Vacuum energy and relativistic invariance

E. Kh. Akhmedov

REVIEW

Vacuum energy and relativistic invariance

E. Kh. Akhmedov

High Energy Physics - Theory · Submitted 4 Apr 2002 · 12 references

Pith's verdict: The trace-identity argument is correct and clean, but the novelty is moderate and the framing oversells the surprise.

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arXiv IDhep-th/0204048 (version 2)

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CitationE. Kh. Akhmedov. "Vacuum energy and relativistic invariance." arXiv:hep-th/0204048 (2002). https://pith.science/paper/GAKWUCCW

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Editor's note

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The central claim holds up: with a Lorentz-invariant regulator, the zero-point energy of a free field goes at most as m²Λ², not Λ⁴, and a massless field contributes nothing. The cleanest part is the trace-identity route (Eqs. 19–22). ρ = (m²/4)⟨φ²⟩, and ⟨φ²⟩ is quadratically sensitive by power counting. That argument doesn't depend on which invariant regulator you pick, which is exactly what you want. The dim-reg check (Eqs. 10–12) is correct, and the observation that a 3-momentum cutoff violates ρ = −p is correct and worth stating clearly. The paper is also honest in comment (1) that the unrenormalized ρ is scheme-dependent, though that honesty sits awkwardly with the abstract's flat claim that the divergence is quadratic and not quartic. If ρ is scheme-dependent, then 'at most quadratic' is a statement about invariant schemes, not about the theory. The paper could say that more precisely. The stress-test worry about Eq. (22) is a fair point about the constructive demonstration but doesn't land on the central claim. The paper uses a Euclidean 4-cutoff to get Eq. (22) after criticizing that same regulator around Eq. (16). The defense — that Eq. (21) was derived without m²-differentiation, so the Eq. (17) objection doesn't apply — is correct as far as it goes, and the power counting on Eq. (20) is regulator-independent anyway. So Eq. (22) is an illustration, not the load-bearing step. Still, the paper would be stronger if it checked ρ = −p explicitly for the 4-cutoff on Eq. (21), or just dropped Eq. (22) and relied on the power counting. The bigger soft spot is framing. 'As is generally believed' is a bit of a strawman: Weinberg's review and the effective-potential literature already note that the quartic divergence is scheme-dependent. The paper's real contribution is the clean trace-identity statement, not the discovery that the quartic term is an artifact. Minor: the field-independent constant (the a₀ heat-kernel term) isn't really addressed. The trace identity only captures the m-dependent part, so 'massless fields contribute nothing' is true for the field-dependent piece but not for a pure cosmological constant. Worth a sentence. Who it's for: people who want a clean, short statement of why the Λ⁴ divergence is a regulator artifact. Not a solution to the CC problem, and the paper doesn't claim to be. Recommendation: worth a serious referee, but expect pushback on novelty and on engagement with the existing effective-potential literature.

Referee report

Significance

The paper makes a genuinely interesting and non-trivial point: the quartic divergence of the zero-point energy is regulator-dependent and can be removed by an invariant subtraction, as the dimensional-regularization result (Eq. 12) explicitly demonstrates. The trace-identity argument (Eqs. 19–22) is elegant and, if correct, would sharpen the cosmological-constant discussion by showing that the leading divergence is quadratic rather than quartic. The algebra in Eqs. (5)–(9) and (10)–(12) is correct, and the paper is candid in comment (1) that the unrenormalized ρ is scheme-dependent and that the real question is whether any invariant regularization produces O(Λ⁴) terms. The result is not a solution of the cosmological-constant problem, and the paper does not claim otherwise.

Major objections (3)

[Four-momentum cutoff, Eqs. (21)–(22)] The paper derives Eq. (22) by applying a Euclidean 4-momentum cutoff to the integral in Eq. (21), but it does not demonstrate that this cutoff preserves the invariance condition ρ = -p for the un-differentiated integral. The paper itself argues, in the paragraph around Eqs. (16)–(18), that the Euclidean 4-cutoff violates condition (3) when combined with m²-differentiation, and concludes that Eq. (16) 'cannot be trusted.' The defense that Eq. (21) was obtained without m²-differentiation is asserted rather than shown: no calculation verifies that the regulated ⟨φ²⟩ satisfies the trace identity and that the resulting ρ satisfies ρ = -p. As written, Eq. (22) is an illustration of the power-counting argument, not an independent proof of it. The paper should either demonstrate explicitly that the Euclidean 4-cutoff preserves both the trace identity and ρ = -p for Eq. (21), or use a regulator that manifestly does so (dimensional regularization, already employed for Eq. (12), is the natural candidate) and present Eq. (22) as a consistency check rather than as the derivation.
[Four-momentum cutoff, Eqs. (16)–(18)] The argument that Eq. (16) 'cannot be trusted' because it conflicts with Eq. (18) is circular as presented. Eq. (18) follows from Eq. (17) only under the assumption that the m²-differentiation is justified, and the paper's own point is that this differentiation is illegitimate for the Euclidean 4-cutoff. If Eq. (17) is not valid for that regulator, then Eq. (18) is not a valid consistency check for Eq. (16), and the conflict with Eq. (18) does not by itself establish that Eq. (16) is wrong. The paper should either justify the m²-differentiation for the Euclidean 4-cutoff (in which case Eq. (16) is a legitimate result and the conflict with Eq. (18) must be resolved differently), or replace the appeal to Eq. (18) with a direct argument that the m²-differentiated integral does not commute with the cutoff. The same issue applies to Eq. (22): the paper does not check whether Eq. (22) satisfies Eq. (18), and if it does not, the paper owes an explanation of why Eq. (22) is trustworthy while Eq. (16) is not.
[Four-momentum cutoff, Eq. (22)] The paper does not compute the pressure independently for the regulated theory and verify that p = -ρ. Instead, ρ = -p is assumed in deriving Eq. (20), and Eq. (20) is then used to obtain Eq. (22). This is circular unless the trace identity is independently valid for the regulated quantities. The paper should either compute p directly from Eq. (4) with a regulator that preserves the invariance condition and show that p = -ρ, or state explicitly that the trace identity is the assumption that defines the invariant regularization and that Eq. (22) is a consequence of that assumption rather than an independent check of it.

Minor objections (4)

[Abstract and comment (5)] The statement that 'the zero-point energies of free massless fields vanish' is a statement about the unrenormalized quantity in a particular regularization, not a physical statement about the Casimir energy. Footnote [7] distinguishes the free vacuum energy from the Casimir energy inside cavities, but the main text would be clearer if it said explicitly that the vanishing refers to the unrenormalized free-field vacuum energy in an invariant scheme, and that boundary-dependent Casimir energies are finite and non-zero.
[Eq. (12)] The paper trades the parameter μ for a mass scale Λ using the MS convention but does not specify the renormalization scale or the scheme in the equation itself. A short parenthetical stating that μ is the MS scale and that the finite part is scheme-dependent would help the reader.
[Eq. (17)] The relation (1 - 2m² ∂/∂m²) ρ = (d-1) p is quoted as following from comparing integrands of Eqs. (1) and (4), but the intermediate steps are not shown. Since this relation is used to derive Eq. (18) and to diagnose the failure of Eq. (16), a one-line derivation would make the logic easier to follow.
[Eqs. (10)–(11)] The factor μ^{4-d} is introduced to keep the dimensions of ρ and p correct, but the text does not explain why the same factor appears in both equations with the same power. A brief clarification would be useful for readers not familiar with dimensional regularization of vacuum-energy integrals.

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The paper

The paper claims

The quartic divergence of the vacuum energy is an artifact of a symmetry-breaking cutoff, not a property of free quantum fields, the paper argues.

The reading

The paper attacks a textbook number: the claim that the zero-point energies of free quantum fields diverge as the fourth power of an ultraviolet cutoff, giving a 'natural' vacuum energy density of order $M_{Pl}^4$ and a cosmological constant problem roughly 120 orders of magnitude wide. Akhmedov's argument is that the quartic divergence is not a fact about the fields but about the regulator. A three-momentum cutoff explicitly breaks relativistic invariance, and that invariance forces the vacuum energy-momentum tensor to have the form $T^{vac}_{mu nu} = const cdot g_{mu nu}$, which means the energy density and pressure must satisfy $rho = -p$. Checked against a scheme that respects the symmetry, dimensional regularization, only the logarithmic pieces survive, and those do satisfy $rho = -p$. Re-deriving the energy density from the trace of the energy-momentum tensor instead, $rho = frac{1}{4} m^2 langle 0 | varphi^2 | 0 rangle$, gives an integral whose leading divergence is quadratic by simple power counting, so free massless fields contribute exactly zero. If the argument holds, the Planck-scale 'natural value' loses its footing, although the paper is explicit that it does not solve the cosmological constant problem.

What carries the argument

The load-bearing identity is $rho = -p$ for the vacuum, which follows from the requirement that the vacuum energy-momentum tensor be proportional to the metric, $T^{vac}_{mu nu} = const cdot g_{mu nu}$; the paper calls this the relativistic invariance condition and treats it as the filter every candidate regularization must pass. The second piece of machinery is the trace relation for a free scalar field, $T^mu_{ mu} = m^2 varphi^2$, which converts the vacuum expectation of the trace into $4rho$ on one side and $m^2 langle 0 | varphi^2 | 0 rangle$ on the other, handing the author an expression for $rho$ that never required differentiating a divergent integral with respect to the mass. Together these two identities turn the divergence question into a power-counting question about a single propagator at coincident points, $D_F(x,x)$.

Extended reading notes

Core claim

The central claim is that the zero-point energy density of a free quantum field diverges at most quadratically, not quartically, because relativistic invariance forbids the quartic term. The author first shows that the standard sharp three-momentum cutoff produces a term $Lambda^3 sqrt{Lambda^2+m^2}$ in the pressure integral that survives as an off-integral boundary piece and violates the required relation $rho = -p$; the same happens with smooth three-momentum cutoffs. Dimensional regularization, which respects the symmetry, yields $rho = -p = -frac{m^4}{64pi^2}(ln(Lambda^2/m^2) + 3/2)$ and reproduces the coefficients of the logarithmic terms found with the naive cutoff, so the only pieces the cutoff got right are the ones that already obey invariance. The decisive step avoids differentiating with respect to $m^2$ (which the author shows is illegitimate for momentum cutoffs, since imposing $rho = -p$ there would force $rho propto m^4$ in conflict with the explicit cutoff result) and instead uses the free-field trace equation $T^mu_{ mu} = m^2 varphi^2$, giving $rho = frac{1}{4} m^2 langle 0 | varphi^2 | 0 rangle$. Power counting on that integral gives a leading quadratic divergence, and an explicit Euclidean four-momentum cutoff yields $rho = frac{1}{64pi^2}[Lambda^2 m^2 - m^4 ln((Lambda^2+m^2)/m^2)]$. The author notes that the interacting case is modified by the flat-space trace anomaly and requires separate study, and that in curved spacetimes the conformal anomaly adds finite terms without changing the divergence structure.

Load-bearing premise

The argument rests on the trace relation $T^mu_{ mu} = m^2 varphi^2$ holding exactly as an operator equation for a free field in flat spacetime, with no quantum correction; if that relation were modified even in this setting, the derivation of the quadratic divergence from $rho = frac{1}{4} m^2 langle 0 | varphi^2 | 0 rangle$ would not go through.

Editorial extensions

If this is right

The zero-point energy of any free massless field vanishes in flat spacetime, including the free electromagnetic field, so the free-field contribution to the cosmological constant is zero rather than Planck-scale.
The 'natural' estimate $rho sim M_{Pl}^4$ is not a consequence of free-field quantum theory but of a regulator that breaks relativistic invariance, so it should not be quoted as the generic expectation.
In a curved spacetime of curvature radius $a$, a free massless field contributes vacuum energy only of order $a^{-4}$, which for the present universe is negligible against the critical density.
Supersymmetry is not needed to remove quartic divergences; the cancellations between bosons and fermions that were credited with doing the job are, on this argument, already present field by field.
Casimir-type zero-point energies computed with an invariant regularization come out finite, so no subtraction of a free-vacuum energy is required when boundaries are present.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the argument is right, the cosmological constant problem changes character: the question is no longer why a Planck-scale number cancels to 120 digits, but why the surviving quadratic and logarithmic pieces are as small as they are, which is a different and perhaps more tractable fine-tuning question.
The reasoning suggests a sharp criterion for any proposed vacuum-energy calculation: check whether the regulator preserves $rho = -p$ before believing its leading divergence. Calculations that fail this test are not merely scheme-dependent, they are answering a different question.
Because the trace route gives $rho$ in terms of $langle 0 | varphi^2 | 0 rangle$, the divergence structure of the vacuum energy is tied to the coincidence limit of the propagator; in interacting theories the trace anomaly adds a term proportional to the beta function, which is where the author's flat-space caveat would need to be reopened.
A useful cross-check would be a lattice or Pauli-Villars computation of the full vacuum stress tensor in a covariant scheme, looking directly for an $O(Lambda^4)$ piece that survives the $rho = -p$ constraint.

Original abstract

It is argued that the zero-point energies of free quantum fields diverge at most quadratically and not quartically, as is generally believed. This is a consequence of the relativistic invariance which requires that the energy density of the vacuum $\rho$ and its pressure $p$ satisfy $\rho=-p$. The usually obtained quartic divergence is an artifact of the use of a noninvariant regularization which violates this relation. One consequence of our results is that the zero-point energies of free massless fields vanish. Implications for the cosmological constant problem are briefly discussed.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.17 · human participants 0.02 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.63 · compares with prior results 0.93 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

It is argued that the zero-point energies of free quantum fields diverge at most quadratically and not quartically, as is generally believed. This is a consequence of the relativistic invariance which requires that the energy density of the vacuum $\rho$ and its pressure $p$ satisfy $\rho=-p$. The usually obtained quartic divergence is an artifact of the use of a noninvariant regularization which violates this relation. One consequence of our results is that the zero-point energies of free massless fields vanish. Implications for the cosmological constant problem are briefly discussed.

background contribution method result interpretation main claim

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	It is argued that the zero-point energies of free quantum fields diverge at most quadratically and not quartically, as is generally believed.	result 0.92	0.42	0.87	asserted 0.82
s1	This is a consequence of the relativistic invariance which requires that the energy density of the vacuum $\rho$ and its pressure $p$ satisfy $\rho=-p$.	interpretation 0.85	0.28	0.15	asserted 1.00
s2	The usually obtained quartic divergence is an artifact of the use of a noninvariant regularization which violates this relation.	interpretation 0.88	0.24	0.12	asserted 1.00
s3	One consequence of our results is that the zero-point energies of free massless fields vanish.	result 0.64	0.85	0.13	asserted 0.99
s4	Implications for the cosmological constant problem are briefly discussed.	interpretation 0.93	0.41	0.03	unclear 0.89

What kind of paper the classifier reports this is: theoretical result 0.84 · evidence from argument 0.81.

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References

Works cited (12)

[1]

S. Perlmutter et al. Astrophys.\ J.\ 517 , 565 (1999)

View on Pith 1999
[2]

A. G. Riess et al. Astron.\ J.\ 116 , 1009 (1998)

View on Pith 1998
[3]

For reviews see, e.g.,

View on Pith
[4]

S. Weinberg, Rev.\ Mod.\ Phys.\ 61 , 1 (1989)

View on Pith 1989
[5]

V. Sahni and A. Starobinsky, Int.\ J.\ Mod.\ Phys.\ D 9 , 373 (2000)

View on Pith 2000
[6]

S. M. Carroll, Living Rev.\ Rel.\ 4 , 1 (2001)

View on Pith 2001
[7]

There are several ways to arrive at this result. One may note that the vacuum can be considered a perfect fluid; using the general form of the energy-momentum tensor of perfect fluids T_ =-p g_ +( +p)U_ U_ where U_ is the four-velocity vector, one obtains ( rel1 ). Alternatively, one may start from a flat spacetime with Minkowski metric g_ = _ diag(1, -1,...

View on Pith
[8]

C. G. Callan, S. R. Coleman and R. Jackiw, Annals Phys.\ 59 , 42 (1970)

View on Pith 1970
[9]

D. M. Capper and M. J. Duff, Nuovo Cim.\ A 23 , 173 (1974)

View on Pith 1974
[10]

S. Deser, M. J. Duff and C. J. Isham, Nucl.\ Phys.\ B 111 , 45 (1976)

View on Pith 1976
[11]

Note that the Pauli-Villars regularization cannot be applied directly to eq. ( rho1 ) because when the regularization is lifted (i.e. in the limit of the infinite masses of the regulator fields) the original divergent expression is not recovered in that case

View on Pith
[12]

This does not apply to the zero-point energies inside cavities or when the boundaries are present, and so the Casimir energies do not vanish, of course. However, when calculated in a covariant way and with a relativistically invariant regularization, these energies must be finite, and no subtraction of the energy of the free vacuum is necessary. reference...

View on Pith

Forward citations

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How the literature uses it

42 arXiv papers cite this in 64 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 · 49 sentences without a confident label

Contests it 3 papers

The physical justification of this last assumption will be discussed further in <Ref>, but we will summarize this discussion here: * The high-energy cutoff $Λ$ used to determine $ρ_vac$ in <Ref> inherently violates local Lorentz invariance already, so using this in combination with <Ref> is inconsistent [ref] .

Can the fluctuations of the quantum vacuum solve the cosmological constant problem? · Cosmological Dynamics under Semiclassical Stochastic Gravity · contest 0.75
Different from the assertions of [ref] , the result [ref] stays infinite also in this case, for the following reason.

The zero-point energy of elementary quantum fields · Appendix A: Regularization · contest 0.80
Different from the assertions of [ref] , the result [ref] stays infinite also in this case, for the following reason.

Canonical quantization of elementary fields · Appendix A: Regularization · contest 0.85

Uses its data or values 1 paper

For a massless field we get zero, while for a scalar field with mass $m$ we get (for 3+1 dimensions) [ref] ρ=-p=-m^464π^2 ( log( M^2m^2)+32)

Three puzzles in cosmology · The cosmological constant · 0.83

Compares with it 2 papers

The above result agrees with the vacuum zero point energy density in flat background [ref] .

Vacuum Zero Point Energy and its Statistical Correlations in dS Background · Heavy Field with Imaginary $ν$ · 0.93
This result for a massless field is also obtained using a dimensional regularization scheme [ref] in a flat space-time.

Vacuum energy as dark matter · UV regime · 0.91

Cites it as support 1 paper

And vacuum energy develops both quartic and quadratic sensitivities to the cutoff scale [ref] .

Gauge and Poincare properties of the UV cutoff and UV completion in quantum field theory · Poincare-Breaking UV Cutoff · 0.96

Cites it as background 7 papers

A number of approaches have been proposed to eliminate Lorentz-violating divergences from a single field by modifying the regulator [ref] or reinterpreting the divergences [ref] .

Logarithmically Divergent Vacuum Energy in Effective Field Theory · Introduction · 0.88
Many other approaches and renormalization prescriptions have been advocated to deal with the vacuum energy, see [ref] , for instance.

Dark energy in quantum field theory: Implications on modern cosmology · Minimal Subtraction scheme and the fine-tuning problem · 0.98
The contribution of a quantum matter field to the vacuum energy density is given by the following expression (see, e.g., Refs. [ref] ): [equation] where $M$ is the mass of the matter field considered and $E_uv$ is the ultraviolet energy cutoff, which is typically assumed to be of the order of the energy scale $E_Planck$ defined above.

Dynamics of the quantum vacuum: Cosmology as relaxation to the equilibrium state · Introduction · 0.93

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

S. Perlmutter et al. Astrophys.\ J.\ 517 , 565 (1999)

View on Pith 1999

[2] [2]

A. G. Riess et al. Astron.\ J.\ 116 , 1009 (1998)

View on Pith 1998

[3] [3]

For reviews see, e.g.,

View on Pith

[4] [4]

S. Weinberg, Rev.\ Mod.\ Phys.\ 61 , 1 (1989)

View on Pith 1989

[5] [5]

V. Sahni and A. Starobinsky, Int.\ J.\ Mod.\ Phys.\ D 9 , 373 (2000)

View on Pith 2000

[6] [6]

S. M. Carroll, Living Rev.\ Rel.\ 4 , 1 (2001)

View on Pith 2001

[7] [7]

There are several ways to arrive at this result. One may note that the vacuum can be considered a perfect fluid; using the general form of the energy-momentum tensor of perfect fluids T_ =-p g_ +( +p)U_ U_ where U_ is the four-velocity vector, one obtains ( rel1 ). Alternatively, one may start from a flat spacetime with Minkowski metric g_ = _ diag(1, -1,...

View on Pith

[8] [8]

C. G. Callan, S. R. Coleman and R. Jackiw, Annals Phys.\ 59 , 42 (1970)

View on Pith 1970

[9] [9]

D. M. Capper and M. J. Duff, Nuovo Cim.\ A 23 , 173 (1974)

View on Pith 1974

[10] [10]

S. Deser, M. J. Duff and C. J. Isham, Nucl.\ Phys.\ B 111 , 45 (1976)

View on Pith 1976

[11] [11]

Note that the Pauli-Villars regularization cannot be applied directly to eq. ( rho1 ) because when the regularization is lifted (i.e. in the limit of the infinite masses of the regulator fields) the original divergent expression is not recovered in that case

View on Pith

[12] [12]

This does not apply to the zero-point energies inside cavities or when the boundaries are present, and so the Casimir energies do not vanish, of course. However, when calculated in a covariant way and with a relativistically invariant regularization, these energies must be finite, and no subtraction of the energy of the free vacuum is necessary. reference...

View on Pith