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The Quantum Interest Conjecture

L. H. Ford, Thomas A. Roman

TL;DR

The paper proves the quantum interest conjecture for delta-function negative/positive energy pulses of massless scalar fields in 2D and 4D Minkowski spacetime by leveraging quantum inequalities with compactly supported sampling functions. It demonstrates that a negative-energy loan must be repaid by a compensating positive pulse, with the overcompensation factor $\\epsilon$ monotonically increasing with the pulse separation $T$, and shows there is a maximum permissible separation. The moving-mirror example provides an explicit mechanism for quantum interest via Doppler shift, while a general formalism and small-$T$ and numerical analyses establish robust lower bounds on $\\epsilon$ across dimensions. These results constrain hypothetical exotic spacetime effects and inform the role of negative energy in quantum field theory.

Abstract

Although quantum field theory allows local negative energy densities and fluxes, it also places severe restrictions upon the magnitude and extent of the negative energy. The restrictions take the form of quantum inequalities. These inequalities imply that a pulse of negative energy must not only be followed by a compensating pulse of positive energy, but that the temporal separation between the pulses is inversely proportional to their amplitude. In an earlier paper we conjectured that there is a further constraint upon a negative and positive energy delta-function pulse pair. This conjecture (the quantum interest conjecture) states that a positive energy pulse must overcompensate the negative energy pulse by an amount which is a monotonically increasing function of the pulse separation. In the present paper we prove the conjecture for massless quantized scalar fields in two and four-dimensional flat spacetime, and show that it is implied by the quantum inequalities.

The Quantum Interest Conjecture

TL;DR

The paper proves the quantum interest conjecture for delta-function negative/positive energy pulses of massless scalar fields in 2D and 4D Minkowski spacetime by leveraging quantum inequalities with compactly supported sampling functions. It demonstrates that a negative-energy loan must be repaid by a compensating positive pulse, with the overcompensation factor monotonically increasing with the pulse separation , and shows there is a maximum permissible separation. The moving-mirror example provides an explicit mechanism for quantum interest via Doppler shift, while a general formalism and small- and numerical analyses establish robust lower bounds on across dimensions. These results constrain hypothetical exotic spacetime effects and inform the role of negative energy in quantum field theory.

Abstract

Although quantum field theory allows local negative energy densities and fluxes, it also places severe restrictions upon the magnitude and extent of the negative energy. The restrictions take the form of quantum inequalities. These inequalities imply that a pulse of negative energy must not only be followed by a compensating pulse of positive energy, but that the temporal separation between the pulses is inversely proportional to their amplitude. In an earlier paper we conjectured that there is a further constraint upon a negative and positive energy delta-function pulse pair. This conjecture (the quantum interest conjecture) states that a positive energy pulse must overcompensate the negative energy pulse by an amount which is a monotonically increasing function of the pulse separation. In the present paper we prove the conjecture for massless quantized scalar fields in two and four-dimensional flat spacetime, and show that it is implied by the quantum inequalities.

Paper Structure

This paper contains 9 sections, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Simple Example: The Moving Mirror
  3. A General Constraint on Pulse Separations
  4. Necessity of Quantum Interest
  5. A Simple Argument in Four Dimensions
  6. A More General Approach to Quantum Interest
  7. The Small $T$ Approximation
  8. Numerical Bounds for General $T$
  9. Conclusions

Figures (3)

  • Figure 1: A moving mirror which emits delta function pulses of negative and positive energy. The mirror is initially at rest at $x =x_0$, and emits a negative energy pulse as it begins to accelerate with constant proper acceleration $a$. At time $t_f$ and position $x =x_f$, it ceases accelerating and emits a pulse of positive energy. An inertial observer at $x=0$ receives these pulses at $t=t_1$ and $t=t_2$, respectively.
  • Figure 2: The lower bound on $\epsilon$ in two dimensions for different choices of $b$. Here the pulse separation $T$ is in units of $|\Delta E|^{-1}$.
  • Figure 3: The lower bound on $\epsilon$ in four dimensions for different choices of $b$. Here the pulse separation $T$ is in units of $(A/|\Delta E|)^{\frac{1}{3}}$.