First-class constraint: Difference between revisions

Revision as of 13:01, 9 March 2010

Revision as of 20:24, 24 June 2010

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For a more elegant way, suppose given a [[vector bundle]] over M, with ''n''-dimensional [[fiber]] ''V''. Equip this vector bundle with a [[connection form|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 function|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 <math>C^{\infty}(M)</math>, 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,

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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 [[orbit (group theory)|orbits]] of the constrained subspace under the action of the [[symplectic flow]]s generated by the ''f'''s. This gives a local [[foliation]] of the subspace because it satisfies [[integrability condition]]s ([[Frobenius theorem (differential topology)|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 restrained subspace, then their Poisson brackets {A₁, B₁} and {A₂, B₂} are also constant over orbits and agree over the constrained subspace.

In general, we{{Who|date=March 2010}} 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.

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In general, the quotient space is a bit "nasty" to work with when doing concrete calculations (not to mention nonlocal when working with [[diffeomorphism constraint]]s), 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 anomaly|global anomalies]] comes in, for example. See [[Gribov ambiguity]]. This is a flaw in quantizing [[gauge theory|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 whichthat vanishes at the zeros of ''f'' is the fiberwise contraction of ''f'' with (a non-unique) smooth section of a <math>\bar{V}</math>-vector bundle where <math>\bar{V}</math> 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 (physics)|action]] is the integral of a local [[Lagrangian]] whichthat 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 whichthat hold off shell are called primary constraints while those whichthat only hold on shell are called secondary constraints.

==Examples==

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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'' &minus; 1) + 1 [[Minkowski spacetime]]. For l in ''L'', we write

:&rho;(l)[&sigma;]