Grassmann variables and pseudoclassical Nuclear Magnetic Resonance

• Main Author: Damion, Robin A.
• Other Authors: Universität Leipzig, Fakultät für Physik und Geowissenschaften
• Format: Article
• Language: English
• Published: Universitätsbibliothek Leipzig 2016
• Subjects: Magnetresonanztomographie; Graßmann; Hermann; pseudoklassische Mechnaik; Diffusion; Wegintegrail; NMR; Nuclear magnetization; propagator; Grassmann numbers; pseudoclassical mechanics; diffusion; path integral; ddc:530
• Online Access: http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-214290; http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-214290; http://www.qucosa.de/fileadmin/data/qucosa/documents/21429/diff_fund_26%282016%291.pdf

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 magnetization or the Green’s function of the Torrey-Bloch equations. The magnetization 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 magnetization 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 magnetization. The trajectories of the pseudoclassical model also provide some insight into the nature of the end-points.