Wightman Fields, Reconstruction, and Structural Theorems

The Wightman framework begins with smeared fields or their complete vacuum correlation functions. It connects these data to Hilbert-space reconstruction, analytic continuation, and structural theorems without assuming a Lagrangian.

Enter this chapter

Read the first three topics in order; they establish the objects used by the later proofs. Helpful background. Quantum fields as operator-valued distributions motivates smearing and common domains; the Källén–Lehmann representation motivates positive spectral measures; and CPT: hypotheses, content, and limits gives the physical structural theorem.

Topics

1. Wightman Fields, Domains, and Axioms — Specifies the common invariant domain and relativistic field axioms.
2. Wightman Functions and Spectral Support — Translates field axioms into properties of the full vacuum hierarchy.
3. The Wightman Reconstruction Theorem — Builds a cyclic field theory by quotienting null vectors and completing.
4. Tube Domains, Complex Lorentz Covariance, and Analyticity — Turns positive-energy support into holomorphic tube functions.
5. Jost Points, Edge-of-the-Wedge, and Locality — Uses locality to join analytic functions with different field orderings.
6. Analyticity, CPT, and Spin–Statistics — Separates the shared analytic ingredients from the two distinct conclusions.
7. CPT Theorem Variants and Their Hypotheses — Compares CPT statements for different field classes and geometries.
8. Spin–Statistics Theorems and Failure Modes — Locates the positivity and exchange-topology assumptions behind statistics.
9. Haag’s Theorem and Inequivalent Representations — Explains the obstruction to a naïve unitary free–interacting identification.
10. Haag-Theorem Variants, Domains, and Proposed Evasions — Tests proposed evasions by identifying which assumption changes.
11. Wightman-Framework Models, Counterexamples, and Scope — Distinguishes constructed models from formal examples and counterexamples.

How the ideas fit together

There are two connected arguments. Positivity of the entire hierarchy supplies a positive inner product and reconstruction; spectrum and locality supply analytic continuation and the structural consequences. Positivity of only the two-point function cannot replace the first argument. Neither argument by itself proves scattering completeness or a mass gap. The table follows these distinctions in Streater and Wightman 2016, Chapters 3–4, pp. 96–174.

Enter the Wightman framework

Objects, hypotheses, licensed conclusions, and stopping conditions

Starting object	Domain and regularity	Essential input	Construction or theorem	Licensed conclusion	What is not implied	Boundary test
Smeared fields	Operator-valued tempered distributions on one common invariant dense domain	Vacuum, Poincaré covariance, positive energy, adjoints, and local or graded-local exchange	Take all vacuum expectation values in their stated order	A complete Wightman hierarchy with the inherited distributional and symmetry properties	Pointwise field operators or positivity from covariance alone	Attempt to multiply unsmeared fields or change domains between factors
Vacuum hierarchy	All tempered distributions Wn, not only the two-point function	Normalization, Hermiticity, covariance, spectral support, locality, and positivity of every polynomial	Quotient the Borchers algebra by null vectors and complete	A cyclic Hilbert-space theory, vacuum, Poincaré action, and fields, unique up to unitary equivalence	Reconstruction from a positive two-point kernel with arbitrary higher functions	Keep W2 positive but choose an incompatible W4
Relative-momentum distributions	Tempered distributions in translation-invariant difference variables	Support in the appropriate products of the closed future cone and polynomial growth control	Fourier–Laplace transformation	Holomorphic functions in primitive tubes with the original distributions as tempered boundary values	The correct tube direction without fixing the Fourier sign, or arbitrary complex continuation	Reverse the Fourier exponential and check that the imaginary cone reverses
Ordered analytic functions	Primitive and extended tubes with real Jost configurations on their boundaries	Local or graded-local equality on a nonempty real open set, plus the relevant analytic continuation theorem	Edge-of-the-wedge continuation and complex Lorentz covariance	Equality of the specified orderings in their joined analytic domain	Equality of arbitrary time-ordered functions or permutations outside the proved domain	Choose a spacelike-looking configuration that fails the full Jost cone criterion
Covariant local field multiplet	Positive-metric Wightman theory with the representation and adjoint structure used by the theorem	Spectrum condition, locality, analyticity, and theorem-specific transformation laws	CPT or spin–statistics argument	An antiunitary CPT implementation or the permitted statistics grading for that field class	The same formula for nonlocal fields, indefinite-metric gauge potentials, or braid statistics in low dimension	Remove positive metric or replace four-dimensional exchange by braid-group exchange
Two time-slice field theories	Wightman fields satisfying the domain, covariance, irreducibility, and vacuum assumptions of the chosen Haag theorem	A unitary identification at one time together with the theorem's other hypotheses	Haag–Hall–Wightman comparison	If one field is free, the identified field has the corresponding free vacuum functions under the stated scope	Nonexistence of interacting QFT, or a no-go result for regulated or merely formal interaction pictures	Introduce a finite cutoff and identify exactly which Wightman assumption no longer holds

Structured table data (JSON).

Review the chapter

Why is cyclicity needed when vacuum correlations are said to determine a theory uniquely?

Solution

The correlations determine the closure of vectors obtained by acting on the vacuum with field polynomials. An orthogonal spectator sector contributes nothing to those correlations. Cyclicity excludes that invisible sector, allowing uniqueness up to a vacuum-preserving unitary.

Where to go next

For reconstruction from Euclidean data, continue to Euclidean fields and Osterwalder–Schrader reconstruction.

References

• Streater, Raymond F., and Arthur S. Wightman. 2016. PCT, Spin and Statistics, and All That. Princeton Landmarks in Physics. Princeton University Press. DOI.