OS–Wightman Comparison Directions and Failure Modes

“OS–Wightman equivalence” is shorthand for several directional theorems with different inputs. A Euclidean measure, a Schwinger hierarchy, an analytic hierarchy, a Wightman hierarchy, and a reconstructed field representation are not the same object. The correct comparison records which operation connects them, what growth and positivity hypotheses it uses, and in what sense the result is unique.

Required background. Counterexamples, nonconverses, and hypothesis stress tests supplies directional reasoning. The Wightman reconstruction theorem supplies the Lorentzian hierarchy-to-field step. Osterwalder–Schrader reconstruction supplies the corrected Euclidean-to-Lorentzian theorem. Analytic continuation between Euclidean and Lorentzian domains supplies the boundary-value operation.

Helpful background. Wightman, Euclidean, local, algebraic, constructive, and perturbative frameworks compares what the frameworks take as primary.

Directional comparison

Input	Operation and hypotheses	Output	Uniqueness	Main failure boundary
Euclidean probability law on a distribution space	Take moments; require their existence and continuity	Schwinger hierarchy	The law fixes all existing moments	Moments may not determine the law; some moments may diverge
Characteristic functional on a nuclear test space	Minlos extension: normalization, continuity, positive definiteness	Euclidean probability law	Unique law with that characteristic functional	A formal moment list is not a characteristic functional
Complete Schwinger hierarchy	Corrected OS II growth, Euclidean covariance, reflection positivity, symmetry, clustering	Tempered Wightman hierarchy satisfying the Wightman axioms	Wightman distributions uniquely determined	Fixed-order temperedness or two-point positivity is insufficient
Complete Wightman hierarchy	Spectrum condition, locality, analytic continuation, and the regularity assumed by the chosen Euclidean theorem	Ordered analytic functions and Euclidean boundary restrictions	Unique within the connected analytic domains	The output need not be presented by a probability measure; sharp Euclidean restrictions require control
Wightman hierarchy with positivity	Wightman reconstruction	Fields, Hilbert space, cyclic vacuum, Poincaré representation	Vacuum representation unique up to unitary equivalence	Correlators outside the complete hierarchy are not reconstructed
One Euclidean or Lorentzian two-point function	Spectral or complex-time continuation	One two-point boundary-value relation	Unique only in its analytic class	Does not fix interacting higher functions or prove complete-theory equivalence

The first OS paper formulated E0–E4 and presented both directions Osterwalder and Schrader 1973, §§3–6, pp. 87–105. The second paper corrected the Euclidean-to-Wightman proof, gave a revised equivalence result under one regularity system, and a practical reconstruction theorem under the E0′ or E0″ growth condition Osterwalder and Schrader 1975, §§III–IV, pp. 285–289. Therefore the unqualified statement “E0–E4 are equivalent to the Wightman axioms” is not the corrected theorem used here.

What uniqueness does and does not mean

Analytic uniqueness says that two analytic functions agreeing on an appropriate open real-analytic set agree throughout the connected common domain. Distributional uniqueness says the resulting boundary-value hierarchy is fixed. Wightman reconstruction then says two cyclic field realizations of the same complete hierarchy are related by a vacuum-preserving unitary map.

None of these statements says that two different Euclidean measures with the same moments are identical. That requires moment determinacy or equality of characteristic functionals. Nor does unitary equivalence of vacuum representations identify gauge-fixed auxiliary fields, thermal representations, or distinct superselection sectors not generated from the chosen vacuum.

Clustering is another directional qualifier. With it, the reconstructed translation-invariant vacuum is unique in the relevant sector. Without it, OS positivity and analytic continuation can still provide a positive-energy representation, but a mixture or direct integral of phases may remain.

First QFT application: the free massive scalar in both directions

Start Lorentzianly with

$$W_2(x)=\int\frac{d^{d-1}\mathbf p}{(2\pi)^{d-1}2\omega_{\mathbf p}} e^{-i\omega_{\mathbf p}x^0+i\mathbf p\cdot\mathbf x}.$$

Positive mass-shell support gives analyticity for $\operatorname{Im}z^0<0$. At $z^0=-i\tau$, $\tau>0$, its value is the massive Euclidean covariance. Wick’s rule continues every $W_n$ and produces the Gaussian Schwinger hierarchy.

In the reverse direction, start with that Gaussian Euclidean measure. The covariance is Euclidean invariant and reflection positive; Wick’s rule supplies symmetry and hierarchy-wide positivity; $m>0$ supplies clustering and straightforward factorial bounds. The OS quotient gives the Fock space, $H=\sqrt{\mathbf P^2+m^2}$ on the one-particle sector, and the displayed $W_2$. The two routes recover the same complete Gaussian hierarchy and therefore the same cyclic free-field representation. This is the exact case underlying the broader framework comparison.

Growth stress test: a generalized free spectral measure

Replace the single mass by a positive spectral measure:

$$W_2(x)=\int_0^\infty d\rho(M^2)\,\Delta_+^{(M)}(x), \qquad S_2(x_E)=\int_0^\infty d\rho(M^2)\,C_M(x_E).$$

If $d\rho$ is polynomially bounded, the Lorentzian two-point function is tempered, the Euclidean covariance is a positive superposition of reflection-positive massive covariances, and Wick’s rule defines a generalized free hierarchy. Generalized free fields and their local Wightman properties were analyzed in Greenberg 1961, pp. 158–176.

Take, for example, $d\rho_N(M^2)=\mathbf 1_{[m^2,\infty)}(M^2)(1+M^2)^N dM^2$ with $m>0$. For noncoincident Euclidean time the factor $e^{-M\lvert\tau\rvert}$ makes the integral finite, but increasing $N$ raises the distributional singularity at $\tau=0$. For each fixed $N$, one can seek a Schwartz seminorm of sufficiently high order and then use Gaussian pairings to obtain factorial hierarchy bounds. The check is model-dependent: reflection positivity of $S_2$ does not itself prove E0′ or E0″ for all orders.

If one instead chooses a suitable superpolynomial but subexponential spectral weight, the Euclidean kernel can still be finite at every strictly positive $\lvert\tau\rvert$ because of exponential damping, while its real-time boundary need not be tempered. Then the strongest surviving claim is a positive Euclidean kernel on the separated domain—not a Wightman theory on Schwartz space. The missing hypothesis is the spectral/distributional growth bound. This example shows why growth must be carried along the arrow rather than inferred from formal continuation.

Adversarial test: one successful continuation

Suppose an interacting proposal has exactly the free massive two-point function, and that it is continued correctly between signatures. Nothing follows about $S_4$. It could be the Wick sum, an interacting connected four-point function, or an inconsistent distribution. Choosing $S_4=0$ already violates polynomial positivity for a smearing with nonzero variance, as

$$\mathbb E[(X^2-\mathbb E[X^2])^2]<0$$

under that prescription. Thus a successful two-point continuation proves only a two-point boundary-value relation. Complete-theory equivalence requires compatible all-$n$ data, reflection positivity, symmetry, uniform growth, and the hypotheses of the selected reconstruction theorem.

Independent comparison checks

• Identify whether the input is a law, characteristic functional, moment hierarchy, analytic hierarchy, or field representation.
• State the arrow direction and do not use the converse without its hypotheses.
• Track the entire hierarchy; mark a Gaussian assumption explicitly when $S_2$ is used to fix all orders.
• Separate analytic uniqueness, moment determinacy, and unitary equivalence.
• State the test space and growth system on both sides of a boundary-value map.
• Keep exact reconstruction separate from numerical analytic-continuation stability and from continuum-limit existence.

References

• Greenberg, Oscar W. “Generalized Free Fields and Models of Local Field Theory.” Annals of Physics 16 (1961): 158–176. doi:10.1016/0003-4916(61)90032-X.
• Minlos, R. A. “Generalized Random Processes and Their Extension to a Measure.” Trudy Moskovskogo Matematicheskogo Obshchestva 8 (1959): 497–518. Stable record and English metadata.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. doi:10.1007/BF01645738. Open PDF.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. doi:10.1007/BF01608978. Open PDF.