Note: Nonuniqueness of generalized quantum master equations for a single observable

TL;DR

This paper explores the nonuniqueness of generalized quantum master equations for a single observable, highlighting how different derivation methods, such as projections and conservation laws, lead to distinct forms of these equations.

Contribution

It demonstrates that generalized master equations are not unique and compares the effects of using projections versus conservation laws in their derivation.

Findings

Different derivation methods produce distinct master equations.

Relationships between memory kernels ensure equivalent dynamics.

Projection and conservation law approaches are fundamentally different.

Abstract

When deriving exact generalized master equations for the evolution of a reduced set of degrees of freedom, one is free to choose what quantities are relevant by specifying projection operators. However, obtaining a reduced description does not always need to be achieved through projections--one can also use conservation laws for this purpose. Such an operation should be considered as distinct from any kind of projection; that is, projection onto a single observable yields a different form of master equation compared to that resulting from a projection followed by the application of a constraint. We give a simple example to show this point and give relationships that the different memory kernels must satisfy to yield the same dynamics.