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.

The standard development of Hamiltonian mechanics is inadequate in several specific situations:

An example in classical mechanics is a particle with charge q and mass m confined to the x - y plane with a strong constant, homogeneous perpendicular magnetic field, so then pointing in the z-direction with strength B.

The Lagrangian for this system with an appropriate choice of parameters is

where →A is the vector potential for the magnetic field, →B; c is the speed of light in vacuum; and V(→r) is an arbitrary external scalar potential; one could easily take it to be quadratic in x and y, without loss of generality. We use

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