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Covariant phase space, constraints, gauge and the Peierls formula

Igor Khavkine

Department of Mathematics, Trento University, Via Sommarive, 14-38123 Povo (TN), Italy

TIFPA-INFN, Trento, Italy

https://doi.org/10.1142/S0217751X14300099Cited by:24

Abstract

It is well known that both the symplectic structure and the Poisson brackets of classical field theory can be constructed directly from the Lagrangian in a covariant way, without passing through the noncovariant canonical Hamiltonian formalism. This is true even in the presence of constraints and gauge symmetries. These constructions go under the names of the covariant phase space formalism and the Peierls bracket. We review both of them, paying more careful attention, than usual, to the precise mathematical hypotheses that they require, illustrating them in examples. Also an extensive historical overview of the development of these constructions is provided. The novel aspect of our presentation is a significant expansion and generalization of an elegant and quite recent argument by Forger and Romero showing the equivalence between the resulting symplectic and Poisson structures without passing through the canonical Hamiltonian formalism as an intermediary. We generalize it to cover theories with constraints and gauge symmetries and formulate precise sufficient conditions under which the argument holds. These conditions include a local condition on the equations of motion that we call hyperbolizability, and some global conditions of cohomological nature. The details of our presentation may shed some light on subtle questions related to the Poisson structure of gauge theories and their quantization.

Keywords:

PACS: 02.30.Jr, 03.50.-z, 04.20.Fy, 11.15.-q

References

P. A. M. Dirac , Lectures on Quantum Mechanics ( Dover Publications , 2001 ) . Google Scholar
M. Henneaux and C. Teitelboim , Quantization of Gauge Systems ( Princeton University Press , 1994 ) . Google Scholar
J. Lee and R. M. Wald , J. Math. Phys. 31 , 725 ( 1990 ) . Crossref, ISI, ADS, Google Scholar
Č. Crnković and E. Witten, Three Hundred Years of Gravitation, eds. S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1987) pp. 676–684. Google Scholar
G. J. Zuckerman, Mathematical Aspects of String Theory, ed. S. T. Yau (World Scientific, Singapore, 1987) pp. 259–284. Link, Google Scholar
R. E. Peierls , Proc. R. Soc. Lond. A 214 , 143 ( 1952 ) . Crossref, ISI, ADS, Google Scholar
B. DeWitt , The Global Approach to Quantum Field Theory I & II ( Oxford University Press , 2003 ) . Google Scholar
G. Barnich , M. Henneaux and C. Schomblond , Phys. Rev. D 44 , R939 ( 1991 ) . Crossref, ISI, ADS, Google Scholar
M. Henneaux , Phys. Lett. B 238 , 299 ( 1990 ) . Crossref, ISI, ADS, Google Scholar
M. Forger and S. V. Romero , Commun. Math. Phys. 256 , 375 ( 2005 ) , arXiv:math-ph/0408008 . Crossref, ISI, ADS, Google Scholar
E. Binz , J. Śniatycki and H. Fischer , Geometry of Classical Fields , Notas de Matemática 154 ( North-Holland , 1988 ) . Google Scholar
K. Fredenhagen and K. Rejzner , Commun. Math. Phys. 314 , 93 ( 2012 ) , arXiv:1101.5112v1 . Crossref, ISI, ADS, Google Scholar
K. Rejzner, Batalin-Vilkovisky formalism in locally covariant field theory, Ph.D. thesis, Hamburg (2011) . Google Scholar
R. Brunetti, K. Fredenhagen and P. L. Ribeiro, Algebraic structure of classical field theory I: Kinematics and linearized dynamics for real scalar fields , arXiv:1209.2148 . Google Scholar
E. T. Whittaker , A Treatise on the Analytical Dynamics of Particles and Rigid Bodies , 1st edn. ( Cambridge University Press , 1904 ) . Google Scholar
H. Goldstein , Classical Mechanics , 2nd edn. ( Addison-Wesley , Reading, MA , 1980 ) . Google Scholar
R. Abraham and J. E. Marsden , Foundations of Mechanics ( Benjamin-Cummings , Reading , 1978 ) . Google Scholar
V. I. Arnold , Mathematical Methods of Classical Mechanics , 2nd edn. , Graduate Texts in Mathematics 60 ( Springer , 1997 ) . Google Scholar
J. M. Souriau , Structure des Systèmes Dynamiques ( Dunod , Paris , 1970 ) . Google Scholar
L. Rosenfeld , Z. Phys. 65 , 589 ( 1930 ) . Crossref, ADS, Google Scholar
J. L. Lagrange , Mémoires de Classe des Sciences Mathématiques et Physiques de l'Institut National de France 1 ( 1808 ) . Google Scholar
V. Glaser , H. Lehmann and W. Zimmermann , Il Nuovo Cimento 6 , 1122 ( 1957 ) . Crossref, ISI, ADS, Google Scholar
I. E. Segal , J. Math. Phys. 1 , 468 ( 1960 ) . Crossref, ISI, ADS, Google Scholar
I. E. Segal , Mathematical Problems of Relativistic Physics , Lectures in Applied Mathematics 1960 ( American Mathematical Society , Providence, RI , 1963 ) . Google Scholar
I. E. Segal, Application of Mathematics to Problems in Theoretical Physics, ed. F. Lurçat (Gordon and Breach, New York, 1967) pp. 107–170. Google Scholar
B. S. DeWitt, Relativity, Groups and Topology, eds. C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1964) pp. 587–820. Google Scholar
O. Steinmann , Perturbation Expansions in Axiomatic Field Theory , Lecture Notes in Physics 11 ( Springer-Verlag , Berlin , 1971 ) . Google Scholar
D. M. Marolf, Green's bracket algebras and their quantization, Ph.D. thesis, The University of Texas at Austin (1992) . Google Scholar
D. Marolf , Ann. Phys. 236 , 374 ( 1994 ) , arXiv:hep-th/9308141 . Crossref, ISI, ADS, Google Scholar
D. Marolf , Ann. Phys. 236 , 392 ( 1994 ) , arXiv:hep-th/9308150 . Crossref, ISI, ADS, Google Scholar
M. Duetsch and K. Fredenhagen , Commun. Math. Phys. 243 , 275 ( 2003 ) , arXiv:hep-th/0211242 . Crossref, ISI, ADS, Google Scholar
F. Brennecke and M. Duetsch , Rev. Math. Phys. 20 , 119 ( 2008 ) , arXiv:0705.3160 . Link, ISI, ADS, Google Scholar
K. Rejzner , Rev. Math. Phys. 23 , 1009 ( 2011 ) , arXiv:1101.5126 . Link, ISI, ADS, Google Scholar
T.-P. Hack and A. Schenkel , Gen. Relativ. Gravit. 45 , 877 ( 2013 ) , arXiv:1205.3484 . Crossref, ISI, ADS, Google Scholar
P. Bergmann and R. Schiller , Phys. Rev. 89 , 4 ( 1953 ) . Crossref, ISI, ADS, Google Scholar
A. Komar , Phys. Rev. 134 , B1430 ( 1964 ) . Crossref, ISI, ADS, Google Scholar
A. Ashtekar and A. Magnon , Proc. R. Soc. Lond. A 346 , 375 ( 1975 ) . Crossref, ISI, ADS, Google Scholar
A. Ashtekar, private communication (2012) . Google Scholar
A. Ashtekar , Phys. Rev. Lett. 46 , 573 ( 1981 ) . Crossref, ISI, ADS, Google Scholar
J. L. Friedman , Commun. Math. Phys. 62 , 247 ( 1978 ) . Crossref, ISI, ADS, Google Scholar
J. L. Friedman and B. F. Schutz , Astrophys. J. 221 , 937 ( 1978 ) . Crossref, ISI, ADS, Google Scholar
J. L. Friedman, private communication (2012) . Google Scholar
A. Trautman, General Relativity, Papers in Honour of J. L. Synge (Clarenden Press, Oxford, 1972) pp. 85–99. Google Scholar
E. Noether , Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse 1918 , 235 ( 1918 ) . Google Scholar
P. Dedecker, Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Mathematics 570, eds. K. Bleuler and A. Reetz (Springer, 1977) pp. 395–456. Crossref, Google Scholar
F. Takens, Geometry and Topology, Lecture Notes in Mathematics 597, eds. J. Palis and M. do Carmo (Springer, 1977) pp. 581–604. Crossref, Google Scholar
H. Goldschmidt and S. Sternberg , Ann. Inst. Fourier 23 , 203 ( 1973 ) . Crossref, Google Scholar
M. Henneaux , Ann. Phys. 140 , 45 ( 1982 ) . Crossref, ISI, ADS, Google Scholar
M. Henneaux, private communication (2013) . Google Scholar
E. Witten , Nucl. Phys. B 276 , 291 ( 1986 ) . Crossref, ISI, ADS, Google Scholar
S. Sternberg, Differential Geometrical Methods in Mathematical Physics II, Lecture Notes in Mathematics 676, eds. K. Bleuler, A. Reetz and H. Petry (Springer, 1978) pp. 399–407. Crossref, Google Scholar
P. Deligne and D. S. Freed, Quantum Fields and Strings: A Course for Mathematicians, eds. P. Deligneet al. (AMS, Providence, RI, 1999) pp. 137–225. Google Scholar
Č. Crnković , Class. Quantum Grav. 5 , 1557 ( 1988 ) . Crossref, ISI, ADS, Google Scholar
A. Ashtekar, L. Bombelli and O. Reula, Mechanics, Analysis and Geometry: 200 Years after Lagrange, eds. M. Francaviglia and D. Holm (North-Holland, Amsterdam, 1991) pp. 417–450. Crossref, Google Scholar
S. Hollands and D. Marolf , Class. Quantum Grav. 24 , 2301 ( 2007 ) , arXiv:gr-qc/0611044 . Crossref, ISI, ADS, Google Scholar
G. Barnich and F. Brandt , Nucl. Phys. B 633 , 3 ( 2002 ) , arXiv:hep-th/0111246 . Crossref, ISI, ADS, Google Scholar
P. Bergmann et al. , Phys. Rev. 103 , 807 ( 1956 ) . Crossref, ISI, ADS, Google Scholar
C. Baer , N. Ginoux and F. Pfaeffle , Wave Equations on Lorentzian Manifolds and Quantization , ESI Lectures in Mathematics and Physics 2 ( European Mathematical Society , 2007 ) . Crossref, Google Scholar
C. Baer, Green-hyperbolic operators on globally hyperbolic spacetimes , arXiv:1310.0738 . Google Scholar
I. Khavkine, Characteristics, conal geometry and causality in locally covariant field theory , arXiv:1211.1914 . Google Scholar
N. Ginoux , Quantum Field Theory on Curved Spacetimes: Concepts and Methods , Lecture Notes in Physics 786 , eds. C. Baer and K. Fredenhagen ( Springer , 2009 ) . Crossref, Google Scholar
H. Ringström , The Cauchy Problem in General Relativity , ESI Lectures in Mathematics and Physics 6 ( European Mathematical Society , 2009 ) . Crossref, Google Scholar
S. Waldmann, Geometric wave equations , arXiv:1208.4706 . Google Scholar
R. M. Wald , Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics , 1st edn. , Chicago Lectures in Physics ( University of Chicago Press , 1994 ) . Google Scholar
K. Sanders , Class. Quantum Grav. 30 , 115014 ( 2012 ) , arXiv:1211.2469 . Crossref, ISI, ADS, Google Scholar
H. Goldschmidt , J. Diff. Geom. 1 , 269 ( 1967 ) . Crossref, Google Scholar
H. Goldschmidt , Ann. Math. 86 , 246 ( 1967 ) . Crossref, ISI, Google Scholar
J.-F. Pommaret , Systems of Partial Differential Equations and Lie Pseudogroups ( Gordon and Breach , New York , 1978 ) . Google Scholar
R. S. Palais , Seminar on the Atiyah-Singer Index Theorem , Annals of Mathematics Studies 57 ( Princeton University Press , 1965 ) . Crossref, Google Scholar
R. Courant and D. Hilbert , Methods of Mathematical Physics 1 ( John Wiley & Sons , 2008 ) . Google Scholar
I. M. Anderson, The variational bicomplex (1989), informally distributed book draft (unpublished) . Google Scholar
G. Giachetta , L. Mangiarotti and G. Sardanashvily , J. Math. Phys. 42 , 4272 ( 2001 ) , arXiv:math/0006074 . Crossref, ISI, ADS, Google Scholar
I. M. Anderson, Mathematical Aspects of Classical Field Theory, Contemporary Mathematics 132, eds. M. J. Gotay, J. E. Marsden and V. Moncrief (American Mathematical Society, Providence, 1992) pp. 51–73. Crossref, Google Scholar
I. Khavkine , J. Math. Phys. 54 , 111502 ( 2013 ) , arXiv:1210.0802 . Crossref, ISI, ADS, Google Scholar
R. Geroch , Partial differential equations of physics , Proc. Forty-Sixth Scottish Summer School in Physics , eds. G. S. Hall and J. R. Pulham ( SUSSP Publications , Edinburgh , 1996 ) , arXiv:gr-qc/9602055 . Google Scholar
K. Sanders, C. Dappiaggi and T.-P. Hack, Electromagnetism, local covariance, the Aharonov–Bohm effect and Gauss' law , arXiv:1211.6420 . Google Scholar
C. Dappiaggi , Rev. Math. Phys. 23 , 1035 ( 2011 ) . Link, ISI, ADS, Google Scholar
R. M. Wald , General Relativity , 1st edn. ( University of Chicago Press , Chicago , 1984 ) . Crossref, Google Scholar
C. J. Fewster and D. S. Hunt , Rev. Math. Phys. 25 , 1330003 ( 2013 ) , arXiv:1203.0261 . Link, ISI, ADS, Google Scholar
E. Calabi, Differential Geometry, Proceedings of Symposia in Pure Mathematics 3, ed. C. B. Allendoerfer (AMS, Providence, 1961) pp. 155–180. Crossref, Google Scholar
I. Khavkine, Cohomology with causally restricted supports (in preparation) . Google Scholar
G. Grubb , Distributions and Operators , Graduate Texts in Mathematics 252 ( Springer , Berlin , 2009 ) . Google Scholar
P. J. Olver , Applications of Lie Groups to Differential Equations , 2nd edn. , Graduate Texts in Mathematics 107 ( Springer-Verlag , New York , 1993 ) . Crossref, Google Scholar
I. Kolař , P. W. Michor and J. Slovák , Natural Operations in Differential Geometry ( Springer , 1993 ) . Crossref, Google Scholar
D. Spring , Convex Integration Theory: Solutions to the H-Principle in Geometry and Topology , Monographs in Mathematics 92 ( Birkhäuser , 1998 ) . Crossref, Google Scholar
W. M. Seiler , Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra , Algorithms and Computation in Mathematics 24 ( Springer , 2010 ) . Crossref, Google Scholar
R. L. Bryant et al. , Exterior Differential Systems , Mathematical Sciences Research Institute Publications 18 ( Springer , 2011 ) . Google Scholar
A. M. Vinogradov and I. S. Krasilshchik (eds.) , Symmetries and Conservation Laws for Differential Equations of Mathematical Physics , Translations of Mathematical Monographs 182 ( American Mathematical Society , Providence, RI , 1999 ) . Google Scholar
N. N. Tarkhanov , Complexes of Differential Operators , Mathematics and Its Applications 340 ( Springer , Netherlands, Dordrecht , 1995 ) . Crossref, Google Scholar
L. Hörmander , The Analysis of Linear Partial Differential Operators III , Grundlehren Der Mathematischen Wissenschaften 274 ( Springer , 1985 ) . Google Scholar
R. T. Smith , Bull. Amer. Math. Soc. 82 , 297 ( 1976 ) . Crossref, ISI, Google Scholar
S. W. Hawking and G. F. R. Ellis , The Large Scale Structure of Space-Time ( Cambridge University Press , Cambridge , 1973 ) . Crossref, Google Scholar
R. Beig, Analytical and Numerical Approaches to Mathematical Relativity, Lecture Notes in Physics 692, eds. J. Frauendiener, D. Giulini and V. Perlick (Springer, Berlin, 2006) pp. 101–116. Crossref, Google Scholar
J. D. Lawson , Forum Mathematicum 1 , 273 ( 1989 ) . Crossref, Google Scholar
K. H. Neeb , Inventiones Mathematicae 104 , 467 ( 1991 ) . Crossref, ISI, ADS, Google Scholar
R. Penrose , Techniques in Differential Topology in Relativity , CBMS-NSF Regional Conference Series in Applied Mathematics 7 ( Society for Industrial and Applied Mathematics , Philadelphia, PA , 1972 ) . Crossref, Google Scholar
R. Geroch , J. Math. Phys. 11 , 437 ( 1970 ) . Crossref, ISI, ADS, Google Scholar
A. García-Parrado and J. M. M. Senovilla , Class. Quantum Grav. 22 , R1 (2005 , arXiv:gr-qc/0501069 . Google Scholar
U. Bannier , Commun. Math. Phys. 118 , 163 ( 1988 ) . Crossref, ISI, ADS, Google Scholar
M. Rainer , J. Math. Phys. 40 , 6589 ( 1999 ) , arXiv:gr-qc/9905106 . Crossref, ISI, ADS, Google Scholar
M. Rainer , Class. Quantum Grav. 17 , 1935 ( 2000 ) , arXiv:gr-qc/9911076 . Crossref, ISI, ADS, Google Scholar
R. T. Rockafellar , Convex Analysis , Princeton Mathematical Series 28 ( Princeton University Press , 1996 ) . Google Scholar
A. N. Bernal and M. Sánchez , Commun. Math. Phys. 257 , 43 ( 2005 ) , arXiv:gr-qc/0401112 . Crossref, ISI, ADS, Google Scholar
A. Bernal and M. Sánchez , Lett. Math. Phys. 77 , 183 ( 2006 ) , arXiv:gr-qc/0512095 . Crossref, ISI, ADS, Google Scholar
A. Fathi and A. Siconolfi , Math. Proc. Cambr. Philos. Soc. 152 , 303 ( 2011 ) . Crossref, ISI, Google Scholar

Published: 19 February 2014

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History

Received 23 December 2013

Accepted 3 January 2014
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Covariant phase space, constraints, gauge and the Peierls formula

Abstract

References

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