First class constraint

In Hamiltonian mechanics, let's say we have a symplectic manifold M with a smooth Hamiltonian over it (for field theories, M would be infinite-dimensional). Let's also say we have a couple of constraints given as function equations f_i(x)=0 for n smooth functions { f_i}_i=1ⁿ (only defined chartwise in general) such that everywhere on the constrained subspace, the n derivatives of the n functions are all linearly independent and also, the Poisson brackets { f_i, f_j } and { f_i, H } all vanish at the constrained subspace. This means we can write for some smooth functions c_ij^k (there is a mathematical theorem on this) and for some smooth functions v_i^j. This can be done globally using a partition of unity. Then, we say we have an irreducible first-class constraint (this is irreducible in a different sense from representation theory).

That's a real ugly way of looking at it. Let's say we have a vector bundle over M with an n-dimensional fiber V. Let's also equip this vector bundle with a connection. Let's also suppose we have a smooth section of this bundle called f. Then, the covariant derivative of f with respect to this connection is a smooth linear map Δf 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 (Δf)g is the identity map) for all the fibers at the zeros of f. Then, according to the implicit function theorem, the subspace of zeros of f is a submanifold. The ordinary Poisson bracket is only defined over , the space of smooth functions over M. However, using the connection, we can extend it to the space of smooth sections of f if we work with the algebra bundle with the graded algebra of V-tensors as fibers. Assume also that under this Poisson bracket, { f, f } = 0 (note that it's not true that { g, g } = 0 in general for this "extended Poisson bracket" anymore) and { f, H } = 0 at the submanifold of zeros of f (If these brackets also happen to be zero everywhere, then we say the constraints close off shell). It turns out the right invertibility condition and the commutativity of flows conditions are independent of the choice of connection. So, we can drop the connection provided we are working solely with the restricted subspace.

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 constaint flows all bring the point to another point on the constrained subspace.

Since we wish to restrict ourselves to the constrained subspace only, this kind of suggests 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 over the constrained subspace (the quotient algebra by the ideal generated by the f's, in other words). But the catch is, the Hamiltonian flows at the constrained subspace depends on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this.

Let's look at the orbits of the constrained subspace under the action of the symplectic flows generated by the f's. This would most definitely give 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 both agree over the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows would 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 over 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 restrainted subspace, then their Poisson brackets {A₁, B₁} and {A₂, B₂} are also constant over orbits and agree over the constrainted 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 basically that an orbit is dense in some submanifold of dimension greater than the orbit's dimension. Note we can't ever have self-intersecting orbits. But for most "practical" applications of first class constraints, we do not such complications and so, 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 less dimensions. In general, the quotient space is a bit "nasty" to work with when doing concrete calculations (not to mention nonlocal when working with difeomorphism 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 BIG problem is (and I really have to emphasize this, since this is a flaw in quantizing gauge theories which many physicists overlook) this bundle might not have a global section in general (This is where the "problem" of global anomalies come in, for example)!!!!! See Gribov ambiguity.

What we've just described is 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 happens 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: Any smooth function which 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)

Constrained Hamiltonian dynamics from a Lagrangian gauge theory

First of all, we will assume the action is the integral of a local Lagrangian which only depends up to the first derivative of the fields. The analysis of more general cases, while possible is more complicated. When going over to the Hamiltonian formalism, we find there are constraints. Recall that in the action formalism, there are on shell and off shell configurations. The constraints which hold off shell are called primary constraints while those which only hold on shell are called secondary constraints.

Examples

Let's look at the dynamics of a single point particle of mass m with no internal degrees of freedom moving in a pseudo-Riemannian spacetime manifold S with metric g. Let's also assume the parameter τ describing the trajectory of the particle is arbitrary (i.e. we insist upon reparametrization invariance). Then, its symplectic space is the cotangent bundle T*S with the canonical symplectic form ω. If we coordinatize T * S by its position x in the base manifold S and its position within the cotangent space p, then we have a constraint f = m² −g(x)⁻¹(p,p) = 0. The Hamiltonian H, is suprisingly enough H = 0. In light of the observation that the Hamiltonian is only defined up to the equivalence class of smooth functions agreeing on the constrained subspace, we can use a new Hamiltonian H'=f instead. Then, we have the interesting case where the Hamiltonian is the same as a constraint! See Hamiltonian constraint for more details.

Let's consider the case of a Yang-Mills theory over a real simple Lie algebra L (with a negative definite Killing form η) minimally coupled to a real scalar field σ which transforms as an orthogonal representation ρ with the underlying vector space V under L in (d − 1) + 1 Minkowski spacetime. for l in L, we write ρ(l)[σ] as l[σ] for simplicity. Let A be the L-valued connection form of the theory. Note that the A here differs from the A used by physicists by a factor of i and also a g. This agrees with the mathematician's convention. The action S is given by where g is the Minkowski metric, F is the curvature form (no is or gs!) where the second term is a formal shorthand for pretending the Lie bracket is a commutator, D is the covariant derivative Dσ = dσ − A[σ] and α is the orthogonal form for ρ.

I hope I have all the signs and factors right. I can't guarentee it.

What is the Hamiltonian version of this model? Well, first, we have to split A noncovariantly into a time component φ and a spatial part . Then, the resulting symplectic space has the conjugate variables σ, π_σ (taking values in the underlying vector space of , the dual rep of ρ), , , φ and π_φ. for each spatial point, we have the constraints, π_φ=0 and the Gaussian constraint where since ρ is an intertwiner , ρ' is the dualized intertwiner (L is self-dual via η). The Hamiltonian,

note that the last two terms are a linear combination of the Gaussian constraints and we have a whole family of (gauge equivalent)Hamiltonians parametrized by f. In fact, since the last three terms vanish for the constrained states, we can drop them.

See also second class constraints, BRST, analysis of flows

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