Non-Heisenbergian quantum mechanics

MohammadJavad Kazemi; Ghadir Jafari

doi:10.22331/q-2025-11-06-1904

Non-Heisenbergian quantum mechanics

MohammadJavad Kazemi¹ and Ghadir Jafari²

¹Department of Physics, Faculty of Science, University of Qom, Qom, Iran
²Department of Physics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran

Published:	2025-11-06, volume 9, page 1904
Editor:	Leon Loveridge
Eprint:	arXiv:2402.11350v4
Doi:	https://doi.org/10.22331/q-2025-11-06-1904
Citation:	Quantum 9, 1904 (2025).

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Abstract

Relaxing the postulates of an axiomatic theory is a natural way to find more general theories, and historically, the discovery of non-Euclidean geometry is a famous example of this procedure. Here, we use this way to extend quantum mechanics by ignoring the $heart$ of Heisenberg's quantum mechanics – We do not assume the existence of a position operator that satisfies the Heisenberg commutation relation, $[\hat x,\hat p]=i\hbar$. The remaining axioms of quantum theory, besides Galilean symmetry, lead to a more general quantum theory with a free parameter $l_0$ of length dimension, such that as $l_0 \to 0$ the theory reduces to standard quantum theory. Perhaps surprisingly, this non-Heisenberg quantum theory, without a priori assumption of the non-commutation relation, leads to a modified Heisenberg uncertainty relation, $\Delta x \Delta p\geq \sqrt{\hbar^2/4+l_0^2(\Delta p)^2}$, which ensures the existence of a minimal position uncertainty, $l_0$, as expected from various quantum gravity studies. By comparing the results of this framework with some observed data, which includes the first longitudinal normal modes of the bar gravitational wave detector AURIGA and the $1S-2S$ transition in the hydrogen atom, we obtain upper bounds on the $l_0$.

► BibTeX data

@article{Kazemi2025nonheisenbergian,
  doi = {10.22331/q-2025-11-06-1904},
  url = {https://doi.org/10.22331/q-2025-11-06-1904},
  title = {Non-{H}eisenbergian quantum mechanics},
  author = {Kazemi, MohammadJavad and Jafari, Ghadir},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1904},
  month = nov,
  year = {2025}
}

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Cited by

[1] Filippo Giraldi, "Modifying the Lower Bound of the Heisenberg Uncertainty Relation with Special Deformed Boson Algebra", Mathematics 14 9, 1505 (2026).

[2] M. M. Ettefaghi, "Neutrino oscillation with minimal length uncertainty relation via wave packet approach", Physics Letters B 854, 138761 (2024).

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