First-class constraint

In physics, a first-class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space. To calculate the first-class constraint, one assumes that there are no second-class constraints, or that they have been calculated previously, and their Dirac brackets generated.
First- and second-class constraints were introduced by as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate.
The terminology of first- and second-class constraints is confusingly similar to that of primary and secondary constraints, reflecting the manner in which these are generated. These divisions are independent: both first- and second-class constraints can be either primary or secondary, so this gives altogether four different classes of constraints.

Poisson brackets

Consider a Poisson manifold M with a smooth Hamiltonian over it.
Suppose we have some constraints
for n smooth functions
These will only be defined chartwise in general. Suppose that everywhere on the constrained set, the n derivatives of the n functions are all linearly independent and also that the Poisson brackets
and
all vanish on the constrained subspace.
This means we can write
for some smooth functions — there is a theorem showing this; and
for some smooth functions.
This can be done globally, using a partition of unity. Then, we say we have an irreducible first-class constraint.

Geometric theory

For a more elegant way, suppose given a vector bundle over, with -dimensional fiber. Equip this vector bundle with a connection. Suppose too we have a smooth section of this bundle.
Then the covariant derivative of with respect to the connection is a smooth linear map from the tangent bundle to, which preserves the base point. Assume this linear map is right invertible for all the fibers at the zeros of . Then, according to the implicit function theorem, the subspace of zeros of 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 if we work with the algebra bundle with the graded algebra of V-tensors as fibers.
Assume also that under this Poisson bracket, and on the submanifold of zeros of . 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.

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 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 's. This gives a local foliation of the subspace because it satisfies integrability conditions. 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 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 and are also constant over orbits and agree over the constrained subspace.
In general, one cannot rule out "ergodic" flows, or "subergodic" flows. 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 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. 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 difficult to work with when doing concrete calculations, so what is usually done instead is something similar. Note that the restricted submanifold is a bundle 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. A global anomaly is different from the Gribov ambiguity, which is when a gauge fixing doesn't work to fix a gauge uniquely, in a global anomaly, there is no consistent definition of the gauge field. A global anomaly is a barrier to defining a quantum gauge theory discovered by Witten in 1980.
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. 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 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 that 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 that hold off shell are called primary constraints while those that only hold on shell are called secondary constraints.

Examples

Consider the dynamics of a single point particle of mass with no internal degrees of freedom moving in a pseudo-Riemannian spacetime manifold with metric g. Assume also that the parameter describing the trajectory of the particle is arbitrary. Then, its symplectic space is the cotangent bundle T*S with the canonical symplectic form .
If we coordinatize T * S by its position in the base manifold and its position within the cotangent space p, then we have a constraint
The Hamiltonian is, surprisingly enough, = 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 '= instead. Then, we have the interesting case where the Hamiltonian is the same as a constraint! See Hamiltonian constraint for more details.
Consider now the case of a Yang–Mills theory for a real simple Lie algebra minimally coupled to a real scalar field , which transforms as an orthogonal representation with the underlying vector space under in + 1 Minkowski spacetime. For in , we write
as
for simplicity. Let A be the -valued connection form of the theory. Note that the A here differs from the A used by physicists by a factor of and . This agrees with the mathematician's convention.
The action is given by
where g is the Minkowski metric, F is the curvature form
where the second term is a formal shorthand for pretending the Lie bracket is a commutator, is the covariant derivative
and is the orthogonal form for .
What is the Hamiltonian version of this model? Well, first, we have to split A noncovariantly into a time component and a spatial part . Then, the resulting symplectic space has the conjugate variables , , , _A, φ and π_φ. For each spatial point, we have the constraints, π_φ=0 and the Gaussian constraint
where since is an intertwiner
' is the dualized intertwiner
. The Hamiltonian,
The last two terms are a linear combination of the Gaussian constraints and we have a whole family of Hamiltonians parametrized by . In fact, since the last three terms vanish for the constrained states, we may drop them.

Second-class constraints

In a constrained Hamiltonian system, a dynamical quantity is second-class if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a second-class constraint.
See Dirac brackets for diverse illustrations.