  The concept of a propagator is useful and is a well-known object in diffusion
NMR experiments. Here, we investigate the related concept; the propagator for
the magnetisation or the Green's function of the Torrey-Bloch equations. The
magnetisation propagator is constructed by defining functions such as the
Hamiltonian and Lagrangian and using these to define a path integral. It is
shown that the equations of motion derived from the Lagrangian produce
complex-valued trajectories (classical paths) and it is conjectured that the
end-points of these trajectories are real-valued. The complex nature of the
trajectories also suggests that the spin degrees of freedom are also encoded
into the trajectories and this idea is explored by explicitly modeling the spin
or precessing magnetisation by anticommuting Grassmann variables. A
pseudoclassical Lagrangian is constructed by combining the diffusive (bosonic)
Lagrangian with the Grassmann (fermionic) Lagrangian, and performing the path
integral over the Grassmann variables recovers the original Lagrangian that was
used in the construction of the propagator for the magnetisation. The
trajectories of the pseudoclassical model also provide some insight into the
nature of the end-points.
