Phase space structure and the path integral for gauge theories on a cylinder

The physical phase space of gauge field theories on a cylindrical spacetime with an arbitrary compact simple gauge group is shown to be the quotient ${\bf R}^{2r}/W_A,$ $r$ a rank of the gauge group, $W_A$ the affine Weyl group. The PI formula resulting from Dirac's operator method contains a symmetrization with respect to $W_A$ rather than the integration domain reduction. It gives a natural solution to Gribov's problem. Some features of fermion quantum dynamics caused by the nontrivial phase space geometry are briefly discussed.