LSZ Preview

LSZ reduction relates scattering amplitudes to the one-particle poles of time-ordered quantum-field correlations. External-leg amputation removes those poles; overlap factors then convert the field insertions into normalized asymptotic particles. This preview explains that mechanism and checks its normalization. It does not prove the existence of scattering limits or provide the complete reduction formula for every incoming and outgoing species.

Required background. Klein–Gordon Propagators and Dirac Propagators give the free poles; Spectral Representation explains residues and overlaps. Helpful background. From Correlation Functions to QFT Observables owns the broader correlation-to-observable map, while QFT Bridge: S-Matrix defines asymptotic scattering conventions.

An interpolating field and an isolated particle pole

Use $\hbar=c=1$ and metric $(+,-,-,-)$. Consider a Hermitian scalar field $\widehat\phi$ in a translation-invariant vacuum. Suppose it couples to a stable scalar particle with physical mass $m_{\rm phys}>0$ and covariant state norm. Choose its overlap phase so that

$$\langle0|\widehat\phi(0)|p\rangle=\sqrt Z, \qquad Z>0.$$

Assume an isolated simple one-particle pole, with the remaining spectral contribution nonsingular in a neighborhood of that pole. The time-ordered two-point function then has the local form

$$\widetilde C_2(p)= \frac{iZ}{p^2-m_{\rm phys}^2+i0} +\widetilde C_{\rm reg}(p).$$

Inserting the one-particle part of spectral completeness explains the residue $Z$: the field creates the particle with amplitude $\sqrt Z$ and destroys it with its conjugate. The physical mass is the pole location, not necessarily the mass parameter in an unrenormalized Lagrangian.

$Z$ describes this field’s overlap. It is not a universal “probability that the particle exists.” Rescaling an interpolating field changes $Z$ without changing the particle or its scattering probabilities. A local composite operator can also interpolate a stable particle when its quantum numbers and nonzero overlap are appropriate.

Removing one scalar external pole

Let $F(p)$ denote a Fourier-transformed correlator with one distinguished external scalar insertion and all other variables held implicit. Near the selected pole write its singular part as

$$F(p)= \frac{i\sqrt Z}{p^2-m_{\rm phys}^2+i0}\, \mathcal A(p)+F_{\rm reg}(p).$$

Here $\mathcal A$ is defined as the pole-free coefficient for that leg; this single-leg equation does not yet fix the overall $S$-matrix phase or the remaining external legs. Writing $\Delta=p^2-m_{\rm phys}^2$, residue extraction gives

$$\mathcal A_{\rm on} =\frac1{\sqrt Z} \lim_{\Delta\to0}\frac{\Delta}{i}F(p).$$

The limit means the residue of the specified pole, with its boundary prescription understood; it is not evaluation of the singular distribution at $\Delta=0$. In a full reduction, wave packets and asymptotic limits make that operation precise. The factor $\Delta$ amputates the pole, and $1/\sqrt Z$ removes the field’s overlap with the normalized particle.

There is another common bookkeeping convention: amputate with the full two-point function rather than the unit-residue free pole. Define $\Gamma(p)$ off shell by $F(p)=\widetilde C_2(p)\Gamma(p)$. Near the isolated pole,

$$F(p)\sim\frac{iZ}{\Delta+i0}\Gamma(p), \qquad \mathcal A_{\rm on}=\sqrt Z\,\Gamma_{\rm on}.$$

The external normalization is now $\sqrt Z$, not $1/\sqrt Z$. The two rules agree because full-propagator amputation has already removed a factor $Z$ as well as the pole. Beisert’s discussion of amputation makes this distinction explicit.

A field-rescaling check with four external legs

For a connected four-scalar correlator, strip the overall momentum delta and define its fully amputated coefficient $\Gamma_4$ by removing one full two-point function from each leg. For identical external species with residue $Z$, the external normalization gives the pole-free coefficient

$$\mathcal C_4=Z^2\Gamma_4$$

on shell, with the overall amplitude phase fixed by the chosen correlator and $S-I$ convention. This equation isolates the external-leg normalization; it is not a new interaction model.

Now change the interpolating field to $\phi'=a\phi$, with real $a>0$. The same quantum theory has

$$\begin{aligned} Z'&=a^2Z,\\ C_4'&=a^4C_4,\qquad C_2'=a^2C_2. \end{aligned}$$

Amputating four full propagators therefore gives $\Gamma_4'=a^{-4}\Gamma_4$. The normalized coefficient is invariant:

$$(Z')^2\Gamma_4'=Z^2\Gamma_4.$$

For a concrete check, if $Z=1/4$ then $\mathcal C_4=\Gamma_4/16$. Using four factors $1/\sqrt Z$ after full-propagator amputation would instead give $16\Gamma_4$ and fail the field-rescaling check. One must specify which object was amputated before copying an external-leg rule.

For several scalar species the same argument uses $\prod_a\sqrt{Z_a}$ after full amputation. Mixing fields requires the appropriate residue eigenvectors or matrices and is outside this one-species example.

Spinor poles require external spin states

For a stable massive Dirac particle in a Lorentz-invariant, parity-preserving vacuum, a conventional isolated pole has form

$$\widetilde S(p)\sim \frac{iZ_\psi(p\!\!\!/+m_{\rm phys})} {p^2-m_{\rm phys}^2+i0}.$$

On the future mass shell its numerator is

$$p\!\!\!/+m_{\rm phys} =\sum_s u_s(p)\bar u_s(p).$$

It has rank two, not four. There is no ordinary full-rank inverse of that on-shell numerator. A correct spinor reduction amputates off shell and projects onto the selected external $u,\bar u$ or $v,\bar v$ state, with the corresponding overlap normalization. Incoming and outgoing particles and antiparticles require consistent Fourier arguments and fermionic ordering signs. Those signs are part of the full reduction formula.

The useful free algebra is $(p\!\!\!/-m)(p\!\!\!/+m)=(p^2-m^2)I_4$. Applying it before the mass-shell limit cancels the pole. Setting $p^2=m^2$ first discards the inverse information and leaves singular projectors. The spin sums and negative-frequency label conventions are derived on Dirac Propagators.

What wave mechanics supplies, and what LSZ adds

The free Klein–Gordon and Dirac equations supply pole locations, numerators, and normalized free modes. Their Green functions solve inhomogeneous differential equations. An interacting scattering amplitude additionally requires quantum-field correlations, a vacuum and particle spectrum, asymptotic in/out states, and the appropriate scattering limits. Multiplying a free inverse by its differential operator does not establish those ingredients.

The isolated-pole assumption has concrete limits. An unstable resonance is not an exact asymptotic external particle of the simple stable-pole construction. Confined colored fields do not create isolated observable colored asymptotic states. Massless radiation, especially charged sectors of QED, introduces infrared subtleties that can invalidate the naive isolated-pole and finite-particle scattering picture. Inclusive observables or dressed-state constructions require additional work.

Lehmann, Symanzik, and Zimmermann (1955) provide the foundational reduction framework. The algebra here is a preparation for that framework, with its hypotheses retained. After an amplitude has been obtained, Invariant Phase Space Preview and Optical Theorem Preview explain the next normalization and unitarity checks.

Exercises

1. A scalar external leg has $F=i\sqrt Z\,\mathcal A/\Delta+R$, where $R$ is regular at the pole. What happens to $R$ under residue extraction?

Solution

Multiplication by $\Delta/i$ sends $R$ to zero as the pole is approached, while the singular term tends to $\sqrt Z\,\mathcal A$. Dividing by $\sqrt Z$ returns the normalized coefficient. This argument depends on regularity of $R$ at the selected pole; it does not apply unchanged at a continuum threshold.

2. Two external species have residues $Z_1$ and $Z_2$. Each species occurs once as an incoming and once as an outgoing particle. What is the normalization after full amputation?

Solution

The product of four external factors is $(\sqrt{Z_1})^2(\sqrt{Z_2})^2=Z_1Z_2$. This multiplies the full-propagator-amputated coefficient. Amputation with unit-residue poles instead uses the inverse overlap factors before extracting the normalized coefficient.

3. At rest, show explicitly why the positive-energy Dirac numerator cannot be inverted as a four-by-four matrix.

Solution

$p=(m,\mathbf0)$ gives $p\!\!\!/+m=m(\gamma^0+I_4)$. In the Dirac basis this is $\operatorname{diag}(2m,2m,0,0)$. It has two zero eigenvalues and resolves the two positive-energy spin states. External spin projection uses that rank-two structure; an inverse matrix on all four components does not exist.

References

• Beisert, Niklas. Quantum Field Theory I. ETH Zurich, autumn semester 2025, sections 10.3–10.4. Lecture notes. Pole residues, scattering, and full-propagator amputation; metric and correlator conventions differ from those used here.
• Lehmann, Harry, Kurt Symanzik, and Wolfhart Zimmermann. “Zur Formulierung quantisierter Feldtheorien.” Il Nuovo Cimento 1, 205–225 (1955). doi:10.1007/BF02731765.
• McGreevy, John. Physics 215B: Particles and Fields. University of California, San Diego, Winter 2019, section 2. Lecture notes. Correlation functions and scattering reduction.