Dirac bracket

The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac[1] to treat classical systems with second class constraints in Hamiltonian mechanics, and to thus allow them to undergo canonical quantization. It is an important part of Dirac's development of Hamiltonian mechanics to elegantly handle more general Lagrangians; specifically, when constraints are at hand, so that the number of apparent variables exceeds that of dynamical ones.[2] More abstractly, the two-form implied from the Dirac bracket is the restriction of the symplectic form to the constraint surface in phase space.[3]

This article assumes familiarity with the standard Lagrangian and Hamiltonian formalisms, and their connection to canonical quantization. Details of Dirac's modified Hamiltonian formalism are also summarized to put the Dirac bracket in context.

  1. ^ Dirac, P. A. M. (1950). "Generalized Hamiltonian dynamics". Canadian Journal of Mathematics. 2: 129–014. doi:10.4153/CJM-1950-012-1. S2CID 119748805.
  2. ^ Dirac, Paul A. M. (1964). Lectures on quantum mechanics. Belfer Graduate School of Science Monographs Series. Vol. 2. Belfer Graduate School of Science, New York. ISBN 9780486417134. MR 2220894.; Dover, ISBN 0486417131.
  3. ^ See pages 48-58 of Ch. 2 in Henneaux, Marc and Teitelboim, Claudio, Quantization of Gauge Systems. Princeton University Press, 1992. ISBN 0-691-08775-X