First-class constraint: Difference between revisions

Revision as of 17:26, 17 January 2016

Revision as of 17:37, 17 January 2016

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Again, this is ''not'' a new constraint; it only determines that

:<math>

u_3 = -\frac{\vec{r}\cdot\vec{p}}{m r^2}~.

</math>

At this point there are ''no more constraints or consistency conditions'' to check.

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</math>

Before analyzing the Hamiltonian, consider the three constraints:,

:<math>

\phi_1 = p_\lambda, \quad \phi_2 = r^2-R^2, \quad \phi_3 = \vec{p}\cdot\vec{r}.

</math>

NoticeNote the nontrivial [[Poisson bracket]] structure of the constraints. In particular,

:<math>

\{\phi_2, \phi_3\} = 2 r^2 \neq 0.

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The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but ''even on-shell it is nonzero''. Therefore, {{math| ''φ''₂}} and {{math| ''φ''₃}} are '''second class constraints''' while {{math| ''φ''₁}} is a first class constraint. Note that these constraints satisfy the regularity condition.

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

If one wantedwished to canonically quantize this system, then, one needs toneed 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 <math>\lambda</math>{{mvar|λ}}, which has ceased to have any effect on the particle's dynamics. Therefore, that {{mvar|λ}} 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.

Note that it would be more natural not to start with a Lagrangian with a Lagrange multiplier, but instead take <math>r^2-R^2</math> as a primary constraint and proceed through the formalism. The result would the elimination of the extraneous {{mvar|λ}} dynamical quantity. Perhaps, the example is more edifying in its current form.