The Klein-Gordon equation is a useful test arena for quantum cosmological
models described by the Wheeler-DeWitt equation. We use the decoherent
histories approach to quantum theory to obtain the probability that a free
relativistic particle crosses a section of spacelike surface. The decoherence
functional is constructed using path integral methods with initial states
attached using the (positive definite) ``induced'' inner product between
solutions to the constraint equation. The notion of crossing a spacelike
surface requires some attention, given that the paths in the path integral may
cross such a surface many times, but we show that first and last crossings are
in essence the only useful possibilities. Different possible results for the
probabilities are obtained, depending on how the relativistic particle is
quantized (using the Klein-Gordon equation, or its square root, with the
associated Newton-Wigner states). In the Klein-Gordon quantization, the
decoherence is only approximate, due to the fact that the paths in the path
integral may go backwards and forwards in time. We compare with the results
obtained using operators which commute with the constraint (the ``evolving
constants'' method).Comment: 51 pages, plain Te
