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).   Google Scholar

[2]

N. Banerjee, R. Banerjee and S. Ghosh, Quantisation of second class systems in the Batalin-Tyutin formalism,, Annals of Physics, 241 (1995), 237.  doi: 10.1006/aphy.1995.1062.  Google Scholar

[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).   Google Scholar

[4]

M. Castrillón López, M. J. Gotay and J. E. Marsden, Concatenating variational principles and the kinetic stress-energy-momentum tensor,, in, (2009), 117.   Google Scholar

[5]

M. Castrillón López and J. Muñoz Masqué, The geometry of the bundle of connections,, Math. Z., 236 (2001), 797.  doi: 10.1007/PL00004852.  Google Scholar

[6]

P. A. M. Dirac, The Hamiltonian form of field dynamics,, Can. J. Math., 3 (1951), 1.  doi: 10.4153/CJM-1951-001-2.  Google Scholar

[7]

P. A. M. Dirac, "Lectures on Quantum Mechanics,", Academic Press, (1964).   Google Scholar

[8]

D. Freed, Classical Chern-Simons theory. I,, Adv. Math., 113 (1995), 237.  doi: 10.1006/aima.1995.1039.  Google Scholar

[9]

D. Freed, Remarks on Chern-Simons theory,, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 221.   Google Scholar

[10]

P. L. García, Gauge algebras, curvature and symplectic structure,, J. Diff. Geom., 12 (1977), 209.   Google Scholar

[11]

M. J. Gotay, An exterior differential systems approach to the Cartan form,, in, 99 (1991), 160.   Google Scholar

[12]

M. J. Gotay and J. E. Marsden, Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula,, in, 132 (1992), 367.   Google Scholar

[13]

M. J. Gotay and J. E. Marsden, Momentum maps and classical fields,, work in progress., ().   Google Scholar

[14]

C. Isham and K. Kuchař, Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories,, Annals of Physics, 164 (1985), 288.  doi: 10.1016/0003-4916(85)90018-1.  Google Scholar

[15]

B. Janssens, Bundles with a lift of infinitesimal diffeomorphisms,, \arXiv{0911.3532}, ().   Google Scholar

[16]

I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).   Google Scholar

[17]

E. Kretschmann, Über den physikalischen Sinn der Relitivitätstheorie. A. Einsteins neue und seine ursprüngliche Relitivitätstheorie,, Ann. Phys., 53 (1917), 575.   Google Scholar

[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.  doi: 10.1142/S0219887810004488.  Google Scholar

[19]

K. Kuchař, Canonical quantization of gravity,, in, (1973), 237.   Google Scholar

[20]

K. Kuchař, Canonical quantization of generally covariant systems,, in, (1988), 93.   Google Scholar

[21]

C. Lanczos, "The Variational Principles of Mechanics," Fourth edition,, Mathematical Expositions, (1970).   Google Scholar

[22]

M. Leok, "Foundations of Computational Geometric Mechanics,", Dissertation (Ph.D.), (): 03022004.   Google Scholar

[23]

J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity,", Prentice-Hall, (1983).   Google Scholar

[24]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs,, Comm. Math. Phys., 199 (1998), 351.  doi: 10.1007/s002200050505.  Google Scholar

[25]

C. W. Misner, K. Thorne and J. A. Wheeler, "Gravitation,", W. H. Freeman, (1973).   Google Scholar

[26]

J. D. Norton, General covariance and the foundations of general relativity: Eight decades of dispute,, Rep. Prog. Phys., 56 (1993), 791.  doi: 10.1088/0034-4885/56/7/001.  Google Scholar

[27]

P. J. Olver, "Applications of Lie Groups to Differential Equations,", Graduate Texts in Mathematics, 107 (1986).   Google Scholar

[28]

E. J. Post, "Formal Structure of Electromagnetics: General Covariance and Electromagnetics,", Dover, (2007).   Google Scholar

[29]

E. C. G. Stückelberg, Théorie de la radiation de photons de masse arbitrairement petite,, Helv. Phys. Acta, 30 (1957), 209.   Google Scholar

[30]

C. Tejero Prieto, Variational formulation of Chern-Simons theory for arbitrary Lie groups,, J. Geom. Phys., 50 (2004), 138.  doi: 10.1016/j.geomphys.2003.11.005.  Google Scholar

[31]

R. Utiyama, Invariant theoretical interpretation of interaction,, Phys. Rev. (2), 101 (1956), 1597.  doi: 10.1103/PhysRev.101.1597.  Google Scholar

show all references

References:
[1]

J. L. Anderson, "Principles of Relativity Physics,'', Academic Press,, New York, (1967).   Google Scholar

[2]

N. Banerjee, R. Banerjee and S. Ghosh, Quantisation of second class systems in the Batalin-Tyutin formalism,, Annals of Physics, 241 (1995), 237.  doi: 10.1006/aphy.1995.1062.  Google Scholar

[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).   Google Scholar

[4]

M. Castrillón López, M. J. Gotay and J. E. Marsden, Concatenating variational principles and the kinetic stress-energy-momentum tensor,, in, (2009), 117.   Google Scholar

[5]

M. Castrillón López and J. Muñoz Masqué, The geometry of the bundle of connections,, Math. Z., 236 (2001), 797.  doi: 10.1007/PL00004852.  Google Scholar

[6]

P. A. M. Dirac, The Hamiltonian form of field dynamics,, Can. J. Math., 3 (1951), 1.  doi: 10.4153/CJM-1951-001-2.  Google Scholar

[7]

P. A. M. Dirac, "Lectures on Quantum Mechanics,", Academic Press, (1964).   Google Scholar

[8]

D. Freed, Classical Chern-Simons theory. I,, Adv. Math., 113 (1995), 237.  doi: 10.1006/aima.1995.1039.  Google Scholar

[9]

D. Freed, Remarks on Chern-Simons theory,, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 221.   Google Scholar

[10]

P. L. García, Gauge algebras, curvature and symplectic structure,, J. Diff. Geom., 12 (1977), 209.   Google Scholar

[11]

M. J. Gotay, An exterior differential systems approach to the Cartan form,, in, 99 (1991), 160.   Google Scholar

[12]

M. J. Gotay and J. E. Marsden, Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula,, in, 132 (1992), 367.   Google Scholar

[13]

M. J. Gotay and J. E. Marsden, Momentum maps and classical fields,, work in progress., ().   Google Scholar

[14]

C. Isham and K. Kuchař, Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories,, Annals of Physics, 164 (1985), 288.  doi: 10.1016/0003-4916(85)90018-1.  Google Scholar

[15]

B. Janssens, Bundles with a lift of infinitesimal diffeomorphisms,, \arXiv{0911.3532}, ().   Google Scholar

[16]

I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).   Google Scholar

[17]

E. Kretschmann, Über den physikalischen Sinn der Relitivitätstheorie. A. Einsteins neue und seine ursprüngliche Relitivitätstheorie,, Ann. Phys., 53 (1917), 575.   Google Scholar

[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.  doi: 10.1142/S0219887810004488.  Google Scholar

[19]

K. Kuchař, Canonical quantization of gravity,, in, (1973), 237.   Google Scholar

[20]

K. Kuchař, Canonical quantization of generally covariant systems,, in, (1988), 93.   Google Scholar

[21]

C. Lanczos, "The Variational Principles of Mechanics," Fourth edition,, Mathematical Expositions, (1970).   Google Scholar

[22]

M. Leok, "Foundations of Computational Geometric Mechanics,", Dissertation (Ph.D.), (): 03022004.   Google Scholar

[23]

J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity,", Prentice-Hall, (1983).   Google Scholar

[24]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs,, Comm. Math. Phys., 199 (1998), 351.  doi: 10.1007/s002200050505.  Google Scholar

[25]

C. W. Misner, K. Thorne and J. A. Wheeler, "Gravitation,", W. H. Freeman, (1973).   Google Scholar

[26]

J. D. Norton, General covariance and the foundations of general relativity: Eight decades of dispute,, Rep. Prog. Phys., 56 (1993), 791.  doi: 10.1088/0034-4885/56/7/001.  Google Scholar

[27]

P. J. Olver, "Applications of Lie Groups to Differential Equations,", Graduate Texts in Mathematics, 107 (1986).   Google Scholar

[28]

E. J. Post, "Formal Structure of Electromagnetics: General Covariance and Electromagnetics,", Dover, (2007).   Google Scholar

[29]

E. C. G. Stückelberg, Théorie de la radiation de photons de masse arbitrairement petite,, Helv. Phys. Acta, 30 (1957), 209.   Google Scholar

[30]

C. Tejero Prieto, Variational formulation of Chern-Simons theory for arbitrary Lie groups,, J. Geom. Phys., 50 (2004), 138.  doi: 10.1016/j.geomphys.2003.11.005.  Google Scholar

[31]

R. Utiyama, Invariant theoretical interpretation of interaction,, Phys. Rev. (2), 101 (1956), 1597.  doi: 10.1103/PhysRev.101.1597.  Google Scholar

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