Positive Conserved Quantities in the Klein-Gordon Equation

• URL: http://arxiv.org/abs/2410.04666v2
• Date: Mon, 21 Oct 2024 00:13:02 GMT
• Title: Positive Conserved Quantities in the Klein-Gordon Equation
• Authors: Robert Lin,
• Abstract summary: We introduce an embedding of the Klein-Gordon equation into a pair of coupled equations that are first-order in time. These coupled equations provide a more satisfactory reduction of the Klein-Gordon equation to first-order differential equations in time than the Schrodinger equation. We show that there are two positive integrals that are conserved (constant in time) in the Klein-Gordon equation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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