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 Mathematical Physics

Conservation laws in the Dirac theory

Roger Boudet

Published Apr 1, 1985• 18 citations
Publisher

Abstract

It is well known that the Dirac equations determine three conservation laws, for current, for energy momentum, and for angular momentum. Dividing the Dirac equations in two parts, one, DI, which does not contain the density ρ, the other, DII, which contains ρ, it is shown that DI together with the three conservation equations, determine DII. From this result, a model of formulation of the Dirac theory is proposed, following a scheme similar to that of classical mechanics, in which principles regarding the motion of one particle are associated with conservation theorems concerning statistical ensembles of particles.

DOI
10.1063/1.526613

Related papers

Consistency in the formulation of the Dirac, Pauli, and Schrödinger theories

Richard Warren Gurtler, David Hestenes

Properties of observables in the Pauli and Schrödinger theories and first order relativistic approximations to them are derived from the Dirac theory. They are found to be inconsistent with customary interpretations in many respects. For example, failure to identify the ’’Darwin term’’ as the s−state spin−orbit energy in conventional treatments of the hydrogen atom is traced to a failure to distinguish between charge and momentum flow in the theory. Consistency with the Dirac theory is shown to imply that the Schrödinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schrödinger theory is discussed. Conservation laws of the Dirac theory are formulated in terms of relative variables, and used to derive virial theorems and the corresponding conservation laws in the Pauli−Schrödinger theory.

Journal of Mathematical Physics•Mar 1, 1975•102 citations

Trajectory construction of Dirac evolution

Peter Holland

We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the ‘relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.

arXiv:1910.05829•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Apr 1, 2020•2 citations

Classical Theory of the Dirac Electron

Zvi Grossmann, Asher Peres

This paper contains the Hamiltonian theory of a classical particle displaying all the features of the Dirac electron (spin, Zitterbewegung, etc.) except, of course, for the superposition principle. The particle is described by eight internal canonical variables, of which three are the spin angular momentum vector. The five other variables have no simple physical meaning, but are nevertheless necessary for a consistent theory. The equations of motion are not manifestly covariant, and the Lorentz invariance of the theory is proved by constructing the ten generators of the inhomogeneous Lorentz group.

Physical Review•Dec 1, 1963•21 citations

The Conservation Laws in Quantum Mechanics

Richard Joseph Oldani

It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although quantum mechanics correctly predicts what it is possible to observe (the emissions) with remarkable accuracy it ignores the other half of natural phenomena, that which cannot be observed (the absorptions), thereby violating the conservation of energy. Similarly, the conservation of momentum is violated by disregarding unobservable molecular impulses. Despite irrefutable proof that molecular impulses transfer momentum asymmetrically as shown by Einstein’s derivation of the A and B coefficients, the wave function is assumed to apply symmetrically in time. The deficiencies are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows non-relativistic quantum mechanics to be described with a physical model and wave function behavior to be interpreted as the combined action of a particle and its associated quantized field. The calculus of variations is applied to the wave function to show that it is an incomplete equation of motion because it yields twice the allowable action minimum. The concept of reality is redefined by showing that reductionism applies to classical phenomena, but not quantum phenomena.

Qeios•Aug 16, 2023•1 citation

The Dirac equation

Tommy Ohlsson B

In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that had a positive-definite conserved probability density and was first order both in time and space. One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E 2 = m 2 + p 2 via the correspondence principle that includes a term E 2 . Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.

Cambridge University Press eBooks•Sep 22, 2011•0 citations

A uniform description of electromagnetism and wave mechanics

Engel Roza

The paper shows the relationship between the major wave equations in quantum mechanics and electromagnetism, such as Schroedinger's equation, Dirac's equation and the Maxwell equations. It is shown that they can be derived in a striking simple way from a common root. This root is the relativistic fourvector formulation of the momentum conservation law. This is shown to be a more attractive starting-point than Einstein's energy relationship for moving particles, which is commonly used for the purpose. The theory developed gives a new interpretation for the origin of antiparticles.

arXiv:quant-ph/0602021•Feb 2, 2006•0 citations
It is well known that the Dirac equations determine three conservation laws, for current, for energy momentum, and for angular momentum. Dividing the Dirac equations in two parts, one, DI, which does not contain the density ρ, the other, DII, which contains ρ, it is shown that DI together with the three conservation equations, determine DII. From this result, a model of formulation of the Dirac theory is proposed, following a scheme similar to that of classical mechanics, in which principles regarding the motion of one particle are associated with conservation theorems concerning statistical ensembles of particles.
Properties of observables in the Pauli and Schrödinger theories and first order relativistic approximations to them are derived from the Dirac theory. They are found to be inconsistent with customary interpretations in many respects. For example, failure to identify the ’’Darwin term’’ as the s−state spin−orbit energy in conventional treatments of the hydrogen atom is traced to a failure to distinguish between charge and momentum flow in the theory. Consistency with the Dirac theory is shown to imply that the Schrödinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schrödinger theory is discussed. Conservation laws of the Dirac theory are formulated in terms of relative variables, and used to derive virial theorems and the corresponding conservation laws in the Pauli−Schrödinger theory.
We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the ‘relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.
This paper contains the Hamiltonian theory of a classical particle displaying all the features of the Dirac electron (spin, Zitterbewegung, etc.) except, of course, for the superposition principle. The particle is described by eight internal canonical variables, of which three are the spin angular momentum vector. The five other variables have no simple physical meaning, but are nevertheless necessary for a consistent theory. The equations of motion are not manifestly covariant, and the Lorentz invariance of the theory is proved by constructing the ten generators of the inhomogeneous Lorentz group.
It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although quantum mechanics correctly predicts what it is possible to observe (the emissions) with remarkable accuracy it ignores the other half of natural phenomena, that which cannot be observed (the absorptions), thereby violating the conservation of energy. Similarly, the conservation of momentum is violated by disregarding unobservable molecular impulses. Despite irrefutable proof that molecular impulses transfer momentum asymmetrically as shown by Einstein’s derivation of the A and B coefficients, the wave function is assumed to apply symmetrically in time. The deficiencies are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows non-relativistic quantum mechanics to be described with a physical model and wave function behavior to be interpreted as the combined action of a particle and its associated quantized field. The calculus of variations is applied to the wave function to show that it is an incomplete equation of motion because it yields twice the allowable action minimum. The concept of reality is redefined by showing that reductionism applies to classical phenomena, but not quantum phenomena.
In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that had a positive-definite conserved probability density and was first order both in time and space. One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E 2 = m 2 + p 2 via the correspondence principle that includes a term E 2 . Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.
The paper shows the relationship between the major wave equations in quantum mechanics and electromagnetism, such as Schroedinger's equation, Dirac's equation and the Maxwell equations. It is shown that they can be derived in a striking simple way from a common root. This root is the relativistic fourvector formulation of the momentum conservation law. This is shown to be a more attractive starting-point than Einstein's energy relationship for moving particles, which is commonly used for the purpose. The theory developed gives a new interpretation for the origin of antiparticles.