Regularized polysymplectic geometry and first steps towards Floer theory for covariant field theories. (English) Zbl 1516.81121
Summary: It is the goal of this paper to present the first steps for defining the analogue of Hamiltonian Floer theory for covariant field theory, treating time and space relativistically. While there already exist a number of competing geometric frameworks for covariant field theory generalizing symplectic geometry, none of them are readily suitable for variational techniques such as Hamiltonian Floer theory, since the corresponding action functionals are too degenerate. Instead, we show how a regularization procedure introduced by Bridges leads to a new geometric framework for which we can show that the finite energy \(L^2\)-gradient lines of the corresponding action functional, called Floer curves, converge asymptotically to space-time periodic solutions. As a concrete example we prove the existence of Floer curves, and hence also of space-time periodic solutions, for a class of coupled particle-field systems defined in this new framework.
MSC:
| 81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
| 70H05 | Hamilton’s equations |
| 53D05 | Symplectic manifolds (general theory) |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 35B10 | Periodic solutions to PDEs |
| 81V22 | Unified quantum theories |
Keywords:
Floer theory; polysymplectic geometry; covariant field theory; classical field theory; calculus of variationsReferences:
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