Axiomatic QFT: Foundations¶
Job: state the Wightman and Osterwalder–Schrader axioms, outline Haag–Kastler nets, record reconstruction theorems in one-liners, and present the free scalar field as the basic model.
Core¶
Wightman axioms (Minkowski)¶
Data for a scalar Bosonic Wightman theory on \(\mathbb{R}^{1,d-1}\) (spin and multiplets similar):
- Hilbert space \(\mathcal{H}\); unitary representation \(U\) of the (identity component of the) Poincaré group \(\mathcal{P}_+^\uparrow\) on \(\mathcal{H}\).
- Vacuum \(\Omega\in\mathcal{H}\), unit vector, unique (up to phase) Poincaré-invariant vector; cyclic for the field algebra.
- Spectrum condition. Generators \(P^\mu\) of translations have joint spectrum in the closed forward light cone \(\overline{V}_+=\{p:p\cdot p\ge 0,\, p^0\ge 0\}\).
- Fields. Operator-valued tempered distributions \(\varphi:\mathcal{S}(\mathbb{R}^{1,d-1})\to\mathrm{Op}(\mathcal{D})\) on a dense invariant domain \(\mathcal{D}\ni\Omega\), with \(U(a,\Lambda)\varphi(f)U(a,\Lambda)^{-1}=\varphi(f_{(a,\Lambda)})\) (covariance).
- Locality (microcausality). If \(\mathrm{supp}\,f\) and \(\mathrm{supp}\,g\) are spacelike separated, \([\varphi(f),\varphi(g)]=0\) on \(\mathcal{D}\).
- Temperedness / continuity of matrix elements as distributions.
Wightman functions (vacuum expectation values) $$ \mathcal{W}_n(f_1,\ldots,f_n)=\bigl(\Omega,\,\varphi(f_1)\cdots\varphi(f_n)\Omega\bigr) $$ are tempered distributions. Reconstruction (Wightman): a sequence of distributions \(\mathcal{W}_n\) satisfying Poincaré invariance, spectrum (support of Fourier transforms), locality (spacelike commutativity), positive-definiteness of the GNS-type sesquilinear form, and clustering, determines a unique (up to unitary equiv.) Wightman theory with those VEVs.
| Axiom package | Analytic consequence |
|---|---|
| Spectrum | holomorphic extension to forward tubes |
| Locality + spectrum | PCT, spin-statistics (with spin) |
| Clustering | uniqueness of vacuum \(\Leftrightarrow\) irreducibility |
Osterwalder–Schrader (Euclidean)¶
Work on Euclidean \(\mathbb{R}^d\) with Schwinger functions \(S_n\) (moments of a hypothetical Euclidean measure).
OS axioms (schematic):
| Axiom | Content |
|---|---|
| Temperedness / growth | \(S_n\) distributions of controlled growth |
| Euclidean invariance | \(E(d)\) (or \(\mathrm{SO}(d)\ltimes\mathbb{R}^d\)) invariance |
| Reflection positivity | for time-ordered supports, \(\sum_{i,j}\Theta S(F_i^*\times F_j)\ge 0\) (\(\Theta=\) time reflection) |
| Symmetry | \(S_n\) symmetric under permutations (Bosons) |
| Clustering | factorization at large separations |
Reconstruction (OS \(\Rightarrow\) Wightman), one-liner. Reflection positivity builds a Hilbert space from half-space Euclidean functionals; analytic continuation in time (Wick rotation reverse) produces Wightman functions satisfying spectrum and locality under standard technical growth hypotheses (OS0–OS5 in the literature).
Constructive route. Build Euclidean measures (e.g.\ \(P(\varphi)_2\)) satisfying OS, then reconstruct Minkowski QFT — Glimm–Jaffe program.
Haag–Kastler nets (algebraic QFT)¶
Assign to each open bounded region \(\mathcal{O}\subset\mathbb{R}^{1,d-1}\) a unital \(C^*\)-algebra \(\mathcal{A}(\mathcal{O})\) (or von Neumann algebra in a representation) such that:
- Isotony. \(\mathcal{O}_1\subset\mathcal{O}_2\Rightarrow\mathcal{A}(\mathcal{O}_1)\subset\mathcal{A}(\mathcal{O}_2)\).
- Locality. If \(\mathcal{O}_1\subset\mathcal{O}_2'\), then \([\mathcal{A}(\mathcal{O}_1),\mathcal{A}(\mathcal{O}_2)]=\{0\}\).
- Poincaré covariance. Automorphisms \(\alpha_{(a,\Lambda)}\) with \(\alpha(\mathcal{A}(\mathcal{O}))=\mathcal{A}((a,\Lambda)\mathcal{O})\).
- Vacuum representation. Irreducible rep on \(\mathcal{H}\) with invariant \(\Omega\) and spectrum condition on translation unitaries.
- Weak additivity / continuity variants as needed.
Fields optional. Local algebras may arise as von Neumann algebras generated by Wightman fields smeared in \(\mathcal{O}\); conversely, AQFT studies superselection, charges, and statistics via DHR theory without choosing fields.
| Wightman | Haag–Kastler |
|---|---|
| concrete operators \(\varphi(f)\) | abstract net \(\mathcal{A}(\mathcal{O})\) |
| VEVs primary | inclusion structure + states |
| reconstruction from \(\mathcal{W}_n\) | reconstruction of fields harder (when possible) |
Reeh–Schlieder (one-liner). Vacuum is cyclic (and separating) for local algebras of nonempty open \(\mathcal{O}\) — vacuum entanglement across regions; cornerstone of AQFT.
Free scalar field (model)¶
Klein–Gordon mass \(m\ge 0\): classical solution space with symplectic form; quantize via Fock space \(\mathcal{F}(L^2(\mathbb{R}^{d-1}))\).
- Two-point function $$ \mathcal{W}_2(x,y)=W_2(x-y)=\int\frac{d^{d-1}k}{(2\pi)^{d-1}2\omega_k}\,e^{-ik\cdot(x-y)},\quad \omega_k=\sqrt{|k|^2+m^2}. $$
- Higher \(\mathcal{W}_n\) by Wick’s theorem (pairings); odd \(n\) vanish.
- Locality from support properties of Pauli–Jordan commutator function \(\Delta(x-y)=W_2(x-y)-W_2(y-x)\).
- Spectrum from positive frequencies in the mode expansion.
- Euclidean free field: Gaussian measure with covariance \((-\Delta+m^2)^{-1}\); OS positivity holds; Schwinger two-point is the Yukawa potential / Bessel kernel.
Wick products \(:\varphi^n:\) renormalize powers; free field is the starting point for perturbations (\(P(\varphi)\), Yukawa, …) in constructive QFT.
Reconstruction theorems (summary table)¶
| Theorem | Input | Output |
|---|---|---|
| Wightman reconstruction | \(\{\mathcal{W}_n\}\) with positivity, … | \((\mathcal{H},U,\Omega,\varphi)\) |
| OS reconstruction | \(\{S_n\}\) with reflection positivity, … | Wightman theory |
| GNS | state \(\omega\) on \(C^*\)-net | Hilbert rep \((\mathcal{H}_\omega,\pi,\Omega)\) |
| DHR (pointer) | localized endomorphisms of net | superselection sectors / statistics |
Modern pointers (not full axioms here)¶
- Factorization algebras (Costello–Gwilliam): prefactorization on opens with Weiss codescent — perturbative AQFT / observables.
- Functorial QFT / cobordism hypotheses: TFT as functors on bordisms (higher categories); relates more to topological than Wightman QFT.
- Conformal nets / vertex algebras: 2d CFT axiomatizations.
See it¶
Haag’s theorem (informal). There is no interaction picture unitary intertwining free and nontrivial interacting Wightman fields with the same vacuum Poincaré representation in the expected way — rigorous versions say free and interacting representations of the CCR are disjoint. Consequence: constructive and axiomatic approaches cannot naively recycle free Fock space for interacting dynamics; Euclidean measures and algebraic nets encode interactions differently.
PCT from axioms. Combining spectrum condition (analyticity) with locality yields the PCT theorem: a theory is invariant under the antiunitary PCT map — a structural theorem without Lagrangian input (Streater–Wightman).
Check¶
Sketch why spacelike commutativity of free fields follows from the commutator function.
\([\varphi(f),\varphi(g)]=i\Delta(f,g)\mathbf{1}\) with \(\Delta\) odd distribution supported in the light cone (propagator support). If \(\mathrm{supp}\,f\) and \(\mathrm{supp}\,g\) are spacelike separated, no lightlike/timelike difference vector connects them, so \(\Delta(f,g)=0\).
What fails in OS reconstruction if reflection positivity is dropped?
No positive semidefinite inner product on the Euclidean half-space test functions \(\Rightarrow\) no Hilbert space; analytic continuation may still exist formally but not as VEVs of operators on a Hilbert space with positive metric.
Pitfalls¶
- Test function space: Schwartz vs compact support vs Gevrey — locality and time-zero fields need compatible choices.
- Uniqueness of vacuum vs cluster property: essentially equivalent under standard assumptions; multiple vacua \(\leftrightarrow\) superselection / phases.
- Indefinite metric (Gupta–Bleuler, BRST): gauge QFT may leave Hilbert-space Wightman form; AQFT uses observables’ nets.
- Mass gap vs \(m=0\): IR issues, continuous spectrum down to \(0\), long-range sectors (QED).
Refs¶
- Streater–Wightman, PCT, Spin and Statistics, and All That
- Haag, Local Quantum Physics
- Osterwalder–Schrader, Axioms for Euclidean Green’s functions I–II
- Glimm–Jaffe, Quantum Physics; Reed–Simon Vol.\ II (Fourier / free fields)
Next¶
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