Abstract
Starting from the concept of involution of field equations, a universal method is proposed for constructing consistent interactions between the fields. The method equally well applies to the Lagrangian and non-Lagrangian equations and it is explicitly covariant. No auxiliary fields are introduced. The equations may have (or have no) gauge symmetry and/or second class constraints in Hamiltonian formalism, providing the theory admits a Hamiltonian description. In every case the method identifies all the consistent interactions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York U.S.A. (1964).
S. Lyakhovich and A. Sharapov, Normal forms and gauge symmetries of local dynamics, J. Math. Phys. 50 (2009) 083510 [arXiv:0812.4914] [INSPIRE].
M. Henneaux, Consistent interactions between gauge fields: the Cohomological approach, Contemp. Math. 219 (1998) 93.
G. Barnich and M. Henneaux, Consistent couplings between fields with a gauge freedom and deformations of the master equation, Phys. Lett. B 311 (1993) 123 [hep-th/9304057] [INSPIRE].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000) 439 [hep-th/0002245] [INSPIRE].
M. Henneaux and B. Knaepen, All consistent interactions for exterior form gauge fields, Phys. Rev. D 56 (1997) 6076 [hep-th/9706119] [INSPIRE].
N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Inconsistency of interacting, multigraviton theories, Nucl. Phys. B 597 (2001) 127 [hep-th/0007220] [INSPIRE].
M. Henneaux, G. Lucena Gomez and R. Rahman, Higher-Spin Fermionic Gauge Fields and Their Electromagnetic Coupling, JHEP 08 (2012) 093 [arXiv:1206.1048] [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].
J. Kluson, Comments About Hamiltonian Formulation of Non-Linear Massive Gravity with Stuckelberg Fields, JHEP 06 (2012) 170 [arXiv:1112.5267] [INSPIRE].
S. Hassan, A. Schmidt-May and M. von Strauss, Proof of Consistency of Nonlinear Massive Gravity in the Stúckelberg Formulation, Phys. Lett. B 715 (2012) 335 [arXiv:1203.5283] [INSPIRE].
Y. Zinoviev, On massive spin 2 interactions, Nucl. Phys. B 770 (2007) 83 [hep-th/0609170] [INSPIRE].
W.M. Seiler, Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra, Algorithms and computations in Mathematics, Volume 24, Springer-Verlag, Berlin Heidelberg (2010).
A. Einstein, The Meaning of the Relativity, 5th. edition, Princeton University Press, Princeton U.S.A. (1955).
K.H. Mariwalla, Application of the concept of strength of a system of partial differential equations, J. Math. Phys. 15 (1974) 468.
B.F. Schutz, On the strenght of a system of partial differential equations, J. Math. Phys. 16 (1975) 855.
N.F.J. Matthews, On the strenght of Maxwell’s equations, J. Math. Phys. 28 (1987) 810.
M. Sué, Involutive systems of differential equations: Einstein’s strenght versus Cartan’s degré d’arbitraire, J. Math. Phys. 32 (1991) 392.
M. Henneaux, C. Teitelboim and J. Zanelli, Gauge invariance and degree of freedom count, Nucl. Phys. B 332 (1990) 169 [INSPIRE].
X. Bekaert, S. Cnockaert, C. Iazeolla and M.A. Vasiliev, Nonlinear higher spin theories in various dimensions, IHES-P-04-47, ULB-TH-04-26, ROM2F-04-29, FIAN-TD-17-04, hep-th/0503128 [INSPIRE].
V. Lopatin and M.A. Vasiliev, Free massless bosonic fields of arbitrary spin in d-dimensional de Sitter space, Mod. Phys. Lett. A 3 (1988) 257 [INSPIRE].
O. Shaynkman and M.A. Vasiliev, Scalar field in any dimension from the higher spin gauge theory perspective, Theor. Math. Phys. 123 (2000) 683 [hep-th/0003123] [INSPIRE].
E. Skvortsov, Gauge fields in (A)dS(d) within the unfolded approach: algebraic aspects, JHEP 01 (2010) 106 [arXiv:0910.3334] [INSPIRE].
S. Lyakhovich and A. Sharapov, BRST theory without Hamiltonian and Lagrangian, JHEP 03 (2005) 011 [hep-th/0411247] [INSPIRE].
P. Kazinski, S. Lyakhovich and A. Sharapov, Lagrange structure and quantization, JHEP 07 (2005) 076 [hep-th/0506093] [INSPIRE].
D. Kaparulin, S. Lyakhovich and A. Sharapov, Local BRST cohomology in (non-)Lagrangian field theory, JHEP 09 (2011) 006 [arXiv:1106.4252] [INSPIRE].
M. Henneaux, Space-time locality of the BRST formalism, Commun. Math. Phys. 140 (1991) 1 [INSPIRE].
M. Fierz and W. Pauli, On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field, Proc. R. Soc. London A 173 (1939) 211.
L. Singh and C. Hagen, Lagrangian formulation for arbitrary spin. 1. The boson case, Phys. Rev. D 9 (1974) 898 [INSPIRE].
S. Lyakhovich and A. Sharapov, Schwinger-Dyson equation for non-Lagrangian field theory, JHEP 02 (2006) 007 [hep-th/0512119] [INSPIRE].
S. Lyakhovich and A. Sharapov, Quantizing non-Lagrangian gauge theories: an augmentation method, JHEP 01 (2007) 047 [hep-th/0612086] [INSPIRE].
D. Kaparulin, S. Lyakhovich and A. Sharapov, Rigid Symmetries and Conservation Laws in Non-Lagrangian Field Theory, J. Math. Phys. 51 (2010) 082902 [arXiv:1001.0091] [INSPIRE].
S. Hassan and R.A. Rosen, On Non-Linear Actions for Massive Gravity, JHEP 07 (2011) 009 [arXiv:1103.6055] [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].
D. Eisenbud, The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra, Graduate Texts in Mathematics, Volume 229, Springer-Verlag New York Inc., NY U.S.A. (2005).
S.L. Lyakhovich and A.A. Sharapov, work in progress.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1210.6821
Rights and permissions
About this article
Cite this article
Kaparulin, D.S., Lyakhovich, S.L. & Sharapov, A.A. Consistent interactions and involution. J. High Energ. Phys. 2013, 97 (2013). https://doi.org/10.1007/JHEP01(2013)097
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2013)097