# American Institute of Mathematical Sciences

December  2011, 3(4): 487-506. doi: 10.3934/jgm.2011.3.487

## Covariantizing classical field theories

 1 ICMAT (CSIC, UAM, UC3M, UCM), Departamento de Geometría y Topología, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain 2 Paciﬁc Institute for the Mathematical Sciences, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

Received  August 2010 Revised  January 2011 Published  February 2012

We show how to enlarge the covariance groups of a wide variety of classical field theories in such a way that the resulting covariantized'' theories are `essentially equivalent' to the originals. In particular, our technique will render many classical field theories generally covariant, that is, the covariantized theories will be spacetime diffeomorphism-covariant and free of absolute objects. Our results thus generalize the well-known parametrization technique of Dirac and Kuchař. Our constructions apply equally well to internal covariance groups, in which context they produce natural derivations of both the Utiyama minimal coupling and Stückelberg tricks.
Citation: Marco Castrillón López, Mark J. Gotay. Covariantizing classical field theories. Journal of Geometric Mechanics, 2011, 3 (4) : 487-506. doi: 10.3934/jgm.2011.3.487
##### References:
 [1] J. L. Anderson, "Principles of Relativity Physics,'', Academic Press, New York, 1967. [2] N. Banerjee, R. Banerjee and S. Ghosh, Quantisation of second class systems in the Batalin-Tyutin formalism, Annals of Physics, 241 (1995), 237-257. doi: 10.1006/aphy.1995.1062. [3] M. Castrillón López, M. J. Gotay and J. E. Marsden, Parametrization and stress-energy-momentum tensors in metric field theories, J. Phys. A, 41 (2008), 344002, 10 pp. [4] M. Castrillón López, M. J. Gotay and J. E. Marsden, Concatenating variational principles and the kinetic stress-energy-momentum tensor, in "Variations, Geometry and Physics" (eds. O. Krupková and D. J. Saunders), Nova Science Publishers, New York, (2009), 117-128. [5] M. Castrillón López and J. Muñoz Masqué, The geometry of the bundle of connections, Math. Z., 236 (2001), 797-811. doi: 10.1007/PL00004852. [6] P. A. M. Dirac, The Hamiltonian form of field dynamics, Can. J. Math., 3 (1951), 1-23. doi: 10.4153/CJM-1951-001-2. [7] P. A. M. Dirac, "Lectures on Quantum Mechanics," Academic Press, New York, 1964. [8] D. Freed, Classical Chern-Simons theory. I, Adv. Math., 113 (1995), 237-303. doi: 10.1006/aima.1995.1039. [9] D. Freed, Remarks on Chern-Simons theory, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 221-254. [10] P. L. García, Gauge algebras, curvature and symplectic structure, J. Diff. Geom., 12 (1977), 209-227. [11] M. J. Gotay, An exterior differential systems approach to the Cartan form, in "Symplectic Geometry and Mathematical Physics" (eds. P. Donato, C. Duval, J. Elhadad and G. M. Tuynman) (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, (1991), 160-188. [12] M. J. Gotay and J. E. Marsden, Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula, in "Mathematical Aspects of Classical Field Theory" (Seattle, WA, 1991), Contemp. Math., 132, Amer. Math. Soc., (1992), 367-392. [13] M. J. Gotay and J. E. Marsden, Momentum maps and classical fields, work in progress. [14] C. Isham and K. Kuchař, Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories, Annals of Physics, 164 (1985), 288-315. doi: 10.1016/0003-4916(85)90018-1. [15] B. Janssens, Bundles with a lift of infinitesimal diffeomorphisms, arXiv:0911.3532, 39 pp. [16] I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, 1993. [17] E. Kretschmann, Über den physikalischen Sinn der Relitivitätstheorie. A. Einsteins neue und seine ursprüngliche Relitivitätstheorie, Ann. Phys., Leipzig, 53 (1917), 575-614. [18] D. Krupka, O. Krupková and D. Saunders, The Cartan form and its generalizations in the calculus of variations, Int. J. of Geometric Methods in Modern Physics, 7 (2010), 631-654. doi: 10.1142/S0219887810004488. [19] K. Kuchař, Canonical quantization of gravity, in "Relativity, Astrophysics and Cosmology" (ed. W. Israel), Reidel, Dordrecht, (1973), 237-288. [20] K. Kuchař, Canonical quantization of generally covariant systems, in "Highlights in Gravitation and Cosmology: Proceedings of the International Conference" (eds. B. R. Iyer, A. Kembhavi, J. V. Narlikar and C. V. Vishveshwara), Cambridge University Press, Cambridge, (1988), 93-120. [21] C. Lanczos, "The Variational Principles of Mechanics," Fourth edition, Mathematical Expositions, No. 4, University of Toronto Press, Toronto, Ont., 1970. [22] M. Leok, "Foundations of Computational Geometric Mechanics," Dissertation (Ph.D.), California Institute of Technology. Available from: http://resolver.caltech.edu/CaltechETD:etd-03022004-000251. [23] J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity," Prentice-Hall, Englewood Cliffs, New Jersey, 1983. [24] J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505. [25] C. W. Misner, K. Thorne and J. A. Wheeler, "Gravitation," W. H. Freeman, San Francisco, 1973. [26] J. D. Norton, General covariance and the foundations of general relativity: Eight decades of dispute, Rep. Prog. Phys., 56 (1993), 791-858. doi: 10.1088/0034-4885/56/7/001. [27] P. J. Olver, "Applications of Lie Groups to Differential Equations," Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986. [28] E. J. Post, "Formal Structure of Electromagnetics: General Covariance and Electromagnetics," Dover, New York, 2007. [29] E. C. G. Stückelberg, Théorie de la radiation de photons de masse arbitrairement petite, Helv. Phys. Acta, 30 (1957), 209-215. [30] C. Tejero Prieto, Variational formulation of Chern-Simons theory for arbitrary Lie groups, J. Geom. Phys., 50 (2004), 138-161. doi: 10.1016/j.geomphys.2003.11.005. [31] R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. (2), 101 (1956), 1597-1607. doi: 10.1103/PhysRev.101.1597.

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##### References:
 [1] J. L. Anderson, "Principles of Relativity Physics,'', Academic Press, New York, 1967. [2] N. Banerjee, R. Banerjee and S. Ghosh, Quantisation of second class systems in the Batalin-Tyutin formalism, Annals of Physics, 241 (1995), 237-257. doi: 10.1006/aphy.1995.1062. [3] M. Castrillón López, M. J. Gotay and J. E. Marsden, Parametrization and stress-energy-momentum tensors in metric field theories, J. Phys. A, 41 (2008), 344002, 10 pp. [4] M. Castrillón López, M. J. Gotay and J. E. Marsden, Concatenating variational principles and the kinetic stress-energy-momentum tensor, in "Variations, Geometry and Physics" (eds. O. Krupková and D. J. Saunders), Nova Science Publishers, New York, (2009), 117-128. [5] M. Castrillón López and J. Muñoz Masqué, The geometry of the bundle of connections, Math. Z., 236 (2001), 797-811. doi: 10.1007/PL00004852. [6] P. A. M. Dirac, The Hamiltonian form of field dynamics, Can. J. Math., 3 (1951), 1-23. doi: 10.4153/CJM-1951-001-2. [7] P. A. M. Dirac, "Lectures on Quantum Mechanics," Academic Press, New York, 1964. [8] D. Freed, Classical Chern-Simons theory. I, Adv. Math., 113 (1995), 237-303. doi: 10.1006/aima.1995.1039. [9] D. Freed, Remarks on Chern-Simons theory, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 221-254. [10] P. L. García, Gauge algebras, curvature and symplectic structure, J. Diff. Geom., 12 (1977), 209-227. [11] M. J. Gotay, An exterior differential systems approach to the Cartan form, in "Symplectic Geometry and Mathematical Physics" (eds. P. Donato, C. Duval, J. Elhadad and G. M. Tuynman) (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, (1991), 160-188. [12] M. J. Gotay and J. E. Marsden, Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula, in "Mathematical Aspects of Classical Field Theory" (Seattle, WA, 1991), Contemp. Math., 132, Amer. Math. Soc., (1992), 367-392. [13] M. J. Gotay and J. E. Marsden, Momentum maps and classical fields, work in progress. [14] C. Isham and K. Kuchař, Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories, Annals of Physics, 164 (1985), 288-315. doi: 10.1016/0003-4916(85)90018-1. [15] B. Janssens, Bundles with a lift of infinitesimal diffeomorphisms, arXiv:0911.3532, 39 pp. [16] I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, 1993. [17] E. Kretschmann, Über den physikalischen Sinn der Relitivitätstheorie. A. Einsteins neue und seine ursprüngliche Relitivitätstheorie, Ann. Phys., Leipzig, 53 (1917), 575-614. [18] D. Krupka, O. Krupková and D. Saunders, The Cartan form and its generalizations in the calculus of variations, Int. J. of Geometric Methods in Modern Physics, 7 (2010), 631-654. doi: 10.1142/S0219887810004488. [19] K. Kuchař, Canonical quantization of gravity, in "Relativity, Astrophysics and Cosmology" (ed. W. Israel), Reidel, Dordrecht, (1973), 237-288. [20] K. Kuchař, Canonical quantization of generally covariant systems, in "Highlights in Gravitation and Cosmology: Proceedings of the International Conference" (eds. B. R. Iyer, A. Kembhavi, J. V. Narlikar and C. V. Vishveshwara), Cambridge University Press, Cambridge, (1988), 93-120. [21] C. Lanczos, "The Variational Principles of Mechanics," Fourth edition, Mathematical Expositions, No. 4, University of Toronto Press, Toronto, Ont., 1970. [22] M. Leok, "Foundations of Computational Geometric Mechanics," Dissertation (Ph.D.), California Institute of Technology. Available from: http://resolver.caltech.edu/CaltechETD:etd-03022004-000251. [23] J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity," Prentice-Hall, Englewood Cliffs, New Jersey, 1983. [24] J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505. [25] C. W. Misner, K. Thorne and J. A. Wheeler, "Gravitation," W. H. Freeman, San Francisco, 1973. [26] J. D. Norton, General covariance and the foundations of general relativity: Eight decades of dispute, Rep. Prog. Phys., 56 (1993), 791-858. doi: 10.1088/0034-4885/56/7/001. [27] P. J. Olver, "Applications of Lie Groups to Differential Equations," Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986. [28] E. J. Post, "Formal Structure of Electromagnetics: General Covariance and Electromagnetics," Dover, New York, 2007. [29] E. C. G. Stückelberg, Théorie de la radiation de photons de masse arbitrairement petite, Helv. Phys. Acta, 30 (1957), 209-215. [30] C. Tejero Prieto, Variational formulation of Chern-Simons theory for arbitrary Lie groups, J. Geom. Phys., 50 (2004), 138-161. doi: 10.1016/j.geomphys.2003.11.005. [31] R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. (2), 101 (1956), 1597-1607. doi: 10.1103/PhysRev.101.1597.
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