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I have split off reduced phase space from covariant phase space and started to expand a bit.
In particular I tried to highlight a bit the important point that the exact presymplectic form which is induced by any local action functional on its covariant phase space (as discussed there), still has to be equipped with equivariant structure as a U(1)-prinicipal connection in order to pass to the reduced phase space.
This is an obvious point that however I find is glossed over in much of the literature and leads to some confusion in some places: some literature fond of the covariant phase space-construction from local action functionals will highlight that this always has exact presymplectic form and will take this as reason to disregard all the subtleties of geometric quantization, which pretty much disappear for exact (pre-)symplectic forms. The point missed in such discussion is that there is non-trivial equivarint structure on the prequantization of this presymplectic form.
This subtlety as such is of course treated correctly in all of the mathematical literature listed at qauzntization commutes with reduction, of course. But that literature in turn doesn’t mention the important construction of covariant phase spaces from local Lagrangians.
Therefore, if anyone can point me to references that do BOTH of the following:
discuss the covariant presymplectic phase space induced form a local Lagrangian;
discuss the need to put equivariant connection structure on the canonically induced globally defined presymplectic potential;
I’d be grateful.
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