Energy Density-Flux Correlations in an Unusual Quantum State and in the Vacuum

TL;DR

This paper investigates the relationship between negative energy density and energy flux in a special quantum state called the Helfer state, and compares it to vacuum fluctuations, revealing that vacuum fluctuations may encode the mechanisms enforcing quantum inequalities.

Contribution

It demonstrates that the correlation between energy density and flux in the Helfer state is similar to that in the vacuum, suggesting vacuum fluctuations encode quantum inequality mechanisms.

Findings

Negative energy density in Helfer state is correlated with positive energy flux.

Vacuum fluctuations exhibit similar energy density-flux correlations as Helfer state.

Quantum inequalities may be inherently encoded in vacuum fluctuation behavior.

Abstract

In this paper we consider the question of the degree to which negative and positive energy are intertwined. We examine in more detail a previously studied quantum state of the massless minimally coupled scalar field, which we call a ``Helfer state''. This is a state in which the energy density can be made arbitrarily negative over an arbitrarily large region of space, but only at one instant in time. In the Helfer state, the negative energy density is accompanied by rapidly time-varying energy fluxes. It is the latter feature which allows the quantum inequalities, bounds which restrict the magnitude and duration of negative energy, to hold for this class of states. An observer who initially passes through the negative energy region will quickly encounter fluxes of positive energy which subsequently enter the region. We examine in detail the correlation between the energy density and flux in the Helfer state in terms of their expectation values. We then study the correlation function between energy density and flux in the Minkowski vacuum state, for a massless minimally coupled scalar field in both two and four dimensions. In this latter analysis we examine correlation functions rather than expectation values. Remarkably, we see qualitatively similar behavior to that in the Helfer state. More specifically, an initial negative energy vacuum fluctuation in some region of space is correlated with a subsequent flux fluctuation of positive energy into the region. We speculate that the mechanism which ensures that the quantum inequalities hold in the Helfer state, as well as in other quantum states associated with negative energy, is, at least in some sense, already ``encoded'' in the fluctuations of the vacuum.