# Bekenstein Bound and Non-Commutative Canonical Variables

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. HUP and Bekenstein Bound

- (i)
- The system is in a regime where the equipartition theorem holds, namely, on average, the energy $\mu $ of each component of the system is approximately given by$$\mu \phantom{\rule{0.166667em}{0ex}}\simeq \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{k}_{B}\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}.$$
- (ii)
- As our system is quantum, the momentum p of each component should satisfy the de Broglie relation$$p\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{\hslash}{\lambda}\phantom{\rule{0.166667em}{0ex}},$$

## 3. Non-Commutative Variables

## 4. Concluding Remarks

#### Appendix

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | For completeness, we emphasize that the ansatz $S(E=0)=0$ contains the hidden assumption of a unique ground state. |

2 | For instance, an electron can be in two possible states (spin up and spin down) and, therefore, its entropy is given by $S={k}_{B}log2\sim \mathcal{O}\left({k}_{B}\right).$ |

## References

- Bekenstein, J.D. Universal upper bound on the entropy-to-energy ratio for bounded systems. Phys. Rev. D
**1981**, 23, 287. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black holes and the second law. Nuovo Cim. Lett.
**1972**, 4, 737–740. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black Holes and Entropy. Phys. Rev. D
**1973**, 7, 2333. [Google Scholar] [CrossRef] - Bekenstein, J.D. Generalized second law of thermodynamics in black-hole physics. Phys. Rev. D
**1974**, 9, 3292. [Google Scholar] [CrossRef] [Green Version] - Bardeen, J.M.; Carter, B.; Hawking, S.W. The four laws of black hole mechanics. Commun. Math. Phys.
**1973**, 31, 161. [Google Scholar] [CrossRef] - Hawking, S.W. Particle creation by black holes. Commun. Math. Phys.
**1975**, 43, 199. [Google Scholar] [CrossRef] - Bekenstein, J.D. Energy Cost of Information Transfer. Phys. Rev. Lett.
**1981**, 46, 623. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black Holes And Everyday Physics. Gen. Rel. Grav.
**1982**, 14, 355. [Google Scholar] [CrossRef] - Bekenstein, J.D. Holographic bound from second law of thermodynamics. Phys. Lett. B
**2000**, 481, 339. [Google Scholar] [CrossRef] [Green Version] - Schiffer, M.; Bekenstein, J.D. Proof of the Quantum Bound on Specific Entropy for Free Fields. Phys. Rev. D
**1989**, 39, 1109. [Google Scholar] [CrossRef] - Bekenstein, J.D.; Schiffer, M. Quantum limitations on the storage and transmission of information. Int. J. Mod. Phys. C
**1990**, 1, 355. [Google Scholar] [CrossRef] [Green Version] - Unruh, W.G.; Wald, R.M. Acceleration radiation and the generalized second law of thermodynamics. Phys. Rev. D
**1982**, 25, 942. [Google Scholar] [CrossRef] - Unwin, S.D. Possible Violations Of The Entropy To Energy Ratio Bound. Phys. Rev. D
**1982**, 26, 944. [Google Scholar] [CrossRef] - Page, D.N. Comment on a Universal Upper Bound on the Entropy to Energy Ratio for Bounded Systems. Phys. Rev. D
**1982**, 26, 947. [Google Scholar] [CrossRef] - Bekenstein, J.D. Entropy Bounds And The Second Law For Black Holes. Phys. Rev. D
**1983**, 27, 2262. [Google Scholar] [CrossRef] - Unruh, W.G.; Wald, R.M. Entropy Bounds, Acceleration Radiation, And The Generalized Second Law. Phys. Rev. D
**1983**, 27, 2271. [Google Scholar] [CrossRef] - Pelath, M.A.; Wald, R.M. Comment on entropy bounds and the generalized second law. Phys. Rev. D
**1999**, 60, 104009. [Google Scholar] [CrossRef] [Green Version] - Page, D.N. Defining entropy bounds. J. High Energy Phys.
**2008**, 10, 7. [Google Scholar] [CrossRef] [Green Version] - Gonzalez-Diaz, P. BOUNDS ON THE ENTROPY. Phys. Rev. D
**1983**, 27, 3042. [Google Scholar] [CrossRef] [Green Version] - Hooft, G. Dimensional reduction in quantum gravity. Conf. Proc. C
**1993**, 930308, 284. [Google Scholar] - Susskind, L. The world as a hologram. J. Math. Phys.
**1995**, 36, 6377. [Google Scholar] [CrossRef] - Bousso, R. A Covariant entropy conjecture. J. High Energy Phys.
**1999**, 7, 4. [Google Scholar] [CrossRef] [Green Version] - Brustein, R.; Veneziano, G. Causal entropy bound for a spacelike region. Phys. Rev. Lett.
**2000**, 84, 5695. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Casini, H. Relative entropy and the Bekenstein bound. Class. Quant. Grav.
**2008**, 25, 205021. [Google Scholar] [CrossRef] - Bekenstein, J.D. Information in the holographic universe. Scientific American
**2003**, 289, 58. [Google Scholar] [CrossRef] [PubMed] - Smolin, J.; Oppenheim, J. Locking information in black holes. Phys. Rev. Lett.
**2006**, 96, 081302. [Google Scholar] [CrossRef] [Green Version] - Hänggi, E.; Wehner, S. A violation of the uncertainty principle implies a violation of the second law of thermodynamics. Nat. Commun.
**2013**, 4, 1670. [Google Scholar] [CrossRef] [Green Version] - Fischler, W.; Susskind, L. Holography and Cosmology. arXiv
**1998**, arXiv:hep-th/9806039. [Google Scholar] - Banks, T.; Fischler, W. Cosmological Implications of the Bekenstein Bound. arXiv
**2018**, arXiv:1810.01671. [Google Scholar] - Veneziano, G. Entropy Bounds and String Cosmology. arXiv
**1999**, arXiv:hep-th/9907012. [Google Scholar] - Bousso, R.; Engelhardt, N. Generalized second law for cosmology. Phys. Rev. D
**2016**, 93, 024025. [Google Scholar] [CrossRef] [Green Version] - Dvali, G. Entropy Bound and Unitarity of Scattering Amplitudes. J. High Energy Phys.
**2021**, 3, 126. [Google Scholar] [CrossRef] - Acquaviva, G.; Iorio, A.; Smaldone, L. Bekenstein bound from the Pauli principle. Phys. Rev. D
**2020**, 102, 106002. [Google Scholar] [CrossRef] - Buoninfante, L.; Luciano, G.G.; Petruzziello, L.; Scardigli, F. Bekenstein bound and uncertainty relations. Phys. Lett. B
**2022**, 824, 136818. [Google Scholar] [CrossRef] - Ivanov, M.G.; Volovich, I.V. Entropy Bounds, Holographic Principle and Uncertainty Relation. Entropy
**2001**, 3, 66. [Google Scholar] [CrossRef] [Green Version] - Custodio, P.S.; Horvath, J.E. The generalized uncertainty principle, entropy bounds and black-hole (non-) evaporation in a thermal bath. Class. Quant. Grav.
**2003**, 20, L197. [Google Scholar] [CrossRef] - Bousso, R. Flat space physics from holography. J. High Energy Phys.
**2004**, 5, 50. [Google Scholar] [CrossRef] [Green Version] - Chaichian, M.; Tureanu, A.; Zet, G. Corrections to Schwarzschild solution in noncommutative gauge theory of gravity. Phys. Lett. B
**2008**, 660, 573–578. [Google Scholar] [CrossRef] [Green Version] - Kanazawa, T.; Lambiase, G.; Vilasi, G.; Yoshioka, A. Noncommutative Schwarzschild geometry and generalized uncertainty principle. Eur. Phys. J. C
**2019**, 79, 95. [Google Scholar] [CrossRef]

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Scardigli, F.
Bekenstein Bound and Non-Commutative Canonical Variables. *Universe* **2022**, *8*, 645.
https://doi.org/10.3390/universe8120645

**AMA Style**

Scardigli F.
Bekenstein Bound and Non-Commutative Canonical Variables. *Universe*. 2022; 8(12):645.
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**Chicago/Turabian Style**

Scardigli, Fabio.
2022. "Bekenstein Bound and Non-Commutative Canonical Variables" *Universe* 8, no. 12: 645.
https://doi.org/10.3390/universe8120645