  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.
