  We extend the definition of the Jordan-Wigner transformation to three
dimensions using the generalization of ideas that were used in the
two-dimensional case by one of the present authors. Under this transformation,
the 3D XY Hamiltonian is transformed into a system of spinless fermions coupled
to a gauge field with only two components. We calculate the flux per plaquette
for the 3 elementary perpendicular plaquettes of a cubic lattice, and find that
it is nonzero for only two of the plaquettes. We provide a simple
interpretation for the average phase-per-plaquette being $\pi$ on the
plaquettes where it is nonzero. Then we apply these findings to the
investigation of the Heisenberg bilayer antiferromagnet.
