Positive Conserved Quantities in the Klein-Gordon Equation

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Autore principale:	Lin, Robert
Natura:	Preprint
Pubblicazione:	2024
Soggetti:	Quantum Physics High Energy Physics - Theory
Accesso online:	Acceder al recurso
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author	Lin, Robert
author_facet	Lin, Robert
contents	We introduce an embedding of the Klein-Gordon equation into a pair of coupled equations that are first-order in time. The existence of such an embedding is based on a positivity property exhibited by the Klein-Gordon equation. These coupled equations provide a more satisfactory reduction of the Klein-Gordon equation to first-order differential equations in time than the Schrodinger equation. Using this embedding, we show that the "negative probabilities" associated with the Klein-Gordon equation do not need to be resolved by introducing matrices as Dirac did with his eponymous equation. For the case of the massive Klein-Gordon equation, the coupled equations are equivalent to a forward Schrodinger equation in time and a backward Schrodinger equation in time, respectively, corresponding to a particle and its antiparticle. We show that there are two positive integrals that are conserved (constant in time) in the Klein-Gordon equation and thus provide a concrete resolution of the historical puzzle regarding the previously supposed lack of a probabilistic interpretation for the field governed by the Klein-Gordon equation.
format	Preprint
id	arxiv_https___arxiv_org_abs_2410_04666
institution	arXiv
publishDate	2024
record_format	arxiv
spellingShingle	Positive Conserved Quantities in the Klein-Gordon Equation Lin, Robert Quantum Physics High Energy Physics - Theory We introduce an embedding of the Klein-Gordon equation into a pair of coupled equations that are first-order in time. The existence of such an embedding is based on a positivity property exhibited by the Klein-Gordon equation. These coupled equations provide a more satisfactory reduction of the Klein-Gordon equation to first-order differential equations in time than the Schrodinger equation. Using this embedding, we show that the "negative probabilities" associated with the Klein-Gordon equation do not need to be resolved by introducing matrices as Dirac did with his eponymous equation. For the case of the massive Klein-Gordon equation, the coupled equations are equivalent to a forward Schrodinger equation in time and a backward Schrodinger equation in time, respectively, corresponding to a particle and its antiparticle. We show that there are two positive integrals that are conserved (constant in time) in the Klein-Gordon equation and thus provide a concrete resolution of the historical puzzle regarding the previously supposed lack of a probabilistic interpretation for the field governed by the Klein-Gordon equation.
title	Positive Conserved Quantities in the Klein-Gordon Equation
topic	Quantum Physics High Energy Physics - Theory
url	https://arxiv.org/abs/2410.04666

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