Jordan in The Church of The Higher Hilbert Space: Entanglement and Thermal Fluctuations

TL;DR

This paper revisits Jordan's derivation of energy fluctuations in black body radiation, emphasizing the role of entanglement and pure states in explaining thermal fluctuations without explicitly using thermal states.

Contribution

It introduces a simple derivation showing that thermal fluctuations can be understood as reductions of pure entangled states in a higher Hilbert space, linking temperature to entanglement.

Findings

Fluctuations in small volumes are due to entanglement with the rest of the black body.

Pure states can produce thermal fluctuations through entanglement.

Temperature corresponds to the degree of entanglement in the system.

Abstract

I revisit Jordan's derivation of Einstein's formula for energy fluctuations in the black body in thermal equilibrium. This formula is usually taken to represent the unification of the wave and the particle aspects of the electromagnetic field since the fluctuations can be shown to be the sum of wave-like and particle-like contributions. However, in Jordan's treatment there is no mention of the Planck distribution and all averages are performed with respect to pure number states of radiation (mixed states had not yet been discovered!). The chief reason why Jordan does reproduce Einstein's result despite not using thermal states of radiation is that he focuses on fluctuations in a small (compared to the whole) volume of the black body. The state of radiation in a small volume is highly entangled to the rest of the black body which leads to the correct fluctuations even though the overall state might, in fact, be assumed to be pure (i.e. at zero temperature). I present a simple derivation of the fluctuations formula as an instance of mixed states being reductions of higher level pure states, a representation that is affectionately known as ``Church of the Higher Hilbert Space". According to this view of mixed states, temperature is nothing but the amount of entanglement between the system and its environment.