Asymptotic States, LSZ, and Scattering Observables

A scattering amplitude compares physical states in the distant past and future. This chapter connects that comparison to correlation-function poles through LSZ reduction, then converts the normalized amplitude into phase-space rates and local-operator matrix elements.

Enter this chapter

Begin with In and Out States to determine whether the proposed external states exist. Then fix S-matrix normalization before using reduction or rate formulas.

Topics

1. In and Out States — States the stable-state and separation assumptions behind asymptotic particle descriptions.
2. S-Matrix and T-Matrix Normalization — Separates the identity, transition operator, and momentum delta function.
3. Relativistic Scattering Kinematics — Relates invariant masses and Mandelstam variables to physical channels.
4. LSZ Reduction: Poles, Residues, and Stable External States — Extracts amplitudes from isolated poles with the correct external residues.
5. LSZ for Spinor and Vector External States — Adds spin wave functions, polarizations, and physical-state restrictions.
6. Lorentz-Invariant Phase Space — Constructs and reduces the invariant multiparticle measure.
7. Cross Sections and Decay Rates — Combines amplitudes with flux, spin sums, and identical-particle factors.
8. Form Factors and Local Operator Insertions — Distinguishes operator-insertion matrix elements from ordinary scattering amplitudes.

How the ideas fit together

LSZ removes external propagator poles and their operator-dependent overlaps, leaving a transition between normalized particle states. The pole must belong to a suitable stable isolated shell with nonzero overlap; a resonance or a charged infraparticle is not an ordinary external state merely because a propagator denominator can be written. See Weinberg 1995, §§ 10.2–10.3.

The result is still an amplitude, not a rate. Flux and phase space convert it to a probability per incident flux or per time, while spin averages and identical-particle factors encode how states are prepared and counted.

Review the chapter

If an interpolating field is multiplied by a nonzero constant, should the physical scattering amplitude change?

Solution

No. The correlator’s external overlaps acquire the corresponding factors, but LSZ divides by the matched pole residues. With consistent normalization these changes cancel. Omitting residue factors would make a field-coordinate choice appear to change a physical prediction.

Where to go next

Continue to Tree Amplitudes and Gauge Consistency to compute and test complete examples.

References

• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995. DOI.