets is developed. We have found a wave equation representing the fermionic particle. In this formalism, the continuity equation mixes the Klein-Gordon and Schrodinger probability density while keeping the Klein-Gordon and Schrodinger current unaltered. We have found time and space transformations under which Dirac’s equation is invariant. The invariance of Maxwell’s equations under these transformations shows that the electric and magnetic fields of a moving charged particle are perpendicular to the velocity of the propagating particle. This formulation agrees with the quaternionic formulation recently developed by Arbab.

arXiv:1002.4709•Journal of Modern Physics•Jan 1, 2012•10 citations

Relativistic Schrödinger equation and probability currents for free particles

David Carfì

In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.

Proceedings of the International Geometry Center•Jan 13, 2024•2 citations
This paper presents a rigorous mathematical derivation of the continuity equation within a reformulated quantum mechanical framework-Unified Quantum Mechanics (UQM)-where the fundamental imaginary unit iii is systematically replaced by the temporal Hilbert transform operator Ht\mathcal{H}_tHt​. This substitution enables a complete description of quantum dynamics using exclusively real-valued wavefunctions while preserving the essential conservation laws of quantum theory. We demonstrate that this real-valued formalism naturally gives rise to a conserved quantity analogous to the probability density in conventional quantum mechanics. By constructing a coupled system of evolution equations for the primary wavefunction ψ\psiψ and its Hilbert transform ϕ=Ht[ψ]\phi = \mathcal{H}_t[\psi]ϕ=Ht​[ψ], we identify an underlying global SO(2) symmetry that is mathematically isomorphic to the U(1) gauge symmetry responsible for charge conservation in standard complex-valued quantum mechanics. Through direct computation, we derive the continuity equation ∂ρ∂t+∇⋅j⃗=0\frac{\partial\rho}{\partial t} + \nabla \cdot \vec{j} = 0∂t∂ρ​+∇⋅j​=0 and obtain explicit expressions for the conserved density ρ\rhoρ and current density j⃗\vec{j}j​. This result establishes that the fundamental principle of local charge conservation emerges naturally within the UQM dynamics, thereby validating its mathematical consistency and physical completeness. Moreover, we extend the result to demonstrate the local conservation of mass, deriving the corresponding continuity equation for mass density.
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 PT\mathcal{P}\mathcal{T}PT-symmetric quantum mechanics and paraxial optics.
The concept of the generalized continuity equation (GCE) was recently introduced in [J. Phys. A: Math. and Theor. {\bf 52}, 1552034 (2019)], and was derived in the context of NNN independent Schrödinger systems. The GCE is induced by a symmetry transformation which mixes the states of these systems, even though the NNN-system Lagrangian does not. As the NNN-system Schrödinger Lagrangian is not invariant under such a transformation, the GCE will involve source terms which, under certain conditions vanish and lead to conserved currents. These conditions may hold globally or locally in a finite domain, leading to globally or locally conserved currents, respectively. In this work, we extend this idea to the case of arbitrary SU(N)SU(N)SU(N)-transformations and we show that a similar GCE emerges for NNN systems in the Dirac dynamics framework. The emerging GCEs and the conditions which lead to the attendant conservation laws provide a rich phenomenology and potential use for the preparation and control of fermionic states.
A new formulation of quantum mechanics based on differential commutator brackets is developed. We have found a wave equation representing the fermionic particle. In this formalism, the continuity equation mixes the Klein-Gordon and Schrodinger probability density while keeping the Klein-Gordon and Schrodinger current unaltered. We have found time and space transformations under which Dirac’s equation is invariant. The invariance of Maxwell’s equations under these transformations shows that the electric and magnetic fields of a moving charged particle are perpendicular to the velocity of the propagating particle. This formulation agrees with the quaternionic formulation recently developed by Arbab.
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.