First Class Constraint - Geometric 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 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

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.