Gaussian Euclidean Fields as Measures

A Euclidean free field is not a random function at every point. It is a Gaussian random tempered distribution whose law is fixed by a positive covariance form. For the massive scalar field this construction is exact, reflection positive, and exponentially clustering; it is the reference measure against which interacting densities are defined.

Required background. Euclidean random fields and Schwinger hierarchies supplies moments of random distributions; Osterwalder–Schrader axioms and reflection positivity supplies the time-reflection quadratic form.

Helpful background. Euclidean growth, regularity, and temperedness explains the distribution topology; domains, signatures, supports, and regularity helps distinguish a kernel, an operator, and its quadratic form.

The Gaussian measure on distributions

Let $C:\mathcal S(\mathbb R^d)\to\mathcal S'(\mathbb R^d)$ be continuous, symmetric, real, and positive:

$$C(f,g)=\langle f,Cg\rangle=C(g,f),\qquad C(f,f)\ge0.$$

If $f\mapsto\exp[-C(f,f)/2]$ is continuous at the origin, the Bochner–Minlos theorem gives a unique probability measure $\mu_C$ on $\mathcal S'(\mathbb R^d)$ with characteristic functional

$$\int e^{i\phi(f)}d\mu_C(\phi)=e^{-C(f,f)/2}.$$

Differentiating at the origin gives mean zero, two-point function $S_2(f,g)=C(f,g)$, and Wick’s rule: odd moments vanish and every even moment is the sum over pairings of products of $C(f_i,f_j)$. Thus positivity and continuity of the covariance, not formal multiplication of infinitely many Lebesgue measures, construct the field.

For $m>0$, set $C_m=(-\Delta+m^2)^{-1}$ on $\mathbb R^d$. In Fourier variables,

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

The denominator proves positivity and continuity on Schwartz space. The resulting measure is supported on distributions of negative regularity, not generally on ordinary functions. Point values $\phi(x)$ are therefore mnemonic; $\phi(f)$ is the defined random variable. Summers 2016, §3, pp. 9–10 gives this massive Gaussian law and its Schwinger two-point function in the OS setting.

Reflection and decay come from the covariance

Write $\theta(t,\mathbf x)=(-t,\mathbf x)$ and let $f$ be supported in $t>0$. Gaussian reflection positivity reduces to

$$C_m(\theta f,f)\ge0.$$

After spatial Fourier transformation, the time kernel is

$$\frac{e^{-\omega_{\mathbf p}|t-s|}}{2\omega_{\mathbf p}}, \qquad \omega_{\mathbf p}=\sqrt{|\mathbf p|^2+m^2}.$$

For reflected arguments $t,s>0$, $|(-t)-s|=t+s$, so the quadratic form becomes an integral of absolute squares weighted by $1/(2\omega_{\mathbf p})$. It is nonnegative. Wick’s rule then lifts this positivity from linear fields to polynomial cylinder functions.

The same spectral denominator controls clustering. Translating the support of $g$ a large distance $a$ away makes $C_m(f,g_a)$ decay at the massive rate, up to the familiar dimension-dependent power. Every connected Gaussian correlation except the two-point function is zero, so the two-point decay supplies the full truncated-correlation statement. The exact mass parameter is consequently visible both in the covariance pole and in Euclidean decay.

The first application is the Gaussian field with sources. For a real source $J\in\mathcal S$, completion of the Gaussian square gives

$$Z[J]=\mathbb E_{\mu_{C_m}}e^{\phi(J)} =\exp\!\left(\frac12 C_m(J,J)\right).$$

Functional derivatives reproduce all Schwinger functions. This equality is a theorem about the moment-generating functional where it is finite; it is not a definition by a nonexistent flat measure on field space.

Infrared failure and an independent check

Set $m=0$ on a periodic box. The constant Fourier mode has denominator $p^2=0$, so $(-\Delta)^{-1}$ is not a covariance on all test functions. One may restrict to zero-mean tests, fix the zero mode, or introduce an infrared regulator, but each changes the domain. Claiming the massive construction unchanged would hide a failed hypothesis. On infinite space, the massless covariance also has dimension-dependent infrared behavior; it cannot be inferred from the massive case merely by substituting $m=0$.

As an independent normalization check, choose one real test function $f$. Then $X=\phi(f)$ must be an ordinary normal random variable of variance $C_m(f,f)$. Its fourth moment computed from the characteristic functional is $3C_m(f,f)^2$. Any cutoff implementation that does not approach this value has the wrong covariance normalization or Fourier measure.

The converse boundary is equally important: a positive translation-invariant covariance defines a Gaussian measure, but it need not be reflection positive. Reflection positivity is a condition on how the kernel couples positive times to their reflected copies, not a consequence of ordinary positive definiteness.

There is also a useful support check. If a sequence of mollifiers approaches a delta function, the variances $C_m(\rho_\varepsilon,\rho_\varepsilon)$ diverge in the dimensions where the field has no pointwise realization. This is not a pathology of the probability measure; it confirms that the correct sample space is distributional. Conversely, smearing at a fixed physical scale keeps the variance finite and gives a genuine Gaussian random variable. Any numerical discretization should reproduce both behaviors: stable smeared observables and regulator-dependent pointlike variance.

Exercises

1. Wick’s fourth moment. Compute $\mathbb E[\phi(f_1)\phi(f_2)\phi(f_3)\phi(f_4)]$.

Solution

Differentiate the joint characteristic functional four times. The result is $C(f_1,f_2)C(f_3,f_4)+C(f_1,f_3)C(f_2,f_4)+C(f_1,f_4)C(f_2,f_3)$, one term for each pairing.

2. Source differentiation. Show that $\delta^2 Z[J]/\delta J(f)\delta J(g)$ at $J=0$ equals $C_m(f,g)$.

Solution

Differentiate $\exp[C_m(J,J)/2]$. The first derivative vanishes at zero; the second leaves the bilinear cross term $C_m(f,g)$. This independently fixes the factor $1/2$ in the exponent.

References

• Minlos, Robert A. “Generalized Random Processes and Their Extension to a Measure.” Trudy Moskovskogo Matematicheskogo Obshchestva 8 (1959): 497–518. MathNet record.
• 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.