Euclidean Random Fields and Schwinger Hierarchies

A Euclidean random field is a probability law on generalized functions, not usually a probability density over ordinary functions. Its moments form the Schwinger hierarchy. Starting from the law automatically supplies the hierarchy’s compatibility and positivity properties; starting from a list of distributions does not, and even a compatible moment list need not determine its law uniquely.

Required background. Existence, construction, reconstruction, and continuum claims distinguishes a constructed measure from a formal moment prescription. Probability, random variables, and conditional expectation supplies expectations and laws. Gaussian measures and Wick’s rule supplies Gaussian characteristic functionals and moments.

Helpful background. Euclidean correlators and Schwinger functions motivates the correlation functions used below.

Random tempered distributions

Let $\mathcal S(\mathbb R^d)$ be real Schwartz space and $\mathcal S'(\mathbb R^d)$ its tempered-distribution dual. A scalar Euclidean field is an $\mathcal S'$-valued random variable $\Phi$ with law $\mu$. Smearing is fundamental:

$$\Phi(f)=\langle \Phi,f\rangle,\qquad f\in\mathcal S(\mathbb R^d).$$

Each finite list $f_1,\ldots,f_n$ gives an ordinary random vector $(\Phi(f_1),\ldots,\Phi(f_n))$. These finite-dimensional laws are the cylinder distributions. They must be consistent under deleting, permuting, and linearly recombining the test functions. A merely formal collection of cylinder densities is not yet a countably additive probability measure on $\mathcal S'$.

The characteristic functional packages all cylinder laws:

$$\chi(f)=\int_{\mathcal S'}e^{i\Phi(f)}\,d\mu(\Phi).$$

It obeys $\chi(0)=1$, $\chi(-f)=\overline{\chi(f)}$, and positive definiteness,

$$\sum_{j,k=1}^N \overline{c_j}c_k\,\chi(f_k-f_j)\geq 0.$$

For a nuclear test-function space, continuity at the origin together with these conditions is the decisive extension input: the Minlos theorem turns the cylindrical prescription into a probability measure on the dual. This is an infinite-dimensional statement; ordinary finite-dimensional Bochner theory alone does not establish countable additivity. The original extension theorem is given in Minlos 1959, pp. 497–518.

Euclidean covariance means invariance of the law under the Euclidean group. For a scalar field and $g=(a,R)\in E(d)$, define $f_g(x)=f(R^{-1}(x-a))$. Then

$$\chi(f_g)=\chi(f),$$

or equivalently the push-forward of $\mu$ by $\Phi(f)\mapsto\Phi(f_g)$ equals $\mu$. Internal indices require the corresponding finite-dimensional representation; no Lorentzian signature has entered.

Moments, cumulants, and the hierarchy

When the necessary moments exist, the smeared Schwinger functions are

$$S_n(f_1,\ldots,f_n) =\mathbb E_\mu\!\left[\Phi(f_1)\cdots\Phi(f_n)\right].$$

Thus $S_n$ is a symmetric multilinear functional. If it is jointly continuous on $\mathcal S(\mathbb R^d)^n$, the Schwartz kernel theorem identifies it with a tempered distribution on $(\mathbb R^d)^n$. Euclidean covariance of $\mu$ gives covariance of every $S_n$, but the converse requires enough information to recover the law.

Connected, or truncated, Schwinger functions are the cumulants. They are defined by the partition identity

$$S_n(f_1,\ldots,f_n) =\sum_{\pi\in\Pi_n}\prod_{B\in\pi} S^{\mathrm T}_{|B|}\bigl((f_j)_{j\in B}\bigr).$$

This identity separates independent components and makes clustering statements natural. It does not assert convergence of a generating-functional power series: differentiability of $\chi$ at the origin and analyticity in a neighborhood of the origin are different regularity claims.

A moment hierarchy coming from a measure has a strong positivity property. For every finite polynomial functional

$$P(\Phi)=\sum_{r=0}^N \Phi^{\otimes r}(F_r),$$

with symmetric test kernels $F_r$, one has

$$\mathbb E_\mu\!\left[\lvert P(\Phi)\rvert^2\right]\geq0.$$

Written only in terms of the $S_n$, this says that every finite moment matrix is positive semidefinite. It is ordinary probability positivity, and is logically weaker than reflection positivity, which tests a reflected positive-time algebra.

First QFT application: the massive Gaussian free field

For $m>0$, define the covariance on real test functions by the formula below. This measure-theoretic construction supplies the exact hierarchy used in the physical discussion of Euclidean correlators and Schwinger functions.

$$C_m(f,g)=\int_{\mathbb R^d}\frac{d^dp}{(2\pi)^d} \frac{\overline{\widetilde f(p)}\widetilde g(p)}{p^2+m^2}.$$

It is continuous, symmetric, and nonnegative. Hence

$$\chi_m(f)=\exp\!\left[-\frac12 C_m(f,f)\right]$$

defines a centered Gaussian measure on $\mathcal S'$. The Euclidean transform used here is $\widetilde f(p)=\int d^dx\,e^{-ip\cdot x}f(x)$; the Euclidean metric is positive definite.

The hierarchy is fixed by Wick’s rule:

$$S_{2n+1}=0, \qquad S_{2n}(f_1,\ldots,f_{2n}) =\sum_{\text{pairings }P}\prod_{\{i,j\}\in P}C_m(f_i,f_j).$$

In particular $S_2=C_m$, while every truncated function above order two vanishes. Translation invariance follows because the Fourier multiplier depends only on $p$; rotation invariance follows because it depends only on $p^2$. These are quick checks that the proposed covariance has the claimed Euclidean symmetry. Osterwalder and Schrader 1973, §3, pp. 87–90 formulate the corresponding hierarchy-level Euclidean conditions.

What a formal hierarchy can fail to provide

There are two distinct obstructions.

First, invalid moment matrices rule out any probability law. For a real random variable $X=\Phi(f)$, positivity of $\mathbb E[(a+bX)^2]$ requires

$$S_0=1,\qquad S_2(f,f)-S_1(f)^2\geq0.$$

A prescription with $S_1(f)=0$ and $S_2(f,f)<0$ is symmetric and may be Euclidean covariant, but cannot be a moment hierarchy of a probability measure.

Second, positivity and existence need not give uniqueness. Moment-indeterminate one-dimensional laws already show that every polynomial moment can agree while the probability laws differ. Consequently an infinite-dimensional Schwinger hierarchy, without a determinacy condition such as suitable exponential bounds, need not identify a unique underlying measure. Conversely, a measure can exist even when not all polynomial moments are finite; then $\chi$ is primary and the full hierarchy is unavailable.

These failure boundaries matter later. Osterwalder–Schrader reconstruction can be formulated for a suitably regular hierarchy, but it does not repair an inconsistent moment problem and it should not be described as constructing a Euclidean measure unless a measure theorem has actually been applied.

Independent checks

For the massive Gaussian example, four checks are immediate.

• Positivity: $p^2+m^2>0$ makes $C_m(f,f)\geq0$.
• Infrared control: $m>0$ bounds the multiplier at $p=0$; the massless case needs dimension- and observable-dependent treatment.
• Normalization: $\chi_m(0)=1$ and $S_0=1$.
• Combinatorics: $S_4(f_1,f_2,f_3,f_4)$ contains exactly the three pairings, which verifies the partition convention.

Exercise

Show that the connected four-point function of the centered Gaussian field vanishes.

Solution

Wick’s rule gives

$$S_4(1,2,3,4)=C_{12}C_{34}+C_{13}C_{24}+C_{14}C_{23}.$$

The partition formula for $S_4$ has these same three products of connected two-point functions, while $S_1^{\mathrm T}=0$. Since $S_2^{\mathrm T}=C$, subtracting all nontrivial partition contributions leaves $S_4^{\mathrm T}=0$.

References

• 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.