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.
Inadequacy of the standard Hamiltonian procedure
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.
Example of a Lagrangian linear in velocity
An example in classical mechanics is a particle with charge and mass confined to the - plane with a strong constant, homogeneous perpendicular magnetic field, so then pointing in the -direction with strength .
The Lagrangian for this system with an appropriate choice of parameters is
where is the vector potential for the magnetic field, ; is the speed of light in vacuum; and is an arbitrary external scalar potential; one could easily take it to be quadratic in and , without loss of generality. We use
as our vector potential; this corresponds to a uniform and constant magnetic field B in the z direction. Here, the hats indicate unit vectors. Later in the article, however, they are used to distinguish quantum mechanical operators from their classical analogs. The usage should be clear from the context.
Explicitly, the Lagrangian amounts to just
which leads to the equations of motion
For a harmonic potential, the gradient of amounts to just the coordinates, .
Now, in the limit of a very large magnetic field, . One may then drop the kinetic term to produce a simple approximate Lagrangian,
with first-order equations of motion
Note that this approximate Lagrangian is linear in the velocities, which is one of the conditions under which the standard Hamiltonian procedure breaks down. While this example has been motivated as an approximation, the Lagrangian under consideration is legitimate and leads to consistent equations of motion in the Lagrangian formalism.
Following the Hamiltonian procedure, however, the canonical momenta associated with the coordinates are now
which are unusual in that they are not invertible to the velocities; instead, they are constrained to be functions of the coordinates: the four phase-space variables are linearly dependent, so the variable basis is overcomplete.
A Legendre transformation then produces the Hamiltonian
Note that this "naive" Hamiltonian has no dependence on the momenta, which means that equations of motion (Hamilton's equations) are inconsistent.
The Hamiltonian procedure has broken down. One might try to fix the problem by eliminating two of the components of the -dimensional phase space, say and , down to a reduced phase space of dimensions, that is sometimes expressing the coordinates as momenta and sometimes as coordinates. However, this is neither a general nor rigorous solution. This gets to the heart of the matter: that the definition of the canonical momenta implies a constraint on phase space (between momenta and coordinates) that was never taken into account.
Generalized Hamiltonian procedure
In Lagrangian mechanics, if the system has holonomic constraints, then one generally adds Lagrange multipliers to the Lagrangian to account for them. The extra terms vanish when the constraints are satisfied, thereby forcing the path of stationary action to be on the constraint surface. In this case, going to the Hamiltonian formalism introduces a constraint on phase space in Hamiltonian mechanics, but the solution is similar.