The Quantum Interest Conjecture

L.H. Ford; Thomas A. Roman

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The Quantum Interest Conjecture

L.H. Ford, Thomas A. Roman

General Relativity and Quantum Cosmology · Submitted 26 Jan 1999

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.41 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.20 · compares with prior results 0.66 · mentions code or data availability 0.02.

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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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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.	background 1.00	0.06	0.03	asserted 0.98
s1	The restrictions take the form of quantum inequalities.	background 1.00	0.07	0.03	asserted 1.00
s2	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.	background 0.97	0.13	0.05	asserted 0.99
s3	In an earlier paper we conjectured that there is a further constraint upon a negative and positive energy delta-function pulse pair.	background 0.86	0.56	0.04	speculative 0.78
s4	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.	background 0.74	0.28	0.20	asserted 0.64
s5	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.	result 0.53	0.96	0.95	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper

Original 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.

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References

Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Proof of the Quantum Null Energy Condition
hep-th 2015-09 conditional novelty 7.0 of 10

The Quantum Null Energy Condition, a conjectured lower bound on the stress tensor in terms of the second variation of entropy, is proven for free and superrenormalizable bosonic fields on stationary null surfaces.
Holographic Proof of the Quantum Null Energy Condition
hep-th 2015-12 accept novelty 6.0 of 10

Holography proves the Quantum Null Energy Condition at leading order in large N for conformal field theories and their relevant deformations with Einstein gravity duals.
Vector wormholes as conduits for matter interaction
gr-qc 2024-12 conditional novelty 5.0 of 10

Massive one-form fields, with a reconstructed self-interacting potential, can support traversable wormholes while ordinary matter in a specially tuned case satisfies all classical energy conditions.
Lectures on quantum energy inequalities
gr-qc 2012-08 accept novelty 3.0 of 10

Quantum fields can have negative local energy density, but their averaged energy density along timelike curves is bounded below by state-independent quantum energy inequalities.
Modave lectures on energy conditions in quantum field theory and semi-classical gravity
hep-th 2026-05 accept novelty 2.0 of 10

Review of classical energy conditions, their quantum violations, and information-theoretic bounds for semi-classical gravity, based on Modave lectures.

Show 2 more citing papers

Exotic solutions in General Relativity: Traversable wormholes and 'warp drive' spacetimes
gr-qc 2007-10 accept novelty 2.0 of 10

Traversable wormholes, warp drives, and Krasnikov tubes all require stress-energy that violates the null energy condition, and quantum-inequality bounds push macroscopic versions of them out of reach.
Wormhole geometries in modified gravity
gr-qc 2024-12

How the literature uses it

27 arXiv papers cite this in 38 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 26 sentences without a confident label

Compares with it 2 papers

This bound is also similar in flavor to the so-called quantum inequalities that have been proposed by Ford & Roman for theories without gravity [ref] .

Upper and Lower Bounds on the Integrated Null Energy in Gravity · Introduction and Summary · 0.97
These integral variants of the NEC are also qualitatively very different from the standard averaged null energy condition (ANEC), see for instance [ref] , or the Ford–Roman quantum inequalities [ref] , or their variants [ref] , or even the recent non-linear energy conditions explored in [ref] .

Clausius entropy for arbitrary bifurcate null surfaces · Generalized second law · 0.92

Cites it as background 10 papers

According to [ref] , the quantum interest conjecture states that an energy loan, namely the negative energy, must always be repaid by the positive energy, with an interest which is determined by the magnitude and duration of the loan.

A Holographic Description of Negative Energy States · Introduction · 0.95
In [ref] , ANEC-type inequalities were initially proposed to hold for two and four-dimensional, free, massless quantized scalar fields.

A Holographic Dual of the Quantum Inequalities · Introduction and Summary · 0.97
The role of squeezed states (originally studied in the context of quantum optics [ref] ) in realizing negative energy densities in quantum field theory is well known [ref] .

Semi-local Bounds on Null Energy in QFT · A series of squeezed states · 0.92

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