Abstract
We provide a full analysis of ghost free higher derivative field theories with coupled degrees of freedom. Assuming the absence of gauge symmetries, we derive the degeneracy conditions in order to evade the Ostrogradsky ghosts, and analyze which (non)trivial classes of solutions this allows for. It is shown explicitly how Lorentz invariance avoids the propagation of “half” degrees of freedom. Moreover, for a large class of theories, we construct the field redefinitions and/or (extended) contact transformations that put the theory in a manifestly first order form. Finally, we identify which class of theories cannot be brought to first order form by such transformations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Ostrogradsky, Mémoires sur les equations differentielle relatives au probleme des isopérimètres, Mem. Ac. St. Petersbourg VI 4 (1850) 385.
R.P. Woodard, Ostrogradsky’s theorem on Hamiltonian instability, Scholarpedia 10 (2015) 32243 [arXiv:1506.02210] [INSPIRE].
E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective, John Wiley and Son, New York, U.S.A. (1974).
H.J. Rothe and K.D. Rothe, Classical and Quantum Dynamics of Constrained Hamiltonian Systems, World Scientific Publishing, Singapore (2010) https://doi.org/10.1142/9789814299657.
A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].
C. Deffayet, S. Deser and G. Esposito-Farese, Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors, Phys. Rev. D 80 (2009) 064015 [arXiv:0906.1967] [INSPIRE].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [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 Stúckelberg language, Phys. Lett. B 711 (2012) 190 [arXiv:1107.3820] [INSPIRE].
L. Heisenberg, Generalization of the Proca Action, JCAP 05 (2014) 015 [arXiv:1402.7026] [INSPIRE].
G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].
T. Kobayashi, M. Yamaguchi and J. Yokoyama, Generalized G-inflation: Inflation with the most general second-order field equations, Prog. Theor. Phys. 126 (2011) 511 [arXiv:1105.5723] [INSPIRE].
C. Deffayet, G. Esposito-Farese and A. Vikman, Covariant Galileon, Phys. Rev. D 79 (2009) 084003 [arXiv:0901.1314] [INSPIRE].
C. Deffayet, X. Gao, D.A. Steer and G. Zahariade, From k-essence to generalised Galileons, Phys. Rev. D 84 (2011) 064039 [arXiv:1103.3260] [INSPIRE].
G. Tasinato, Cosmic Acceleration from Abelian Symmetry Breaking, JHEP 04 (2014) 067 [arXiv:1402.6450] [INSPIRE].
M. Hull, K. Koyama and G. Tasinato, Covariantized vector Galileons, Phys. Rev. D 93 (2016) 064012 [arXiv:1510.07029] [INSPIRE].
A. Chatzistavrakidis, F.S. Khoo, D. Roest and P. Schupp, Tensor Galileons and Gravity, JHEP 03 (2017) 070 [arXiv:1612.05991] [INSPIRE].
M. Zumalacárregui and J. García-Bellido, Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian, Phys. Rev. D 89 (2014) 064046 [arXiv:1308.4685] [INSPIRE].
J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Healthy theories beyond Horndeski, Phys. Rev. Lett. 114 (2015) 211101 [arXiv:1404.6495] [INSPIRE].
J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Exploring gravitational theories beyond Horndeski, JCAP 02 (2015) 018 [arXiv:1408.1952] [INSPIRE].
C. Deffayet, G. Esposito-Farese and D.A. Steer, Counting the degrees of freedom of generalized Galileons, Phys. Rev. D 92 (2015) 084013 [arXiv:1506.01974] [INSPIRE].
D. Langlois and K. Noui, Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability, JCAP 02 (2016) 034 [arXiv:1510.06930] [INSPIRE].
D. Langlois and K. Noui, Hamiltonian analysis of higher derivative scalar-tensor theories, JCAP 07 (2016) 016 [arXiv:1512.06820] [INSPIRE].
M. Crisostomi, M. Hull, K. Koyama and G. Tasinato, Horndeski: beyond, or not beyond?, JCAP 03 (2016) 038 [arXiv:1601.04658] [INSPIRE].
M. Crisostomi, K. Koyama and G. Tasinato, Extended Scalar-Tensor Theories of Gravity, JCAP 04 (2016) 044 [arXiv:1602.03119] [INSPIRE].
J. Ben Achour, D. Langlois and K. Noui, Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations, Phys. Rev. D 93 (2016) 124005 [arXiv:1602.08398] [INSPIRE].
C. de Rham and A. Matas, Ostrogradsky in Theories with Multiple Fields, JCAP 06 (2016) 041 [arXiv:1604.08638] [INSPIRE].
J.M. Ezquiaga, J. García-Bellido and M. Zumalacárregui, Field redefinitions in theories beyond Einstein gravity using the language of differential forms, Phys. Rev. D 95 (2017) 084039 [arXiv:1701.05476] [INSPIRE].
J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui and G. Tasinato, Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order, JHEP 12 (2016) 100 [arXiv:1608.08135] [INSPIRE].
L. Heisenberg, R. Kase and S. Tsujikawa, Beyond generalized Proca theories, Phys. Lett. B 760 (2016) 617 [arXiv:1605.05565] [INSPIRE].
R. Kimura, A. Naruko and D. Yoshida, Extended vector-tensor theories, JCAP 01 (2017) 002 [arXiv:1608.07066] [INSPIRE].
H. Motohashi, K. Noui, T. Suyama, M. Yamaguchi and D. Langlois, Healthy degenerate theories with higher derivatives, JCAP 07 (2016) 033 [arXiv:1603.09355] [INSPIRE].
R. Klein and D. Roest, Exorcising the Ostrogradsky ghost in coupled systems, JHEP 07 (2016) 130 [arXiv:1604.01719] [INSPIRE].
H. Motohashi and T. Suyama, Third order equations of motion and the Ostrogradsky instability, Phys. Rev. D 91 (2015) 085009 [arXiv:1411.3721] [INSPIRE].
M. Henneaux, A. Kleinschmidt and G. Lucena Gómez, Remarks on Gauge Invariance and First-Class Constraints, Proc. Steklov Inst. Math. 272 (2011) 141 [arXiv:1004.3769] [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].
C. Deffayet, S. Deser and G. Esposito-Farese, Arbitrary p-form Galileons, Phys. Rev. D 82 (2010) 061501 [arXiv:1007.5278] [INSPIRE].
A. Padilla, P.M. Saffin and S.-Y. Zhou, Bi-galileon theory I: Motivation and formulation, JHEP 12 (2010) 031 [arXiv:1007.5424] [INSPIRE].
K. Hinterbichler, M. Trodden and D. Wesley, Multi-field galileons and higher co-dimension branes, Phys. Rev. D 82 (2010) 124018 [arXiv:1008.1305] [INSPIRE].
A. Padilla and V. Sivanesan, Covariant multi-galileons and their generalisation, JHEP 04 (2013) 032 [arXiv:1210.4026] [INSPIRE].
E. Allys, New terms for scalar multi-Galileon models and application to SO(N) and SU(N) group representations, Phys. Rev. D 95 (2017) 064051 [arXiv:1612.01972] [INSPIRE].
V. Sivanesan, Generalized multiple-scalar field theory in Minkowski space-time free of Ostrogradski ghosts, Phys. Rev. D 90 (2014) 104006 [arXiv:1307.8081] [INSPIRE].
D.B. Fairlie and A.N. Leznov, General solutions of the Monge-Ampere equation in n-dimensional space, J. Geom. Phys. 16 (1995) 385 [hep-th/9403134] [INSPIRE].
D. Comelli, F. Nesti and L. Pilo, Massive gravity: a General Analysis, JHEP 07 (2013) 161 [arXiv:1305.0236] [INSPIRE].
X. Gracia and J.M. Pons, Gauge Generators, Dirac’s Conjecture and Degrees of Freedom for Constrained Systems, Annals Phys. 187 (1988) 355 [INSPIRE].
J.M. Pons, New Relations Between Hamiltonian and Lagrangian Constraints, J. Phys. A 21 (1988) 2705 [INSPIRE].
M. Henneaux, C. Teitelboim and J. Zanelli, Gauge Invariance and Degree of Freedom Count, Nucl. Phys. B 332 (1990) 169 [INSPIRE].
B. Díaz, D. Higuita and M. Montesinos, Lagrangian approach to the physical degree of freedom count, J. Math. Phys. 55 (2014) 122901 [arXiv:1406.1156] [INSPIRE].
P. Dirac, Lectures on Quantum Mechanics, Dover Books on Physics, Dover Publications (2001).
C. de Rham, M. Fasiello and A.J. Tolley, Galileon Duality, Phys. Lett. B 733 (2014) 46 [arXiv:1308.2702] [INSPIRE].
C. De Rham, L. Keltner and A.J. Tolley, Generalized galileon duality, Phys. Rev. D 90 (2014) 024050 [arXiv:1403.3690] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1703.01623
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Crisostomi, M., Klein, R. & Roest, D. Higher derivative field theories: degeneracy conditions and classes. J. High Energ. Phys. 2017, 124 (2017). https://doi.org/10.1007/JHEP06(2017)124
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2017)124