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Integrable Relativistic QFT

In suitable massive quantum field theories in 1+1 dimensions, higher conserved charges reduce scattering to consistent two-body processes. This chapter follows that reduction into exact S-matrices, finite-volume spectra, local-operator matrix elements, and thermodynamics, keeping the spectrum and analytic assumptions explicit.

Enter this chapter

Section titled “Enter this chapter”

Begin with Classical and Quantum Conserved Charges to distinguish quantum conserved charges from a classical integrability claim.

Helpful background. Review In and Out States and Analyticity and Crossing of Amplitudes for asymptotic states and analytic scattering amplitudes.

Topics

Section titled “Topics”
  1. Classical and Quantum Conserved Charges — Identifies charges capable of constraining quantum scattering.
  2. Elasticity, Factorization, and Their Hypotheses — Derives elasticity and factorization under stated hypotheses.
  3. The Yang–Baxter Equation — Checks independence of the order of pairwise collisions.
  4. Exact S-Matrix Bootstrap, CDD Freedom, and Completeness — Tracks analytic ambiguity and the assumed particle spectrum.
  5. Bound-State Poles, Fusion, and Coleman–Thun Alternatives — Distinguishes bound states from other on-shell singularities.
  6. Bethe Quantization and Finite-Volume Spectra — Quantizes particles on a large spatial circle.
  7. Form-Factor Bootstrap and Correlator Expansions — Constructs operator matrix elements compatible with scattering.
  8. Thermodynamic Bethe Ansatz and Finite-Size Ground-State Energy — Relates scattering data to finite-size ground-state energies.
  9. Integrable Deformations and Renormalization-Group Flows — Tests which relevant deformations retain conserved charges.
  10. Integrable-QFT Casebook, Exact Data, and Limits — Compares exact model data and the scope of their conclusions.

How the ideas fit together

Section titled “How the ideas fit together”

Factorization reduces a multiparticle process to two-body amplitudes, and the Yang–Baxter equation ensures compatible collision orderings. It does not uniquely determine the scalar amplitude: unitarity and crossing can permit additional CDD factors. Pole interpretation, ultraviolet behavior, and completeness supply further tests before an amplitude describes the intended model. Form-factor calculations add operator-specific data, and thermodynamic equations add a chosen particle content. Thus consistency of one scattering equation is only one stage of an exact solution. Zamolodchikov and Zamolodchikov 1979, §§ 1–3.

Where to go next

Section titled “Where to go next”

Continue to Evidence, Validation, and Nonperturbative Status to separate each scientific proposition from the method that produced it.

References

Section titled “References”
  • Zamolodchikov, Alexander B., and Alexey B. Zamolodchikov. “Factorized S-Matrices in Two Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models.” Annals of Physics 120 (1979): 253–291. DOI.