First-class constraint: Difference between revisions

Revision as of 11:02, 15 February 2023

Revision as of 12:14, 9 July 2023

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{{distinguish|Primary constraint}}

{{Main|Dirac bracket}}

A '''first class constraint''' is a dynamical quantity in a constrained [[Hamiltonian mechanics|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 bracket]]s generated.<ref name=FysikSuSePDF>{{cite web|author1=Ingemar Bengtsson, Stockholm University|title=Constrained Hamiltonian Systems|url=http://3dhouse.se/ingemar/Nr13.pdf|publisher=Stockholm University|access-date=29 May 2018|format=PDF|quote=We start from a Lagrangian L ( q, ̇ q ), derive the canonical momenta, postulate the naive Poisso n brackets, and compute the Hamiltonian. For simplicity, one assumes that no second class constraints occur, or if they do, that they have been dealt with already and the naive brackets replaced with Dirac brackets. There remain a set of constraints [...]}}</ref>

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|title=Generalized Hamiltonian dynamics|year=1950|last1=Dirac|first1=Paul A. M.|author1-link=Paul Dirac|journal=[[Canadian Journal of Mathematics]]|volume=2|pages=129–148|doi=10.4153/CJM-1950-012-1|issn=0008-414X|mr=0043724|s2cid=119748805 }}</ref><ref>{{Citation|title=Lectures on Quantum Mechanics|url=https://books.google.com/books?id=GVwzb1rZW9kC|year=1964|last1=Dirac|first1=Paul A. M.|series=Belfer Graduate School of Science Monographs Series|volume=2|publisher=Belfer Graduate School of Science, New York| isbn=9780486417134 |mr=2220894}}. Unabridged reprint of original, Dover Publications, New York, NY, 2001. </ref>

The terminology of first and second class constraints is confusingly similar to that of [[primary constraint|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.

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==Further reading==

* {{Cite journal | last1 = Falck | first1 = N. K. | last2 = Hirshfeld | first2 = A. C. | doi = 10.1088/0143-0807/4/1/003 | title = Dirac-bracket quantisation of a constrained nonlinear system: The rigid rotator | journal = European Journal of Physics | volume = 4 | pages = 55–9 | year = 1983 | issue = 1 |bibcode = 1983EJPh....4....5F | s2cid = 250845310 }}

* {{Cite journal | last1 = Homma | first1 = T. | last2 = Inamoto | first2 = T. | last3 = Miyazaki | first3 = T. | doi = 10.1103/PhysRevD.42.2049 | title = Schrödinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space | journal = Physical Review D | volume = 42 | issue = 6 | pages = 20492049–2056 | year = 1990 | pmid = 10013054 |bibcode = 1990PhRvD..42.2049H }}

{{DEFAULTSORT:First Class Constraint}}