First-class constraint: Difference between revisions

Revision as of 18:41, 13 January 2020

Revision as of 13:59, 27 February 2020

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In general, the quotient space is a bit difficult 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 (mathematics)|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 (category theory)|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. SeeA global anomaly is different from the [[Gribov ambiguity]]., Thiswhich is when a flawgauge fixing doesn't work to fix a gauge uniquely, in quantizinga global anomaly, there is no consistent definition of the gauge field. A global anomaly is a barrier to defining a quantum [[gauge theory|gauge theories]] manydiscovered physicistsby Witten in overlooked1980.

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.