Spectra of Harmonic Oscillators with GUP and Extra Dimensions

Benrong Mu, Rong Yu, Dongmei Wang^*

Physics Teaching and Research Section, College of Medical Technology, Chengdu University of Traditional Chinese Medicine, Chengdu, China.

DOI: 10.4236/jhepgc.2019.51015                

Abstract

In this paper, we address the spectra of simple harmonic oscillators based on the generalized uncertainty principle (GUP) with a Kaluza-Klein compactified extra dimension. We show that in this scenario, to make the results compatible with experiments, the minimal length scale equals to the radius of compact extra dimension.

Keywords

Compactified Extra Dimension, GUP, Harmonic Oscillator

Conflicts of Interest

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

References

[1]	Veneziano, G. (1986) A Stringy Nature Needs Just Two Constants. Europhysics Letters, 2, 199. https://doi.org/10.1209/0295-5075/2/3/006
[2]	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
[3]	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
[4]	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
[5]	Guida, R., Konishi, K. and Provero, P. (1991) On the Short Distance Behavior of String Theories. Modern Physics Letters A, 6, 1487-1503. https://doi.org/10.1142/S0217732391001603
[6]	Maggiore, M. (1993) A Generalized Uncertainty Principle in Quantum Gravity. Physics Letters B, 304, 65-69. https://doi.org/10.1016/0370-2693(93)91401-8
[7]	Zhao, R., Zhang, L.C., Wu, Y.Q., et al. (2010) Generalized Uncertainty Principle and Tunneling Radiation of the SAdS5 Black Hole. Chinese Physics B, 19, 010402. https://doi.org/10.1088/1674-1056/19/1/010402
[8]	Corda, C., Chakraborty, S. and Saha, S. (2015) Light from Black Holes and Uncertainty in Quantum Gravity. Electronic Journal of Theoretical Physics, 12, 107.
[9]	Haldar, S., Corda, C. and Chakraborty, S. (2018) Tunnelling Mechanism in Noncommutative Space with Generalized Uncertainty Principle and Bohr-Like Black Hole. Advances in High Energy Physics, 2018, Article ID: 9851598. https://doi.org/10.1155/2018/9851598
[10]	Mu, B., Wang, P. and Yang, H. (2015) Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes. Advances in High Energy Physics, 2015, Article ID: 898916. https://doi.org/10.1155/2015/898916
[11]	Mu, B.R., Wang, P. and Yang, H.T. (2015) Minimal Length Effects on Schwinger Mechanism. Communications in Theoretical Physics, 63, 715. https://doi.org/10.1088/0253-6102/63/6/715
[12]	Gecim, G. and Sucu, Y. (2018) Quantum Gravity Effect on the Hawking Radiation of Charged Rotating BTZ Black Hole. General Relativity and Gravitation, 50, 152. https://doi.org/10.1007/s10714-018-2478-x
[13]	Kempf, A., Mangano, G. and Mann, R.B. (1995) Hilbert Space Representation of the Minimal Length Uncertainty Relation. Physical Review D, 52, 1108.
[14]	Kempf, A. (1997) Non-Pointlike Particles in Harmonic Oscillators. Journal of Physics A, 30, 2093. https://doi.org/10.1088/0305-4470/30/6/030
[15]	Brau, F. (1999) Minimal Length Uncertainty Relation and Hydrogen Atom. Journal of Physics A, 32, 7691-7696. https://doi.org/10.1088/0305-4470/32/44/308
[16]	Brau, F. and Buisseret, F. (2006) Minimal Length Uncertainty Relation and Gravitational Quantum Well. Physical Review D, 74, Article ID: 036002. https://doi.org/10.1103/PhysRevD.74.036002
[17]	Chang, L.N., Minic, D., Okamura, N. and Takeuchi, T. (2002) Exact Solution of the Harmonic Oscillator in Arbitrary Dimensions with Minimal Length Uncertainty Relations. Physical Review D, 65, Article ID: 125027. https://doi.org/10.1103/PhysRevD.65.125027
[18]	Dadic, I., Jonke, L. and Meljanac, S. (2003) Physical Review D, 67, Article ID: 087701. https://doi.org/10.1103/PhysRevD.67.087701
[19]	Nozari, K. and Azizi, T. (2006) Some Aspects of Gravitational Quantum Mechanics. General Relativity and Gravitation, 38, 735-742. https://doi.org/10.1007/s10714-006-0262-9
[20]	Stetsko, M.M. and Tkachuk, V.M. (2006) Perturbation Analysis for Competing Reactions with Initially Separated Components. Physical Review A, 74, Article ID: 012101. https://doi.org/10.1103/PhysRevA.74.012101
[21]	Stetsko, M.M. (2006) Corrections to the ns-Levels of Hydrogen Atom in Deformed Space with Minimal Length. Physical Review A, 74, Article ID: 062105.
[22]	Benczik, S.Z. (2007) Investigations on the Minimal-Length Uncertainty Relation. PhD Thesis, Virginia Polytechnic Institute and State University, Blacksburg.
[23]	Battisti, M.V. and Montani, G. (2007) The Big-Bang Singularity in the Framework of a Generalized Uncertainty Principle. Physics Letters B, 656, 96-101. https://doi.org/10.1016/j.physletb.2007.09.012
[24]	Battisti, M.V. and Montani, G. (2008) Quantum Dynamics of the Taub Universe in a Generalized Uncertainty Principle Framework. Physical Review D, 77, Article ID: 023518. https://doi.org/10.1103/PhysRevD.77.023518
[25]	Das, S. and Vagenas, E.C. (2008) Universality of Quantum Gravity Corrections. Physical Review Letters, 101, Article ID: 221301. https://doi.org/10.1103/PhysRevLett.101.221301
[26]	Mu, B., Wu, H. and Yang, H. (2011) The Generalized Uncertainty Principle in the Presence of Extra Dimensions. Chinese Physics Letters, 28, Article ID: 091101.
[27]	Antoniadis, I. (1990) A Possible New Dimension at a Few TeV. Physics Letters B, 246, 377-384. https://doi.org/10.1016/0370-2693(90)90617-F
[28]	Arkani-Hamed, N., Dimopoulos, S. and Dvali, G.R. (1998) The Hierarchy Problem and New Dimensions at a Millimeter. Physics Letters B, 429, 263-272. https://doi.org/10.1016/S0370-2693(98)00466-3
[29]	Antoniadis, I., Arkani-Hamed, N., Dimopoulos, S. and Dvali, G.R. (1998) New Dimensions at a Millimeter to a Fermi and Superstrings at a TeV. Physics Letters B, 436, 257-263. https://doi.org/10.1016/S0370-2693(98)00860-0
[30]	Randall, L. and Sundrum, R. (1999) A Large Mass Hierarchy from a Small Extra Dimension. Physical Review Letters, 83, 3370-3373. https://doi.org/10.1103/PhysRevLett.83.3370
[31]	Randall, L. and Sundrum, R. (1999) An Alternative to Compactification. Physical Review Letters, 83, 4690-4693. https://doi.org/10.1103/PhysRevLett.83.4690
[32]	Dvali, G.R., Gabadadze, G. and Porrati, M. (2000) 4D Gravity on a Brane in 5D Minkowski Space. Physics Letters B, 485, 208-214. https://doi.org/10.1016/S0370-2693(00)00669-9
[33]	Appelquist, T., Cheng, H.C. and Dobrescu, B.A. (2001) Bounds on Universal Extra Dimensions. Physical Review D, 64, Article ID: 035002. https://doi.org/10.1103/PhysRevD.64.035002
[34]	Cremades, D., Ibanez, L.E. and Marchesano, F. (2002) Standard Model at Intersecting D5-Branes: Lowering the String Scale. Nuclear Physics B, 643, 93-130. https://doi.org/10.1016/S0550-3213(02)00746-0
[35]	Kokorelis, C. (2004) Exact Standard Model Structures from Intersecting D5-Branes. Nuclear Physics B, 677, 115-163. https://doi.org/10.1016/j.nuclphysb.2003.11.007
[36]	Li, Z.G. and Ni, W.T. (2008) Extra Dimensions and Atomic Transition Frequencies. Chinese Physics B, 17, 70-75.
[37]	Bezerra, V.B. and Rego-Monteiro, M.A. (2004) Some Boundary Effects in Quantum Field Theory. Physical Review D, 70, Article ID: 065018.
[38]	Takenaga, K. (2000) Quantization Ambiguity and Supersymmetric Ground State Wave Functions. Physical Review D, 62, Article ID: 065001. https://doi.org/10.1103/PhysRevD.62.065001
[39]	Doncheski, M.A. and Robinett, R.W. (2003) Wave Packet Revivals and the Energy Eigenvalue Spectrum of the Quantum Pendulum. Annals of Physics, 308, 578-598. https://doi.org/10.1016/S0003-4916(03)00171-4
[40]	McLachlan, N.W. (1947) Theory and Application of Mathieu Functions. Clarendon Press, Oxford.
[41]	Hradil, et al. (2006) Minimum Uncertainty Measurements of Angle and Angular Momentum. Physical Review Letters, 97, Article ID: 243601.
[42]	Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, 724.