leads to (note the equality in this case is not a weak one in that no constraint need be imposed after the Poisson bracket is worked out) (see Todorov, and Wong and Crater ) with the same defined above. Quantization In addition to replacing classical dynamical variables by their quantum counterparts, quantization of the constraint mechanics takes place by replacing the constraint on the dynamical variables with a restriction on the wave function , . The first set of equations for i = 1, 2 play the role for spinless particles that the two Dirac equations play for spin-one-half particles. The classical Poisson brackets are replaced by commutators Thus and we see in this case that the constraint formalism leads to the vanishing commutator of the wave operators for the two particles. This is the analogue of the claim stated earlier that the two Dirac operators commute with one another. Covariant elimination of the relative energy The vanishing of the above commutator ensures that the dynamics is independent of the relative time in the c.m. frame. In order to covariantly eliminate the relative energy, introduce the relative momentum defined by The above definition of the relative momentum forces the orthogonality of the total momentum and the relative momentum, , which follows from taking the scalar product of either equation with . From Eqs.() and (), this relative momentum can be written in terms of and as where are the projections of the momenta and along the direction of the total momentum . Subtracting the two constraints and , gives Thus on these states . The equation describes both the c.m. motion and the internal relative motion. To characterize the former motion, observe that since the potential depends only on the difference of the two coordinates . (This does not require that since the .) Thus, the total momentum is a constant of motion and is an eigenstate state characterized by a total momentum . In the c.m. system with the invariant center of momentum (c.m.) energy. Thus and so is also an eigenstate of c.m. energy operators for each of the two particles, . The relative momentum then satisfies , so that , , The above set of equations follow from the constraints and the definition of the relative momenta given in Eqs.() and (). If instead one chooses to define (for a more general choice see Horwitz), independent of