Asymptotic States, LSZ, and Scattering Observables

This chapter follows one chain from a prepared beam to a measurable rate. Stable, widely separated wave packets define in and out states; their overlap defines the S-matrix; translation invariance isolates an invariant amplitude; LSZ extracts that amplitude from one-particle poles of time-ordered correlators; and Lorentz-invariant phase space converts it into cross sections and decay rates. The same reduction applied around one declared local insertion gives form factors rather than an ordinary S-matrix element.

The chain is conditional. Ordinary LSZ requires sharp, stable asymptotic particles and isolated mass-shell poles. It does not make a resonance, a charged infraparticle, a confined colored excitation, or a state in a background without suitable past and future particle regions into an external particle.

Enter this chapter

There is no hard prerequisite for using this overview. Choose an entry by asking what you need to control:

If you need to…	Start with…	You are ready to continue when…
decide whether an S-matrix exists at all	In and Out States	you can state the wave-packet, stability, and asymptotic-separation assumptions
decode a convention-dependent amplitude	S-Matrix and T-Matrix Normalization	you can separate the identity term, the connected delta function, and $\mathcal M$
translate masses and angles into invariants	Relativistic Scattering Kinematics	you can compute $s,t,u$ , thresholds, and the physical angular interval
obtain amplitudes from correlators	LSZ Reduction	you can identify every external pole, residue, and amputation factor
attach fermions or spin-one particles	LSZ for Spinor and Vector External States	you can contract an amputated object with the correct spinor or physical polarization
integrate final-state momenta	Lorentz-Invariant Phase Space	you can reduce $\mathrm d\Phi_n$ and reproduce the two-body normalization
turn an amplitude into an observable rate	Cross Sections and Decay Rates	you can supply flux, spin sums or averages, and identical-particle factors
insert a current or composite operator	Form Factors and Local Operator Insertions	you can distinguish an injected momentum from the S-matrix delta function and leave the insertion unamputated

If the invariant mass-shell measure $\mathrm d^3\mathbf p/[(2\pi)^3 2E_{\mathbf p}]$ or relativistic state normalization is unfamiliar, repair that first on One-Particle States: Mass, Spin, and Relativistic Normalization. If a pole need not represent a stable particle, use Resonances, Infraparticles, and Limits of Particle Language before applying LSZ.

The chapter’s organizing map

The logical dependencies are more important than the reading order:

Stage	Requires	Produces	Characteristic failure
asymptotic construction	stable one-particle sectors and separating packets	$	\alpha,\mathrm{in}\rangle $,$
scattering normalization	two asymptotic bases and translation symmetry	$S=1+iT$ and a stripped $\mathcal M$	mixing state or delta-function conventions
kinematics	on-shell momenta and momentum conservation	invariants, thresholds, angular domains	evaluating an amplitude outside the intended physical region
LSZ reduction	isolated real poles with nonzero residues	amputated on-shell amplitudes	resonance poles, branch-point mass shells, confined fields
phase space and rates	normalized amplitudes and allowed final states	$\mathrm d\Phi_n$ , $\mathrm d\sigma$ , $\mathrm d\Gamma$	missing flux, averages, or permutation factors
operator insertion	the same external-state assumptions plus one defined operator	form factors at momentum transfer $q$	confusing a bare insertion with a finite renormalized operator

The scalar thread keeps these interfaces visible. For a real scalar with a stable mass $m$ , a two-point pole fixes the LSZ residue $Z$ , a connected four-point correlator yields $\mathcal M$ , $s,t,u$ locate the physical process, and $\mathrm d\Phi_2$ supplies the rate normalization. A gauge-theory thread follows the same structure but replaces scalar external factors by spinors or physical polarizations and later checks the complete amplitude with Ward identities.

Conventions that must travel together

The site’s four-dimensional Lorentzian baseline uses metric signature $(+---)$ , $p^2=m^2$ on shell, future-directed external momenta, and

\langle \mathbf p',r'|\mathbf p,r\rangle =(2\pi)^3 2E_{\mathbf p}\, \delta^{(3)}(\mathbf p'-\mathbf p)\delta_{r'r}.

For a connected process we use

\langle f|S|i\rangle_c =i(2\pi)^4\delta^{(4)}(P_f-P_i)\,\mathcal M_{fi}.

These choices determine the reciprocal phase-space measure, every power of $2\pi$ , the invariant flux, and the residue factors in LSZ. A source that normalizes sharp states with $\delta^{(3)}$ alone must be translated as a whole; preserving a cross section after the translation is the decisive check. Weinberg’s scattering discussion uses a different sharp-state normalization, so its state and rate formulas must be rescaled before comparison with the convention above Weinberg 1995, §§ 3.1–3.4, pp. 107–141.

Exact chapter guide

In and Out States states when interacting histories admit free-particle labels in the distant past and future. Continue to S-matrix normalization, or to the rigorous Haag–Ruelle handoff when existence rather than physical orientation is the question.
S-Matrix and T-Matrix Normalization separates no scattering, connected scattering, and the momentum-conserving distribution. Continue to kinematics or LSZ.
Relativistic Scattering Kinematics derives $s,t,u$ , Källén functions, thresholds, and two-body angular ranges. Continue to phase space for integration or to tree-level crossing for analytic channel relations.
LSZ Reduction: Poles, Residues, and Stable External States gives the complete scalar reduction, including wave packets and failure conditions. Continue to spinning external states, rates, or the theorem-first treatment.
LSZ for Spinor and Vector External States replaces scalar residues by spin wave functions and physical polarization projectors. Continue to vector-amplitude Ward checks.
Lorentz-Invariant Phase Space constructs and recursively factors $\mathrm d\Phi_n$ , with two- and three-body checks. Continue to observable rates or numerical integration.
Cross Sections and Decay Rates combines $|\mathcal M|^2$ with flux, phase space, degeneracy averages, and identical-particle factors. Continue to the optical theorem or infrared-safe observables.
Form Factors and Local Operator Insertions reduces external legs around one local insertion and derives elementary Lorentz decompositions. Continue to renormalized composite insertions, exact integrable form factors, or model-specific currents according to the question.

Synthesis: what is definition and what is dynamics?

The state normalization, $S=1+iT$ , the definition of $\mathcal M$ , and $\mathrm d\Phi_n$ are conventions or kinematic constructions. The existence of wave operators, a sharp one-particle pole, its residue, and the value of $\mathcal M$ are dynamical facts about the theory. LSZ is the bridge: under its hypotheses, it identifies the residue of the simultaneous external poles of a time-ordered correlator with the scattering amplitude. The original LSZ formulation makes precisely this connection between field matrix elements and the S-matrix Lehmann, Symanzik, and Zimmermann 1955, pp. 205–225; modern derivations and normalization checks are given in Schwartz 2014, § 6.1, pp. 70–74 and Srednicki 2007, § 5, pp. 49–56.

The category error to avoid is simple: a pole-like feature is not automatically an external state. A real isolated pole belonging to a stable positive-norm state supports ordinary reduction. A complex resonance pole describes unstable dynamics, a branch point can replace a charged particle pole in a massless gauge theory, and a gauge-fixed colored propagator does not establish a physical asymptotic state.

Review the chapter

Use the questions below to identify topics worth revisiting.

Reconstruct the chain. Starting from two normalized incoming packets, write the sequence of objects needed to obtain a differential $2\to2$ cross section. A successful answer names the wave operators, connected S-matrix element, invariant amplitude, LSZ residues, physical $s,t$ range, invariant flux, and $\mathrm d\Phi_2$ without changing normalization midway.

Diagnose a failure. A two-point function has no delta-function spectral weight at $p^2=m^2$ , only a continuum beginning there. Explain which LSZ step fails. The check is whether your answer identifies the missing isolated simple pole rather than trying to repair the calculation with a finite $Z$ .

Translate a convention. Rescale covariantly normalized kets to kets with a bare $\delta^{(3)}$ norm. Track the compensating factors in completeness and in an external leg. The invariant checkpoint is that the final cross section is unchanged.

Separate two insertions. Compare $\langle f|S|i\rangle_c$ with $\langle f|\mathcal O(0)|i\rangle_c$ . A successful answer explains why the first carries $\delta^{(4)}(P_f-P_i)$ while a fixed local insertion can transfer momentum and is not amputated.

Derive a benchmark. Starting from the positive-energy shell measure, reduce two-body phase space in the center-of-mass frame. A successful derivation obtains $\mathrm d\Phi_2=|\mathbf p_*|\,\mathrm d\Omega/(16\pi^2\sqrt{s})$ and checks both its dimension and its threshold limit.

Transfer the reduction. Replace one outgoing scalar leg by a stable massive vector. State what remains unchanged in LSZ and what replaces the scalar endpoint. The check is a contraction with a physical polarization satisfying $p\cdot\varepsilon=0$ , not the inclusion of all four components of a gauge-fixed field.

Complete a normalization loop. For massless distinguishable scalar $2\to2$ scattering with constant $\mathcal M_0$ , use $\Phi_2=1/(8\pi)$ and the center-of-mass flux $\mathcal F=2s$ to obtain $\sigma=|\mathcal M_0|^2/(16\pi s)$ . Reaching a different answer diagnoses a mismatch among the stripped-amplitude, state, flux, or phase-space conventions.

If a check fails at the state or pole step, repair it with In and Out States and LSZ Reduction. If it fails through a factor of $2E$ , $2\pi$ , flux, or a permutation count, return to S-Matrix and T-Matrix Normalization, Lorentz-Invariant Phase Space, and Cross Sections and Decay Rates before continuing.

Where the boundaries lead

For explicit tree amplitudes, continue to Connected Tree Diagrams and Amputated Amplitudes.
For unitarity rather than normalization, continue to S-Matrix Unitarity and The Optical Theorem and Cut Interpretation.
For resonance poles and line shapes, continue to Resonance Poles, Riemann Sheets, and Unstable States.
For massless charged sectors, continue to Dressed States and Infrared-Finite Scattering only after retaining the observable and dressing assumptions.
For theorem-level existence and distributional reduction, continue to Haag–Ruelle Scattering-State Construction and LSZ Reduction and Amputated Distributions.

References

Lehmann, Harry, Kurt Symanzik, and Wolfhart Zimmermann. “Zur Formulierung quantisierter Feldtheorien.” Il Nuovo Cimento 1, no. 1 (1955): 205–225. DOI.
Schwartz, Matthew D. Quantum Field Theory and the Standard Model. 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. Cambridge: Cambridge University Press, 1995. DOI.