Geometric formulation of the Covariant Phase Space methods with boundaries

Kavli Affiliate: J. S. Villasenor

| First 5 Authors: Juan Margalef-Bentabol, Eduardo J. S. VillaseƱor, , ,

| Summary:

We analyze in full-detail the geometric structure of the covariant phase
space (CPS) of any local field theory defined over a space-time with boundary.
To this end, we introduce a new frame: the "relative bicomplex framework". It
is the result of merging an extended version of the "relative framework"
(initially developed in the context of algebraic topology by R.~Bott and
L.W.~Tu in the 1980s to deal with boundaries) and the variational bicomplex
framework (the differential geometric arena for the variational calculus). The
relative bicomplex framework is the natural one to deal with field theories
with boundary contributions, including corner contributions. In fact, we prove
a formal equivalence between the relative version of a theory with boundary and
the non-relative version of the same theory with no boundary. With these tools
at hand, we endow the space of solutions of the theory with a (pre)symplectic
structure canonically associated with the action and which, in general, has
boundary contributions. We also study the symmetries of the theory and
construct, for a large group of them, their Noether currents, and charges.
Moreover, we completely characterize the arbitrariness (or lack thereof for
fiber bundles with contractible fibers) of these constructions. This clarifies
many misconceptions about the role of the boundary terms in the CPS description
of a field theory. Finally, we provide what we call the CPS-algorithm to
construct the aforementioned (pre)symplectic structure and apply it to some
relevant examples.

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