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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.04666 |
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| _version_ | 1866914980118396928 |
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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 |