dorsal/arxiv
View SchemaFoundations of a spacetime path formalism for relativistic quantum mechanics
| Authors | Ed Seidewitz |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0507115 |
| URL | https://arxiv.org/abs/quant-ph/0507115 |
| DOI | 10.1063/1.2375033 |
| Journal | J.Math.Phys. 47 (2006) 112302 |
Abstract
Quantum field theory is the traditional solution to the problems inherent in melding quantum mechanics with special relativity. However, it has also long been known that an alternative first-quantized formulation can be given for relativistic quantum mechanics, based on the parametrized paths of particles in spacetime. Because time is treated similarly to the three space coordinates, rather than as an evolution parameter, such a spacetime approach has proved particularly useful in the study of quantum gravity and cosmology. This paper shows how a spacetime path formalism can be considered to arise naturally from the fundamental principles of the Born probability rule, superposition, and Poincar\'e invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches in the literature, relating, in particular, "off-shell" theories to traditional on-shell quantum field theory. It reproduces the results of perturbative quantum field theory for free and interacting particles, but provides intriguing possibilities for a natural program for regularization and renormalization. Further, an important consequence of the formalism is that a clear probabilistic interpretation can be maintained throughout, with a natural reduction to non-relativistic quantum mechanics.
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"abstract": "Quantum field theory is the traditional solution to the problems inherent in\nmelding quantum mechanics with special relativity. However, it has also long\nbeen known that an alternative first-quantized formulation can be given for\nrelativistic quantum mechanics, based on the parametrized paths of particles in\nspacetime. Because time is treated similarly to the three space coordinates,\nrather than as an evolution parameter, such a spacetime approach has proved\nparticularly useful in the study of quantum gravity and cosmology. This paper\nshows how a spacetime path formalism can be considered to arise naturally from\nthe fundamental principles of the Born probability rule, superposition, and\nPoincar\\\u0027e invariance. The resulting formalism can be seen as a foundation for\na number of previous parametrized approaches in the literature, relating, in\nparticular, \"off-shell\" theories to traditional on-shell quantum field theory.\nIt reproduces the results of perturbative quantum field theory for free and\ninteracting particles, but provides intriguing possibilities for a natural\nprogram for regularization and renormalization. Further, an important\nconsequence of the formalism is that a clear probabilistic interpretation can\nbe maintained throughout, with a natural reduction to non-relativistic quantum\nmechanics.",
"arxiv_id": "quant-ph/0507115",
"authors": [
"Ed Seidewitz"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.2375033",
"journal_ref": "J.Math.Phys. 47 (2006) 112302",
"title": "Foundations of a spacetime path formalism for relativistic quantum mechanics",
"url": "https://arxiv.org/abs/quant-ph/0507115"
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