Abstract
We propose and study a new first order version of the ghost-free massive gravity. Instead of metrics or tetrads, it uses a connection together with Plebanski’s chiral 2-forms as fundamental variables, rendering the phase space structure similar to that of SU(2) gauge theories. The chiral description simplifies computations of the constraint algebra, and allows us to perform the complete canonical analysis of the system. In particular, we explicitly compute the secondary constraint and carry out the stabilization procedure, thus proving that in general the theory propagates 7 degrees of freedom, consistently with previous claims. Finally, we point out that the description in terms of 2-forms opens the door to an infinite class of ghost-free massive bi-gravity actions. Our results apply directly to Euclidean signature. The reality conditions to be imposed in the Lorentzian signature appear to be more complicated than in the usual gravity case and are left as an open issue.
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References
K. Hinterbichler, Theoretical aspects of massive gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].
D. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368 [INSPIRE].
C. Deffayet and J.-W. Rombouts, Ghosts, strong coupling and accidental symmetries in massive gravity, Phys. Rev. D 72 (2005) 044003 [gr-qc/0505134] [INSPIRE].
A. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett. B 39 (1972) 393 [INSPIRE].
H. van Dam and M. Veltman, Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B 22 (1970) 397 [INSPIRE].
N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys. 305 (2003) 96 [hep-th/0210184] [INSPIRE].
C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of massive gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Ghost free massive gravity in the Stückelberg language, Phys. Lett. B 711 (2012) 190 [arXiv:1107.3820] [INSPIRE].
C. Burrage, N. Kaloper and A. Padilla, Strong coupling and bounds on the graviton mass in massive gravity, arXiv:1211.6001 [INSPIRE].
S. Hassan and R.A. Rosen, Resolving the ghost problem in non-linear massive gravity, Phys. Rev. Lett. 108 (2012) 041101 [arXiv:1106.3344] [INSPIRE].
S. Hassan, R.A. Rosen and A. Schmidt-May, Ghost-free massive gravity with a general reference metric, JHEP 02 (2012) 026 [arXiv:1109.3230] [INSPIRE].
S. Hassan and R.A. Rosen, Bimetric gravity from ghost-free massive gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [INSPIRE].
S. Hassan and R.A. Rosen, Confirmation of the secondary constraint and absence of ghost in massive gravity and bimetric gravity, JHEP 04 (2012) 123 [arXiv:1111.2070] [INSPIRE].
L. Alberte, A.H. Chamseddine and V. Mukhanov, Massive gravity: exorcising the ghost, JHEP 04 (2011) 004 [arXiv:1011.0183] [INSPIRE].
A.H. Chamseddine and V. Mukhanov, Massive gravity simplified: a quadratic action, JHEP 08 (2011) 091 [arXiv:1106.5868] [INSPIRE].
J. Kluson, Note about hamiltonian structure of non-linear massive gravity, JHEP 01 (2012) 013 [arXiv:1109.3052] [INSPIRE].
D. Comelli, M. Crisostomi, F. Nesti and L. Pilo, Spherically symmetric solutions in ghost-free massive gravity, Phys. Rev. D 85 (2012) 024044 [arXiv:1110.4967] [INSPIRE].
A. Golovnev, On the hamiltonian analysis of non-linear massive gravity, Phys. Lett. B 707 (2012) 404 [arXiv:1112.2134] [INSPIRE].
J. Kluson, Comments about hamiltonian formulation of non-linear massive gravity with Stückelberg fields, JHEP 06 (2012) 170 [arXiv:1112.5267] [INSPIRE].
I. Buchbinder, D. Pereira and I. Shapiro, One-loop divergences in massive gravity theory, Phys. Lett. B 712 (2012) 104 [arXiv:1201.3145] [INSPIRE].
J. Kluson, Remark about hamiltonian formulation of non-linear massive gravity in Stückelberg formalism, Phys. Rev. D 86 (2012) 124005 [arXiv:1202.5899] [INSPIRE].
J. Kluson, Non-linear massive gravity with additional primary constraint and absence of ghosts, Phys. Rev. D 86 (2012) 044024 [arXiv:1204.2957] [INSPIRE].
S. Hassan, A. Schmidt-May and M. von Strauss, Metric formulation of ghost-free multivielbein theory, arXiv:1204.5202 [INSPIRE].
S. Hassan, A. Schmidt-May and M. von Strauss, On consistent theories of massive spin-2 fields coupled to gravity, JHEP 05 (2013) 086 [arXiv:1208.1515] [INSPIRE].
J. Kluson, Note about hamiltonian formalism for general non-linear massive gravity action in Stückelberg formalism, arXiv:1209.3612 [INSPIRE].
K. Hinterbichler and R.A. Rosen, Interacting spin-2 fields, JHEP 07 (2012) 047 [arXiv:1203.5783] [INSPIRE].
C. Deffayet, J. Mourad and G. Zahariade, Covariant constraints in ghost free massive gravity, JCAP 01 (2013) 032 [arXiv:1207.6338] [INSPIRE].
C. Deffayet, J. Mourad and G. Zahariade, A note on ’symmetric’ vielbeins in bimetric, massive, perturbative and non perturbative gravities, JHEP 03 (2013) 086 [arXiv:1208.4493] [INSPIRE].
S. Hassan, A. Schmidt-May and M. von Strauss, Proof of consistency of nonlinear massive gravity in the Stückelberg formulation, Phys. Lett. B 715 (2012) 335 [arXiv:1203.5283] [INSPIRE].
J.F. Plebanski, On the separation of Einsteinian substructures, J. Math. Phys. 18 (1977) 2511 [INSPIRE].
R. Capovilla, T. Jacobson, J. Dell and L. Mason, Selfdual two forms and gravity, Class. Quant. Grav. 8 (1991) 41 [INSPIRE].
K. Krasnov, Plebanski formulation of general relativity: a practical introduction, Gen. Rel. Grav. 43 (2011) 1 [arXiv:0904.0423] [INSPIRE].
H. Urbantke, On integrability properties of SU(2) Yang-Mills fields. I. Infinitesimal part, J. Math. Phys. 25 (1984) 2321 [INSPIRE].
K. Krasnov, Plebanski gravity without the simplicity constraints, Class. Quant. Grav. 26 (2009) 055002 [arXiv:0811.3147] [INSPIRE].
A. Ashtekar, Lectures on nonperturbative canonical gravity, Adv. Ser. Astrophys. Cosmol. 6 (1991) 1 [INSPIRE].
T. Thiemann, Modern canonical quantum general relativity, gr-qc/0110034 [INSPIRE].
S. Hojman, K. Kuchar and C. Teitelboim, Geometrodynamics regained, Annals Phys. 96 (1976) 88 [INSPIRE].
S.Y. Alexandrov and D. Vassilevich, Path integral for the Hilbert-Palatini and Ashtekar gravity, Phys. Rev. D 58 (1998) 124029 [gr-qc/9806001] [INSPIRE].
T. Damour and I.I. Kogan, Effective lagrangians and universality classes of nonlinear bigravity, Phys. Rev. D 66 (2002) 104024 [hep-th/0206042] [INSPIRE].
J. Kluson, Hamiltonian formalism of particular bimetric gravity model, arXiv:1211.6267 [INSPIRE].
S. Alexandrov, Degenerate Plebanski sector and spin foam quantization, Class. Quant. Grav. 29 (2012) 145018 [arXiv:1202.5039] [INSPIRE].
K. Krasnov, On deformations of Ashtekar’s constraint algebra, Phys. Rev. Lett. 100 (2008) 081102 [arXiv:0711.0090] [INSPIRE].
K. Krasnov, Effective metric lagrangians from an underlying theory with two propagating degrees of freedom, Phys. Rev. D 81 (2010) 084026 [arXiv:0911.4903] [INSPIRE].
D. Comelli, M. Crisostomi, F. Nesti and L. Pilo, Degrees of freedom in massive gravity, Phys. Rev. D 86 (2012) 101502 [arXiv:1204.1027] [INSPIRE].
S. Dubovsky, Phases of massive gravity, JHEP 10 (2004) 076 [hep-th/0409124] [INSPIRE].
M.P. Reisenberger, Classical Euclidean general relativity from ’left-handed area = right-handed area’, gr-qc/9804061 [INSPIRE].
R. De Pietri and L. Freidel, SO(4) Plebanski action and relativistic spin foam model, Class. Quant. Grav. 16 (1999) 2187 [gr-qc/9804071] [INSPIRE].
A. Perez, Spin foam models for quantum gravity, Class. Quant. Grav. 20 (2003) R43 [gr-qc/0301113] [INSPIRE].
S. Alexandrov, M. Geiller and K. Noui, Spin foams and canonical quantization, SIGMA 8 (2012) 055 [arXiv:1112.1961] [INSPIRE].
E. Buffenoir, M. Henneaux, K. Noui and P. Roche, Hamiltonian analysis of Plebanski theory, Class. Quant. Grav. 21 (2004) 5203 [gr-qc/0404041] [INSPIRE].
S. Alexandrov, E. Buffenoir and P. Roche, Plebanski theory and covariant canonical formulation, Class. Quant. Grav. 24 (2007) 2809 [gr-qc/0612071] [INSPIRE].
S. Alexandrov and K. Krasnov, Hamiltonian analysis of non-chiral Plebanski theory and its generalizations, Class. Quant. Grav. 26 (2009) 055005 [arXiv:0809.4763] [INSPIRE].
S. Speziale, Bi-metric theory of gravity from the non-chiral Plebanski action, Phys. Rev. D 82 (2010) 064003 [arXiv:1003.4701] [INSPIRE].
D. Beke, G. Palmisano and S. Speziale, Pauli-Fierz Mass Term in Modified Plebanski Gravity, JHEP 03 (2012) 069 [arXiv:1112.4051] [INSPIRE].
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ArXiv ePrint: 1212.3614
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Alexandrov, S., Krasnov, K. & Speziale, S. Chiral description of massive gravity. J. High Energ. Phys. 2013, 68 (2013). https://doi.org/10.1007/JHEP06(2013)068
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DOI: https://doi.org/10.1007/JHEP06(2013)068