Abstract
We review in detail the Hamiltonian dynamics for constrained systems. Emphasis is put on the total Hamiltonian system rather than on the extended Hamiltonian system. We provide a systematic analysis of (global and local) symmetries in total Hamiltonian systems. In particular, in analogy to total Hamiltonians, we introduce the notion of total Noether charges. Grassmannian degrees of freedom are also addressed in detail.
Similar content being viewed by others
References
H. Weyl, Symmetry (Princeton University Press, Princeton, 1952)
S. Chandrasekhar, Truth and Beauty: Aesthetics and Motivations in Science (University of Chicago Press, Chicago, 1990)
L. O’Raifeartaigh, The Dawning of Gauge Theory (Princeton University Press, Princeton, 1997)
P.A.M. Dirac, Generalized Hamiltonian dynamics. Can. J. Math. 2, 129 (1950)
P.A.M. Dirac, The Hamiltonian form of field dynamics. Can. J. Math. 3, 1 (1951)
P.A.M. Dirac, Generalized Hamiltonian dynamics and the theory of gravitation in Hamiltonian form. Proc. R. Soc. Lond. A 246, 326–333 (1958)
P.A.M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, Yeshiva, 1964)
A. Hanson, T. Regge, C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale dei Lincei, Rome, 1976)
D.M. Gitman, I.V. Tyutin, Quantization of Fields with Constraints (Springer, Berlin, 1990)
M. Henneaux, C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, 1992)
K.B. Marathe, Constrained Hamiltonian Systems, Lecture Notes in Physics, vol. 180 (Springer, Berlin, 1983)
J. Govaerts, Hamiltonian Quantisation and Constrained Dynamics (Leuven University, Leuven, 1991)
M. Blagojevic, Gravitation and Gauge Symmetries (Institute of Physics Publishing, London, 2001)
P. Spindel, Mécanique Analytique (Scientifiques GB, Paris, 2002)
G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory (World Scientific, Singapore, 1995)
I.L. Buchbinder, S.M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity: A Walk through Superspace (Institute of Physics Publishing, London, 1998)
G. Barnich, F. Brandt, M. Henneaux, Local BRST cohomology in gauge theories. Phys. Rep. 338, 439 (2000). hep-th/0002245
M. Henneaux, C. Teitelboim, J. Zanelli, Gauge invariance and degree of freedom count. Nucl. Phys. B 332, 169 (1990)
J.M. Souriau, Structure des Systèmes Dynamiques (Dunod, Paris, 1970)
J. Butterfield, On symplectic reduction in classical mechanics, in Philosophy of Physics, ed. by J. Butterfield, J. Earman (North Holland, Amsterdam, 2006), p. 1. physics/0507194
A. Dresse, P. Gregoire, M. Henneaux, Path integral equivalence between the extended and nonextended Hamiltonian formalisms. Phys. Lett. B 245, 192 (1990)
J.-H. Park, Superfield theories and dual supermatrix models. J. High Energy Phys. 0309, 046 (2003). hep-th/0307060
L.D. Faddeev, V.N. Popov, Feynman diagrams for the Yang–Mills field. Phys. Lett. B 25, 29 (1967)
I.R. Klebanov, String theory in two-dimensions. hep-th/9108019
J. Conway, A Course in Functional Analysis (Springer, Berlin, 1990)
C. Becchi, A. Rouet, R. Stora, Renormalization of the Abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127 (1975)
C. Becchi, A. Rouet, R. Stora, Renormalization of gauge theories. Annals Phys. 98, 287 (1976)
I.V. Tyutin, Gauge invariance in field theory and statistical physics in operator formalism. Preprint LEBEDEV-75-39
S. Weinberg, The Quantum Theory of Fields, Modern Applications, vol. 2 (Cambridge University Press, Cambridge, 1996)
M. Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom. Phys. Rep. 126, 1 (1985)
L. Baulieu, Perturbative gauge theories. Phys. Rep. 129, 1 (1985)
I.A. Batalin, G.A. Vilkovisky, Quantization of gauge theories with linearly dependent generators. Phys. Rev. D 28, 2567 (1983) [Erratum: Phys. Rev. D 30 (1984) 508]
I.A. Batalin, G.A. Vilkovisky, Closure of the gauge algebra, generalized Lie equations and Feynman rules. Nucl. Phys. B 234, 106 (1984)
M. Henneaux, Lectures on the antifield-BRST formalism for gauge theories. Nucl. Phys. Proc. Suppl. A 18, 47 (1990)
J. Gomis, J. Paris, S. Samuel, Antibracket, antifields and gauge theory quantization. Phys. Rep. 259, 1 (1995). hep-th/9412228
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bekaert, X., Park, JH. Symmetries and dynamics in constrained systems. Eur. Phys. J. C 61, 141–183 (2009). https://doi.org/10.1140/epjc/s10052-009-0973-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjc/s10052-009-0973-7