From One-Particle Poles to the Scattering Handoff

For the conventional isolated-shell scalar route, scattering reduction may begin only when every proposed external particle is represented by a stable, positive-norm one-particle sector on a real mass shell, a chosen interpolating operator has finite nonzero overlap with that sector, the shell has the separation required by the standard construction, suitable wave-packet in/out limits exist, and long-range infrared or gauge obstructions have been controlled. The pole data are necessary spectral input, not a proof of scattering theory. More refined no-gap constructions can replace open shell isolation by stronger regularity estimates; this page identifies that boundary but works the standard isolated-shell test. The reduction formula itself belongs to Scattering.

Required background. Poles, Cuts, Thresholds, and Stable Particles supplies the physical-sheet pole, residue, threshold, and finite-volume distinctions used here. Fields, Observables, and Interpolating Operators supplies the nonzero vacuum-to-one-particle overlap, covariant state normalization, and operator-versus-particle distinction.

The stable scalar data

Work in a positive physical Hilbert space with a Poincaré-invariant vacuum $|\Omega\rangle$. Let $|\mathbf p,r\rangle$ be a stable scalar one-particle state of mass $m_r>0$, normalized by

$$\langle\mathbf p',r|\mathbf p,r\rangle =(2\pi)^3 2E_{\mathbf p,r} \delta^{(3)}(\mathbf p'-\mathbf p), \qquad E_{\mathbf p,r}=\sqrt{\mathbf p^2+m_r^2}.$$

The displayed template assumes one nondegenerate neutral-scalar direction at this mass in the selected channel. With several degenerate states and interpolators, the overlaps instead form the positive-semidefinite residue matrix

$$(Z_r)_{ab} =\sum_\sigma \kappa_{a\sigma}^*\kappa_{b\sigma}, \qquad \kappa_{a\sigma} :=\langle\mathbf p,r,\sigma|O_a(0)|\Omega\rangle.$$

On the nonzero residue subspace, one resolves the stable degenerate sector and normalizes with the positive matrix inverse square root $Z_r^{-1/2}$. The scalar square root used below is the one-dimensional special case.

For a centered Hermitian scalar operator $O_{\mathrm s}$ with the right quantum numbers, choose the one-particle phase so that

$$\langle\Omega|O_{\mathrm s}(0)|\mathbf p,r\rangle =\sqrt{Z_{O,r}}, \qquad 0<Z_{O,r}<\infty.$$

Translation covariance makes this overlap independent of $\mathbf p$ apart from the plane-wave factor at nonzero position. In the corresponding exact time-ordered two-point function, an isolated spectral atom appears as

$$\widetilde D_{F,O}(p) =\frac{iZ_{O,r}}{p^2-m_r^2+i0} +D_{\mathrm{less\ singular}}(p).$$

Here “isolated” means more than “a denominator becomes small.” For some $\varepsilon>0$, the rest of the spectral measure has no support in

$$(m_r^2-\varepsilon,m_r^2+\varepsilon).$$

This establishes an isolated pole in this operator’s two-point function. It does not show that the operator sees every nearby state. The conventional isolated-shell construction separately assumes that the one-particle hyperboloid is isolated in the joint energy–momentum spectrum of the relevant physical sector. Under those two distinct conditions, the pole lies on the physical sheet, its location is real, and its numerator is $iZ_{O,r}$ in the site convention. A local polynomial contact term cannot change this conclusion. The completeness-to-pole argument is developed in Schwartz 2014, § 24.3, pp. 471–474, while its validity for arbitrary local operators with the required nonzero matrix elements appears in Weinberg 1995, § 10.2, p. 430.

The number $Z_{O,r}$ belongs to the chosen operator. Rescaling $O_{\mathrm s}$ rescales $Z_{O,r}$, so it is not a universal probability and need not obey $Z_{O,r}\leq1$. The unit-overlap field

$$\phi_r:=Z_{O,r}^{-1/2}O_{\mathrm s}$$

satisfies

$$\langle\Omega|\phi_r(0)|\mathbf p,r\rangle=1, \qquad \widetilde D_{F,\phi_r}(p) =\frac{i}{p^2-m_r^2+i0} +\text{less singular terms}.$$

This normalization does not turn an interacting operator into a free field or remove its continuum component. It only fixes the external one-particle overlap. A composite operator or bound-state interpolator is equally acceptable when the same conditions hold; whether a field is “elementary” is irrelevant to this test Weinberg 1995, § 10.3, pp. 437–439. Coleman constructs the corresponding normalized stable-particle wave-packet operator in Coleman 2019, § 13.5, pp. 279–281.

Spectral readiness is not scattering readiness

The pole data certify that a chosen operator sees a stable one-particle sector. They do not construct remote-past or remote-future states. That second step requires limits of smooth wave packets, not sharp plane waves.

For a smooth compactly supported profile $f\in C_c^\infty(\mathbb R^3)$ parametrizing the positive-energy mass shell, write

$$|f,r\rangle =\int \mathrm d\Pi_{\mathbf p,r}\, f(\mathbf p)|\mathbf p,r\rangle, \qquad \mathrm d\Pi_{\mathbf p,r} =\frac{\mathrm d^3\mathbf p}{(2\pi)^3 2E_{\mathbf p,r}}.$$

A standard sufficient Haag–Ruelle construction first chooses products of such packets with disjoint velocity supports. These packets form a dense construction class; disjoint support is not asserted as a necessary property of every scattering state. The remaining hypotheses include a fixed vacuum sector with a unique invariant vector, the spectrum condition, local or sufficiently almost-local interpolating operators, an isolated stable mass shell, and suitable spectral and operator regularity. Locality then suppresses commutators between widely separated packets, while the required spectral separation or regularity controls unwanted components. Srednicki gives a scalar packet and dephasing argument in Srednicki 2007, § 5, pp. 49–55; the structural construction appears in Ruelle 1962, § 4, pp. 157–159.

It is useful to keep the two stages separate:

Stage	Required evidence	What it does not yet establish
isolated-shell spectral readiness	real isolated physical-sheet pole, positive-norm state, finite nonzero overlap	existence of large-time in/out limits or coverage of specialized no-gap constructions
asymptotic readiness	convergent packet limits, separation of packet velocities, controlled long-range interactions	that every physical state is a scattering state
reduction readiness	the two stages above for every external leg, plus the declared time-ordered connected correlator and conventions	the LSZ formula, amplitudes, or cross sections themselves

Asymptotic completeness is stronger than what is needed for one selected matrix element. The relevant in/out states must exist, but this page does not assume that they exhaust the Hilbert space. Treating the full in and out families as complete is an extra step in the standard $S$-matrix construction Weinberg 1995, § 3.2, pp. 113–114.

How much of a mass gap is needed

For the conventional route used in the scalar test below, the direct external-leg assumption is an isolated mass shell in the relevant sector, not automatically a gap above the vacuum throughout the whole theory. Within the other local, covariant, spectral, and regularity hypotheses, a vacuum gap together with an isolated one-particle shell is the conventional sufficient spectral setup for the massive Haag–Ruelle route. The vacuum gap does not replace shell isolation, and this setup is not a theorem of universal necessity.

For this standard scalar test, first require an open neighborhood of $m_r^2$ in which the chosen operator’s measure contains only its one-particle atom. Its visible continuum may begin at $s_{0,r}>m_r^2$ and coexist with the pole. Then impose the stronger sufficient scattering assumption: the same mass hyperboloid is isolated from the rest of the relevant physical-sector spectrum, including states that this operator might miss. This guarantees that a packet can be localized near the external shell without also selecting other spectral states.

Massless sectors require a separate analysis. Their presence does not by itself forbid scattering, but soft quanta can make continuum support touch the nominal charged mass shell and destroy its isolation. When that happens, this page’s isolated-shell test fails before any reduction formula is applied; failure of this test is not a general impossibility theorem. Under Herbst regularity, Dybalski constructs Haag–Ruelle states for stable massive sharp-dispersion particles amid massless excitations without an open mass gap, with neutral atoms as motivating examples. That result neither solves the charged infraparticle problem nor by itself proves correlator reduction Dybalski 2005, abstract and § 1, manuscript pp. 1–3 (Open PDF).

Infrared, gauge, and boundary conditions

Ordinary LSZ compares the interacting theory with freely propagating asymptotic particles. That comparison needs all of the following qualifications.

• Infrared control. Residual interactions must become negligible in the relevant packet limit. An unscreened long-range gauge field can prevent this. Dressed states or inclusive observables may still define useful scattering quantities, but they are not the ordinary sharp-particle route certified here.
• Physical-state control. The pole must belong to the positive physical spectrum. A pole of a gauge-fixed quark, ghost, or nonphysical vector component is not by itself an admissible external particle. Spinning particles also require the correct on-shell projectors and physical polarizations.
• Boundary-condition control. The correlator used downstream is the connected, time-ordered in–out correlator with the Feynman $+i0$ boundary value. Retarded response functions and in–in expectation values answer different questions.
• Infinite-volume control. A discrete pole at finite spatial volume is only a level. Its separation and overlap must survive the infinite-volume limit before it is interpreted as an external mass shell.
• Stability control. A pole reached only after continuation through a cut is a resonance pole, not a stable external state. A normalizable exact energy eigenstate has only a phase under time evolution and cannot possess a decay width.

These conditions concern the proposed external legs. Resonances may occur internally in an amplitude, and massless particles may admit specialized scattering constructions. Neither fact relaxes the readiness test by itself.

First application: a scalar LSZ readiness test

Suppose the exact physical spectral measure of a centered scalar operator has the form

$$\rho_O(\mathrm d\sigma) =Z_O\,\delta(\sigma-m^2)\,\mathrm d\sigma +\rho_{\mathrm{rest}}(\mathrm d\sigma), \qquad 0<Z_O<\infty,$$

with

$$\operatorname{supp}\rho_{\mathrm{rest}} \subseteq[s_0,\infty), \qquad m^2<s_0.$$

Then the two-point function has the isolated term

$$\widetilde D_{F,O}(p) =\frac{iZ_O}{p^2-m^2+i0} +\widetilde D_{F,\mathrm{rest}}(p),$$

and $\phi_R=Z_O^{-1/2}O_{\mathrm s}$ has unit one-particle overlap. This passes the operator-level pole and normalization checks. Sector-level shell isolation and every asymptotic condition remain separate.

Before handing the correlator to LSZ, check each row:

Question	Passing condition	Failure diagnosis
Is there a particle state?	the atom belongs to a stable positive-norm one-particle sector	an auxiliary-space or purely analytic pole is insufficient
Is the external shell isolated?	an open neighborhood separates $m^2$ from the rest of the relevant physical-sector joint spectrum	a threshold touching $m^2$ fails this ordinary isolated-shell route and needs separate analysis
Does the operator interpolate it?	$0<Z_O<\infty$	$Z_O=0$ means only that this operator misses the sector
Are the external states asymptotic?	the chosen smooth packets possess in/out limits	a two-point pole alone gives no such limit
Do packets separate?	locality or the relevant decay estimates suppress residual interactions at large times	persistent long-range forces require a modified framework
Is the state physical?	it survives the physical-state or gauge constraint with the correct norm	a gauge-dependent denominator is not enough
Is the limit the desired observable?	the theory admits an in–out scattering question with the chosen boundary conditions	nonstationary or in–in problems require different observables

If every row passes for every external species, the connected time-ordered correlator is ready for the downstream reduction. The downstream LSZ treatment may then divide by one factor $\sqrt{Z_{O,r}}$ per external overlap, amputate the corresponding pole, and take the on-shell packet limit. Those operations are stated here only to identify the required inputs; their derivation and overall $S$-matrix normalization are deliberately deferred.

For the canonically normalized free scalar field $O=\phi$, $Z_\phi=1$, the field creates the isolated one-particle state, and the free in and out fields coincide. The readiness conditions hold, although the connected scattering amplitude is trivial. An interacting stable bound state can pass the same test through a composite operator with nonzero overlap; elementarity is not an extra requirement.

What a pole does not authorize

Observation	Readiness conclusion
sector-isolated stable state and positive physical-sheet pole, but no established in/out packet limits	spectrally ready; scattering readiness remains open
continued-sheet complex pole	resonance; not an ordinary external ket
continuous support reaches the nominal mass with no atom	infraparticle-type obstruction; no ordinary external pole
pole only at finite volume or nonzero infrared regulator	take the limits in the declared order and test whether isolation survives
zero pole residue for one operator	try another compatible interpolator; particle absence has not been proved
pole in an indefinite-metric gauge-fixed component	identify a positive physical state and its projector before proceeding
stable isolated bound-state pole with nonzero composite overlap	eligible in principle, subject to the same asymptotic and infrared tests

The pole criterion also does not imply asymptotic completeness, a unique scattering theory, perturbative convergence, or the existence of an $S$-matrix in a nonstationary background.

Check your understanding

Rescale the interpolator. If $O'=cO$ with $c\in\mathbb R\setminus\{0\}$ so that Hermiticity is preserved, how do the pole residue and the readiness verdict change?

Check

The overlap becomes $c\sqrt{Z_O}$ up to the chosen state phase, and the two-point residue becomes $c^2Z_O$. The existence, mass, and isolation of the one-particle sector do not change. Normalizing by the positive square root of the new residue returns $\operatorname{sgn}(c)\phi_r$, the same one-particle direction up to phase.

Separate the two stages. A physical correlator has an isolated real pole with positive residue, but no large-time packet limit has been proved. Is ordinary reduction ready?

Check

No. The pole passes the spectral test, but the in/out states are an independent assumption or theorem. Reduction begins only after the relevant asymptotic packet limits and infrared conditions are also available.

Diagnose a threshold at the mass. The spectral measure has continuous support from $m^2$ upward and no atom at $m^2$. Can multiplying by $p^2-m^2$ isolate an ordinary external leg?

Check

No. There is no isolated simple pole with finite nonzero residue. This is the spectral pattern that obstructs the ordinary sharp-particle route; it is not repaired by declaring a field-strength factor.

Distinguish isolation from completeness. Selected stable packets have well-defined in/out limits, but it is unknown whether every finite-energy state is generated by scattering states. Which claim is still unavailable?

Check

Asymptotic completeness remains unavailable. The selected matrix elements may still be meaningful because existence of those particular in/out states is weaker than exhaustion of the physical Hilbert space.

Cross into scattering theory

The Foundations handoff is now complete: a stable real pole and finite overlap supply the external spectral data; isolated packet support, in/out limits, and infrared and physical-state control supply the asymptotic data. None can be inferred from the word “pole” alone.

Enter Perturbative QFT and Scattering for the developed scattering sequence. Continue directly to LSZ Reduction: Poles, Residues, and Stable External States for the scalar wave-packet derivation, connected amputation, residue normalization, and convention-normalized amplitude. In and Out States develops the Møller-operator viewpoint, while LSZ for Spinor and Vector External States adds spin projectors and physical polarizations.

If the scalar test fails, return to Resonances, Infraparticles, and Limits of Particle Language for the diagnosis. Long-range charged sectors continue to Dressed States and Infrared-Finite Scattering. The theorem-level massive construction belongs to Haag–Ruelle Scattering-State Construction, and the distributional reduction hypotheses belong to LSZ Reduction and Amputated Distributions.

References

• Coleman, Sidney. Lectures of Sidney Coleman on Quantum Field Theory. Edited by Bryan Gin-ge Chen, David Derbes, David Griffiths, Brian Hill, Richard Sohn, and Yuan-Sen Ting. Singapore: World Scientific, 2019. DOI.

• Dybalski, Wojciech. “Haag–Ruelle Scattering Theory in Presence of Massless Particles.” Letters in Mathematical Physics 72 (2005): 27–38. DOI. Open PDF.

• Ruelle, David. “On the Asymptotic Condition in Quantum Field Theory.” Helvetica Physica Acta 35 (1962): 147–163. DOI.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. First ed. Cambridge: Cambridge University Press, 2014. DOI.

• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.

• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. First ed. Cambridge: Cambridge University Press, 1995; 2005 paperback, 2012 printing consulted. DOI.