Classical paths in systems of fermions

TL;DR

This paper extends the pseudoclassical path formalism to fermionic systems, enabling a classical-like description of quantum states and fluctuations of fermions that respects their algebraic properties.

Contribution

It introduces a pseudoclassical path approach for fermions, allowing classical stochastic descriptions of quantum fluctuations while maintaining fermionic algebra.

Findings

Fermionic quantum states can be represented as non-interfering paths obeying Dirac equations.

Classical stochastic models approximate the dynamics of collective fermionic observables.

The formalism effectively describes local fluctuations of conserved fermion numbers.

Abstract

We implement in systems of fermions the formalism of pseudoclassical paths that we recently developed for systems of bosons and show that quantum states of fermionic fields can be described, in the Heisenberg picture, as linear combinations of randomly distributed paths that do not interfere between themselves and obey classical Dirac equations. Every physical observable is assigned a time-dependent value on each path in a way that respects the anticommutative algebra between quantum operators and we observe that these values on paths do not necessarily satisfy the usual algebraic relations between classical observables. We use these pseudoclassical paths to define the dynamics of quantum fluctuations in systems of fermions and show that, as we found for systems of bosons, the dynamics of fluctuations of a wide class of observables that we call "collective" observables can be approximately described in terms of classical stochastic concepts. Finally, we apply this formalism to describe the dynamics of local fluctuations of globally conserved fermion numbers.