Axiomatic Quantum Field Theory

Job: organize non-perturbative QFT via precise axiom systems — Wightman, Osterwalder–Schrader, Haag–Kastler — so that reconstruction theorems connect Euclidean, Minkowski, and algebraic formulations; free fields as the basic model.

Core

Axiomatic QFT replaces Lagrangian heuristics by operator / distribution / \(C^*\)-algebra axioms on Minkowski (or Euclidean) spacetime. Three classical pillars:

Framework Data Reconstruction
Wightman Hilbert space, fields as operator-valued distributions, Poincaré covariance, spectrum, vacuum, locality Wightman functions \(\leftrightarrow\) fields
Osterwalder–Schrader Euclidean Schwinger functions; reflection positivity OS \(\Rightarrow\) Wightman
Haag–Kastler net of \(C^*\) / von Neumann algebras on regions algebraic QFT; fields optional
Layer Content
Wightman axioms tempered distributions; spectrum condition; locality
OS axioms Euclidean invariance; OS positivity; clustering
Haag–Kastler isotony; locality; covariance; vacuum / spectrum
Free field GNS from two-point function; Wick products
Interfaces constructive QFT; CFT axioms; functorial QFT (pointers)

Reading map

Page Focus
Foundations Wightman; OS; Haag–Kastler; reconstruction one-liners; free field
Constructive QFT \(P(\varphi)_2\) / \(\varphi^4_3\) construction; Aizenman–Fröhlich triviality \(d\ge4\); gauge / Millennium
Algebraic QFT & Modular Theory Haag–Kastler nets; Reeh–Schlieder; Tomita–Takesaki; Bisognano–Wichmann; split / III\(_1\); DHR
CFT & TFT Axioms Segal CFT; VOAs; BPZ minimal models; Atiyah–Segal TFT; cobordism hypothesis; factorization algebras
Scattering Theory LSZ reduction; Haag–Ruelle; asymptotic completeness; infraparticle; Bros–Epstein–Glaser

Dependency order

  1. Functional analysis / distributions; mathematical physics.
  2. Representation theory of Poincaré / Lorentz (representation theory).
  3. Operator algebras (\(C^*\), von Neumann) for Haag–Kastler.
  4. Free fields \(\to\) interacting constructive program; optional homotopy / higher cats for modern TFT.

Interfaces

Area Contact
Mathematical physics ambient physical context; gauge theory
Mathematical physics quantum Hilbert space QM baseline
Representation theory unitary reps of Poincaré
PDE / analysis wave / Klein–Gordon propagators
Category theory nets as functors; AQFT as cosheaves of algebras
Higher categories FQFT / factorization algebras (Costello–Gwilliam)

Notation (paper-standard)

  • \(\varphi(f)\), \(f\in\mathcal{S}(\mathbb{R}^{1,d-1})\); \(\mathcal{W}_n(x_1,\ldots,x_n)\) Wightman functions
  • \(S_n\) Schwinger functions; \(\theta\) Euclidean time reflection
  • \(\mathcal{A}(\mathcal{O})\) local algebra; \(\mathcal{O}''\) causal complement
  • \(\Omega\) vacuum; \(U(a,\Lambda)\) Poincaré representation; \(\overline{V}_+\) forward light cone

See it

Why axioms? Perturbative Feynman diagram expansions are formal and often divergent; continuum interacting measures are hard. Axiom systems isolate structural properties any continuum QFT should satisfy (locality, covariance, positive energy), enable no-go theorems (Haag, Coleman–Mandula with care, …), and give a target for constructive existence proofs (e.g.\ \(P(\varphi)_2\), \(\varphi^4_3\) in classical constructive QFT).

Check

What does the spectrum condition buy you analytically?

Joint spectrum of four-momentum in \(\overline{V}_+\) \(\Rightarrow\) Wightman functions extend to holomorphic functions in tube domains; edge-of-the-wedge and PCT theorems flow from this analyticity + locality.

How does reflection positivity replace unitary time evolution in the Euclidean setting?

OS positivity produces a pre-Hilbert space from Euclidean correlators; transfer matrix / modular reconstruction yields a unitary Minkowski dynamics after analytic continuation — the Euclidean avatar of positive energy.

Pitfalls

  • Distributions not functions: fields \(\varphi(x)\) are never pointwise operators in interacting Wightman theory (UV); always smear.
  • AQFT vs Lagrangian fields: algebraic nets may not come from Wightman fields; fields are a presentation, not the structure.
  • Dimension dependence: constructive results and super-renormalizability depend sharply on \(d\).
  • Gauge theories: local gauge-invariant nets need care (charge superselection, indefinite metric BRST) — naive Wightman for \(A_\mu\) fails.

Refs

  • Streater–Wightman, PCT, Spin and Statistics, and All That
  • Haag, Local Quantum Physics
  • Glimm–Jaffe, Quantum Physics: A Functional Integral Point of View
  • Ruelle, Statistical Mechanics; Simon, The \(P(\varphi)_2\) Euclidean (Quantum) Field Theory

Next

Foundations · Constructive QFT · Algebraic QFT & Modular Theory · CFT & TFT Axioms · Scattering Theory · Mathematical physics · Quantum · Gauge theory · Representation theory · Functional analysis · Category theory