Quantum statistical mechanical gauge invariance

We address gauge invariance in the statistical mechanics of quantum many-body systems. The gauge transformation acts on the position and momentum degrees of freedom and it is represented by a quantum shifting superoperator that maps quantum observables onto each other. The shifting superoperator is anti-self-adjoint and it has noncommutative Lie algebra structure. These properties induce exact equilibrium sum rules that connect locally-resolved force and hyperforce densities for any given observable. We argue that the framework is amenable to tight integration into quantum hyperdensity functional theory and that it generalizes naturally to nonequilibrium.