Perturbative Quantum Gravity on de Sitter Spacetime ()

Ashaq Hussain Sofi^{1}, Shabir Ahmad Akhoon^{1}, Asloob Ahmad Rather^{2}, Anil Maini^{3}

^{1}Department of Physics, National Institute of Technology, Srinagar, India.

^{2}Department of Physics, Aligarh Muslim University, Aligarh, India.

^{3}Department of Applied Sciences, College of Engineering and Technology, BGSB University, Rajouri, India.

**DOI: **10.4236/oalib.1101422
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We will analyse perturbative quantum gravity on de Sitter spacetime. We
propose a new type of inner product for modes on de Sitter spacetime. This
inner product is used to mode decompose perturbations of the metric on de
Sitter spacetime. Using this inner product, it is possible to calculate the
two-point function for perturbative quantum gravity on de Sitter spacetime.
This two-point function will be written in terms of a mode sum for various
modes on de Sitter spacetime.

Share and Cite:

Sofi, A. , Akhoon, S. , Rather, A. and Maini, A. (2015) Perturbative Quantum Gravity on de Sitter Spacetime. *Open Access Library Journal*, **2**, 1-9. doi: 10.4236/oalib.1101422.

**Subject Areas:** **Applied Physics, Modern Physics**

1. Introduction

Quantum field theory has been one of the greatest scientific achievements of the last centenary. The perturbative quantum theory has led to many important developments in high energy physics and condensed matter physics. These can include things like ghost condensation which causes the breaking of Lorentz symmetry [1] . It has also been used for analyzing the BRST and anti-BRST symmetries [2] . Most quantum field theories that are used in high energy physics are, gauge theories. Gravity can also be considered as a gauge theory of coordinate transformations [3] - [20] . As not all the degrees of freedom, in a theory with gauge symmetry are physical, so it such theories cannot be quantized without fixing a gauge. A gauge is fixed at the quantum level by adding a new term to the original action. This new term is called a gauge fixing term. However, apart from the gauge fixing term, we also need to add a ghost term to the original action. This ensures that the theory unitary to all orders in the perturbation theory. This term is called the ghost term. This term is composed of ghosts fields, which are not physical fields. But it is important to include these fields into the calculation to keep the theory unitarity [21] - [28] . Even though the perturbative quantum gravity is not renormalizable, it can be used in the framework of effective field theories. From an effective theory point of view there is no fundamental difference between renormalizable and non-renormalizable theories, except the way these theories depend on lower energy scale. The universe can be approximated by de Sitter spacetime in the inflationary era. The universe may also be approaching de Sitter spacetime asymptotically. So, we will study perturbative quantum gravity on de Sitter spacetime. Quantum field theory on has had some interesting applications [29] - [92] . The Hawking radiation was arrived at by studding quantum field theory in the back ground of a black hole. We will define an inner product for perturbative quantum gravity in de Sitter spacetime. The de Sitter spacetime is defined to be a spacetime of constant curvature. This curvature is always positive for de Sitter spacetime. This is because it is generated from a positive cosmological constant. This cosmological constant sets the rate of expansion of the universe.

2. de Sitter Spacetime

As this rate is measured by the Hubble’s constant H, so we can write

(1)

The Hubble’s constant can be written in terms of the radius of de Sitter spacetime r,

(2)

Furthermore, we can write

(3)

So we get

(4)

(5)

Now we can also write

(6)

As we know,

(7)

So we can also write

(8)

The metric in de Sitter spacetime can be written as

(9)

Furthermore, we can write it using the metric on a three-sphere,

(10)

Here is the metric on a three dimensional sphere,

(11)

Now using, which is given by

(12)

we can write

(13)

3. Perturbative Quantum Gravity

We now start with pure gravity with a cosmological constant. The Lagrangian for pure gravity with a cosmological constant is

(14)

We adopt units, such that

(15)

We now split the metric into a background metric and small perturbations of of that metric. Then we treat the perturbation as a classical field and try to quantize it,

(16)

Now we have

(117)

and

(18)

If is the original Riemann tensor and is the background Riemann tensor then we have up to second order in,

(19)

where we have define

(20)

and

(21)

along with

(22)

We can now write,

(23)

So we have

(24)

This operator has a zero eigenvalue and it is not invertible. However, we can fix a gauge. This will be done by adding a ghost term and a gauge fixing term to the original action. Now if the sum of this term is given by, then we can define as the momentum current,

(25)

for perturbative quantum gravity are and. So, we can define as,

(26)

We have

(27)

Furthermore, the inner product on a space-like hyper-surface is gain by

(28)

Now we have

(29)

This inner product is conserved. Now by using a complete set of solutions and, we get

(30)

So we have

(31)

We also have

(32)

and

(33)

So, we can write

(34)

Now we can write

(35)

where

(36)

Thus we have

(37)

and

(38)

In Matrix notation, this can be written as

(39)

So we have

(40)

and

(41)

4. Conclusion

In this paper, we analysed the inner product for perturbative quantum gravity in de Sitter spacetime. In doing our calculations, we proposed a general kind of inner product for modes on de Sitter spacetime. Using this general inner product, we were able to calculate the two-point function on de Sitter spacetime. The de Sitter spacetime is important as the universe is expected to be approaching de Sitter spacetime. This two-point function can be written as a mode sum of the graviton modes on de Sitter spacetime. It may be noted that quantum gravity correction to quantum field theory has been studied [93] - [107] . It will be interesting to analyse such effects in de Sitter spacetime.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
Faizal, M. (2011) Spontaneous Breaking of Lorentz Symmetry by Ghost Condensation in Perturbative Quantum Gravity. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 402001. http://dx.doi.org/10.1088/1751-8113/44/40/402001 |

[2] |
Faizal, M. (2011) BRST and Anti-BRST Symmetries in Perturbative Quantum Gravity. Foundations of Physics, 41, 270-277. http://dx.doi.org/10.1007/s10701-010-9511-6 |

[3] |
Fabbri, L. (2011) From the Torsion Tensor for Spinors to the Weak Forces for Leptons. International Journal of Theoretical Physics, 50, 3616-3620. http://dx.doi.org/10.1007/s10773-011-0870-3 |

[4] |
Kugo, T. and Ojima, I. (1978) Subsidiary Conditions and Physical S-Matrix Unitarity in Indefinite-Metric Quantum Gravitation Theory. Nuclear Physics B, 144, 234-252. http://dx.doi.org/10.1016/0550-3213(78)90504-7 |

[5] |
Nishijima, K. and Okawa, M. (1978) The Becchi-Rouet-Stora Transformation for the Gravitational Field. Progress of Theoretical Physics, 60, 272-283. http://dx.doi.org/10.1143/PTP.60.272 |

[6] |
Nakanishi, N. and Ojima, I. (1990) Covariant Operator Formalism of Gauge Theories and Quantum Gravity. World Scientific, The Singapore City, 298. http://dx.doi.org/10.1142/0362 |

[7] | Lesov, A. (2009) The Weak Force: From Fermi to Feynman. arXiv preprint arXiv:0911.0058. |

[8] |
Faizal, M. and Upadhyay, S. (2014) Spontaneous Breaking of the BRST Symmetry in the ABJM Theory. Physics Letters B, 736, 288-292. http://dx.doi.org/10.1016/j.physletb.2014.07.040 |

[9] |
Faizal, M. and Tsun, T.S. (2014) Monopoles in Superloop Space. EPL (Europhysics Letters), 107, Article ID: 20008. http://dx.doi.org/10.1209/0295-5075/107/20008 |

[10] | Faizal, M. and Kruglov, S.I. (2014) Deformation of the Dirac Equation. arXiv preprint arXiv:1406.2653. |

[11] |
Takook, M.V. (2002) Negative Norm States in de Sitter Space and QFT without Renormalization Procedure. International Journal of Modern Physics E, 11, 509-518. http://dx.doi.org/10.1142/S0218301302001071 |

[12] | Ali, A.F., Faizal, M. and Majumder, B. (2015) Absence of an Effective Horizon for Black Holes in Gravity’s Rainbow. EPL (Europhysics Letters), 109, Article ID: 20001. |

[13] | Metsaev, R.R. (2014) The BRST-Invariant Effective Action of Shadows, Conformal Fields, and the AdS/CFT Correspondence. Theoretical and Mathematical Physics, 181, 1548-1565. |

[14] | Moshin, P.Yu. and Reshetnyak, A.A. (2015) Int. J. Mod. Phys. A, 30, Article ID: 1550021 |

[15] | MacDonald, J. and Mullan, D.J. (2009) Big Bang Nucleosynthesis: The Strong Nuclear Force Meets the Weak Anthropic Principle. Physical Review D, 80, Article ID: 043507. |

[16] |
Sinev, A.M. (2008) On a Quantum Model of a Laser-Interferometer Measuring a Weak Classical Force. arXiv:0806.3212 |

[17] | Pagliarone, C. (2006) Results on QCD Physics from the CDF-II Experiment. arXiv:hep-ex/0612037 |

[18] |
Faizal, M. (2012) Covariant Graviton Propagator in Anti-De Sitter Spacetime. Classical and Quantum Gravity, 29, Article ID: 035007. http://dx.doi.org/10.1088/0264-9381/29/3/035007 |

[19] |
Pakman, A. (2003) BRST Quantization of String Theory in AdS3. Journal of High Energy Physics, 2003, 053. http://dx.doi.org/10.1088/1126-6708/2003/06/053 |

[20] |
Izumi, K. and Tanaka, T. (2009) Particle Production in Models with Helicity-0 Graviton Ghost in de Sitter Spacetime. Progress of Theoretical Physics, 121, 427-436. http://dx.doi.org/10.1143/PTP.121.427 |

[21] |
Batalin, I.A. and Vilkovisky, G.A. (1981) Gauge Algebra and Quantization. Physics Letters B, 102, 27-31. http://dx.doi.org/10.1016/0370-2693(81)90205-7 |

[22] |
Batalin, I.A. and Vilkovisky, G.A. (1983) Quantization of Gauge Theories with Linearly Dependent Generators. Physical Review D, 28, Article ID: 2567. http://dx.doi.org/10.1103/PhysRevD.28.2567 |

[23] |
Bizdadea, C. and Saliu, S.O. (1998) Irreducible Antifield Formalism for Reducible Constrained Hamiltonian Systems. Journal of Physics A: Mathematical and General, 31, 8805. http://dx.doi.org/10.1088/0305-4470/31/44/009 |

[24] |
Bizdadea, C., Negru, I. and Saliu, S.O. (1999) Irreducible Antifield BRST-Anti-BRST Formalism for Reducible Gauge Theories. International Journal of Modern Physics A, 14, 359-386. http://dx.doi.org/10.1142/S0217751X9900018X |

[25] |
Faizal, M. (2011) Perturbative Quantum Gravity on Complex Space-Time. Physics Letters B, 705, 120-123. http://dx.doi.org/10.1016/j.physletb.2011.09.062 |

[26] |
Moffat, J.W. (2001) Noncommutative and Non-Anticommutative Quantum Field Theory. Physics Letters B, 506, 193-199. http://dx.doi.org/10.1016/S0370-2693(01)00409-9 |

[27] |
Moffat, J.W. (2000) Noncommutative Quantum Gravity. Physics Letters B, 491, 345-352. http://dx.doi.org/10.1016/S0370-2693(00)01064-9 |

[28] |
Faizal, M. (2012) Noncommutativity and Non-Anticommutativity Perturbative Quantum Gravity. Modern Physics Letters A, 27, Article ID: 1250075. http://dx.doi.org/10.1142/S0217732312500757 |

[29] |
Ahmad, S. (2013) General Formalism for the BRST Symmetry. Communications in Theoretical Physics, 59, 439. http://dx.doi.org/10.1088/0253-6102/59/4/09 |

[30] |
Faizal, M. (2011) M Theory on Deformed Superspace. Physical Review D, 84, Article ID: 106011 http://dx.doi.org/10.1103/PhysRevD.84.106011 |

[31] |
Faizal, M. and Smith, D.J. (2012) Supersymmetric Chern-Simons Theory in the Presence of a Boundary. Physical Review D, 85, c105007. http://dx.doi.org/10.1103/PhysRevD.85.105007 |

[32] |
Mader, V., Schaden, M., Zwanziger, D. and Alkofer, R. (2014) Infrared Saturation and Phases of Gauge Theories with BRST Symmetry. The European Physical Journal C, 74, 2881. http://dx.doi.org/10.1140/epjc/s10052-014-2881-8 |

[33] |
Gustavsson, A. (2012) Deconstructing Graviphoton from Mass-Deformed ABJM. Journal of High Energy Physics, 2012, 174. http://dx.doi.org/10.1007/JHEP06(2012)174 |

[34] |
Faizal, M. (2012) M-Theory in the Gaugeon Formalism. Communications in Theoretical Physics, 57, 637-640. http://dx.doi.org/10.1088/0253-6102/57/4/20 |

[35] |
Bianchi, M.S., Leoni, M. and Penati, S. (2012) An All Order Identity between ABJM and Mathcal N = 4 SYM Four-Point Amplitudes. Journal of High Energy Physics, 2012, 45. http://dx.doi.org/10.1007/JHEP04(2012)045 |

[36] |
Faizal, M. (2012) Deformation of the ABJM Theory. EPL (Europhysics Letters), 98, Article ID: 31003. http://dx.doi.org/10.1209/0295-5075/98/31003 |

[37] |
Marino, M. and Putrov, P. (2012) ABJM Theory as a Fermi Gas. Journal of Statistical Mechanics: Theory and Experiment, 2012, Article ID: P03001. http://dx.doi.org/10.1088/1742-5468/2012/03/P03001 |

[38] |
Okuyama, K. (2012) A Note on the Partition Function of ABJM Theory on S3. Progress of Theoretical Physics, 127, 229-242. http://dx.doi.org/10.1143/PTP.127.229 |

[39] |
Faizal, M. (2012) Gauge and Supersymmetric Invariance of a Boundary Bagger-Lambert-Gustavsson Theory. Journal of High Energy Physics, 2012, 17. http://dx.doi.org/10.1007/JHEP04(2012)017 |

[40] |
Belhaj, A. (2011) On Fractional Quantum Hall Solitons in ABJM-like Theory. Physics Letters B, 705, 539-542. http://dx.doi.org/10.1016/j.physletb.2011.10.060 |

[41] |
Faizal, M. (2014) Boundary Effects in the BLG Theory. Modern Physics Letters A, 29, Article ID: 1450154. http://dx.doi.org/10.1142/S0217732314501545 |

[42] | Zwanziger, D. (2007) On the Equation of State of the Gluon Plasma. AIP Conference Proceedings, 892. |

[43] |
Faizal, M. (2013) Noncommutative Quantum Gravity. Modern Physics Letters A, 28, Article ID: 1350034. http://dx.doi.org/10.1142/S021773231350034X |

[44] |
Zwanziger, D. (2007) Equation of State of Gluon Plasma from Local Action. Physical Review D, 76, Article ID: 125014. http://dx.doi.org/10.1103/PhysRevD.76.125014 |

[45] |
Faizal, M. (2013) Chern-Simons-Matter Theory. International Journal of Modern Physics A, 28, Article ID: 1350012. http://dx.doi.org/10.1142/S0217751X13500127 |

[46] |
Golterman, M. and Zimmerman, L. (2005) Masslessness of Ghosts in Equivariantly Gauge-Fixed Yang-Mills Theories. Physical Review D, 71, Article ID: 117502. http://dx.doi.org/10.1103/PhysRevD.71.117502 |

[47] |
Faizal, M. (2013) Aspects of ABJ Theory. Journal of High Energy Physics, 2013, 156. http://dx.doi.org/10.1007/JHEP01(2013)156 |

[48] |
Polyakov, D. (2005) GL(1) Charged States in Twistor String Theory. Physics Letters B, 611, 173-181. http://dx.doi.org/10.1016/j.physletb.2005.02.014 |

[49] |
Faizal, M. (2013) Superloop Space. EPL (Europhysics Letters), 103, Article ID: 21003. http://dx.doi.org/10.1209/0295-5075/103/21003 |

[50] |
Imaanpur, A. (2005) Supersymmetric D3-Branes in Five-Form Flux. Journal of High Energy Physics, 2005, 030. http://dx.doi.org/10.1088/1126-6708/2005/03/030 |

[51] |
Faizal, M. (2013) Non-Anticommutative ABJ Theory. Nuclear Physics B, 869, 598-607. http://dx.doi.org/10.1016/j.nuclphysb.2012.12.018 |

[52] | Lee, J.C. (1998) SO(2, C) invariant Discrete Gauge States in Liouville Gravity Coupled to Minimal Conformal Matter. The European Physical Journal C-Particles and Fields, 1, 739-741. |

[53] |
Faizal, M. and Smith, D.J. (2013) Nonanticommutativity in the Presence of a Boundary. Physical Review D, 87, Article ID: 025019. http://dx.doi.org/10.1103/PhysRevD.87.025019 |

[54] |
Kapustin, A. and Li, Y. (2005) Open-String BRST Cohomology for Generalized Complex Branes. Advances in Theoretical and Mathematical Physics, 9, 559-574. http://dx.doi.org/10.4310/ATMP.2005.v9.n4.a2 |

[55] |
Faizal, M. (2013) Deformed Super-Yang-Mills in Batalin-Vilkovisky Formalism. International Journal of Theoretical Physics, 52, 392-403. http://dx.doi.org/10.1007/s10773-012-1344-y |

[56] |
Chan, C.T. and Lee, J.C. (2005) Anatomy of Zero-Norm States in String Theory. Physical Review D, 71, Article ID: 086005. http://dx.doi.org/10.1103/PhysRevD.71.086005 |

[57] |
Faizal, M. (2012) Super-Group Field Cosmology. Classical and Quantum Gravity, 29, Article ID: 215009. http://dx.doi.org/10.1088/0264-9381/29/21/215009 |

[58] |
Malik, R.P. (2004) Time-Space Noncommutativity and Symmetries for a Massive Relativistic Particle. arXiv:hep-th/0412333 |

[59] |
Faizal, M. (2012) The BV Formalization of Chern-Simons Theory on Deformed Superspace. Communications in Theoretical Physics, 58, 704. http://dx.doi.org/10.1088/0253-6102/58/5/14 |

[60] |
Boulanger, N. (2005) A Weyl-Covariant Tensor Calculus. Journal of Mathematical Physics, 46, 3508. http://dx.doi.org/10.1063/1.1896381 |

[61] |
Faizal, M. (2012) Harmonic Superspace Gaugeon Formalism for the ABJM Theory. Modern Physics Letters A, 27, Article ID: 1250147. http://dx.doi.org/10.1142/S0217732312501477 |

[62] | Huang, W.H. (2011) M2-KK6 System in ABJM Theory: Fuzzy S3 and Wrapped KK6. arXiv:1107.2030 |

[63] |
Faizal, M. and Khan, M. (2011) A Superspace Formulation of the BV Action for Higher Derivative Theories. The European Physical Journal C-Particles and Fields, 71, 1603. http://dx.doi.org/10.1140/epjc/s10052-011-1603-8 |

[64] | Liu, J.T. and Zhao, Z. (2011) A Holographic C-Theorem for Higher Derivative Gravity. arXiv:1108.5179 |

[65] |
Fontanini, M. and Trodden, M. (2011) Tackling Higher Derivative Ghosts with the Euclidean Path Integral. Physical Review D, 83, Article ID: 103518. http://dx.doi.org/10.1103/PhysRevD.83.103518 |

[66] |
Rivelles, V.O. (2003) Triviality of Higher Derivative Theories. Physics Letters B, 577, 137-142. http://dx.doi.org/10.1016/j.physletb.2003.10.039 |

[67] |
DeWitt, B.S. (1967) Quantum Theory of Gravity. I. The Canonical Theory. Physical Review, 160, 1113. http://dx.doi.org/10.1103/PhysRev.160.1113 |

[68] |
Faizal, M. (2012) Some Aspects of Virtual Black Holes. Journal of Experimental and Theoretical Physics, 114, 400-405. http://dx.doi.org/10.1134/S1063776112020045 |

[69] |
Ohkuwa, Y. (1998) Third Quantization of Kaluzaklein Cosmology and Compactification. International Journal of Modern Physics A, 13, 4091-4100. http://dx.doi.org/10.1142/S0217751X98001918 |

[70] |
Faizal, M. (2012) Multiverse in the Third Quantized Horavalifshitz Theory of Gravity. Modern Physics Letters A, 27, Article ID: 1250007. http://dx.doi.org/10.1142/S0217732312500071 |

[71] | Durham, I.T. (2013) Contextuality: Wheeler’s Universal Regulating Principle. arXiv:1307.3691 |

[72] |
Faizal, M. (2013) Fourth Quantization. Physics Letters B, 727, 536-540. http://dx.doi.org/10.1016/j.physletb.2013.10.069 |

[73] |
Bonzom, V. (2011) Spin Foam Models and the Wheeler-DeWitt Equation for the Quantum 4-Simplex. Physical Review D, 84, Article ID: 024009. http://dx.doi.org/10.1103/PhysRevD.84.024009 |

[74] |
Faizal, M. (2014) Noethers Charge in the Super-Group Field Cosmology. Gravitation and Cosmology, 20, 132-137. http://dx.doi.org/10.1134/S0202289314020030 |

[75] |
Chowdhury, D., Raju, S., Sachdev, S., Singh, A. and Strack, P. (2013) Multipoint Correlators of Conformal Field Theories: Implications for Quantum Critical Transport. Physical Review B, 87, Article ID: 085138. http://dx.doi.org/10.1103/PhysRevB.87.085138 |

[76] |
Singh, S., Ganguly, C. and Padmanabhan, T. (2013) Quantum Field Theory in de Sitter and Quaside Sitter Spacetimes Revisited. Physical Review D, 87, Article ID: 104004. http://dx.doi.org/10.1103/PhysRevD.87.104004 |

[77] |
Faizal, M., Ali, A.F. and Nassar, A. (2014) AdS/CFT Correspondence beyond Its Supergravity Approximation. arXiv:1405.4519 |

[78] | Faizal, M. (2015) Consequences of Deformation of the Heisenberg Algebra. International Journal of Geometric Methods in Modern Physics, 12, Article ID: 1550022. |

[79] | Takook, M.V. (2014) Quantum Field Theory in de Sitter Universe: Ambient Space Formalism. arXiv:1403.1204 |

[80] | Faizal, M. (2014) Multiverse in the Third Quantized Formalism. Communications in Theoretical Physics, 62, 697. |

[81] |
Faizal, M. and Tsun, T.S. (2014) Monopoles in Superloop Space. EPL (Europhysics Letters), 107, Article ID: 20008. http://dx.doi.org/10.1209/0295-5075/107/20008 |

[82] | Faizal, M. and Tsun, T.S. (2014) Polyakov Loops for the ABJ Theory. International Journal of Theoretical Physics, 1-14. |

[83] | Huang, R.N. (2013) The Wheeler-DeWitt equation of f (R, Lm) Gravity in Minisuperspace. arXiv:1304.5309 |

[84] |
Kruglov, S.I. and Faizal, M. (2014) Wave Function of the Universe from a Matrix Valued First-Order Formalism. arXiv:1408.3794 |

[85] |
Garattini, R. and Majumder, B. (2014) Naked Singularities Are Not Singular in Distorted Gravity. Nuclear Physics B, 884, 125-141. http://dx.doi.org/10.1016/j.nuclphysb.2014.04.014 |

[86] |
Awad, A., Ali, A.F. and Majumder, B. (2013) Nonsingular Rainbow Universes. Journal of Cosmology and Astroparticle Physics, 2013, 052. http://dx.doi.org/10.1088/1475-7516/2013/10/052 |

[87] |
Ali, A.F., Faizal, M. and Majumder, B. (2015) Absence of an Effective Horizon for Black Holes in Gravity’s Rainbow. EPL (Europhysics Letters), 109, Article ID: 20001. http://dx.doi.org/10.1209/0295-5075/109/20001 |

[88] |
Faizal, M. (2014) Deformation of the Wheeler-DeWitt Equation. International Journal of Modern Physics A, 29, Article ID: 1450106. http://dx.doi.org/10.1142/S0217751X14501061 |

[89] |
Awad, A. and Ali, A.F. (2014) Minimal Length, Friedmann Equations and Maximum Density. Journal of High Energy Physics, 2014, 93. http://dx.doi.org/10.1007/JHEP06(2014)093 |

[90] |
Faizal, M. (2014) Absence of Black Holes Information Paradox in Group Field Cosmology. International Journal of Geometric Methods in Modern Physics, 11, Article ID: 1450010. http://dx.doi.org/10.1142/S0219887814500108 |

[91] |
Faizal, M. and Majumder, B. (2014) Incorporation of Generalized Uncertainty Principle into Lifshitz Field Theories. arXiv:1408.3795 |

[92] | Wald, R.M., Ed. (1994) Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press, Chicago. |

[93] |
Faizal, M. and Awad, A. (2015) Four Dimensional Supersymmetric Theories in Presence of a Boundary. arXiv:1502.07717 |

[94] | Das, S., Robbins, M.P. and Walton, M.A. (2014) Generalized Uncertainty Principle Corrections to the Simple Harmonic Oscillator in Phase Space. arXiv:1412.6467 |

[95] |
Balasubramanian, V., Das, S. and Vagenas, E.C. (2014) Generalized Uncertainty Principle and Self-Adjoint Operators. arXiv:1404.3962 |

[96] | Majumder, B. (2013) Singularity Free Rainbow Universe. International Journal of Modern Physics D, 22. |

[97] |
Majumder, B. (2013) Quantum Rainbow Cosmological Model with Perfect Fluid. International Journal of Modern Physics D, 22, Article ID: 1350079. http://dx.doi.org/10.1142/S021827181350079X |

[98] | Faizal, M., Khalil, M.M. and Das, S. (2014) Time Crystals from Minimum Time Uncertainty. arXiv:1501.03111 |

[99] | Gangopadhyay, S., Dutta, A. and Faizal, M. (2015) Constraints on the Generalized Uncertainty Principle from Black Hole Thermodynamics. arXiv:1501.01482 |

[100] | Faizal, M. and Tsun, T.S. (2014) Supersymmetric Duality in Superloop Space. arXiv:1412.7594 |

[101] | Faizal, M., Ali, A.F. and Das, S. (2014) Discreteness of Time in the Evolution of the Universe. arXiv:1411.5675 |

[102] | Pramanik, S., Faizal, M., Moussa, M. and Ali, A.F. (2014) The Path Integral Quantization Corresponding to the Deformed Heisenberg Algebra. arXiv:1411.4979 |

[103] | Faizal, M. and Khalil, M.M. (2014) GUP-Corrected Thermodynamics for All Black Objects and the Existence of Remnants. arXiv:1411.4042 |

[104] | Ali, A.F., Faizal, M. and Khalil, M.M. (2015) Absence of Black Holes at LHC Due to Gravity’s Rainbow. Physics Letters B, 743, 295-300. |

[105] |
Ali, A.F., Faizal, M. and Khalil, M.M. (2014) Remnants of Black Rings from Gravity’s Rainbow. Journal of High Energy Physics, 2014, 159. http://dx.doi.org/10.1007/JHEP12(2014)159 |

[106] | Ali, A.F., Faizal, M. and Khalil, M.M. (2015) Remnant for All Black Objects Due to Gravity’s Rainbow. Nuclear Physics B, 894, 341-360. |

[107] |
Majumder, B. and Sen, S. (2012) Do the Modified Uncertainty Principle and Polymer Quantization Predict Same Physics? Physics Letters B, 717, 291-294. http://dx.doi.org/10.1016/j.physletb.2012.09.035 |

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