Dirac bracket

The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac 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. More abstractly, the two-form implied from the Dirac bracket is the restriction of the symplectic form to the constraint surface in phase space. { f , g } D B = { f , g } P B − ∑ a , b { f , ϕ ~ a } P B M a b − 1 { ϕ ~ b , g } P B   , {displaystyle {f,g}_{DB}={f,g}_{PB}-sum _{a,b}{f,{ ilde {phi }}_{a}}_{PB}M_{ab}^{-1}{{ ilde {phi }}_{b},g}_{PB}~,} The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac 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. More abstractly, the two-form implied from the Dirac bracket is the restriction of the symplectic form to the constraint surface in phase space. 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.