Revisiting the Algebraic and Analytic Descriptions of Quantum Mechanics
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915799627726848 |
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| author | Fromm, Ortwin Ehlen, Felicitas |
| author_facet | Fromm, Ortwin Ehlen, Felicitas |
| contents | We study Heisenberg's matrix mechanics within an algebraic pre-Hilbert framework of arbitrary finite dimension. The commutator of the position and momentum matrices naturally generates a third Hermitian operator whose unbounded character originates from boundary contributions and whose structure induces a discrete analogue of the Cauchy-Hilbert kernel. Compared with the separable Hilbert-space completion, the algebraic framework reproduces the standard spectra, canonical commutation relations, and Heisenberg uncertainty relation for finite-energy states, while the discrete kernel is absorbed into its continuous integral counterpart under completion. The comparison shows that both formulations require restrictions on admissible states for effective calculations -- analytic domain restrictions in Hilbert space and finite-energy restrictions in the pre-Hilbert framework. Finally, we discuss to what extent quantum randomness arises from the algebraic structure of the pre-Hilbert framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14126 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Revisiting the Algebraic and Analytic Descriptions of Quantum Mechanics Fromm, Ortwin Ehlen, Felicitas Quantum Algebra We study Heisenberg's matrix mechanics within an algebraic pre-Hilbert framework of arbitrary finite dimension. The commutator of the position and momentum matrices naturally generates a third Hermitian operator whose unbounded character originates from boundary contributions and whose structure induces a discrete analogue of the Cauchy-Hilbert kernel. Compared with the separable Hilbert-space completion, the algebraic framework reproduces the standard spectra, canonical commutation relations, and Heisenberg uncertainty relation for finite-energy states, while the discrete kernel is absorbed into its continuous integral counterpart under completion. The comparison shows that both formulations require restrictions on admissible states for effective calculations -- analytic domain restrictions in Hilbert space and finite-energy restrictions in the pre-Hilbert framework. Finally, we discuss to what extent quantum randomness arises from the algebraic structure of the pre-Hilbert framework. |
| title | Revisiting the Algebraic and Analytic Descriptions of Quantum Mechanics |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2602.14126 |