Mixed states for reference frames transformations

TL;DR

This paper explores how classical and quantum reference frame transformations can be mixed states, leading to new insights on how states appear differently depending on the reference frame, especially in the context of Galilei transformations.

Contribution

It introduces the concept of mixed states for reference frame transformations in both classical and quantum settings, revealing how states can appear pure or mixed depending on the frame.

Findings

Quantum reference frames can be in mixed states, affecting system state perceptions.

A thermal state of a reference frame induces a thermal state in a particle observed from another frame.

The framework relates to time/energy uncertainty principles.

Abstract

We discuss the concept of transformations among reference frames (classical or quantum). Usually transformations among classical reference frames have sharply defined parameters; geometrically they can be considered as {pure states in the parameters' space, and they form a group. It is however possible that the distributions in the parameters' space are mixed states; such states form a semigroup. Similarly, transformations among quantum reference frames can be either pure or mixed. This gives rise to interesting consequences: the state of a system can be pure with respect to a reference frame and mixed with respect to another; we concretely discuss this in the framework of Galilei transformations in 1+1 dimensions. In particular, if the state of a reference frame with respect to another frame is thermal at some temperature, a quantum particle in the pure (improper) rest state with respect to the first frame will appear in a thermal state with a related nonzero temperature with respect to the other. This can also be discussed in relation to the time/energy uncertainty relation.