First-class constraint: Difference between revisions

Revision as of 14:33, 18 August 2012

Revision as of 14:35, 18 August 2012

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In a constrained Hamiltonian system, a dynamical quantity is called a '''first class constraint''' if its Poisson bracket with all the other constraints vanishes on the '''constraint surface''' (the surface implicitly defined by the simultaneous vanishing of all the constraints). A '''second class constraint''' is one that is not first class.

First and second class constraints were introduced by {{harvs|txt|last=Dirac|authorlink=Paul Dirac|yearyear1=19641950|loc=p.136|year2=1964|loc2=p.17}} 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 constraint|primary and secondary constraints]]. 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.