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, when constraints and thus more apparent than dynamical variables are at hand. 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.

Read more about Dirac BracketInadequacy of The Standard Hamiltonian Procedure, Generalized Hamiltonian Procedure, The Dirac Bracket, Illustration On The Example Provided

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Dirac Bracket - Illustration On The Example Provided
... Thus, the Dirac brackets are defined to be If one always uses the Dirac bracket instead of the Poisson bracket then there is no issue about the order of applying constraints and evaluating ... This means that one can just use the naive Hamiltonian with Dirac brackets, and get the correct equations of motion, which one can easily confirm ... To quantize the system, the Dirac brackets between all of the phase space variables are needed ...