2015/05/31 by Alberto Ibort, Ibort, Alberto, Amelia Spivak +1 · 1 citation
Mathematics · Physics and Astronomy · #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research #Black Holes and Theoretical Physics
paper · pdf · doi:10.48550/arxiv.1506.00338
The multisymplectic formalism of field theories developed by many\nmathematicians over the last fifty years is extended in this work to deal with\nmanifolds that have boundaries. In particular, we develop a multisymplectic\nframework for first order covariant Hamiltonian field theories on manifolds\nwith boundaries. This work is a geometric fulfillment of Fock's\ncharacterization of field theories as it appears in recent work by Cattaneo,\nMnev and Reshetikhin [Ca14]. This framework leads to a true geometric\nunderstanding of conventional choices for boundary conditions. For example, the\nboundary condition that the pull-back of the 1-form on the cotangent space of\nfields at the boundary vanish, i.e. \π * \α = 0 , is shown to be a\nconsequence of our finding that the boundary fields of the theory lie in the\n0-level set of the moment map of the gauge group of the theory.\n It is also shown that the natural way to interpret Euler-Lagrange equations\nas an evolution system near the boundary is as a presymplectic system in an\nextended phase space containing the natural configuration and momenta fields at\nthe boundary together with extra degrees of freedom corresponding to the\ntransversal components at the boundary of the momenta fields of the theory. The\nconsistency conditions at the boundary are analyzed and the reduced phase space\nof the system is determined to be a symplectic manifold with a distinguished\nisotropic submanifold corresponding to the boundary data of the solutions of\nEuler-Lagrange equations. This setting makes it possible to define well-posed\nboundary conditions, and provides the adequate setting for the canonical\nquantization of the system.\n The notions of the theory will be tested against three significant examples:\nscalar fields, Poisson\σ-model and Yang-Mills theories.\n