First-class constraint: Difference between revisions

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Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace. But [[Paul Dirac|Dirac]] noticed that we can turn the underlying [[differential manifold]] of the [[symplectic manifold|symplectic space]] into a [[Poisson manifold]] using a different bracket, called the [[Dirac bracket]], such that the Dirac bracket of any (smooth) function with any of the second class constraints always vanishes and a couple of other nice properties.

If one wanted to canonically quantize this system, then, one needs to promote the canonical Dirac brackets<ref>{{citeCite journal doi| last1 = Corrigan | first1 = E. | last2 = Zachos | first2 = C. K. | doi = 10.1016/0370-2693(79)90465-9 |noedit 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 <math>\lambda</math> dropped out of the extended Hamiltonian. Since <math>\phi_1</math> 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 <math>\lambda</math> which has ceased to have any effect on the particle's dynamics. Therefore, that <math>\lambda</math> dropped out of the Hamiltonian, that <math>u_1</math> is undetermined, and that <math>\phi_1 = p_\lambda</math> 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 = }}

* {{cite doi|10.1088/0143-0807/4/1/003|noedit}}

* {{cite doi| 10.1103/PhysRevD.42.2049|noedit}}