First-class constraint: Difference between revisions

Revision as of 21:55, 17 January 2016

Revision as of 06:28, 18 January 2016

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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]] vanishes on the '''constraint surface''' (the surface implicitly defined by the simultaneous vanishing of all the constraints) with all the other constraints. To calculate the first class constraint, we assume 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://www.fysik.su.se/~ingemar/Nr13.pdf|publisher=Stockholm University|accessdate=18 September 2015|page=7|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, we assume 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>

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and

:<math>\{f_i,H\}</math>

all vanish on the constrained subspace.

This means we can write

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Then the [[covariant derivative]] of {{mvar|f}} with respect to the connection is a smooth [[linear map]] Δ''f'' from the [[tangent bundle]] ''TM'' to ''V'', which preserves the [[base point]]. Assume this linear map is right [[invertible]] (i.e. there exists a linear map ''g'' such that (Δ''f'')''g'' is the [[identity function|identity map]]) for all the fibers at the zeros of {{mvar|f}}. Then, according to the [[implicit function theorem]], the subspace of zeros of {{mvar|f}} is a [[submanifold]].

The ordinary [[Poisson bracket]] is only defined over <math>C^{\infty}(M)</math>, the space of smooth functions over ''M''. However, using the connection, we can extend it to the space of smooth sections of {{mvar|f}} if we work with the [[algebra bundle]] with the [[graded algebra]] of ''V''-tensors as fibers.

Assume also that under this Poisson bracket,

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Likewise, from this secondary constraint, we get the tertiary constraint,

<math>\vec{p}\cdot\vec{r}=0</math>,

by demanding, for consistency, that <math>\{r^2-R^2,\, H\}_{PB} = 0</math> on-shell.

Again, one should add this constraint into the Hamiltonian, since, on-shell, no one can tell the difference. Therefore, so far, the Hamiltonian looks like

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H = \frac{p^2}{2m} + mgz - \frac{\lambda}{2}(r^2-R^2) + u_1 p_\lambda + u_2 (r^2-R^2) + u_3 \vec{p}\cdot\vec{r}~,

</math>

where {{mvar|u}}₁, {{mvar|u}}₂, and {{mvar|u}}₃ are still completely undetermined.

Note that, frequently, all constraints that are found from consistency conditions are referred to as "secondary constraints" and secondary, tertiary, quaternary, etc., constraints are not distinguished.

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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. However, [[Paul Dirac|Dirac]] noticed that we can turn the underlying [[differential manifold]] of the [[symplectic manifold|symplectic space]] into a [[Poisson manifold]] using his eponymous modified bracket, called the [[Dirac bracket]], such that this ''Dirac bracket of any (smooth) function with any of the second class constraints always vanishes''. 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.