Rigorous operator identities for the Zwanzig projection

Establish rigorous proofs for the operator identities—particularly the Dyson operator identity and any associated identities used to handle orthogonal dynamics—when the generalized Langevin equation is derived or applied with the Zwanzig projection operator of infinite rank, so that these identities are justified under mathematically precise conditions.

Background

The paper shows that the generalized Langevin equation (GLE) and the second fluctuation-dissipation theorem can be derived from Volterra equations without invoking the Dyson identity or orthogonal dynamics. However, many works in statistical physics rely on the Dyson identity together with the Zwanzig projection operator, which is of infinite rank, to manage orthogonal dynamics.

Because the Zwanzig projection is infinite rank, familiar semigroup and operator arguments become subtle. The authors caution that in such settings, the operator identities typically assumed in the literature lack a rigorous foundation, motivating a precise proof of these identities for the Zwanzig projection context.