First-class constraint: Difference between revisions

Revision as of 00:41, 6 October 2015

Revision as of 15:20, 17 January 2016

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Before going on to the general theory, consider a specific example step by step to motivate the general analysis.

Start with the [[action (physics)|action]] describing a [[Newtonian dynamics|Newtonian]] particle of [[mass]] {{mvar|m}} constrained to a surface of radius {{mvar|R}} within a uniform [[gravitational field]] ''{{mvar|g''}}. When one works in Lagrangian mechanics, there are several ways to implement a constraint: one can switch to generalized coordinates that manifestly solve the constraint, or one can use a Lagrange multiplier while retaining the redundant coordinates so constrained.

In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice). For pedagogical reasons, instead, consider the problem in Cartesian coordinates, with a Lagrange multiplier term enforcing the constraint.

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Of course, as indicated, we could have just used different [[coordinates]] and written it as

:<math>S=\int dt \left[\frac{mR^2}{2}(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)+mgR\cos(\theta)\right]</math>

instead, without extra constraints, but we look at the former coordinatization to illustrate constraints.

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

</math>

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''<sub>\phi_22</mathsub>}} and < {{math| ''φ''<sub>\phi_33</mathsub>}} are '''second class constraints''' while < {{math| ''φ''<sub>\phi_11</mathsub>}} 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. 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.

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If one wanted to canonically quantize this system, then, one needs to 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 <math>\lambda</math> {{mvar|λ}} dropped out of the extended Hamiltonian. Since < {{math| ''φ''<sub>\phi_11</mathsub>}} 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> {{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.