First Class Constraint - Intuitive Meaning

Intuitive Meaning

What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other on the constrained subspace; or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constraint flows all bring the point to another point on the constrained subspace.

Since we wish to restrict ourselves to the constrained subspace only, this suggests that the Hamiltonian, or any other physical observable, should only be defined on that subspace. Equivalently, we can look at the equivalence class of smooth functions over the symplectic manifold, which agree on the constrained subspace (the quotient algebra by the ideal generated by the f's, in other words).

The catch is, the Hamiltonian flows on the constrained subspace depend on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this.

Look at the orbits of the constrained subspace under the action of the symplectic flows generated by the f's. This gives a local foliation of the subspace because it satisfies integrability conditions (Frobenius theorem). It turns out if we start with two different points on a same orbit on the constrained subspace and evolve both of them under two different Hamiltonians, respectively, which agree on the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows will always lie in the same orbit at equal times. It also turns out if we have two smooth functions A₁ and B₁, which are constant over orbits at least on the constrained subspace (i.e. physical observables) (i.e. {A₁,f}={B₁,f}=0 over the constrained subspace)and another two A₂ and B₂, which are also constant over orbits such that A₁ and B₁ agrees with A₂ and B₂ respectively over the restrained subspace, then their Poisson brackets {A₁, B₁} and {A₂, B₂} are also constant over orbits and agree over the constrained subspace.

In general, we can't rule out "ergodic" flows (which basically means that an orbit is dense in some open set), or "subergodic" flows (which an orbit dense in some submanifold of dimension greater than the orbit's dimension). We can't have self-intersecting orbits.

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 constraints), 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 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 anomalies comes in, for example. See Gribov ambiguity. This is a flaw in quantizing gauge theories many physicists overlooked.

What have been described are irreducible first-class constraints. Another complication is that Δf might not be right invertible on subspaces of the restricted submanifold of codimension 1 or greater (which violates the stronger assumption stated earlier in this article). This happens, for example in the cotetrad formulation of general relativity, at the subspace of configurations where the cotetrad field and the connection form happen to be zero over some open subset of space. Here, the constraints are the diffeomorphism constraints.

One way to get around this is this: For reducible constraints, we relax the condition on the right invertibility of Δf into this one: Any smooth function that vanishes at the zeros of f is the fiberwise contraction of f with (a non-unique) smooth section of a -vector bundle where is the dual vector space to the constraint vector space V. This is called the regularity condition.