Grassmann Variables and Pseudoclassical Nuclear Magnetic Resonance

TL;DR

This paper introduces a pseudoclassical approach using Grassmann variables to model the propagator of magnetisation in diffusion NMR, linking complex trajectories with spin dynamics.

Contribution

It develops a novel pseudoclassical Lagrangian combining bosonic and fermionic variables to analyze magnetisation propagation in NMR.

Findings

Complex-valued trajectories from the Lagrangian

Grassmann variables encode spin degrees of freedom

Path integral over Grassmann variables recovers original propagator

Abstract

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.