Generalized uncertainty principle

The Generalized Uncertainty Principle (GUP) represents a pivotal extension of the Heisenberg Uncertainty Principle, incorporating the effects of gravitational forces to refine the limits of measurement precision within quantum mechanics. Rooted in advanced theories of quantum gravity, including string theory and loop quantum gravity, the GUP introduces the concept of a minimal measurable length. This fundamental limit challenges the classical notion that positions can be measured with arbitrary precision, hinting at a discrete structure of spacetime at the Planck scale. The mathematical expression of the GUP is often formulated as:

${\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}}+\beta \Delta p^{2}}$

In this equation, ${\displaystyle \Delta x}$ and ${\displaystyle \Delta p}$ denote the uncertainties in position and momentum, respectively. The term ${\displaystyle \hbar }$ represents the reduced Planck constant, while ${\displaystyle \beta }$ is a parameter that embodies the minimal length scale predicted by the GUP. The GUP is more than a theoretical curiosity; it signifies a cornerstone concept in the pursuit of unifying quantum mechanics with general relativity. It posits an absolute minimum uncertainty in the position of particles, approximated by the Planck length, underscoring its significance in the realms of quantum gravity and string theory where such minimal length scales are anticipated. ^{[1] [2]} Various quantum gravity theories, such as string theory, loop quantum gravity, and quantum geometry, propose a generalized version of the uncertainty principle (GUP), which suggests the presence of a minimum measurable length. In earlier research, multiple forms of the GUP have been introduced ^{[3] [4] [5] [6] [7] [8] [9] [10] [11] [12]}

Observable consequences

The GUP's phenomenological and experimental implications have been examined across low and high-energy contexts, encompassing atomic systems, ^{[13] [14]} quantum optical systems, ^[15] gravitational bar detectors, ^[16] gravitational decoherence, ^[17] and macroscopic harmonic oscillators, ^[18] further extending to composite particles, ^[19] astrophysical systems ^[20]

See also

• Uncertainty principle

References

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8. Kempf, A.; Mangano, G.; Mann, R. B. (1995). "Hilbert space representation of the minimal length uncertainty relation". Physical Review D. 52 (2): 1108–1118. arXiv: hep-th/9412167 . Bibcode:1995PhRvD..52.1108K. doi:10.1103/PhysRevD.52.1108. PMID   10019328.
9. Maggiore, M. (1993). "A Generalized Uncertainty Principle in Quantum Gravity". Physics Letters B. 304 (1–2): 65–69. arXiv: hep-th/9301067 . Bibcode:1993PhLB..304...65M. doi:10.1016/0370-2693(93)91401-8.
10. Capozziello, S.; Lambiase, G.; Scarpetta, G. (1999). "The Generalized Uncertainty Principle from Quantum Geometry". International Journal of Theoretical Physics. 39 (1): 15–22. doi:10.1023/A:1003634814685.
11. Todorinov, V.; Bosso, P.; Das, S. (2019). "Relativistic Generalized Uncertainty Principle". Annals of Physics. 405: 92–100. arXiv: 1810.11761 . Bibcode:2019AnPhy.405...92T. doi:10.1016/j.aop.2019.03.014.
12. Ali, A. F.; Das, S.; Vagenas, E. C. (2009). "Discreteness of Space from the Generalized Uncertainty Principle". Physics Letters B. 678 (5): 497–499. arXiv: 0906.5396 . Bibcode:2009PhLB..678..497A. doi:10.1016/j.physletb.2009.06.061.
13. Ali, A. F., Das, S., & Vagenas, E. C. (2011). "Proposal for testing quantum gravity in the lab". Physical Review D, 84(4), 044013. http://dx.doi.org/10.1103/PhysRevD.84.044013
14. Das, S., & Vagenas, E. C. (2008). "Universality of Quantum Gravity Corrections". Physical Review Letters, 101(22), 221301. http://dx.doi.org/10.1103/PhysRevLett.101.221301
15. Pikovski, I., Vanner, M. R., Aspelmeyer, M., Kim, M. S., & Brukner, Č. (2012). "Probing Planck-scale physics with quantum optics". Nature Physics, 8(5), 393-397. http://dx.doi.org/10.1038/nphys2262
16. Marin, F., et al. (2014). "Gravitational bar detectors set limits to Planck-scale physics on macroscopic variables". Nature Physics, 9(1), 71-73. http://dx.doi.org/10.1038/nphys2503
17. Petruzziello, L., and Illuminati, F. (2021). "Quantum gravitational decoherence from fluctuating minimal length and deformation parameter at the Planck scale". Nature Communications, 12, 4449, arXiv:2011.01255 [gr-qc],http://dx.doi.org/10.1038/s41467-021-24711-7.
18. Bawaj, M., et al. (2015). "Probing deformed commutators with macroscopic harmonic oscillators". Nature Communications, 6, 7503 http://dx.doi.org/10.1038/ncomms8503.
19. Kumar, S. P., and Plenio, M. B. (2020). "Composite particle dynamics in quantum optical systems". Nature Communications, 11, 3900, arXiv:1908.11164 [quant-ph], http://dx.doi.org/10.1038/s41467-020-17518-5 .
20. Moradpour, H., Ziaie, A. H., Ghaffari, S., and Feleppa, F. (2019). "The generalized and extended uncertainty principles and their implications on the Jeans mass". Monthly Notices of the Royal Astronomical Society, 488, L69, arXiv:1907.12940 [gr-qc], http://dx.doi.org/10.1093/mnrasl/slz098 .

External links

• Research papers on Generalized Uncertainty Principle