Brazilian Journal of Physics

, Volume 47, Issue 3, pp 350–365 | Cite as

BRST Quantization of Unimodular Gravity

  • Sudhaker Upadhyay
  • Markku Oksanen
  • Rodrigo Bufalo
Particles and Fields


We study the quantization of two versions of unimodular gravity, namely fully diffeomorphism-invariant unimodular gravity and unimodular gravity with fixed metric determinant, utilizing standard path integral approach. We derive the BRST symmetry of effective actions corresponding to several relevant gauge conditions. We observe that for some gauge conditions, the restricted gauge structure may complicate the formulation and effective actions, in particular, if the chosen gauge conditions involve the canonical momentum conjugate to the induced metric on the spatial hypersurface. The BRST symmetry is extended further to the finite field-dependent BRST transformation, in order to establish the mapping between different gauge conditions in each of the two versions of unimodular gravity.


Unimodular gravity BRST symmetry 



M.O. gratefully acknowledges support from the Emil Aaltonen Foundation. R.B. acknowledges FAPESP and CNPq for partial support, FAPESP Project No. 2011/20653-3 and CNPq Project No. 304241/2016-4.


  1. 1.
    S. Weinberg, The cosmological constant problem. Rev. Mod. Phys. 61, 1 (1989)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    T. Padmanabhan, Cosmological constant: the weight of the vacuum. Phys. Rept. 380, 235 (2003). arXiv:hep-th/0212290 ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    R. Bousso, TASI lectures on the cosmological constant. Gen. Rel. Grav. 40, 607 (2008). arXiv:0708.4231 [hep-th]ADSCrossRefMATHGoogle Scholar
  4. 4.
    A. Einstein, The foundation of the general theory of relativity. Annalen Phys. 49, 769 (1916). Translated and included in The Principle of Relativity, by H.A. Lorentz et al. (Dover Press, New York, 1923)ADSCrossRefGoogle Scholar
  5. 5.
    A. Einstein, Do gravitational fields play an essential part in the structure of the elementary particles of matter? Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1919, 433 (1919). Translated and included in The Principle of Relativity, by H.A. Lorentz et al. (Dover Press, New York, 1923)Google Scholar
  6. 6.
    L. Smolin, Quantization of unimodular gravity and the cosmological constant problems. Phys. Rev. D. 80, 084003 (2009). arXiv:0904.4841 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    W. Buchmuller, N. Dragon, Einstein gravity from restricted coordinate invariance. Phys. Lett. B. 207, 292 (1988)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    W.G. Unruh, A unimodular theory of canonical quantum gravity. Phys. Rev. D. 40, 1048 (1989)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    K.V. Kuchar, Does an unspecified cosmological constant solve the problem of time in quantum gravity? Phys. Rev. D. 43, 3332 (1991)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    M Henneaux, C. Teitelboim, The cosmological constant and general covariance. Phys. Lett. B. 222, 195 (1989)ADSCrossRefGoogle Scholar
  11. 11.
    R. Bufalo, M. Oksanen, A. Tureaun, How unimodular gravity theories differ from general relativity at quantum level. Eur. Phys. J. C. 75, 477 (2015). arXiv:1505.04978 [hep-th]ADSCrossRefGoogle Scholar
  12. 12.
    Y.J. Ng, H. van Dam, Possible solution to the cosmological constant problem. Phys. Rev. Lett. 65, 1972 (1990)Google Scholar
  13. 13.
    R. D. Sorkin, On the role of time in the sum over histories framework for gravity. Int. J. Theor. Phys. 33, 523 (1994). Originally presented at the conference, The History of Modern Gauge Theories, held at Logan, Utah, July 1987MathSciNetCrossRefGoogle Scholar
  14. 14.
    W. G. Unruh, R. M. Wald, Time and the interpretation of canonical quantum gravity. Phys. Rev. D. 40, 2598 (1989)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Álvarez, S. González-Martín, M. Herrero-Valea, C. P. Martín, Quantum corrections to unimodular gravity. JHEP. 1508, 078 (2015). arXiv:1505.01995 [hep-th]MathSciNetCrossRefGoogle Scholar
  16. 16.
    S. D. Joglekar, B. P. Mandal, Finite field dependent BRS transformations. Phys. Rev. D. 51, 1919 (1995)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    S. D. Joglekar, A. Misra, Correct treatment of 1/(.k)p singularities in the axial gauge propagator. Int. J. Mod. Phys. A. 15, 1453 (2000). arXiv:hep-th/9909123 ADSMATHGoogle Scholar
  18. 18.
    S.D. Joglekar, B. P. Mandal, Application of finite field dependent BRS transformations to problems of the Coulomb gauge. Int. J. Mod. Phys. A. 17, 1279 (2002). arXiv:hep-th/0105042 ADSCrossRefMATHGoogle Scholar
  19. 19.
    S. Upadhyay, S. K. Rai, B. P. Mandal, Off-shell nilpotent finite BRST/anti-BRST transformations. J. Math. Phys. 52, 022301 (2011). arXiv:1002.1373 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    V. N. Gribov, Quantization of non-Abelian gauge theories. Nucl. Phys. B. 139, 1 (1978)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    D. Zwanziger, Local and renormalizable action from the gribov horizon. Nucl. Phys. B. 323, 513 (1989)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    D. Zwanziger, Renormalizability of the critical limit of lattice gauge theory by BRS invariance. Nucl. Phys. B. 399, 477 (1993)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    S. Upadhyay, B. P. Mandal, Generalized BRST symmetry for arbitrary spin conformal field theory. Phys. Lett. B. 744, 231 (2015). arXiv:1409.1735 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    S. Upadhyay, B. P. Mandal, Gaugeon formalism in the framework of generalized BRST symmetry. Prog. Theor. Exp. Phys. 053B04, 1 (2014). arXiv:1403.6194 [hep-th]MATHGoogle Scholar
  25. 25.
    S. Upadhyay, B. P. Mandal, Field dependent nilpotent symmetry for gauge theories. Eur. Phys. J. C. 72, 2065 (2012). arXiv:1201.0084 [hep-th]ADSCrossRefGoogle Scholar
  26. 26.
    S. Upadhyay, B. P. Mandal, Finite BRST transformation and constrained systems. Annls. Phys. 327, 2885 (2012). arXiv:1207.6449 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    S. Upadhyay, B. P. Mandal, Relating Gribov-Zwanziger theory to effective Yang-Mills theory. Eur. Phys. Lett. 93, 31001 (2011). arXiv:1101.5448 [hep-th]ADSCrossRefGoogle Scholar
  28. 28.
    S. Upadhyay, B. P. Mandal, Generalized BRST transformation in Abelian rank-2 antisymmetric tensor field theory. Mod. Phys. Lett. A. 25, 3347 (2010). arXiv:1004.0330 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    S. Upadhyay, M. K. Dwivedi, B. P. Mandal, The noncovariant gauges in 3-form theories. Int. J. Mod. Phys. A. 28, 1350033 (2013). arXiv:1301.0222 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    S. Upadhyay, M. K. Dwivedi, B. P. Mandal, Emergence of Lowenstein-Zimmermann mass terms for QED 3. arXiv:1407.2017 [hep-th]
  31. 31.
    M. Faizal, B. P. Mandal, S. Upadhyay, Finite BRST Transformations for the Bagger-Lambert-Gustavasson Theory. Phys. Lett. B. 721, 159 (2013). arXiv:1212.5653 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    S. Upadhyay, D. Das, ABJM theory in Batalin-Vilkovisky formulation. Phys. Lett. B. 733, 63 (2014). arXiv:1404.2633 [hep-th]ADSCrossRefGoogle Scholar
  33. 33.
    M. Faizal, S. Upadhyay, B. P. Mandal, Finite field-dependent BRST symmetry for ABJM theory in \({\mathcal {N}}=1\) superspace. Phys. Lett. B. 738, 201 (2014). arXiv:1410.0671 [hep-th]ADSCrossRefGoogle Scholar
  34. 34.
    M. Faizal, S. Upadhyay, B. P. Mandal, IR finite graviton propagators in de Sitter spacetime. Eur. Phys. J. C. 76, 189 (2016). arXiv:1604.00390 [hep-th]ADSCrossRefGoogle Scholar
  35. 35.
    M. Faizal, S. Upadhyay, B. P. Mandal, Anti-FFBRST Transformations for the BLG Theory in Presence of a Boundary. Int. J. Mod. Phys. A. 30, 1550032 (2015). arXiv:1501.01616 [hep-th]ADSCrossRefMATHGoogle Scholar
  36. 36.
    J. F. Bagger, N. Lambert, Modeling multiple M2-branes. Phys. Rev. D. 75, 045020 (2007). arXiv:hep-th/0611108 ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    J. F. Bagger, N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev. D. 77, 065008 (2008). arXiv:0711.0955 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    J. F. Bagger, N. Lambert, Comments on multiple M2-branes. JHEP. 0802, 105 (2008). arXiv:0712.3738 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    A. Gustavsson, Algebraic structures on parallel M2 branes. Nucl. Phys. B. 811, 66 (2009). arXiv:0709.1260 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    O. Aharony, O. Bergman, D. L. Jafferis, J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals. JHEP. 0810, 091 (2008). arXiv:0806.1218 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    R. Banerjee, S. Upadhyay, Generalized supersymmetry and sigma models. Phys. Lett. B. 734, 369 (2014). arXiv:1310.1168 [hep-th]ADSCrossRefGoogle Scholar
  42. 42.
    B. P. Mandal, S. K. Rai, S. Upadhyay, Finite nilpotent symmetry in Batalin-Vilkovisky formalism. Eur. Phys. Lett. 92, 21001 (2010). arXiv:1009.5859 [hep-th]ADSCrossRefGoogle Scholar
  43. 43.
    S. Upadhyay, Super-group field cosmology in Batalin-Vilkovisky formulation. Int. J. Theor. Phys. (2016). arXiv:1606.09606 [hep-th]
  44. 44.
    S. Upadhyay, P. A. Ganai, Finite field-dependent symmetry in thirring model. Prog. Theor. Exp. Phys. 063B04, 1 (2016). arXiv:1605.04290 [hep-th]Google Scholar
  45. 45.
    S. Upadhyay, The conformal gauge to the derivative gauge for worldsheet gravity. Phys. Lett. B. 740, 341 (2015). arXiv:1412.5911 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    S. Upadhyay, Field-dependent symmetries in Friedmann-Robertson-Walker models. Ann. Phys. 356, 299 (2015). arXiv:1503.04197 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    S. Upadhyay, Nilpotent symmetries in super-group field cosmology. Mod. Phys. Lett. A. 30, 1550072 (2015). arXiv:1502.05217 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    S. Upadhyay, Finite field-dependent symmetries in perturbative quantum gravity. Ann. Phys. 340, 110 (2014). arXiv:1310.8579 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    S. Upadhyay, Generalized BRST symmetry and Gaugeon formalism for perturbative quantum gravity: novel observation. Ann. Phys. 344, 290 (2014). arXiv:1403.6166 [hep-th]ADSCrossRefMATHGoogle Scholar
  50. 50.
    S. Upadhyay, Field-dependent quantum gauge transformation. Eur. Phys. Lett. 105, 21001 (2014). arXiv:1402.3373 [hep-th]ADSCrossRefGoogle Scholar
  51. 51.
    S. Upadhyay, N = 1 super-Chern-Simons theory in Batalin-Vilkovisky formulation. Eur. Phys. Lett. 104, 61001 (2013). arXiv:1401.1968 [hep-th]ADSCrossRefGoogle Scholar
  52. 52.
    S. Upadhyay, Aspects of finite field-dependent symmetry in SU(2) Cho-Faddeev-Niemi decomposition. Phys. Lett. B. 727, 293 (2013). arXiv:1310.2013 [hep-th]ADSCrossRefMATHGoogle Scholar
  53. 53.
    M. Henneaux, C. Teitelboim. Quantization of Gauge Systems (Univ Press, Princeton, 1992)MATHGoogle Scholar
  54. 54.
    P. M. Lavrov, O. Lechtenfeld, Field-dependent BRST transformations in Yang-Mills theory. Phys. Lett. B. 725, 382 (2013). arXiv:1305.0712 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    P. Y. Moshin, A. A. Reshetnyak, Field-dependent BRST-antiBRST transformations in generalized Hamiltonian formalism. Int. J. Mod. Phys. A. 29, 1450159 (2014). arXiv:1405.7549 [hep-th]ADSCrossRefMATHGoogle Scholar
  56. 56.
    P. Y. Moshin, A. A. Reshetnyak, Field-dependent BRST–anti-BRST Lagrangian transformations. Int. J. Mod. Phys. A. 30, 1550021 (2015). arXiv:1406.5086 [hep-th]MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    P. Y. Moshin, A. A. Reshetnyak, Finite BRST-antiBRST Transformations in Lagrangian Formalism. Phys. Lett. B. 739, 110 (2014). arXiv:1406.0179 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    S. Upadhyay, A. Reshetnyak, B. P. Mandal, Comments on interactions in the SUSY models. Eur. Phys. J. C. 76, 391 (2016). arXiv:1605.02973v5 [physics.gen-ph]ADSCrossRefGoogle Scholar
  59. 59.
    P. Y. Moshin, A. A. Reshetnyak, Field-dependent BRST-antiBRST Transformations in Yang-Mills and Gribov-Zwanziger Theories. Nucl. Phys. B. 888, 92 (2014). arXiv:1405.0790 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    P. Y. Moshin, A. A. Reshetnyak, Finite field-dependent BRST-antiBRST transformations: Jacobians and application to the standard model. arXiv:1506.04660 [hep-th]
  61. 61.
    K. Nishijima, M. Okawa, The Becchi-Rouet-Stora transformation for the gravitational field. Prog. Theor. Phys. 60, 272 (1978)ADSMathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    L. D. Faddeev, V. N. Popov, Covariant quantization of the gravitational field. Sov. Phys. Usp. 74, 777 (1974)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedade Brasileira de Física 2017

Authors and Affiliations

  • Sudhaker Upadhyay
    • 1
  • Markku Oksanen
    • 2
  • Rodrigo Bufalo
    • 3
  1. 1.Centre for Theoretical StudiesIndian Institute of Technology KharagpurKharagpurIndia
  2. 2.Department of PhysicsUniversity of HelsinkiHelsinkiFinland
  3. 3.Departamento de FísicaUniversidade Federal de LavrasLavras-MGBrazil

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