Interacting-fermion dynamics in Majorana phase space

Abstract

The problem of fermion dynamics is studied using the $Q$ function for fermions. This is a probabilistic phase-space representation, which we express using Majorana operators and a phase-space variable of real antisymmetric matrices. We consider a general Hamiltonian with quadratic and quartic Majorana interactions and arbitrary couplings. Our model includes the Majorana-Hubbard and Fermi-Hubbard Hamiltonians, as well as general quantum field theories of interacting fermions. The relevant coefficients are calculated for a number of both local and nonlocal Hamiltonian models. We also give a detailed example of a four-Majorana interaction. Using the Majorana $Q$ function we derive a generalized Fokker-Planck equation, with results for the drift and diffusion terms. The diffusion term is shown to be traceless, which has a dynamical interpretation as a forward-backward stochastic process. We prove that an initial pure state has trajectories that remain on a surface of pure states. This approach leads to a model of quantum measurement in terms of an ontology with real vacuum fluctuations.