Scattering matrix and inclusive scattering matrix in algebraic quantum field theory

Albert Schwarz

REVIEW 3 major objections 4 minor 1 cited by

Scattering matrix and inclusive scattering matrix in algebraic quantum field theory

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes an inclusive scattering matrix in algebraic quantum field theory, defined by on-shell generalized Green functions, and proves that inclusive cross-sections are expressed through it by an LSZ-like formula.

desk verdict A clean but sketchy formalization of inclusive scattering in algebraic QFT; the central claim is proved only in the particle-interpretation case and unsupported in the advertised unstable/no-particle regimes. read the letter →

arxiv 1908.09388 v3 pith:A7BX4QVA submitted 2019-08-25 hep-th math-phmath.MP

Albert Schwarz This is my paper · ORCID

classification hep-thmath-phmath.MP MSC 81T0581U20

keywords algebraicquantumfieldtheoryinclusivescatteringmatrixgeneralizedGreenfunctionsLSZreductionformulaquasiparticlescross-sectionsHaag-Ruelleasymptoticcommutativity

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 sets out to make scattering theory work in algebraic quantum field theory even when the conventional unitary scattering matrix does not exist. Its proposal is an inclusive scattering matrix, defined as the on-shell value of generalized Green functions built from one time-ordered and one anti-time-ordered product of operators evaluated in the stationary state. The central claim is that inclusive cross-sections—the probability of seeing specified outgoing excitations together with anything else—are given by matrix elements of this inclusive S-matrix through a formula of LSZ type. That would matter because quasiparticles, the elementary excitations of nontrivial stationary states, are usually unstable, and because actual experiments often measure inclusive rather than exclusive final states. The paper carries out the construction under cluster-property or asymptotic-commutativity assumptions and gives a heuristic treatment of the unstable case for times much shorter than the lifetime.

What carries the argument

The load-bearing object is the generalized Green function in a stationary state $\omega$, defined as $G_n=\omega(MN)$ with $N=T(B_1(x_1,t_1)\cdots B_n(x_n,t_n))$ the chronological product and $M=T^{opp}(B_1^*(x'_1,t'_1)\cdots)$ the antichronological product; equivalently $G_n=(Q\omega)(1)$ where $Q$ is built from the operators $B$ and $\tilde B$ acting on linear functionals. The inclusive scattering matrix is then the on-shell limit of $G_n$, meaning after multiplying by factors $\Lambda_i(p_i)(\epsilon_i\mp\varepsilon(p_i))$ and letting the energies $\epsilon_i$ tend to the one-particle energies, in direct analogy with how ordinary scattering amplitudes are extracted from time-ordered Green functions. That identification is what does the work: it converts inclusive probabilities, which sum over unobserved final particles, into objects that algebraic quantum field theory can compute directly from correlation functions of the state.

What would settle it

Take a solvable model with a resonance, i.e. a two-point Green function whose pole in $\epsilon$ sits at $\varepsilon(p)+i\Gamma(p)$ with small $\Gamma>0$, and compute the inclusive cross-section for producing the resonance plus anything else by two routes: the on-shell generalized Green-function residue defined in Section 9, and the explicit sum over final states in the $t\ll T$ approximation. If the two disagree once terms of order $t/T$ are included, the claimed LSZ-like formula for the inclusive S-matrix fails. A more direct disproof would be a state whose truncated correlation functions decay only as a power law; then the proof of $\int\|\dot\Psi(t)\|dt<\infty$ gives no bound, and an explicit logarithmic divergence of that integral would refute the asymptotic-limit argument.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the inclusive scattering matrix exists and is computable where the ordinary scattering matrix is not available. If $\omega$ is a translation-invariant stationary state and $\Phi_k(p)$ are elementary excitations with strictly convex dispersion laws $\varepsilon_k(p)$, the matrix elements of the inclusive S-matrix are obtained by taking on-shell limits of generalized Green functions $G_n=\omega(MN)$, where $N=T(B_1(x_1,t_1)\cdots B_n(x_n,t_n))$ is chronological and $M=T^{opp}(B_1^*(x'_1,t'_1)\cdots)$ is antichronological. The paper proves that expectation values of out-operators such as $\nu(a^\dagger_{out,k_1}(p_1)a_{out,k_1}(p_1)\cdots)$, which encode the probability density for finding specified outgoing quasiparticles plus unobserved ones, are equal to these on-shell generalized Green functions. When the theory does have particle interpretation, the conventional LSZ formula for the scattering matrix reappears as the special case with no unobserved final particles, so the inclusive object is the more general and more widely applicable one.

Load-bearing premise

The construction assumes that the smeared Heisenberg operators $\hat B_k(f,t)$ have well-defined limits as $t\to\pm\infty$ in the state $\omega$, which in turn rests on strong cluster properties or asymptotic commutativity, strictly convex dispersion laws, and a one-particle spectrum separated from the multi-particle continuum; for genuinely unstable quasiparticles these limits do not exist, and the appendix supplies only a heuristic estimate valid for times much shorter than the lifetime $T$.

Editorial extensions

If this is right

Collision probabilities for specified outgoing particles plus anything else acquire a direct formula from correlation functions, with no need to construct interacting multiparticle states or a unitary S-matrix.
In any theory with particle interpretation, the conventional scattering matrix is recovered as the exclusive special case, so the inclusive formula extends rather than replaces the LSZ formalism.
Quasiparticle scattering in thermal or other stationary states becomes formulable through the same on-shell generalized Green functions whenever the spectral and cluster assumptions hold.
For space-time dimension $d\ge 4$, the non-overlap condition on velocity supports can be dropped, widening the range of collision kinematics covered by the construction.

Reading between the lines

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

This suggests that inclusive S-matrix elements could be read directly from Keldysh or thermo-field diagrammatics, since those formalisms are built on the same generalized Green functions; matching the algebraic residues to standard diagrammatic inclusive rates would be a concrete test.
The heuristic $t\ll T$ treatment implies a falsifiable signature: inclusive cross-sections for long-lived quasiparticles should be approximately time-independent on time scales below their lifetime, with corrections of order $t/T$; a plateau of that kind could be looked for in analog condensed-matter experiments.
If the on-shell prescription remains stable beyond the stated assumptions, it may offer a definition of scattering in theories with infraparticles or long-range interactions where ordinary particles and Møller operators are not defined, although the paper itself excludes infraparticles.

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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

The paper does something genuinely new: it defines an inclusive scattering matrix as an on-shell generalized Green function in the algebraic setting, and argues that inclusive cross-sections can be expressed as its matrix elements. That is a real step beyond the author's earlier perturbative treatment and beyond the standard Keldysh/TFD reviews. The construction of Møller operators for quasiparticles from cluster properties or asymptotic commutativity, and the LSZ-type identification for particles, are laid out carefully and are plausible. The paper is also honest about its limits: it flags infraparticles, and the appendix explicitly says unstable quasiparticles force t << T, with no t → ∞ limit.

That honesty makes the main soft spot all the more visible. Section 9's derivation of the inclusive cross-section from the inclusive S-matrix uses the out-operator limits in formula (10). Those limits are established only on vectors built from the image of S_+ (or S_-), and the argument works when the two images coincide—i.e., in the particle-interpretation case, or at least when a unitary S exists. But the paper's advertised motivation is exactly the opposite: theories without particle interpretation, and unstable quasiparticles. In those regimes the limits are not available, and the paper simply asserts that the inclusive cross-section is still given by the inclusive S-matrix. The appendix offers only a heuristic estimate for t << T. So the central claim is not proved in the two cases named in the abstract. This is a load-bearing gap, not a trivial technicality.

The estimates that support the existence of scattering states are also sketched rather than fully proved, with references to the author's book. That is acceptable for a research paper but it does mean a referee would need to verify a fair amount of machinery.

Who is this for? People working in mathematical QFT and in formal condensed-matter treatments of quasiparticle scattering. They will find a useful conceptual framework and a clear statement of what would need to be proved. The piece is not a finished theorem paper; it is more a research announcement with partial proofs and a serious conjecture.

I would engage with it, and I think a good journal should send it to a referee. A referee should ask for a precise separation of what is proved (the unitary/particle-interpretation case) from what is conjectured (the non-unitary and unstable cases), and for either a proof or an explicit conjecture in those cases. As written, the abstract overstates the proven content, but the core idea is solid and worth taking seriously.

Referee Report

3 major / 4 minor

Summary. The paper develops a framework for scattering theory in algebraic quantum field theory, aiming to cover quasiparticles (elementary excitations of stationary translation-invariant states) and theories without a particle interpretation. It constructs Møller operators S± from time limits of smeared good operators, defines the conventional scattering matrix when the theory has a particle interpretation, and introduces an 'inclusive scattering matrix' defined as the on-shell value of generalized Green functions (GGreen functions). The central claim is that inclusive cross-sections—probabilities of finding specified outgoing quasiparticles plus anything else—are expressed as matrix elements of this inclusive scattering matrix, via a formula similar to the LSZ formula. The paper also discusses unstable quasiparticles in an appendix, arguing heuristically that for times much shorter than the lifetime the stable-particle picture approximately applies.

Significance. If the central claim were fully established, the paper would provide a useful algebraic framework for inclusive scattering that does not require a unitary S-matrix, with applications to quasiparticle collisions and theories without particle interpretation. The construction of Møller maps from asymptotic limits and the connection between inclusive quantities and generalized Green functions are natural and potentially valuable. The paper is also commendably explicit about limitations: it notes infraparticles are not covered, and it acknowledges in the appendix that unstable quasiparticles cannot be handled by taking t to infinity. However, the advertised generality is not matched by the proof: the key Section 9 identification is only sketched for the case where the incoming state lies in the domain of the out-operators, and the non-particle-interpretation and unstable-quasiparticle regimes are asserted rather than derived.

major comments (3)

[Section 9, Eq. (23) and following paragraph] The derivation of the inclusive cross-section formula assumes that the incoming vector (8) lies in the domain of the out-operators a_out, which are defined through the limit (10). Formula (10) is proven only on vectors of the form (7), i.e. on the image of S_+. The incoming state (8) lies in the image of S_-, so the step requires that the images of S_+ and S_- coincide or that a separate argument supplies the needed domain property. In the case the paper advertises as the motivation—the absence of a unitary scattering matrix—this is not established. The paragraph after (23) simply asserts that the inclusive cross-section can still be expressed in terms of the inclusive S-matrix, without proof. Since this is the central claim of the paper, the argument must either be completed or the claim restricted to cases where the domain issue is resolved.
[Appendix, paragraph on unstable particles] The appendix states that for unstable particles 'we cannot take the limit t → ∞; for unstable particles we should always assume that t << T.' This directly undermines the advertised application to quasiparticle collisions, because the inclusive LSZ-like formula in Section 9 relies on the limit t→∞ to replace out-operators by their asymptotic limits (10). The appendix offers only a heuristic estimate for times much smaller than the lifetime, not a derivation of the inclusive cross-section formula. Thus the abstract's claim that the inclusive scattering matrix is 'always necessary if we want to consider collisions of quasiparticles' is not supported by the proof given for the unstable case.
[Section 9, definition of inclusive S-matrix] The paper defines the inclusive S-matrix as the on-shell GGreen function, and then states that inclusive cross-sections 'can be expressed in terms of' this inclusive S-matrix. Part of this statement is true by construction, because the inclusive S-matrix was defined exactly as the expression obtained from the expectation value (23) after taking the on-shell limit. The nontrivial content is the physical identification of (22) as the inclusive cross-section and the derivation showing that (23) equals the on-shell GGreen function. This distinction is not made explicit, and the derivation of the latter identity is incomplete outside the particle-interpretation regime. The authors should separate the definitional identity from the substantive physical claim and prove the substantive part under precisely stated assumptions.

minor comments (4)

[Abstract and Introduction] The abstract contains a missing 'of': 'expressed in terms generalized Green functions' should read 'expressed in terms of generalized Green functions.'
[Section 6, Eq. (15)] The coefficient functions N(p,p'), T1(p), and T2(p') are defined after (15) but their dependence on x and t is not displayed in the notation, which makes the formula harder to follow; adding the arguments would improve clarity.
[Section 9, Eq. (22)] The notation ν(a+_{out,k1}(p1)a_{out,k1}(p1)...) is used for a probability density in momentum space, but the state ν was earlier defined as a linear functional on the algebra A. Since the out-operators are not in general elements of A, the domain of ν in (22) should be clarified, for example by stating that ν is extended to the algebra generated by the out-operators.
[Section 5, Eq. (13)] The text notes that formula (13) is proved only for distinct momenta, but this restriction is not repeated when (13) is used later in Section 8; adding a cross-reference would help the reader track the validity conditions.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-definitional labeling: the inclusive S-matrix is defined as the on-shell GGreen function, so the final identification of inclusive cross-sections with its matrix elements is partly by construction; the intervening (10)-based calculation is independent and not circular.

self definitional [Section 9, 'Generalized Green functions. Inclusive scattering matrix' (definition of inclusive S-matrix and Eqs. (22)-(23) computation)]
"Let us define inclusive S-matrix as on-shell GGreen function. We will show that in the case when the theory has particle interpretation inclusive cross-section can be expressed in terms of inclusive S-matrix. ... Expressing the out-operators by the formula (10) we obtain the expression of (23) in terms of GGreen functions on-shell."

Formula (23), via (22), is the object that encodes inclusive cross-sections. After substituting (10), the paper obtains an on-shell GGreen function. But 'inclusive S-matrix' was introduced, a few sentences earlier, as 'on-shell GGreen function'. Therefore the concluding claim that inclusive cross-sections are expressed through matrix elements of the inclusive S-matrix is obtained by attaching the defined name to the computed expression; no property of an independently characterized S-operator is used at that final step. The nontrivial content is the reduction of (22)/(23) to GGreen functions using the limit theorem (10), which is independent of the definition and not circular. This is a minor self-definitional labeling step, not a collapse of the derivation.

full rationale

The paper's central derivation runs as follows: (22) identifies an out-operator expectation value with an inclusive probability; formula (10) converts out-operators into asymptotic limits of good operators; substituting into (23) yields an on-shell generalized Green function. Since Section 9 defines 'inclusive S-matrix' as 'on-shell GGreen function', the final statement that inclusive cross-sections are expressed by matrix elements of the inclusive S-matrix is, in its last labeling step, an application of that definition. This is a mild presentational circularity, but the physically substantive content lies in the independent derivation of the GGreen expression from the out-operator formula and in the validity of (10), neither of which assumes the target equality. The paper contains no fitted parameters and no numerical prediction that is statistically forced. Self-citations ([4], [10], [12]) are used for background, for perturbation-theory motivation, and for standard Haag-Ruelle-style estimates; they are not used to assume the inclusive-S-matrix/cross-section equality. The Appendix's caveat that for unstable quasiparticles 'we cannot take the limit t → ∞; ... we should always assume that t << T' is a limitation on the regime of validity, not a circular step. Overall, the derivation is self-contained up to standard scattering-theoretic estimates, with only the definitional labeling noted above.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical constants are fitted to data; the derivation is parameter-free in the sense that it holds for arbitrary smooth strictly convex dispersion laws and for arbitrary states satisfying the cluster and spectral assumptions. No new physical entities such as particles, forces, or dimensions are postulated; the inclusive scattering matrix is a new mathematical object rather than a new physical entity.

assumptions (5)

domain assumption The state omega is translation-invariant and satisfies the strong cluster property or strong asymptotic commutativity: truncated correlation functions decay faster than any power in spatial separation, or the algebra A(omega) satisfies the stated commutator bounds.
Invoked in Section 5 to prove the existence of the limits defining the spaces D+ and D- and the Møller operators, and again in Sections 6 and 7 to justify the asymptotic formulas.
domain assumption Dispersion laws epsilon_k(p) are smooth and strictly convex, and for spacetime dimension d < 3 the velocity supports of the wave packets are non-overlapping (NO condition).
Section 5 uses convexity and the NO condition to obtain the stationary-phase estimate (9) and the summability of the time derivative of the scattering vectors; without these, the asymptotic limits may fail.
domain assumption The one-particle spectrum does not overlap the multi-particle spectrum, and good operators B_k exist with B_k Phi = Phi(phi_k) and with B*_k Phi equal to zero or a one-particle state.
Section 8 needs spectral separation to construct a good operator from a local field A with Fourier support avoiding the multi-particle spectrum, and this is used to control LSZ residues and prove formula (20).
domain assumption The two-point Green function has real poles for stable particles; complex poles correspond to unstable particles with lifetime T, and for times t << T the unstable particle behaves approximately like a stable one.
The Appendix uses the pole position epsilon(p) + i Gamma(p) and asserts the approximate validity of the framework for t << T. This is a heuristic physics assumption, not derived from the algebraic axioms.
standard math Poles of the two-point Green function correspond to one-particle states, following the Källén-Lehmann representation.
Section 8 uses this standard QFT result to identify the dispersion law from the pole position and to connect Green function residues to particle data.

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Pith. "Pith review of Scattering matrix and inclusive scattering matrix in algebraic quantum field theory." pith.science (2026). https://pith.science/paper/A7BX4QVA

@misc{pith2026190809388,
  author       = {Pith},
  title        = {Pith review of: Scattering matrix and inclusive scattering matrix in algebraic quantum field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7BX4QVA}},
  note         = {Machine review of arXiv:1908.09388}
}

read the original abstract

We study the scattering of particles and quasiparticles in the framework of algebraic quantum field theory. The main novelty is the construction of inclusive scattering matrix related to inclusive cross-sections. The inclusive scattering matrix can be expressed in terms of generalized Green functions by a formula similar to the LSZ formula for the conventional scattering matrix. The consideration of inclusive scattering matrix is necessary in quantum field theory if a unitary scattering matrix does not exist (if the theory does not have particle interpretation). It is always necessary if we want to consider collisions of quasiparticles.

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Forward citations

Cited by 1 Pith paper

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

Adiabatic definitions of scattering matrix and inclusive scattering matrix
quant-ph 2024-12 conditional novelty 4.0 of 10

Adiabatic dressing with phase factors defines the renormalized scattering matrix and its inclusive counterpart, whose matrix elements are amputated Green functions on shell.

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Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Araki, H., Haag, R. (1967). Collision cross sections in t erms of local observables. Commu- nications in Mathematical Physics, 4(2), 77-91

work page 1967

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work page 2005

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A., Shvarts, A

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