We illustrate, using a simple model, that in the usual formulation the time-component of the Klein-Gordon current is not generally positive definite even if one restricts allowed solutions to those with positive frequencies. Since in de Broglie's theory of particle trajectories the particle follows the current this leads to difficulties of interpretation, with the appearance of trajectories which are closed loops in space-time and velocities not limited from above. We show that at least this pathology can be avoided if one uses a covariant extension of the canonical formulation of relativistic point particle dynamics proposed by Gitman and Tyutin.We illustrate, using a simple model, that in the usual formulation the time-component of the Klein-Gordon current is not generally positive definite even if one restricts allowed solutions to those with positive frequencies. Since in de Broglie's theory of particle trajectories the particle follows the current this leads to difficulties of interpretation, with the appearance of trajectories which are closed loops in space-time and velocities not limited from above. We show that at least this pathology can be avoided if one uses a covariant extension of the canonical formulation of relativistic point particle dynamics proposed by Gitman and Tyutin.

arXiv:quant-ph/0109059Sep 13, 20010 citations

Conserved Current Densities, Localization Probabilities, and a New Global Gauge Symmetry of Klein-Gordon Fields

Mostafazadeh, A., Farhad Zamani

For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities $J_a^μ$ , with $a\in(-1,1)$ , and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density ${\cal J}_a^μ$ for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both $J_a^μ$ and ${\cal J}_a^μ$ tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike $J_a^μ$ the current density ${\cal J}_a^μ$ is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of $J_a^μ$ . The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter $a$ is rational. It is the multiplicative group of positive real numbers if $a$ is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities $J_a^μ$, with $a\in(-1,1)$, and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density ${\cal J}_a^μ$ for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both $J_a^μ$ and ${\cal J}_a^μ$ tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike $J_a^μ$ the current density ${\cal J}_a^μ$ is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of $J_a^μ$. The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter $a$ is rational. It is the multiplicative group of positive real numbers if $a$ is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.

arXiv:quant-ph/0312078Dec 9, 20034 citations

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.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 eBooksSep 22, 20110 citations

On a quantum heavy particle

Dmitrii Valer'evich Treschev

International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. PontryaginJan 1, 20180 citations

For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities

J_a^μ

, with

a\in(-1,1)

, and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density

{\cal J}_a^μ

for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both

J_a^μ

and

{\cal J}_a^μ

tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike

J_a^μ

the current density

{\cal J}_a^μ

is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of

J_a^μ

. The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter

a

is rational. It is the multiplicative group of positive real numbers if

a

is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.