Interacting QFT: Definitions, Limits, and Further Developments
An interacting Lagrangian proposes local dynamics; it does not by itself construct every quantum observable or a continuum theory. This chapter identifies the additional regulator, state, renormalization, and limiting data needed to interpret an interacting calculation and to choose an appropriate method.
Enter this chapter
Section titled “Enter this chapter”Begin with What an Interacting Lagrangian Does and Does Not Specify. Then distinguish a finite regulated system from the continuum before interpreting the interaction picture.
Topics
Section titled “Topics”- What an Interacting Lagrangian Does and Does Not Specify — Separates a classical proposal from a specified quantum construction.
- Regulators, Cutoffs, and Continuum Limits — Distinguishes a regulator, a physical cutoff, and a demonstrated limiting procedure.
- Haag’s Theorem: Physical Meaning and Scope — Explains the obstruction to a naive unitary identification of free and interacting continuum fields.
- Interacting Fields, Asymptotic Observables, and Effective Descriptions — Chooses between amplitudes, local observables, effective descriptions, and nonperturbative constructions.
How the ideas fit together
Section titled “How the ideas fit together”At a finite regulator, interacting evolution can be a well-defined operator problem. Removing that regulator is a further task: parameters, observables, and states must be controlled together. A perturbative series gives coefficientwise information, while a nonperturbative construction establishes an actual theory within its stated setting. The distinction is central to Summers 2016, §§ 1–2.
Haag’s theorem restricts a particular continuum identification of free and interacting fields under specified hypotheses. It does not prohibit regulated dynamics, formal perturbation theory, or all scattering constructions. The comparison of interacting descriptions links each physical question to the method and assumptions it needs.
Review the chapter
Section titled “Review the chapter”Does agreement between two cutoff calculations at one fixed cutoff establish a continuum QFT?
Solution
No. It establishes agreement for that regulated problem. A continuum claim additionally needs control as the regulator is removed, with an identified family of observables and parameter adjustments, and a stated sense of convergence or perturbative validity.
Where to go next
Section titled “Where to go next”Continue to Wick Expansion and Feynman Rules for perturbative calculations, or Regulators, Cutoffs, and Continuum Limits for the qualifications behind regulator removal.
References
Section titled “References”- Stephen J. Summers, “A Perspective on Constructive Quantum Field Theory,” arXiv:1203.3991, submitted 2012, revised 2016. Stable record. Open PDF.