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States, Observables, and Spectra

A two-point function can reveal which states a field creates, but poles, thresholds, resonances, and infraparticles have different meanings. This chapter connects matrix elements to spectral measures and explains when that information supports ordinary particle and scattering language.

Enter this chapter

Section titled “Enter this chapter”

Begin with Operators, Observables, and Matrix Elements, then follow the spectral decomposition and Källén–Lehmann representation before interpreting poles.

Topics

Section titled “Topics”
  1. Operators, Observables, and Matrix Elements — Identifies the state, representation, domain, and normalization behind a matrix element.
  2. Spectral Decomposition of Two-Point Functions — Inserts complete energy–momentum states to construct spectral data.
  3. The Källén–Lehmann Representation — Expresses scalar vacuum correlators through an invariant-mass measure.
  4. Poles, Cuts, Thresholds, and Stable Particles — Distinguishes isolated stable contributions from continuum thresholds.
  5. Resonances, Infraparticles, and Limits of Particle Language — Explains why resonances and infrared-dressed charged sectors need different descriptions.
  6. From One-Particle Poles to Scattering States — Lists the additional assumptions needed to use stable poles as external scattering states.

How the ideas fit together

Section titled “How the ideas fit together”

For a vacuum-subtracted Hermitian scalar in a positive physical Hilbert space, completeness turns squared matrix elements into a nonnegative spectral measure. An isolated atom in that measure produces a stable physical-sheet pole; continuous support produces a cut. This derivation requires the stated vacuum, covariance, spectrum, and regularity conditions. See Weinberg 1995, § 10.2.

Read the upper chain and then the distinct lower branches in the figure. A continuum is not automatically a resonance, and a pole alone does not prove that the asymptotic limits needed for scattering exist.

Vacuum matrix elements and completeness produce a positive scalar spectral measure and its Källén–Lehmann transform; an isolated delta gives a stable physical-sheet pole that can reach LSZ only after extra asymptotic and infrared checks, while continuum support gives a threshold cut and may require separate resonance or infraparticle analysis.

Schematic spectral-content map for a vacuum-subtracted Hermitian scalar operator in a positive-metric physical Hilbert space. The stable delta atom and continuum may coexist. Resonance and infraparticle branches are conditional diagnoses, while the ordinary scattering construction additionally requires an isolated shell, nonzero overlap, suitable in/out limits, and infrared control; the diagram is not a quantitative spectrum and is not to scale.

Review the chapter

Section titled “Review the chapter”

Can two operators have different residues at the same stable-particle pole?

Solution

Yes. Each residue measures that operator’s overlap with the particle sector. The mass is a property of the state, but the overlap changes with the operator and its normalization. Rescaling an operator rescales its residue without changing the particle mass.

Where to go next

Section titled “Where to go next”

Continue to Scattering Theory and LSZ to turn suitable stable-state overlaps into scattering amplitudes.

References

Section titled “References”
  • Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995. DOI.