Exploring the Implications of the Deformation Parameter and Minimal Length in the Generalized Uncertainty Principle

Mahgoub A. Salih, Taysir M. Elmahdi

Department of Physics, Faculty of Science and Arts in Al-Miznab, Qassim University, Al Miznab, Saudi Arabia.

DOI: 10.4236/jqis.2024.141001                

Abstract

The breakdown of the Heisenberg Uncertainty Principle occurs when energies approach the Planck scale, and the corresponding Schwarzschild radius becomes similar to the Compton wavelength. Both of these quantities are approximately equal to the Planck length. In this context, we have introduced a model that utilizes a combination of Schwarzschild’s radius and Compton length to quantify the gravitational length of an object. This model has provided a novel perspective in generalizing the uncertainty principle. Furthermore, it has elucidated the significance of the deforming linear parameter β and its range of variation from unity to its maximum value.

Keywords

Generalized Uncertainty Principle, Deformed Heisenberg Algebra, Minimal Length

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Kubo, R. (1966) The Fluctuation-Dissipation Theorem. Reports on Progress in Physics, 29, 255-284. https://doi.org/10.1088/0034-4885/29/1/306
[2]	Hossenfelder, S. (2013) Minimal Length Scale Scenarios for Quantum Gravity. Living Reviews in Relativity, 16, Article No. 2. https://doi.org/10.12942/lrr-2013-2
[3]	Veneziano, G. (1986) A Stringy Nature Needs Just Two Constants. EPL (Europhysics Letters), 2, Article 199. https://doi.org/10.1209/0295-5075/2/3/006
[4]	Gross, D.J. and Mende, P.F. (1988) String Theory beyond the Planck Scale. Nuclear Physics B, 303, 407-454. https://doi.org/10.1016/0550-3213(88)90390-2
[5]	Amati, D., Ciafaloni, M. and Veneziano, G. (1987) Superstring Collisions at Planckian Energies. Physics Letters B, 197, 81-88. https://doi.org/10.1016/0370-2693(87)90346-7
[6]	Konishi, K., Paffuti, G. and Provero, P. (1990) Minimum Physical Length and the Generalized Uncertainty Principle in String Theory. Physics Letters B, 234, 276-284. https://doi.org/10.1016/0370-2693(90)91927-4
[7]	Pedram, P. (2012) New Approach to Nonperturbative Quantum Mechanics with Minimal Length Uncertainty. Physical Review D, 85, Article ID: 024016. https://doi.org/10.1103/PhysRevD.85.024016
[8]	Bolen, B. and Cavagliá, M. (2005) (Anti-)de Sitter Black Hole Thermodynamics and the Generalized Uncertainty Principle. General Relativity and Gravitation, 37, 1255-1262. https://doi.org/10.1007/s10714-005-0108-x
[9]	Garay, L.J. (1995) Quantum Gravity and Minimum Length. International Journal of Modern Physics A, 10, 145-165. https://doi.org/10.1142/S0217751X95000085
[10]	Das, S. and Vagenas, E.C. (2008) Universality of Quantum Gravity Corrections. arXiv: 0810.5333.
[11]	Bosso, P. and Das, S. (2017) Generalized Uncertainty Principle and Angular Momentum. Annals of Physics, 383, 416-438. https://doi.org/10.1016/j.aop.2017.06.003
[12]	Scardigli, F. (2019) The Deformation Parameter of the Generalized Uncertainty Principle. Journal of Physics: Conference Series, 1275, Article ID: 012004. https://doi.org/10.1088/1742-6596/1275/1/012004
[13]	Wu, X., Wu, B., Li, H. and Wu, Q. (2023) From Generalized Hamilton Principle to Generalized Schrodinger Equation. Journal of Modern Physics, 14, 676-691. https://doi.org/10.4236/jmp.2023.145039
[14]	Salih, M.A. and Almahdi, T.M. (2012) New Approach to Generalize Uncertainty Principle. Journal of Basic and Applied Scientific Research, 2, 9348-9351.
[15]	Beckwith, A. (2016) Is the Cosmological Constant, a “Vacuum” Field? We Explore This by Squeezing Early Universe “Coherent-Semi Classical States”, and Compare This to Energy from the Early Universe Heisenberg Uncertainty Principle. Journal of High Energy Physics, Gravitation and Cosmology, 2, 546-561. https://doi.org/10.4236/jhepgc.2016.24047
[16]	Scardigli, F. (1999) Generalized Uncertainty Principle in Quantum Gravity from Micro-Black Hole Gedanken Experiment. arXiv: hep-th/9904025.
[17]	Amati, D., Ciafaloni, M. and Veneziano, G. (1989) Can Spacetime Be Probed below the String Size? Physics Letters B, 216, 41-47. https://doi.org/10.1016/0370-2693(89)91366-X
[18]	Witten, E. (1996) Duality, Spacetime and Quantum Mechanics. Physics Today, 49, 24-30. https://doi.org/10.1063/1.881493
[19]	Arbab, A.I. (2004) Quantization of Gravitational System and its Cosmological Consequences. arXiv: gr-qc/0309040v2.
[20]	Schwarz, J.H. (1996) The Second Superstring Revolution. arXiv: hep-th/9607067v1.
[21]	Amelino-Camelia, G. (2003) Phenomenology of Doubly Special Relativity. arXiv: gr-qc/0312124v1.
[22]	Masłowski, T., Nowicki, A. and Tkachuk, V.M. (2012) Deformed Heisenberg Algebra and Minimal Length. Journal of Physics A: Mathematical and Theoretical, 45, Article ID: 075309. https://doi.org/10.1088/1751-8113/45/7/075309
[23]	Girelli, F., Livine, E.R. and Oriti, D. (2005) Deformed Special Relativity as an Effective Flat Limit of Quantum Gravity. Nuclear Physics B, 708, 411-433. https://doi.org/10.1016/j.nuclphysb.2004.11.026
[24]	Kempf, A., Mangano, G. and Mann, R.B. (1995) Hilbert Space Representation of the Minimal Length Uncertainty Relation. Physical Review D, 52, 1108-1118. https://doi.org/10.1103/PhysRevD.52.1108
[25]	Kempf, A. (1996) Nonpointlike Particles in Harmonic Oscillators. arXiv: hep-th/9604045v19.
[26]	Kibble, T.W.B. (1976) Topology of Cosmic Domains and Strings. Journal of Physics A: Mathematical and General, 9, 1387. https://doi.org/10.1088/0305-4470/9/8/029
[27]	Vilenkin, A. (1985) Cosmic Strings and Domain Walls. Physics Reports, 121, 263-315. https://doi.org/10.1016/0370-1573(85)90033-X
[28]	Witten, E. (1985) Cosmic Superstrings. Physics Letters B, 153, 243-246. https://doi.org/10.1016/0370-2693(85)90540-4
[29]	Vilenkin, A. (1981) Cosmological Density Fluctuations Produced by Vacuum Strings. Physical Review Letters, 46, 1169-1172. https://doi.org/10.1103/PhysRevLett.46.1169
[30]	Mu, B.R., Wu, H.W. and Yang, H.T. (2009) The Generalized Uncertainty Principle in the Presence of Extra Dimensions. arXiv: 0909.3635v2.