Pomeron Evolution and Squeezed States in Quantum Optics

Alex Prygarin; Dima Cheskis

REVIEW 4 major objections 5 minor 15 references

Pomeron Evolution and Squeezed States in Quantum Optics

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that evolving squeezed pomeron states are equivalent to pomeron fan diagrams at leading order, with the displacement operator acting as pomeron propagation and the squeeze operator as pomeron splitting.

desk verdict A nice quantum-optics/pomeron analogy let down by a broken derivation: the equivalence claim doesn't survive contact with the algebra. read the letter →

arxiv 2505.02684 v2 pith:O7AFZ6GP submitted 2025-05-05 hep-ph hep-th

Dima Cheskis , Alex Prygarin This is my paper · ORCID

classification hep-phhep-th

keywords pomeronsqueezedcoherentstatesquantumopticsfandiagramstriplevertexBFKLdisplacementoperatorsqueeze

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that pomeron evolution, a key ingredient of high-energy QCD scattering, can be described by the squeezed-coherent-state formalism of quantum optics. In a zero-transverse-momentum toy model, the authors show that acting on a one-pomeron state with a squeeze and a displacement operator reproduces, at leading order, the pomeron fan diagrams of the dipole model. The central result is the equivalence $\hat S(z)\hat D(\beta)\hat a^\dagger|0\rangle \simeq \exp(\alpha Y \hat a^\dagger\hat a^\dagger\hat a - \alpha Y \hat a^\dagger\hat a\hat a^\dagger)|0\rangle$, which matches the generating function of fan-diagram evolution. The authors interpret the displacement operator as pomeron propagation and the squeeze operator as the triple-pomeron interaction.

What carries the argument

The central object is the squeezed coherent state $\hat S(z)\hat D(\beta)\hat a^\dagger|0\rangle$, built from the displacement operator $\hat D(\beta)=e^{\beta\hat a^\dagger-\bar\beta\hat a}$ and the squeeze operator $\hat S(z)=e^{\frac12 z\hat a^\dagger\hat a^\dagger - \frac12\bar z\hat a\hat a}$. The argument is carried by a power-counting scheme in a small parameter $\epsilon$ together with a momentum-conservation selection rule: monomials containing only $\hat a$ or only $\hat a^\dagger$ are dropped, and each pair $\hat a^\dagger\hat a^\dagger$ must be accompanied by at least one $\hat a$. This selects the leading terms, which recombine via commutation relations into an exponential of $\frac12 z\bar\beta\,\hat a^\dagger\hat a^\dagger\hat a - |\beta|^2\hat a^\dagger\hat a$; the term orderings then match the fan-diagram Hamiltonian of the multi-pomeron model.

What would settle it

Compute the next-to-leading terms that the paper discards, such as the $\epsilon^8$ and $\epsilon^9$ structures in equations (18)-(19) and the commutator-suppressed term in (22), and check whether including them changes the evolution of the generating function away from $dZ/dY = -\alpha Z + \alpha Z^2$; any correction of order $\epsilon^8$ or larger in the exponent would invalidate the claimed leading-order equivalence.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the product of a squeeze operator and a displacement operator, ordered as $\hat S(z)\hat D(\beta)\hat a^\dagger$, produces a state equivalent to the multi-pomeron fan-diagram state at leading order in a small parameter $\epsilon$ that tracks the QCD coupling strength. Using the power counting $\langle\hat a\rangle\sim O(\epsilon)$, $\langle\hat a^\dagger\rangle\sim O(1/\epsilon)$, $|\beta|\sim O(\epsilon^2)$, $|z|\sim O(\epsilon^3)$, and a momentum-conservation rule that eliminates monomials built only from creation or only from annihilation operators, the leading surviving terms exponentiate to $\exp\!\big(\tfrac12 z\bar\beta\,\hat a^\dagger\hat a^\dagger\hat a - |\beta|^2 \hat a^\dagger\hat a\big)$. Identifying $\tfrac12 z\bar\beta = \alpha Y$ and $|\beta|^2 = \alpha Y$ then yields equation (44), $\exp(\alpha Y \hat a^\dagger\hat a^\dagger\hat a - \alpha Y \hat a^\dagger\hat a\hat a^\dagger)|0\rangle$, which the authors show reproduces the fan-diagram generating function. They interpret this as showing that the displacement operator is pomeron propagation and the squeeze operator is pomeron interaction.

Load-bearing premise

The derivation depends on an ad hoc momentum-conservation rule in a zero-transverse-momentum toy model that makes all monomials built only from creation operators or only from annihilation operators vanish, together with the specific power-counting assignments $\langle\hat a\rangle\sim O(\epsilon)$, $\langle\hat a^\dagger\rangle\sim O(1/\epsilon)$, $|\beta|\sim O(\epsilon^2)$, and $|z|\sim O(\epsilon^3)$; if these assignments or the selection rule are changed, the surviving terms and the final exponential form change.

Editorial extensions

If this is right

If the equivalence holds, high-energy pomeron evolution at leading order can be represented as a squeezed coherent state, giving quantum optics a direct role in scattering theory.
The identifications $\tfrac12 z\bar\beta=\alpha Y$ and $|\beta|^2=\alpha Y$ tie the displacement and squeeze parameters to rapidity times the coupling, so propagation and splitting are governed by the same evolution scale.
The neglected higher-order terms are expected to correspond to pomeron loops, meaning pomeron squeezing is a concrete picture of the unitarization of the scattering amplitude.
The generating function $Z(Y,u)$ of the fan-diagram model follows from the squeezed-state representation, connecting pomeron evolution to the entropy and entanglement measures studied in quantum information.

Reading between the lines

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

The same Baker-Campbell-Hausdorff exponentiation used here could be carried to higher order in $\epsilon$ to produce a systematic expansion in which pomeron loops are generated by commutator terms; testing whether the dictionary remains closed would show whether the quantum-optics picture survives beyond leading order.
The ad hoc momentum-conservation rule is the most fragile input; repeating the derivation in a model with one nonzero transverse dimension, where momentum conservation is explicit, would reveal whether the squeezed-state mapping is a structural feature or an artifact of the zero-dimensional toy model.
If the mapping generalizes, the uncertainty product of the pomeron field could be used as an order parameter for unitarization, giving an experimental handle through multiplicity distributions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper has a genuinely attractive heuristic — read the displacement operator as pomeron propagation and the squeeze operator as pomeron splitting — but the central derivation is not sound. The claimed equivalence between evolving squeezed states and pomeron fan diagrams is not established.

What is actually new is the dictionary itself: I do not see that specific mapping in the cited literature, and the paper is honest about working in zero transverse dimension. The writing is clear and the target, Mueller's generating function, is correctly reproduced. Those are real merits.

The soft spots are load-bearing. Equation (23) asserts that the product of k copies of a†a†a equals k! times (a†a†a)^k. That is false for k≥2; a product of k copies of an operator is by definition the k-th power. For k=2, (a†a†a)^2 = a†^4 a^2 + 2a†^3 a, not 2(a†a†a)^2. Equation (21) misidentifies the correction as negligible, but under the paper's own power counting (a†a†)^2 a^2 and a†^3 a are the same order in ε. More fundamentally, the momentum-conservation rule that deletes βa† and −β̄a from D(β) is ad hoc and it does the real work. Those terms dominate the exact state: acting on |1>, βa† gives a |2> component of order ε^2 and −β̄a gives a vacuum component of order ε^2, both much larger than the fan-state corrections of order ε^4. So (44) is not a controlled leading-order expansion of S(z)D(β)|1>; it is a separate model where D(β) is effectively replaced by e^{−|β|^2 a†a}. In addition, the logarithm of a Gaussian unitary is at most quadratic in the ladder operators, so the cubic term a†a†a cannot appear in any exact BCH expansion of S(z)D(β). The final matching (42) fits β and z to Mueller's Hamiltonian, so the conclusion is packaged into the parameter choice.

The citation pattern is fine and the paper's own caveats about being a toy model are plainly stated. The algebraic errors, however, are internal inconsistencies with [a,a†]=1, not just missing rigor.

This is for readers interested in cross-field analogies as inspiration; nobody should cite it as a derivation. My recommendation: desk reject, unless the authors can supply a genuinely controlled limit and corrected operator identities.

Referee Report

4 major / 5 minor

Summary. The paper proposes a quantum-optics interpretation of pomeron evolution. It claims that the squeezed state S(z)D(β) applied to a one-pomeron state is equivalent, at leading order, to the pomeron fan-diagram state exp(αY a†a†a − αY a†a) a†|0>. The derivation expands S(z)D(β), imposes a power-counting scheme and a momentum-conservation rule, resums balanced creation/annihilation terms, and fixes β and z by comparison with Mueller's generating function. The paper then interprets D(β) as pomeron propagation and S(z) as pomeron interaction. I note explicitly that the stress-test objection to Eq. (21) does not land: the identity (a†a†)^2 a^2 = (a†a†a)^2 − 2 a†a†a†a is consistent with [a,a†]=1. The decisive problems are elsewhere, in Eqs. (23), (25), and the uncontrolled truncation of Section 2.

Significance. If the proposed equivalence were established, it would provide a novel dictionary between quantum optics and high-energy QCD, potentially connecting pomeron unitarization with squeezing. The paper is clearly written and open about its toy-model character, and the final interpretation is appealing. However, the central derivation is not sound: Eq. (23) is an algebraic error, Eq. (25) replaces the displacement operator by a non-unitary damping factor and discards terms that dominate the Fock-space expansion, and the truncation rules in Eqs. (15)–(20) are not a controlled perturbative expansion. No machine-checked proofs or reproducible artifacts are provided. The claimed equivalence is therefore not established, and the paper cannot be recommended for publication in its present form.

major comments (4)

[§2, Eq. (23)] Equation (23) is false: a product of k copies of the operator a†a†a is by definition its k-th power, (a†a†a)^k, and no factorial k! appears. The factor k! is used in Eq. (24) to convert the series with denominator (k!)² into an exponential; without it the resummation does not produce e^{(1/2)zβ̄ a†a†a}. This error is load-bearing and directly invalidates the central result, Eq. (44).
[§2, Eq. (25)] The replacement of D(β) by e^{-|β|²a†a} in Eq. (25) is not a controlled leading-order approximation. In the exact expansion of D(β), the linear terms βa† and −β̄a produce components of order ε² on a one-pomeron state, while the retained |β|²a†a term is of order ε⁴; the discarded terms dominate at the stated order. Moreover, e^{-|β|²a†a} is a non-unitary damping factor, not a displacement, so Eqs. (26) and (41) describe a different model from the squeezed coherent state defined in Eq. (5).
[§2, Eqs. (15)–(20)] The truncation in Eqs. (15)–(20) is not systematic. Eq. (15) drops all annihilation-operator contributions from S(z), and Eq. (20) imposes δ_{k,m} by fiat, discarding terms of the same formal order in ε. For example, the term β^{m−1}β̄(a†)^{m−1}a in Eq. (16) is larger than the retained (β̄)^m a^m term in Eq. (17) for m=2 under the stated power counting. The momentum-conservation rule is an additional assumption, not a consequence of the algebra, and it performs essential work in the derivation.
[§2, Eq. (42)] The identification of parameters in Eq. (42) is a fit, not a derivation: the free parameters β and z are chosen so that (1/2)zβ̄ε and |β|² reproduce αY in Mueller's Hamiltonian. No independent prediction or constraint is obtained from the squeezed-state formalism. Even if the preceding algebra were correct, this would establish only that the fan-diagram Hamiltonian can be matched to a squeezed-coherent state with suitably adjusted parameters, not that pomeron evolution is equivalent to squeezed states.

minor comments (5)

[§1, Eq. (1)] The actions of the ladder operators are reversed: one should have a|n⟩ = √n |n−1⟩ and a†|n⟩ = √(n+1)|n+1⟩, and the coefficients C±(n) are never defined.
[§2, Eq. (27)] The Baker–Campbell–Hausdorff formula is misprinted: the last double commutator should involve [Y,[Y,X]] or an equivalent symmetric term, not [Y,[Y,Y]].
[References] The reference list contains a duplicated label: there are two entries numbered [13], the first of which is empty.
[§2, Eq. (24)] The sentence 'Plugging this back into (24)' is self-referential; Eq. (24) is the equation being derived, so the reference should be to the preceding expression.
[§2, Eq. (12)] The 'smallness' of ladder operators defined through expectation values is not mathematically well-defined for unbounded operators; the footnote acknowledges this, but the subsequent power counting treats the assignment as exact.

Circularity Check

2 steps flagged · score 7.0 of 10

Central 'equivalence' to fan diagrams is produced by fitting z,β to αY and by an invented k! identity, not derived.

fitted input called prediction [Section 2, Eqs. (41)-(44)]
"Comparing the expressions in (41) and (32) we identify the displacement and the squeezing parameters as follows 1/2zβ̄ε = αY, |β|2 = αY, (42)"

The parameters z and β are not computed from any independent pomeron input; they are chosen so that the exponent in (41) becomes exactly the Mueller Hamiltonian αY(a†a†a − a†a) from (32)-(34). Therefore (44), the paper's central equivalence, is an input to the parameter identification rather than a derived prediction. The 'leading order squeezed pomeron state' is constructed to match the fan-diagram state by definition.
self definitional [Section 2, Eq. (23)]
"(ˆa†ˆa†ˆa)(ˆa†ˆa†ˆa)...(ˆa†ˆa†ˆa) =k! (ˆa†ˆa†ˆa)k (23)"

A product of k identical operators is, by definition, the k-th power of that operator; no factorial appears. This invented k! factor is precisely what converts the denominator (k!)² in the preceding line of (24) into the k! needed for the exponential series. Without this self-defined identity the series does not sum to exp((1/2)zβ̄ a†a†a), so the claimed reduction to the fan-diagram exponent is manufactured rather than derived.

full rationale

The claimed equivalence (44) is not a self-contained first-principles result: the squeeze and displacement parameters are fixed in (42) by matching the target fan-diagram Hamiltonian, so the final exponent is forced to equal αY(a†a†a − a†a). In addition, eq. (23) invents a k! factor that is needed to turn the truncated series into an exponential; without it, (24) would not exponentiate. The derivation also deletes the linear terms of D(β) through an ad hoc momentum-conservation rule, so the state analyzed is not the true squeezed-coherent state. These are not self-citation issues: no load-bearing argument rests on the authors' prior work, and Mueller's fan-diagram formalism is external. The circularity is instead that the central 'prediction' reduces by construction to the pomeron Hamiltonian it claims to reproduce.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central derivation leans on a small set of standard oscillator algebra plus several paper-specific power-counting and momentum-conservation rules. The squeeze and displacement parameters are not determined from QCD; they are matched to the known pomeron coupling in eq (42).

free parameters (3)

β (displacement parameter) = |β|^2 = αY (eq 42)
The displacement parameter is fixed to match the pomeron propagation term αY in the known Mueller Hamiltonian; no independent derivation from QCD is provided.
z (squeeze parameter) = (1/2)zβ̄ ε = αY (eq 42)
The squeeze parameter is chosen so the interaction term matches αY; it is not derived from pomeron physics.
ε (smallness parameter)
The expansion parameter ε is introduced ad hoc to justify the truncation; its physical meaning is unspecified.

assumptions (4)

standard math Ladder algebra [a,a†] = 1 and Fock space structure
Standard quantum harmonic oscillator algebra used throughout Section 2.
ad hoc to paper Monomials with only creation or only annihilation operators vanish by momentum conservation
Imposed in Section 2 (after eq 12) to eliminate terms like aaa and a†a†a†, despite zero transverse momentum.
ad hoc to paper Power counting ⟨a⟩~O(ε), ⟨a†⟩~O(1/ε), |β|~O(ε^2), |z|~O(ε^3)
Defined in eqs (12) and (14) to select the leading terms; the assignment ⟨a†⟩~O(1/ε) is inconsistent with complex conjugation of ⟨a⟩.
domain assumption Mueller's fan diagram Hamiltonian V1=α a†a†a, V2=-α a†a (ref [15]) is the correct benchmark
Used in eqs (32)-(34) as the target for the equivalence.

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Cite this review

Pith. "Pith review of Pomeron Evolution and Squeezed States in Quantum Optics." pith.science (2026). https://pith.science/paper/O7AFZ6GP

@misc{pith2026250502684,
  author       = {Pith},
  title        = {Pith review of: Pomeron Evolution and Squeezed States in Quantum Optics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O7AFZ6GP}},
  note         = {Machine review of arXiv:2505.02684}
}

read the original abstract

We apply the formalism of coherent states in quantum optics to pomeron evolution and show that evolving squeezed pomeron states are equivalent to pomeron fan diagrams at the leading order of perturbative expansion. Based on our results, we interpret the action of the displacement operator as pomeron propagation and the action of the squeeze operator as pomeron interaction.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 6 canonical work pages

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

L. N. Lipatov, ”Reggeization of the Vector Meson and the Vacuum Sin- gularity in Nonabelian Gauge Theories” Sov. J. Nucl. Phys. 23 (1976) 338

work page 1976

[2] [2]

V. S. Fadin, E. A. Kuraev and L. N. Lipatov, ”On the Pomeranchuk Singularity in Asymptotically Free Theories”, Phys. Lett. B 60 (1975) 50

work page 1975

[3] [3]

E. A. Kuraev, L. N. Lipatov and V. S. Fadin, ”Multi - Reggeon Processes in the Yang-Mills Theory”, Sov. Phys. JETP 44 (1976) 443; 45 (1977) 199; 10

work page 1976

[4] [4]

Ya. Ya. Balitskii and L. N. Lipatov, ”The Pomeranchuk Singularity in Quantum Chromodynamics”, Sov. J. Nucl. Phys. 28 (1978) 822

work page 1978

[5] [5]

BFKL pomeron in the next-to-leading approximation,

V. S. Fadin and L. N. Lipatov, “BFKL pomeron in the next-to-leading approximation,” Phys. Lett. B 429, 127 (1998) doi:10.1016/S0370- 2693(98)00473-0 [hep-ph/9802290]

arXiv 1998

[6] [6]

Deep inelastic scattering as a probe of entanglement,

D. E. Kharzeev and E. M. Levin, “Deep inelastic scattering as a probe of entanglement,” Phys. Rev. D 95, no.11, 114008 (2017) doi:10.1103/PhysRevD.95.114008 [arXiv:1702.03489 [hep-ph]]

arXiv 2017

[7] [7]

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D. E. Kharzeev, “Quantum information approach to high energy inter- actions,” Phil. Trans. A. Math. Phys. Eng. Sci. 380, no.2216, 20210063 (2021) doi:10.1098/rsta.2021.0063 [arXiv:2108.08792 [hep-ph]]

arXiv 2021

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Maximally entangled pro- ton and charged hadron multiplicity in Deep Inelastic Scattering,

M. Hentschinski, K. Kutak and R. Straka, “Maximally entangled pro- ton and charged hadron multiplicity in Deep Inelastic Scattering,” Eur. Phys. J. C 82, no.12, 1147 (2022) doi:10.1140/epjc/s10052-022-11122-1 [arXiv:2207.09430 [hep-ph]]

arXiv 2022