First class constraint

In Hamiltonian mechanics, consider a symplectic manifold "M" with a smooth Hamiltonian over it (for field theories, "M" would be infinite-dimensional).

Poisson brackets

Suppose we have some constraints :$f\_i(x)=0,$ for "n" smooth functions

:$\{\; f\_i\; \}\_\{i=\; 1\}^n$

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

:{ "f"_"i", "f"_"j" }

and

:{ "f"_"i", "H" }

all vanish on the constrained subspace. This means we can write

:$\{f\_i,f\_j\}=sum\_k\; c\_\{ij\}^k\; f\_k$

for some smooth functions

:"c"_"ij"^"k"

(there is a theorem showing this) and

:$\{f\_i,H\}=sum\_j\; v\_i^j\; f\_j$

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 ("irreducible" here is in a different sense from that used in representation theory).

Geometric theory

For a more elegant way, suppose given a vector bundle over M, with "n"-dimensional fiber "V". Equip this vector bundle with a connection. Suppose too we have a smooth section "f" of this bundle.

Then the covariant derivative of "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 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 $C^\{infty\}(M)$, 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

on 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.

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 "A"₁ and "B"₁ 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 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 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 which many physicists had 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 Δ"f" into this one: 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

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. Assume also that 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" = "m"² −g("x")⁻¹(p,p) = 0.

The Hamiltonian "H" is, surprisingly 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.

Consider now the case of a Yang-Mills theory for 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 "i"s or "g"s!) 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 guarantee 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 $vec\{A\}$. 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), $vec\{A\}$, $vec\{pi\}\_A$, φ and π_φ. for each spatial point, we have the constraints, π_φ=0 and the Gaussian constraint

:$vec\{D\}cdotvec\{pi\}\_A-\; ho\text{'}(pi\_sigma,sigma)=0$

where since ρ is an intertwiner

:$ho:Lotimes\; V\; ightarrow\; V$,

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 Dirac bracket, second class constraints, BRST, analysis of flows