First class constraint

A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishing of all the constraints). To calculate the first class constraint, one assumes that there are no second class constraints, or that they have been calculated previously, and their Dirac brackets generated. First and second class constraints were introduced by as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate. The terminology of first and second class constraints is confusingly similar to that of primary and secondary constraints, reflecting the manner in which these are generated. These divisions are independent: both first and second class constraints can be either primary or secondary, so this gives altogether four different classes of constraints. Consider a Poisson manifold M with a smooth Hamiltonian over it (for field theories, M would be infinite-dimensional). Suppose we have some constraints for n smooth functions These will only be defined chartwise in general. Suppose that everywhere on the constrained set, the n derivatives of the n functions are all linearly independent and also that the Poisson brackets and all vanish on the constrained subspace. This means we can write for some smooth functions −−there is a theorem showing this; and for some smooth functions . This can be done globally, using a partition of unity. Then, we say we have an irreducible first-class constraint (irreducible here is in a different sense from that used in representation theory). For a more elegant way, suppose given a vector bundle over , with -dimensional fiber . Equip this vector bundle with a connection. Suppose too we have a smooth section f of this bundle. Then the covariant derivative of f with respect to the connection is a smooth linear map from the tangent bundle to , which preserves the base point.