Integral quantization based on the Heisenberg-Weyl group
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910923644469248 |
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| author | Pȩdrak, Aleksandra Góźdź, Andrzej Piechocki, Włodzimierz Mach, Patryk Cieślik, Adam |
| author_facet | Pȩdrak, Aleksandra Góźdź, Andrzej Piechocki, Włodzimierz Mach, Patryk Cieślik, Adam |
| contents | We develop a relativistic framework of integral quantization applied to the motion of spinless particles in the four-dimensional Minkowski spacetime. The proposed scheme is based on coherent states generated by the action of the Heisenberg-Weyl group and has been motivated by the Hamiltonian description of the geodesic motion in General Relativity. We believe that this formulation should also allow for a generalization to the motion of test particles in curved spacetimes. A key element in our construction is the use of suitably defined positive operator-valued measures. We show that this approach can be used to quantize the one-dimensional nonrelativistic harmonic oscillator, recovering the standard Hamiltonian as obtained by the canonical quantization. A direct application of our model, including a computation of transition amplitudes between states characterized by fixed positions and momenta, is postponed to a forthcoming article. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral quantization based on the Heisenberg-Weyl group Pȩdrak, Aleksandra Góźdź, Andrzej Piechocki, Włodzimierz Mach, Patryk Cieślik, Adam General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics Quantum Physics We develop a relativistic framework of integral quantization applied to the motion of spinless particles in the four-dimensional Minkowski spacetime. The proposed scheme is based on coherent states generated by the action of the Heisenberg-Weyl group and has been motivated by the Hamiltonian description of the geodesic motion in General Relativity. We believe that this formulation should also allow for a generalization to the motion of test particles in curved spacetimes. A key element in our construction is the use of suitably defined positive operator-valued measures. We show that this approach can be used to quantize the one-dimensional nonrelativistic harmonic oscillator, recovering the standard Hamiltonian as obtained by the canonical quantization. A direct application of our model, including a computation of transition amplitudes between states characterized by fixed positions and momenta, is postponed to a forthcoming article. |
| title | Integral quantization based on the Heisenberg-Weyl group |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2410.23982 |