Skip to main content
SpringerLink
Log in

Quantum gravity from noncommutative spacetime

  • Published:
Journal of the Korean Physical Society Aims and scope Submit manuscript

Abstract

We review a novel and authentic way to quantize gravity. This novel approach is based on the fact that Einstein gravity can be formulated in terms of a symplectic geometry rather than a Riemannian geometry in the context of emergent gravity. An essential step for emergent gravity is to realize the equivalence principle, the most important property in the theory of gravity (general relativity), from U(1) gauge theory on a symplectic or Poisson manifold. Through the realization of the equivalence principle, which is an intrinsic property in symplectic geometry known as the Darboux theorem or the Moser lemma, one can understand how diffeomorphism symmetry arises from noncommutative U(1) gauge theory; thus, gravity can emerge from the noncommutative electromagnetism, which is also an interacting theory. As a consequence, a background-independent quantum gravity in which the prior existence of any spacetime structure is not a priori assumed but is defined by using the fundamental ingredients in quantum gravity theory can be formulated. This scheme for quantum gravity can be used to resolve many notorious problems in theoretical physics, such as the cosmological constant problem, to understand the nature of dark energy, and to explain why gravity is so weak compared to other forces. In particular, it leads to a remarkable picture of what matter is. A matter field, such as leptons and quarks, simply arises as a stable localized geometry, which is a topological object in the defining algebra (noncommutative ★-algebra) of quantum gravity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nevertheless, gravitational phenomena are ubiquitous in our everyday life. The reason is that the gravitational force is only attractive and always additive. As a result, the standard gravitational parameter GM for an astronomical body with mass M is not small. For example, GM e = 4 × 1014 m 3/s 2 for the Earth, where M e = 5.96 × 1024 Kg, which corresponds to 1 cm compared to the Planck length L pl = \(\sqrt G \) ~ 10−33 cm.

  2. Niels Henrik Abel (1802–1829): A problem that seems insurmountable is just seemingly so because we have not asked the right question. You should always ask the right question and then you can solve the problem.

  3. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (W. H. Freeman and Company, New York, 1973).

    Google Scholar 

  4. H. S. Yang, Europhys. Lett. 88, 31002 (2009), [hepth/0608013].

    Article  ADS  Google Scholar 

  5. H. S. Yang, Int. J. Mod. Phys. A 24, 4473 (2009), [hepth/0611174].

    Article  ADS  MATH  Google Scholar 

  6. H. S. Yang, Eur. Phys. J. C 64, 445 (2009), [arXiv:0704.0929].

    Article  ADS  MATH  Google Scholar 

  7. H. S. Yang, J. High Energy Phys. 05, 012 (2009), [arXiv:0809.4728].

    Article  ADS  Google Scholar 

  8. R. Abraham and J. E. Marsden, Foundations of Mechanics (Addison-Wesley, Reading, 1978).

    MATH  Google Scholar 

  9. M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001), [hep-th/0106048]; R. J. Szabo, Phys. Rep. 378, 207 (2003), [hep-th/0109162].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. R. J. Szabo, Class. Quantum Grav. 23, R199 (2006), [hep-th/0606233].

    Article  ADS  MATH  Google Scholar 

  11. K. Becker, M. Becker and J. Schwarz, String Theory and M-Theory: A Modern Introduction (Cambridge University Press, New York, 2007).

    Google Scholar 

  12. There is an interesting peculiarity in D = 6 that the gauge coupling constant e 2 in natural units c = ħ = 1 carries the physical dimension of (length)2. Thus, the gauge coupling constant e 2 in six dimensions operates like a noncommutative parameter |θ| or a string length α′, i.e., e 2 ~ |θ| ~ α′. This may imply that a six-dimensional gauge theory intrinsically behaves like a string theory without assuming any noncommutative spacetime.

  13. The theory on a D-brane also needs an intrinsic length parameter because it is always defined with the action \({\smallint _M}\sqrt {\det \left( {g + 2\pi \alpha 'F} \right)} ,\;so\;\alpha ' \equiv \;l_s^2\) should carry the dimension of (length)2 for a dimensional reason. Because the parameter l s 2 disappears from the theory if F = 0, it is necessary for the field strength F to be nowhere vanishing in order for the parameter l s 2 to have an operational meaning as an intrinsic length scale of the theory. In this case where 〈F〉 r→∞ = B, the two theories will be physically equivalent, as was shown in Ref. [16].

  14. A. Connes, Noncommutative Geometry (Academic Press, San Diego, CA, 1994).

    MATH  Google Scholar 

  15. M. Kontsevich, Lett. Math. Phys. 66, 157 (2003), [qalg/9709040].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. N. Seiberg and E. Witten, J. High Energy Phys. 09, 032 (1999), [hep-th/9908142].

    Article  ADS  MathSciNet  Google Scholar 

  17. T. Banks, W. Fischler, S. H. Shenker and L. Susskind, Phys. Rev. D 55, 5112 (1997), [hep-th/9610043].

    Article  ADS  MathSciNet  Google Scholar 

  18. N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, Nucl. Phys. B 498, 467 (1997), [hep-th/9612115]; H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Prog. Theor. Phys. 99, 713 (1998), [hep-th/9802085]; H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Nucl. Phys. B 565, 176 (2000), [hep-th/9908141].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. L. Motl, [hep-th/9701025]; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 500, 43 (1997), [hepth/9703030].

    Article  ADS  Google Scholar 

  20. W. Taylor, Rev. Mod. Phys. 73, 419 (2001), [hepth/0101126].

    Article  ADS  MATH  Google Scholar 

  21. G. E. Volovik, The Universe in a Helium Droplet (Oxford University Press, Oxford, 2003); C. Barcelo, S. Liberati and M. Visser, Living Rev. Rel. 8, 12 (2005), [gr-qc/0505065].

    MATH  Google Scholar 

  22. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), [hep-th/9711200]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998), [hepth/9802109]; E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998), [hep-th/9802150].

    ADS  MATH  MathSciNet  Google Scholar 

  23. V. O. Rivelles, Phys. Lett. B 558, 191 (2003), [hepth/0212262].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. H. S. Yang, Mod. Phys. Lett. A 21, 2637 (2006), [hepth/0402002]; R. Banerjee and H. S. Yang, Nucl. Phys. B B708, 434 (2005), [hep-th/0404064].

    Article  ADS  MATH  Google Scholar 

  25. H. Steinacker, J. High Energy Phys. 12, 049 (2007), [arXiv:0708.2426]; H. Grosse, H. Steinacker and M. Wohlgenannt, ibid. 04, 023 (2008), [arXiv:0802.0973]; D. Klammer and H. Steinacker, ibid. 08, 074 (2008), [arXiv:0805.1157]; ibid. 02, 074 (2010), [arXiv:0909.5298]; H. Steinacker, Nucl. Phys. B 810, 1 (2009), [arXiv:0806.2032]; J. High Energy Phys. 02, 044 (2009), [arXiv:0812.3761].

    Article  ADS  MathSciNet  Google Scholar 

  26. D. Klammer and H. Steinacker, Phys. Rev. Lett. 102, 221301 (2009), [arXiv:0903.0986].

    Article  ADS  MathSciNet  Google Scholar 

  27. H. Steinacker, J. High Energy Phys. 12, 024 (2009), [arXiv:0909.4621]; D. N. Blaschke and H. Steinacker, Class. Quant. Grav. 27, 165010 (2010), [arXiv:1003.4132].

    Article  ADS  MathSciNet  Google Scholar 

  28. H. Steinacker, Class. Quant. Grav. 27, 133001 (2010), [arXiv:1003.4134].

    Article  ADS  MathSciNet  Google Scholar 

  29. A. D. Sakharov, Sov. Phys. Dokl. 12, 1040 (1968); A. H. Chamseddine and A. Connes, Commun. Math. Phys. 186, 731 (1997), [hep-th/9606001]; A. H. Chamseddine, A. Connes and M. Marcolli, Adv. Theor. Math. Phys. 11, 991 (2007), [hep-th/0610241].

    ADS  Google Scholar 

  30. J. Madore and J. Mourad, J. Math. Phys. 39, 423 (1998), [gr-qc/9607060]; S. I. Vacaru, Phys. Lett. B 498, 74 (2001), [hep-th/0009163]; E. Langmann and R. J. Szabo, Phys. Rev. D 64, 104019 (2001), [hepth/0105094]; R. J. Szabo, Gen. Relativ. Grav. 42, 1 (2010), [arXiv:0906.2913].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. M. Burić, T. Grammatikopoulos, J. Madore and G. Zoupanos, J. High Energy Phys. 04, 054 (2006), [hepth/0603044]; M. Burić, J. Madore and G. Zoupanos, SIGMA 3, 125 (2007), [arXiv:0712.4024]; M. Burić, H. Grosse and J. Madore, J. High Energy Phys. 07, 010 (2010), [arXiv:1003.2284].

    ADS  Google Scholar 

  32. B. Muthukumar, Phys. Rev. D 71, 105007 (2005), [hep-th/0412069]; A. H. Fatollahi, Phys. Lett. B 665, 257 (2008), [arXiv:0805.1159]; A. Stern, Phys. Rev. D 80, 067703 (2009), [arXiv:0907.4532]; R. Banerjee, B. Chakraborty, S. Ghosh, P. Mukherjee and S. Samanta, Found. Phys. 39, 1297 (2009), [arXiv:0909.1000].

    Article  ADS  MathSciNet  Google Scholar 

  33. H. Lin, O. Lunin and J. Maldacena, J. High Energy Phys. 10, 025 (2004), [hep-th/0409174]; H. Lin and J. Maldacena, Phys. Rev. D 74, 084014 (2006), [hep-th/0509235].

    Article  ADS  MathSciNet  Google Scholar 

  34. D. Berenstein, J. High Energy Phys. 01, 125 (2006), [hep-th/0507203]; D. Berenstein and D. H. Correa, ibid. 08, 006 (2006), [hep-th/0511104]; D. Berenstein and R. Cotta, Phys. Rev. D 74, 026006 (2006), [hep-th/0605220]; D. Berenstein, M. Hanada and S. A. Hartnoll, J. High Energy Phys. 02, 010 (2009), [arXiv:0805.4658].

    Article  ADS  MathSciNet  Google Scholar 

  35. S. Iso, Y. Kimura, K. Tanaka and K. Wakatsuki, Nucl. Phys. B 604, 121 (2001), [hep-th/0101102]; R. Delgadillo-Blando, D. O’Connor and B. Ydri, Phys. Rev. Lett. 100, 201601 (2008), [arXiv:0712.3011]; J. High Energy Phys. 05, 049 (2009), [arXiv:0806.0558]; M. Hanada, Y. Hyakutake, J. Nishimura and S. Takeuchi, Phys. Rev. Lett. 102, 191602 (2009), [arXiv:0811.3102]; H. Aoki, J. Nishimura and Y. Susaki, J. High Energy Phys. 04, 055 (2009), [arXiv:0810.5234]; ibid. 09, 084 (2009), [arXiv:0907.2107].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. T. Ishii, G. Ishiki, S. Shimasaki and A. Tsuchiya, Phys. Rev. D 78, 106001 (2008), [arXiv:0807.2352]; G. Ishiki, S.-W. Kim, J. Nishimura and A. Tsuchiya, J. High Energy Phys. 09, 029 (2009), [arXiv:0907.1488].

    Article  ADS  MathSciNet  Google Scholar 

  37. T. Ohl and A. Schenkel, Gen. Rel. Grav. 42, 2785 (2010), [arXiv:0912.2252]; A. Schenkel and C. F. Uhlemann, SIGMA 6, 061 (2010), [arXiv:1003.3190].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. M. Salizzoni, A. Torrielli and H. S. Yang, Phys. Lett. B 634, 427 (2006), [hep-th/0510249]; H. S. Yang and M. Salizzoni, Phys. Rev. Lett. 96, 201602 (2006), [hepth/0512215].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. H. S. Yang, Mod. Phys. Lett. A 22, 1119 (2007), [hepth/0612231].

    Article  ADS  MATH  Google Scholar 

  40. H. S. Yang, Bulg. J. Phys. 35, 323 (2008), [arXiv:0711.0234].

    ADS  MATH  Google Scholar 

  41. H. S. Yang, [arXiv:0711.2797]; Int. J. Mod. Phys. A 23, 2181 (2008), [arXiv:0803.3272]; Int. J. Mod. Phys. Conf. Ser. 1, 266 (2011), [arXiv:0902.0035].

    Article  ADS  Google Scholar 

  42. H. S. Yang and M. Sivakumar, Phys. Rev. D 82, 045004 (2010), [arXiv:0908.2809].

    Article  ADS  Google Scholar 

  43. M. Kontsevich, Homological Algebra of Mirror Symmetry, [alg-geom/9411018].

  44. G. Darboux, Sur le probl‘eme de Pfaff, Bull. Sci. Math. 6, 14–36, 49–68 (1882).

    Google Scholar 

  45. J. Moser, Trans. Amer. Math. Soc. 120, 286 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  46. F. J. Dyson, Am. J. Phys. 58, 209 (1990).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  47. C. R. Lee, Phys. Lett. A 148, 146 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  48. S. Tanimura, Ann. Phys. 220, 229 (1992), [hepth/9306066]; J. F. Carinena, L. A. Ibort, G. Marmo and A. Stern, Phys. Rep. 263, 153 (1995), [hep-th/9306066].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  49. S. Sternberg, Proc. Natl. Acad. Sci. USA 74, 5253 (1977).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  50. M. Karoubi, K-Theory: An Introduction (Springer, Berlin, 1978).

    Book  MATH  Google Scholar 

  51. R. Minasian and G. Moore, J. High Energy Phys. 11, 002 (1997), [hep-th/9710230]; P. Hořava, Adv. Theor. Math. Phys. 2, 1373 (1998), [hep-th/9812135].

    Article  ADS  MathSciNet  Google Scholar 

  52. E. Witten, J. High Energy Phys. 12, 019 (1998), [hepth/9810188]; Int. J. Mod. Phys. A 16, 063 (2001), [hepth/0007175].

    Article  ADS  MathSciNet  Google Scholar 

  53. K. Olsen and R. J. Szabo, Adv. Theor. Math. Phys. 3, 889 (1999), [hep-th/9907140]; J. A. Harvey, Komaba Lectures on Noncommutative Solitons and D-Branes, [hep-th/0102076].

    MATH  MathSciNet  Google Scholar 

  54. M. F. Atiyah, R. Bott and A. Shapiro, Topology 3, 3 (1964).

    Article  MATH  MathSciNet  Google Scholar 

  55. N. Hitchin, Quart. J. Math. Oxford Ser. 54, 281 (2003), [math.DG/0209099]; M. Gualtieri, [math.DG/0401221]; [math.DG/0703298]; G. R. Cavalcanti, [math.DG/0501406].

    Article  MATH  MathSciNet  Google Scholar 

  56. L. Cornalba, Adv. Theor. Math. Phys. 4, 271 (2000), [hep-th/9909081].

    MATH  MathSciNet  Google Scholar 

  57. However, Eq. (56) shows that the SU(2) gauge fields A μ (±) have their own gauge fields, i.e., the vierbeins E a whose degrees of freedom match with those of the U(1) gauge fields D a in the action in Eq. (44) when applying the map in Eq. (12). Therefore, it is sensible that SU(2) gauge theory describing self-dual gravity can be mapped to the U(1) gauge theory described by the action in Eq. (44), as will be shown later.

  58. S. W. Hawking, Phys. Lett. 60A, 81 (1977); T. Eguchi and A. J. Hanson, Phys. Lett. 74B, 249 (1978); G. W. Gibbons and S. W. Hawking, Phys. Lett. 78B, 430 (1978).

    Google Scholar 

  59. J. J. Oh, C. Park and H. S. Yang, J. High Energy Phys. 04, 087 (2011), [arXiv:1101.1357].

    Article  ADS  MathSciNet  Google Scholar 

  60. Y. Hashimoto, Y. Yasui, S. Miyagi and T. Ootsuka, J. Math. Phys. 38, 5833 (1997), [hep-th/9610069]; T. Ootsuka, S. Miyagi, Y. Yasui and S. Zeze, Class. Quant. Grav. 16, 1305 (1999), [gr-qc/9809083].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  61. One can show that the ansatz in Eq. (83) is equivalent to Eq. (69) (up to a sign) by the identification of the volume form ν = λ−2 E 1 ⋀ ··· ⋀ E 4 = λ−2 ν g. See Eq. (95). According to the definition in Eq. (83), we have \({J^{\left( \pm \right)i}} = \frac{1}{2}\eta _{ab}^{\left( \pm \right)i}{\iota _a}{\iota _b}\left( {{\lambda ^{ - 2}}{\upsilon _g}} \right) = \frac{1}{2}\eta _{ab}^{\left( \pm \right)i}{\iota _{\lambda {E_a}}}{\iota _{\lambda {E_b}}}\left( {{\lambda ^{ - 2}}{\upsilon _g}} \right) = \frac{1}{2}\eta _{ab}^{\left( \pm \right)i}{\iota _{{E_a}}}{\iota _{{E_b}}}{\upsilon _g} = \frac{1}{2}\left( {\frac{1}{2}{\varepsilon _{abcd}}\eta _{ab}^{\left( \pm \right)i}} \right){E^c} \wedge {E^d} = \pm \frac{1}{2}\eta _{ab}^{\left( \pm \right)i}{E^a} \wedge {E^b}\).

  62. A. Ashtekar, T. Jabobson and L. Smolin, Commun. Math. Phys. 115, 631 (1988); L. J. Mason and E. T. Newman, Commun. Math. Phys. 121, 659 (1989); S. Chakravarty, L. Mason and E. T. Newman, J. Math. Phys. 32, 1458 (1991); D. D. Joyce, Duke Math. J. 77, 519 (1995).

    Article  ADS  MATH  Google Scholar 

  63. This formula can be easily derived by applying Cartan’s homotopy formula ℒ X β = ι X dβ+dι X β for β = ι Y α and [ℒ X, ι Y] = ι [X, Y] [8].

  64. N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198, 689 (1998), [hep-th/9802068]; K.-Y. Kim, B.-H. Lee, H. S. Yang, J. Korean Phys. Soc. 41, 290 (2002), [hep-th/0003093]; Phys. Lett. B 523, 357 (2001), [hepth/0109121].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  65. There is a perverse sign problem in reversing the map. We still don’t understand how to properly translate the map Γ(TM) → C ∞ (M) to avoid the sign problem. In a way, one may consider some analytic continuation such that every module is defined over ℂ, as was suggested in Ref. [7]. Anyway, the sign issue will be insignificant at this stage.

  66. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Ann. Phys. (N.Y.) 111, 61; 111 (1978); B. V. Fedosov, J. Diff. Geom. 40, 213 (1994).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  67. Here, ħ is a formal deformation parameter that may be identified with either |θ|or the Planck constant ħ, depending on the choice of a Poisson manifold M or P. For the Poisson manifold M with the Poisson structure θ ∈ Γ(Λ2 TM), it would be convenient to take the rescaling θ μν → ħθ μν because ħ is regarded as a plain parameter without any indices. In the end, one may set ħ = 1.

  68. Our notation here may be sloppy. D a in Eq. (157) contains an extra −i compared to Eq. (131). This sloppy notation is for a parallel march with D μ.

  69. R. S. Ward, Class. Quantum Grav. 7, L217 (1990).

    Article  ADS  MATH  Google Scholar 

  70. E. Corrigan, C. Devchand, D. B. Fairlie and J. Nuyts, Nucl. Phys. B 214, 452 (1983); R. S. Ward, Nucl. Phys. B 236, 381 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  71. D. Bak, K. Lee and J.-H. Park, Phys. Rev. D 66, 025021 (2002), [hep-th/0204221].

    Article  ADS  MathSciNet  Google Scholar 

  72. H. Elvang and J. Polchinski, C. R. Physique 4, 405 (2003), [hep-th/0209104].

    Article  ADS  Google Scholar 

  73. In this respect, quoting a recent comment by Zee [74] would be interesting: “The basic equation for the graviton field has the same form g μν = η μν + h μν. This naturally suggests that η μν = 〈g μν〉 and perhaps some sort of spontaneous symmetry breaking.” It turns out that this pattern is not an accidental happening.

  74. A. Zee, Int. J. Mod. Phys. A 23, 1295 (2008), [arXiv:0805.2183].

    Article  ADS  MathSciNet  Google Scholar 

  75. L. Cornalba, J. High Energy Phys. 09, 017 (2000), [hepth/9912293]; Adv. Theor. Math. Phys. 4, 1259 (2002), [hep-th/0006018]; L. Cornalba and R. Schiappa, Commun. Math. Phys. 225, 33 (2002), [hep-th/0101219].

    Article  ADS  MathSciNet  Google Scholar 

  76. Y. M. Cho and P. G. O. Freund, Phys. Rev. D 12, 1711 (1975).

    Article  ADS  MathSciNet  Google Scholar 

  77. I. Vaisman, Lectures on the geometry of Poisson manifolds (Birkhäuser, Basel, 1994).

    Book  MATH  Google Scholar 

  78. A. Weinstein, J. Diff. Geom. 18, 523 (1983).

    MATH  Google Scholar 

  79. S. K. Wong, Nuovo Cimento 65A, 689 (1970).

    Article  ADS  Google Scholar 

  80. V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge University Press, 1984).

    MATH  Google Scholar 

  81. A. Weinstein, Lett. Math. Phys. 2, 417 (1978).

    Article  ADS  MATH  Google Scholar 

  82. P. Hořava, Phys. Rev. Lett. 95, 016405 (2005), [hepth/0503006].

    Article  ADS  Google Scholar 

  83. G. E. Volovik, Lect. Notes Phys. 718, 31 (2007), [condmat/0601372]; Fermi-point scenario of emergent gravity, [arXiv:0709.1258].

    Article  ADS  MathSciNet  Google Scholar 

  84. H. Georgi, Lie Algebras in Particle Physics: From Isospin to Unified Theories (Advanced Book Program, 1999).

    Google Scholar 

  85. One may imagine that the system in Eq. (277) is quantized a la Eq. (5}). Then, the time evolution of a quantum system is derived from a Heisenberg algebra, i.e., \(i\hbar \frac{{d\hat f}}{{dt}} = \left[ {\hat f,\hat H} \right]\;for\;\hat f,\hat H\; \in \;{A_h}\). This is also in line with our philosophy that a geometry is derived from an algebra

  86. H. S. Yang, J. Phys. Conf. Ser. 343, 012132 (2012), [arXiv:1111.0015].

    Article  ADS  Google Scholar 

  87. T. Padmanabhan, Gen. Rel. Grav. 40, 529 (2008), [arXiv:0705.2533]; AIP Conf. Proc. 939, 114 (2007), [arXiv:0706.1654]; Adv. Sci. Lett. 2, 174 (2009), [arXiv:0807.2356].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  88. This argument was corrected in Ref. [86], but we will not delete this incorrect argument to illustrate how a prejudice easily ignores important physics.

  89. S. Minwalla, M. Van Raamsdonk and N. Seiberg, J. High Energy Phys. 02, 020 (2000), [hep-th/9912072].

    Article  ADS  Google Scholar 

  90. A. Raychaudhuri, Relativistic Cosmology. I, Phys. Rev. 98, 1123 (1955).

    MATH  MathSciNet  Google Scholar 

  91. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, 1973).

    Book  MATH  Google Scholar 

  92. M. Blau and S. Theisen, Gen. Relativ. Grav. 41, 743 (2009).

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Daejin University, Pocheon, 487-711, Korea

    Jungjai Lee

  2. School of Physics, Korea Institute for Advanced Study, Seoul, 130-722, Korea

    Hyun Seok Yang

Authors
  1. Jungjai Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hyun Seok Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Jungjai Lee or Hyun Seok Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, J., Yang, H.S. Quantum gravity from noncommutative spacetime. Journal of the Korean Physical Society 65, 1754–1798 (2014). https://doi.org/10.3938/jkps.65.1754

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3938/jkps.65.1754

Keywords

Search

Navigation