Concept
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. The standard development of Hamiltonian mechanics is inadequate in several specific situations: When the Lagrangian is at most linear in the velocity of at least one coordinate; in which case, the definition of the canonical momentum leads to a constraint. This is the most frequent reason to resort to Dirac brackets. For instance, the Lagrangian (density) for any fermion is of this form. When there are gauge (or other unphysical) degrees of freedom which need to be fixed. When there are any other constraints that one wishes to impose in phase space. 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 is the vector potential for the magnetic field, ; c is the speed of light in vacuum; and V() is an arbitrary external scalar potential; one could easily take it to be quadratic in x and y, without loss of generality. We use as our vector potential; this corresponds to a uniform and constant magnetic field B in the z direction.