From Euclidean Measures to Relativistic Models

An interacting Euclidean probability measure becomes a relativistic quantum field theory only when its Schwinger functions satisfy the full reconstruction hypotheses. Positivity of the measure is not reflection positivity, and analytic continuation of a two-point function is not reconstruction of a local field hierarchy.

Required background. The OS reconstruction theorem supplies the theorem; the constructive program and cutoff removal supplies the limiting measure; constructing P(φ)₂ and φ⁴₃ models supplies the worked hierarchy.

Helpful background. OS–Wightman comparison directions and failure modes marks the nonconverses; clustering, vacuum uniqueness, and mass-gap implications supplies the vacuum and spectral consequences.

The reconstruction hypotheses

Let $S_n\in\mathcal S'((\mathbb R^d)^n)$ be the moments of a Euclidean measure. The standard OS route requires a compatible package: permutation symmetry; Euclidean covariance; reflection positivity for polynomial functionals supported at positive Euclidean times; regularity or growth sufficient for analytic continuation; and clustering when a unique vacuum is desired. For a cylinder polynomial

$$F(\phi)=\sum_j c_j\prod_{r=1}^{n_j}\phi(f_{jr}), \qquad \operatorname{supp}f_{jr}\subset\{t>0\},$$

reflection positivity says $\mathbb E[\overline{(\Theta F)}F]\ge0$. Quotienting the positive-time polynomial space by null vectors and completing gives a Hilbert space. Euclidean time translations become a contraction semigroup $e^{-tH}$ with $H\ge0$; spatial symmetries and analytic continuation yield the positive-energy Poincaré representation and local Wightman fields.

Osterwalder and Schrader’s theorem is stated for a hierarchy, not merely for a probability measure. The original construction and the possible mismatch of growth conditions in the reverse direction are explained in Osterwalder and Schrader 1973, pp. 83–112 and Summers 2016, §3, pp. 9–11.

The P(φ)₂ hierarchy

Take the infinite-volume massive $P(\phi)_2$ measure constructed in the weak-coupling regime. Its moments are symmetric because the random variables commute. Euclidean covariance follows after the boundary is removed. Reflection positivity is inherited from reflection-positive finite-volume approximants chosen compatibly with the time-zero plane. Uniform moment bounds provide temperedness and the needed growth; cluster estimates provide exponential clustering.

The OS theorem therefore reconstructs a Hilbert space $\mathcal H$, vacuum $\Omega$, positive Hamiltonian, Poincaré representation, and local scalar Wightman field whose Euclidean boundary values are the constructed Schwinger functions. The isolated-particle and scattering conclusions require the additional spectral estimates described separately. The original $P(\phi)_2$ construction verifies the Wightman axioms and particle structure; Glimm, Jaffe, and Spencer 1974, pp. 585–632.

This worked route returns to reflection positivity within Osterwalder–Schrader reconstruction. Its conclusion is a local relativistic scalar model. It is not an assertion that the Euclidean sample field is an operator-valued Minkowski field, nor that Euclidean time can simply be replaced pointwise by $it$.

An independent check applies the construction to the free covariance. The OS inner product of one-field vectors must yield the positive-energy one-particle norm with factor $1/(2\omega_{\mathbf p})$, and time translation must act by $e^{-t\omega_{\mathbf p}}$. This recovers $H=\omega_{\mathbf p}\ge0$ and fixes the reflection convention.

Adversarial reconstruction failures

First, multiply a legitimate Schwinger two-point function by a translation-invariant momentum-space factor that remains positive definite but produces a negative reflected-time quadratic form. It still defines a Gaussian probability measure; it does not define a positive OS Hilbert space. Ordinary measure positivity cannot repair the negative norm.

Second, suppose only $S_2$ has a good analytic continuation while higher moments violate OS positivity or the growth condition. One can reconstruct a generalized free two-point sector, but not the asserted interacting local hierarchy. Higher Wightman functions, locality, and a common field domain do not follow from $S_2$ alone.

Third, weak convergence of measures without uniform moment bounds need not imply convergence of unbounded polynomial moments. The limiting law may exist while the proposed Schwinger hierarchy does not converge. Uniform integrability is the missing hypothesis.

The converse is also limited: Wightman functions analytically continue to Euclidean distributions under standard assumptions, but their being moments of a probability measure and satisfying a particular OS growth axiom require additional conditions. Reconstruction is a theorem with direction and domain, not a blanket equivalence.

Accordingly, every reconstructed property should be traced to the exact Euclidean hypothesis that supplies it.

Exercises

1. Null vectors. Why must one quotient by $F$ with $\langle F,F\rangle_{OS}=0$ before completing?

Solution

Reflection positivity initially gives only a positive semidefinite form. Null vectors have zero distance from zero; quotienting them makes the form definite and hence a Hilbert-space norm after completion.

2. Clustering and the vacuum. What does failure of clustering leave undecided?

Solution

The basic OS reconstruction can still produce a representation, but the translation-invariant subspace need not be one-dimensional. The state may be a mixture of phases, so vacuum uniqueness is not licensed.

References

• Glimm, James, Arthur Jaffe, and Thomas Spencer. “The Wightman Axioms and Particle Structure in the $P(\phi)_2$ Quantum Field Model.” Annals of Mathematics 100 (1974): 585–632. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI.
• Summers, Stephen J. “A Perspective on Constructive Quantum Field Theory.” arXiv:1203.3991, revised 2016. Open PDF.