First Class Constraint - Examples

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 = m2 −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)

as

l

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 "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

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 . 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,

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.