  In this paper I give overviews of the polysymplectic approach to covariant
Hamiltonian field theory and the simplest geometric quantization of classical
particle theories. I then give a synopsis of a recently proposed toy model for
applying this geometric quantization map to polysymplectic field theory. I show
that no special difficulties arise when this toy model is applied to general
relativity. I then sketch the reasons why the standard tools of covariant
Hamiltonian field theory are not up to the challenge of GR, so that the
resulting quantum theory cannot be taken seriously. A few remarks are given
about prospects for future work.
