CoChat
Sign Up
Cited by 91Black Holes and Theoretical PhysicsRelativity and Gravitational TheoryCosmology and Gravitation Theories

Null energy conditions in quantum field theory

Open access
Fewster, Christopher J., Roman, Thomas A.·2003·Physical Review D
DOI 10.1103/physrevd.67.044003
Open this paper in CoChat — free accountOpen PDF

Abstract

For the quantized, massless, minimally coupled real scalar field in four-dimensional Minkowski space, we show (by an explicit construction) that weighted averages of the null-contracted stress-energy tensor along null geodesics are unbounded from below on the class of Hadamard states. Thus there are no quantum inequalities along null geodesics in four-dimensional Minkowski spacetime. This is in contrast with the case for two-dimensional flat spacetime, where such inequalities do exist. We discuss in detail the properties of the quantum states used in our analysis, and also show that the renormalized expectation value of the stress energy tensor evaluated in these states satisfies the averaged null energy condition (as expected), despite the nonexistence of a null-averaged quantum inequality. However, we also show that in any globally hyperbolic spacetime the null-contracted stress energy averaged over a timelike worldline does satisfy a quantum inequality bound (for both massive and massless fields). We comment briefly on the implications of our results for singularity theorems.

AI snapshot

Written from the paper’s full text.

TL;DR

For the massless minimally coupled real scalar field in four-dimensional Minkowski space, weighted averages of the null-contracted stress-energy tensor along null geodesics are unbounded from below over Hadamard states, so there are no null-worldline quantum inequalities in 4D Minkowski. The paper also shows that null-contracted stress-energy averaged along timelike worldlines does satisfy a quantum inequality in any globally hyperbolic spacetime.

Key findings

  • Theorem II.1: The quantity \(\langle \rho(f)\rangle_\omega\) is unbounded from below as \(\omega\) varies over Hadamard states; the explicit family \(\omega_\alpha\) satisfies \(\langle \rho(f)\rangle_{\omega_\alpha}\to-\infty\) as \(\alpha\to0\). Thus no null-worldline QNEIs exist in four-dimensional Minkowski spacetime.
  • This contrasts with two-dimensional flat spacetime, where such null quantum inequalities do exist; the 4D proof uses field modes almost, but not exactly, parallel to the null vector \(\ell^a\).
  • Despite the nonexistence of a null-averaged quantum inequality, the renormalized expectation value of the stress-energy tensor in the constructed states satisfies the averaged null energy condition (ANEC), as expected.
  • Theorem III.1: In any globally hyperbolic spacetime, the null-contracted stress-energy averaged over a smooth timelike worldline satisfies a quantum inequality lower bound for both massive and massless Klein-Gordon fields; large negative values on one null geodesic must be compensated by positive values on neighbouring geodesics.

Methods

  • Theoretical quantum field theory analysis of the massless minimally coupled real scalar field in 1+3-dimensional Minkowski space, with signature \(+---\) and units \(\hbar=c=1\).
  • The nonexistence proof constructs vector Hadamard states \(\omega_\alpha\) (\(\alpha\in(0,1)\)) as superpositions of the vacuum with multimode two-particle states, with momentum cutoff \(\Lambda=\Lambda_0/\alpha\) and excited modes lying in a cone of opening angle \(\cos^{-1}(1-\alpha)\) about \(\ell\). Parameters satisfy \(2\nu+3/2<\sigma<2\nu+2\); the analysis shows \(\rho_1(f)\to0\) and \(\rho_2(f)\to-\infty\) as \(\alpha\to0\).
  • An explicit example sets \(\nu=1\), \(\sigma=3.75\), \(\Lambda_0=1\) or \(2\), and \(\alpha=0.2\), \(0.05\), or \(0.005\). The timelike-worldline QNEI is derived by adapting the argument of Fewster (2000) and evaluated explicitly in 4D Minkowski space for an inertial observer and constant null vector.

Limitations

  • The paper states that the nonexistence result is proven only for flat spacetime, and the authors have no reason to believe a null-worldline QNEI is more likely to exist in curved spacetime. They consider only the massless field but comment that the results generalise directly to the massive case.
  • The ANEC consistency check is given for sampling functions satisfying the stated hypotheses and \(f(0)=0\); the ANEC itself is cited as established for a dense set of Fock-space states and a large subclass of Hadamard states.
  • The paper notes that no bound strong enough to guarantee the focussing required by the Penrose theorem could hold everywhere in an evaporating black hole spacetime, because that would be inconsistent with Hawking evaporation; whether physically interesting global results can be proved using the timelike-worldline inequality is left open.
  • Evident from the text: the work is a mathematical/theoretical construction with no experimental data; the explicit example’s function \(B\) lacks the joint-continuity property used in the general proof, though the paper states the needed integrals still exist.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevd.67.044003

Cites

50 of 50 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

7 of 18 cited sources checked · 5 supported · 0 flagged · accuracy not yet measured

  • Interpretationown data
    § D. Consistency with the ANEC · p. 10

    The ANEC cannot be obtained from the AWEC without more assumptions than used in Ref. [7].

    “our example shows - even in Minkowski space - that the ANEC cannot be obtained from the AWEC without more assumptions than used in Ref. [7].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [7] L.H. Ford and T.A. Roman, Phys. Rev. D 51 , 4277 (1995), gr-qc/9410043. source
  • Premisecitation
    § III. TIMELIKE WORLDLINE QNEIS · p. 10

    Techniques from microlocal analysis provide sufficient conditions for restricting the differentiated two-point function to the curve γ, which are satisfied for timelike γ owing to the singularity properties of Hadamard states.

    “Techniques drawn from microlocal analysis provide sufficient conditions for this to be accomplished, which are satisfied for timelike γ owing to the singularity properties of Hadamard states - see Ref. [14] for more details on this point.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [14] C.J. Fewster, Class. Quantum Grav. 17 , 1897 (2000), gr-qc/9910060. source
  • Findingcitation
    § Null energy conditions in quantum field theory · p. 1

    In any globally hyperbolic spacetime, the null-contracted stress-energy averaged over a timelike worldline satisfies a quantum inequality bound for both massive and massless fields.

    “However, we also show that in any globally hyperbolic spacetime the null-contracted stress energy averaged over a timelike worldline does satisfy a quantum inequality bound (for both massive and massless fields).”
    • Also stated in: § I. INTRODUCTION · p. 2 · boosted
      “For both massive and massless fields, these quantities do obey lower bounds - which we call timelike worldline QNEIs - as a direct consequence of the arguments used in Ref. [14].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [14] C.J. Fewster, Class. Quantum Grav. 17 , 1897 (2000), gr-qc/9910060. source

      The Introduction (equation 1.1; page not supplied) establishes a worldline quantum inequality for a minimally coupled Klein–Gordon field with mass m ≥ 0 in a globally hyperbolic spacetime, covering massive and massless cases. But the bounded quantity is the energy density contracted with the timelike observer’s velocity, uᵃuᵇTₐᵦ, not the null-contracted stress-energy claimed here. (LLM, conf=0.89)

      Experimental — accuracy not yet measured
  • Premisecitation
    § IV. CONCLUSION · p. 12

    Quantum inequalities along null geodesics do exist in two-dimensional flat spacetime, as shown in Ref. [7].

    “By contrast, it was shown in Ref. [7] that such bounds do exist in two-dimensional flat spacetime.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [7] L.H. Ford and T.A. Roman, Phys. Rev. D 51 , 4277 (1995), gr-qc/9410043. source
  • Premisecitation
    § Null energy conditions in quantum field theory · p. 1

    In two-dimensional flat spacetime, quantum inequalities along null geodesics do exist.

    “This is in contrast to the case for two-dimensional flat spacetime, where such inequalities do exist.”
    • Also stated in: § I. INTRODUCTION · p. 2 · plain
      “For massless fields in two-dimensional Minkowski space, this question was answered affirmatively in Ref. [7], using a Lorentzian sampling function.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [7] L.H. Ford and T.A. Roman, Phys. Rev. D 51 , 4277 (1995), gr-qc/9410043. source
  • May combine two claims. 11
  • Classified as probably not a substantial claim. 3
  • The quote only partly states this claim. 1

  • May combine two claims.

    Interpretationreasoning
    § III. TIMELIKE WORLDLINE QNEIS · p. 10

    The sufficient conditions for restricting the differentiated two-point function to γ would not be satisfied if γ was null, which explains why one cannot derive null worldline QNEIs using the arguments of Ref. [14].

    “However, the sufficient conditions would not be satisfied if γ was null, which explains why one cannot derive null worldline QNEIs using the arguments of Ref. [14] (although this does not in itself demonstrate the nonexistence of such bounds).”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [14] C.J. Fewster, Class. Quantum Grav. 17 , 1897 (2000), gr-qc/9910060. source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given
    § III. TIMELIKE WORLDLINE QNEIS · p. 12

    Both sides of the massless Gaussian expression scale by a factor of σ^2 under l^a → σl^a.

    “We note that both sides of this expression scale by a factor of σ 2 under ℓ a ↦→ σℓ a .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § B. An explicit calculation · p. 7

    Along the timelike curve (t,0,0,0), decreasing α increases both the amplitude and frequency of the oscillations of 〈:T_ab l^a l^b:〉_{ω_α}.

    “Along the timelike curve, however, decreasing α increases both the amplitude and frequency of the oscillations.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingcitation
    § III. TIMELIKE WORLDLINE QNEIS · p. 12

    In the massless case, the quantum inequality takes a simpler expression that takes the same form as the corresponding QWEI derived in [6].

    “In the massless case, we have the simpler expression”
    • Also stated in: § III. TIMELIKE WORLDLINE QNEIS · p. 12 · plain
      “This takes the same form as the corresponding QWEI derived in [6] which reads”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [6] C.J. Fewster and S.P. Eveson, Phys. Rev. D 58 , 084010 (1998), gr-qc/9805024. source

  • May combine two claims.

    Premisereasoning
    § A. Nonexistence result · p. 6

    In two-dimensional spacetime, the one-momenta of any two field modes are either parallel or anti-parallel; modes propagating in the same direction as the chosen null geodesic contribute nothing to the integral, and the only contribution comes from modes moving in the opposite direction and is bounded.

    “The crucial difference is that, in two-dimensional spacetime, the one-momenta of any two field modes are either parallel or anti-parallel. The modes which propagate in the same direction as the chosen null geodesic contribute nothing to the integral (due to factors of the form ℓ a k a ). The only contribution comes from modes moving in the opposite direction, and turns out to be bounded.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning
    § C. Convergence to the vacuum state · p. 8

    The averaged energy density is an unbounded quadratic form, so convergence of a sequence of states in Hilbert space norm does not imply convergence of the corresponding expectation values.

    “The answer to this question resides in the fact that the averaged energy density is an unbounded quadratic form, so the convergence of a sequence of states in the Hilbert space norm does not imply the convergence of the corresponding expectation values.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning
    § C. Convergence to the vacuum state · p. 9

    In the harmonic oscillator example, the expected energy of φ_n diverges as n→∞ while φ_n converges to the ground state |0〉.

    “and therefore diverges as n →∞ , while φ n manifestly converges to the ground state | 0 〉 , because ‖ φ n -| 0 〉‖ = n -1 / 4 → 0.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § B. An explicit calculation · p. 7

    The plots of the energy density show an oscillatory fringe pattern, with dark regions negative and light regions positive, resembling interference patterns because the dominant contribution arises from interference between the vacuum and two-particle components of ψ_α.

    “These plots share the common feature of an oscillatory fringe pattern, with dark regions representing negative values for 〈 ρ ( t, 0 , 0 , z ) 〉 ω α and light regions representing positive values. It is no coincidence that these plots resemble interference patterns: the dominant contribution arises precisely from interference between the vacuum and twoparticle components of ψ α .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § B. An explicit calculation · p. 7

    Averaged against a fixed sampling function, the rapid oscillations along the timelike curve tend to cancel, so that the averages ∫ f(t)〈ρ(t,0,0,0)〉_{ω_α} could be bounded below.

    “Averaged against a fixed sampling function, one might expect that the rapid oscillations would tend to cancel, so that the averages ∫ f ( t ) 〈 ρ ( t, 0 , 0 , 0) 〉 ω α could be bounded below. This is borne out by the results of Sec. III below.”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Interpretationcitation
    § B. An explicit calculation · p. 7

    The behavior of the energy density in the vicinity of the chosen null geodesic is analogous to that found in Ref. [19] for spatially averaged QIs: the central trough is a compactly supported region of the null geodesic where the energy density can be made unboundedly negative, and as one moves off the null geodesic the energy density oscillates rapidly in sign, as required for worldline QIs to be.

    “The behavior of the energy density in the vicinity of our chosen null geodesic, e.g., as exhibited in Fig. 2, is almost exactly analogous to that found in the analysis of Ref. [19] for spatially averaged QIs, in the following sense. There, it was shown that the sampled energy density could be unboundedly negative in a spatially compact region on a t =const surface, in four-dimensional Minkowski sp”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [19]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § D. Consistency with the ANEC · p. 9

    The ANEC was established by Klinkhammer [21] at least for a dense set of states in the Fock space of the Minkowski vacuum, and by Wald and Yurtsever [22] for a large subclass of Hadamard states [41].

    “established by Klinkhammer [21] at least for a dense set of states in the Fock space of the Minkowski vacuum, and by Wald and Yurtsever [22] for a large subclass of Hadamard states [41].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [21] G. Klinkhammer, Phys. Rev. D 43 , 2542 (1991).

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [22] R. Wald and U. Yurtsever, Phys. Rev. D 44 , 403 (1991).

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [41]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § I. INTRODUCTION · p. 2

    The results for the massless field generalise directly to the massive case.

    “Although we consider only the massless field, we comment that our results generalise directly to the massive case.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § B. An explicit calculation · p. 7

    The fringes point along spacelike directions which become more parallel to l^a as α decreases; these directions cannot be timelike, which follows from the existence of the timelike worldline QNEIs.

    “The fringes are centred near the null ray parallel to ℓ a running from the lower left to upper right corners of the figures, and in fact point along spacelike directions which become more parallel to ℓ a as α is decreased (moving from the left-hand to right-hand figure in each row). That these directions cannot be timelike follows from the existence of the timelike worldline QNEIs discussed in Sec”
    Quote in the paperNot a substantial claim
13 of 13

  • The paper relies on this without support.

    For a Hadamard state, the renormalised contracted stress tensor is a smooth function on spacetime, so the integral defining the weighted average converges.

    “If ω is a Hadamard state, the renormalised contracted stress tensor is a smooth function on spacetime, so the above integral will certainly converge.”

  • The paper relies on this without support.

    The expectation of the renormalised null-contracted stress-energy is unbounded from below as a function of the quantum state.

    “In fact, the expectation 〈 T ren ab ℓ a ℓ b 〉 ω of the renormalised null-contracted stress-energy is unbounded from below as a function of the quantum state ω .”
2 of 2

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • There are signs that the weak and null energy conditions may be violated at the quantum-microscopic scale, although they seem to be satisfied at macroscopic scales.

    “Itis worth mentioning that there are signs that WEC and NECmaybeviolated at the quantum-microscopic scale [4049], although they seem to be satisfied at the macroscopic scales.”

    From Alcubierre warp drive in spherical coordinates with some matter configurations (2023) · cites this paper as [45]

    The Introduction (p. 1) says quantum fields can violate both the NEC and WEC, supporting the quantum-scale part of the claim. It says many classical matter models satisfy the NEC, but does not establish that both conditions are satisfied at macroscopic scales; the bounds on averaged energy discussed in the Conclusion (pp. 12–13) are not the same as satisfaction of those conditions. (LLM, conf=0.86)

    Experimental — accuracy not yet measured

Cites

Flagged first, then by citations

Cited by

Sorted by citations

Related

Authors
Fewster, C. J., & Roman, T. A.
Affiliations
Central Connecticut State University (US)
Corresponding author
Roman, Thomas A.
ORCID
Fewster, Christopher J.
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2003-02-10
Volume
67
Issue
4
ISSN
  • 0556-2821 (ISSN-L)
  • 1089-4918
DOI
10.1103/physrevd.67.044003
OpenAlex
W2053550747
Cited by
91
References
33
Citations by year
2026: 4 · 2025: 3 · 2024: 4 · 2023: 5 · 2022: 8 · 2021: 11 · 2020: 3 · 2019: 3 · 2018: 3 · 2017: 2 · 2016: 2 · 2015: 6 · 2014: 5 · 2013: 5
FWCI
1.95
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
Keywords
null energy condition, quantum inequalities, energy-momentum tensor, Hadamard states, null geodesics, renormalized stress-energy tensor, globally hyperbolic spacetimes, singularity theorems, massless scalar field

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-30.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-30.

Cited sources are checked by gpt-6-sol.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review