Duality and Conditional Expectations in the Nakajima-Mori-Zwanzig Formulation

TL;DR

This paper introduces a rigorous operator algebraic framework for the Nakajima-Mori-Zwanzig method, establishing a duality principle and showing that projection operators are conditional expectations, applicable to classical and quantum systems.

Contribution

It develops a new mathematical formulation of the NMZ method based on operator algebras, revealing the duality and the nature of projection operators as conditional expectations.

Findings

Established a duality principle between observable and state space formulations.

Proved that projection operators in NMZ are conditional expectations.

Illustrated the theory with various classical and quantum examples.

Abstract

We develop a new operator algebraic formulation of the Nakajima-Mori-Zwanzig (NMZ) method of projections. The new theory is built upon rigorous mathematical foundations, and it can be applied to both classical and quantum systems. We show that a duality principle between the NMZ formulation in the space of observables and in the state space can be established, analogous to the Heisenberg and Schr\"odinger pictures in quantum mechanics. Based on this duality we prove that, under natural assumptions, the projection operators appearing in the NMZ equation must be conditional expectations. The proposed formulation is illustrated in various examples.