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.
[2] More abstractly, the two-form implied from the Dirac bracket is the restriction of the symplectic form to the constraint surface in phase space.
[3] This article assumes familiarity with the standard Lagrangian and Hamiltonian formalisms, and their connection to canonical quantization.
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.
We use as our vector potential; this corresponds to a uniform and constant magnetic field B in the z direction.
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.
The new procedure works as follows, start with a Lagrangian and define the canonical momenta in the usual way.
Dirac argues that we should generalize the Hamiltonian (somewhat analogously to the method of Lagrange multipliers) to where the cj are not constants but functions of the coordinates and momenta.
To further illuminate the cj, consider how one gets the equations of motion from the naive Hamiltonian in the standard procedure.
One expands the variation of the Hamiltonian out in two ways and sets them equal (using a somewhat abbreviated notation with suppressed indices and sums): where the second equality holds after simplifying with the Euler-Lagrange equations of motion and the definition of canonical momentum.
In the present context, one cannot simply set the coefficients of δq and δp separately to zero, since the variations are somewhat restricted by the constraints.
The equations of motion become more compact when using the Poisson bracket, since if f is some function of the coordinates and momenta then if one assumes that the Poisson bracket with the uk (functions of the velocity) exist; this causes no problems since the contribution weakly vanishes.
If, at the end of this process, the uk are not completely determined, then that means there are unphysical (gauge) degrees of freedom in the system.
We call a function f(q, p) of coordinates and momenta first class if its Poisson bracket with all of the constraints weakly vanishes, that is, for all j.
The first-class constraints are intimately connected with the unphysical degrees of freedom mentioned earlier.
Dirac further postulated that all secondary first-class constraints are generators of gauge transformations, which turns out to be false; however, typically one operates under the assumption that all first-class constraints generate gauge transformations when using this treatment.
For instance, consider second-class constraints ϕ1 and ϕ2 whose Poisson bracket is simply a constant, c, Now, suppose one wishes to employ canonical quantization, then the phase-space coordinates become operators whose commutators become iħ times their classical Poisson bracket.
Assuming there are no ordering issues that give rise to new quantum corrections, this implies that where the hats emphasize the fact that the constraints are on operators.
This example illustrates the need for some generalization of the Poisson bracket which respects the system's constraints, and which leads to a consistent quantization procedure.
It is straightforward to check that the above definition of the Dirac bracket satisfies all of the desired properties, and especially the last one, of vanishing for an argument which is a second-class constraint.
When applying canonical quantization on a constrained Hamiltonian system, the commutator of the operators is supplanted by iħ times their classical Dirac bracket.
Returning to the above example, the naive Hamiltonian and the two primary constraints are Therefore, the extended Hamiltonian can be written The next step is to apply the consistency conditions {Φj, H*}PB ≈ 0, which in this case become These are not secondary constraints, but conditions that fix u1 and u2.
Therefore, there are no secondary constraints and the arbitrary coefficients are completely determined, indicating that there are no unphysical degrees of freedom.
A simple calculation confirms that ϕ1 and ϕ2 are second class constraints since hence the matrix looks like which is easily inverted to where εab is the Levi-Civita symbol.
This means that one can just use the naive Hamiltonian with Dirac brackets, instead, to thus get the correct equations of motion, which one can easily confirm on the above ones.
The nonvanishing Dirac brackets for this system are while the cross-terms vanish, and Therefore, the correct implementation of canonical quantization dictates the commutation relations, with the cross terms vanishing, and This example has a nonvanishing commutator between ∧x and ∧y, which means this structure specifies a noncommutative geometry.
The Dirac brackets add simplicity and elegance, at the cost of excessive (constrained) phase-space variables.
For example, for free motion on a circle, n = 1, for x1 ≡ z and eliminating x2 from the circle constraint yields the unconstrained with equations of motion an oscillation; whereas the equivalent constrained system with H = p2/2 = E yields whence, instantly, virtually by inspection, oscillation for both variables,