First-class constraint: Difference between revisions

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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|year1=1950|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.<ref>{{Citation | last1=Dirac | first1=P. A. M. | author1-link=Paul Dirac | title=Generalized Hamiltonian dynamics | doi=10.4153/CJM-1950-012-1 | idmr={{MR|0043724}} | year=1950 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=2 | pages=129–148}}</ref>

<ref>{{Citation | last1=Dirac | first1=Paul A. M. | title=Lectures on quantum mechanics | url=http://books.google.com/books?id=GVwzb1rZW9kC | publisher=Belfer Graduate School of Science, New York | series=Belfer Graduate School of Science Monographs Series | idmr={{MR|2220894}} Reprinted by Dover in 2001. | year=1964 | volume=2}}</ref>

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.