First-class constraint: Difference between revisions

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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|accessdateaccess-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}}</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|mr=2220894}}. Unabridged reprint of original, Dover Publications, New York, NY, 2001. </ref>

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Effectively, these brackets (illustrated for this spherical surface in the [[Dirac bracket]] article) project the system back onto the constraints surface.

If one then wished to canonically quantize this system, then one need promote the canonical Dirac brackets,<ref>{{Cite journal | last1 = Corrigan | first1 = E. | last2 = Zachos | first2 = C. K. | doi = 10.1016/0370-2693(79)90465-9 | title = Non-local charges for the supersymmetric σ-model | journal = Physics Letters B | volume = 88 | issue = 3–4 | pages = 273 | year = 1979 | pmid = | pmc = |bibcode = 1979PhLB...88..273C }}</ref> ''not'' the canonical Poisson brackets to commutation relations.

Examination of the above Hamiltonian shows a number of interesting things happening. One thing to note is that, on-shell when the constraints are satisfied, the extended Hamiltonian is identical to the naive Hamiltonian, as required. Also, note that {{mvar|λ}} dropped out of the extended Hamiltonian. Since {{math| ''φ''₁}} is a first class primary constraint, it should be interpreted as a generator of a gauge transformation. The gauge freedom is the freedom to choose {{mvar|λ}}, which has ceased to have any effect on the particle's dynamics. Therefore, that {{mvar|λ}} dropped out of the Hamiltonian, that {{mvar|u}}₁ is undetermined, and that {{math| ''φ''₁}} = ''p_λ'' is first class, are all closely interrelated.

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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 = 5 | year = 1983 | pmid = | pmc = |bibcode = 1983EJPh....4....5F }}

* {{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 = 2049 | year = 1990 | pmid = | pmc = |bibcode = 1990PhRvD..42.2049H }}

{{DEFAULTSORT:First Class Constraint}}