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Mixed states for reference frames transformations

Gaetano Fiore, Fedele Lizzi

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.

Mixed states for reference frames transformations

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.

Paper Structure

This paper contains 6 sections, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Reference Frame Transformations
  3. The Geometry of classical RF Transformations
  4. Pure and Mixed Translations in One Dimension
  5. Mixed states of RF (1+1)-dim Galilei transformations and thermal states
  6. Final Remarks

Figures (2)

  • Figure 1: The probability density for the pure and mixed states for the sum of two Gaussians. The chosen parameters are $\upalpha=0.75, a_2=2.5$.
  • Figure 2: The probability density for the pure and mixed states for the difference between two Gaussians. The chosen parameters are $\upalpha=0.75, a_2=2.5$.