A Dynamic Systems Approach to Fermions and Their Relation to Spins

TL;DR

This paper explores the dynamic properties of fermionic quantum systems using Lie theory, analyzing controllability, reachability, and simulability across various symmetry and conservation constraints.

Contribution

It introduces a Lie-theoretical framework for fermionic systems, characterizing their dynamic Lie algebras and reachable states under different physical constraints.

Findings

Identifies dynamic Lie algebras for fermionic systems

Characterizes reachable state sets via orbit manifolds

Analyzes controllability under superselection rules

Abstract

Dynamic properties of fermionic systems, like contollability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. Observing the parity superselection rule, we treat the fully controllable and quasifree cases, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.