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{{Expert-subject|Mathematics|date=November 2008}}
In a constrained Hamiltonian system, a dynamical quantity is called a '''first class constraint''' if its Poisson bracket with all the other constraints vanishes on the '''constraint surface''' (the surface implicitly defined by the simultaneous vanishing of all the constraints).
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For most "practical" applications of first-class constraints, we do not see such complications: the [[quotient space]] of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a [[differentiable manifold]], which can be turned into a [[symplectic manifold]] by projecting the [[symplectic form]] of M onto it (this can be shown to be [[well defined]]). In light of the observation about physical observables mentioned earlier, we can work with this more "physical" smaller symplectic manifold, but with 2n fewer dimensions.
In general, the quotient space is a bit "nasty" to work with when doing concrete calculations (not to mention nonlocal when working with [[diffeomorphism constraint]]s), so what is usually done instead is something similar. Note that the restricted submanifold is a [[bundle]] (but not a [[fiber bundle]] in general) over the quotient manifold. So, instead of working with the quotient manifold, we can work with a [[Section (category theory)|section]] of the bundle instead. This is called [[gauge fixing]].
The ''major'' problem is this bundle might not have a [[global section]] in general. This is where the "problem" of [[global anomaly|global anomalies]] comes in, for example. See [[Gribov ambiguity]]. This is a flaw in quantizing [[gauge theory|gauge theories]] many physicists overlooked.
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