Interacting Measures, Stability, and Wick Ordering

An interacting Euclidean measure is obtained by weighting a Gaussian random distribution only when the interaction is defined on that distribution space and its exponential is integrable. Wick ordering removes covariance-dependent self-contractions; stability prevents the weight from running to infinity along large-field directions. Neither condition can be replaced by a formal action.

Required background. Gaussian Euclidean fields as measures fixes the reference law and Wick contractions; the constructive program and cutoff removal separates fixed-cutoff normalization from uniform removal.

Helpful background. Osterwalder–Schrader axioms and reflection positivity explains the positivity that a regulator should preserve; Euclidean growth, regularity, and temperedness supplies the distributional setting.

Wick ordering defines the local polynomial

Let $\phi_\kappa$ be a mollified massive Gaussian field in two dimensions and let $c_\kappa=\mathbb E[\phi_\kappa(x)^2]$, which diverges as the ultraviolet scale $\kappa\to\infty$. The covariance-dependent Wick powers are

$$:\!\phi_\kappa^2\!:=\phi_\kappa^2-c_\kappa, \qquad :\!\phi_\kappa^4\!:=\phi_\kappa^4-6c_\kappa\phi_\kappa^2+3c_\kappa^2.$$

Their smeared limits exist as random distributions even though the pointwise powers of $\phi$ do not. Equivalently, they are Hermite polynomials with the variance fixed by the chosen covariance. Changing the reference covariance changes lower-order terms, so the convention must remain fixed or be accompanied by the corresponding finite counterterm.

On a two-dimensional torus $\mathbb T_L^2$, the regulated quartic law is

$$d\nu_{L,\kappa}(\phi)=Z_{L,\kappa}^{-1} \exp\!\left[-\lambda\int_{\mathbb T_L^2}:\!\phi_\kappa^4\!:_C dx\right]d\mu_C(\phi), \qquad \lambda\ge0.$$

For fixed $L$, the Wick polynomial converges in every finite $L^p(\mu_C)$ needed in the standard construction, and a stability estimate bounds its negative tail strongly enough that the exponential is integrable. Thus the ultraviolet limit defines a normalizable finite-volume $\lambda\phi^4_2$ measure. The classic finite-volume and cutoff-removal mechanism is reviewed in Summers 2016, §3.1, pp. 10–12; the original no-cutoff construction is Glimm and Jaffe 1970, pp. 362–401.

What stability licenses

For a general even polynomial $P(u)=a_{2n}u^{2n}+\cdots+a_0$, the decisive large-field hypothesis is $a_{2n}>0$. Wick ordering adds lower-degree, cutoff-dependent terms but does not reverse the positive leading behavior. One seeks a bound of the form

$$\int_\Lambda :\!P(\phi_\kappa)\!:_C dx \ge -A|\Lambda|-R_\kappa(\phi),$$

where the remainder is controlled in exponential moments uniformly enough for the intended limit. This licenses $0<Z_{\Lambda,\kappa}<\infty$ and uniform moment estimates. It does not by itself prove an infinite-volume limit, clustering, reflection positivity, or non-Gaussianity; those require further arguments.

For the worked torus example, differentiating the finite-cutoff partition function gives

$$-\frac{\partial}{\partial\lambda}\log Z_{L,\kappa} =\mathbb E_{\nu_{L,\kappa}}\int_{\mathbb T_L^2}:\!\phi_\kappa^4\!:_C dx.$$

This identity checks both normalization and the sign of the interaction. Convexity of $\log Z$ follows because its second derivative is the variance of the integrated Wick polynomial. These finite-volume facts survive a limit only under uniform integrability.

The physical status and the precise low-dimensional scope return to rigorous status, construction, and open problems. The theorem constructs a two-dimensional scalar measure; it does not transfer unchanged to $d=4$, where additional renormalization and the nontrivial continuum-limit problem intervene.

Adversarial stability tests

Reverse the sign of the leading coefficient: $P(u)=-u^4$. Restricting a finite-dimensional ultraviolet approximation to its constant mode $u$ makes the density proportional to $e^{+\lambda |\Lambda|u^4}$ times a Gaussian. The quartic growth dominates the Gaussian decay, so $Z=\infty$. This failure occurs before any limit.

A less obvious error Wick-orders successive cutoffs with unrelated covariances but keeps the same bare quadratic coefficient. Since

$$:\!\phi^4\!:_{C'}=: \!\phi^4\!:_C-6(C'-C)(0):\!\phi^2\!:_C+3(C'-C)(0)^2,$$

the sequence has silently changed its mass and vacuum-energy counterterms. Apparent convergence can then describe different theories. The independent diagnostic is to translate every approximation to one common Wick convention and verify that the induced lower-order coefficients converge.

The converse also fails: normalizability at each cutoff does not imply a uniform lower bound or a nontrivial continuum measure. A sequence can be perfectly integrable while concentrating at a point or drifting with the regulator.

One should also distinguish vacuum-energy renormalization from observable normalization. Adding a cutoff-dependent constant $E_\kappa|\Lambda|$ to the action multiplies both the numerator and partition function by the same factor, so normalized correlation functions are unchanged at fixed cutoff. A quadratic counterterm is different: it changes the relative weight of field configurations and hence the two-point function. This gives a practical check on a counterterm calculation. Constants may be fixed by a pressure convention, whereas mass terms must be fixed by a physical or correlation-length condition. Confusing the two can make a finite partition function look correct while sending the renormalized mass to an unintended value. Repeat this comparison after every change of ultraviolet regulator: finite counterterms may change even when the continuum normalization condition does not.

Exercises

1. Centering check. Show that $\mathbb E_{\mu_C}[:\!\phi_\kappa^4\!:]=0$ at a fixed point of the regulated field.

Solution

For a centered Gaussian $X$ of variance $c$, $\mathbb E X^2=c$ and $\mathbb E X^4=3c^2$. Hence $3c^2-6c^2+3c^2=0$.

2. Partition-function convexity. Prove $\partial_\lambda^2\log Z_{L,\kappa}\ge0$.

Solution

Let $V=\int:\!\phi_\kappa^4\!:$ and $Z(\lambda)=\mathbb E e^{-\lambda V}$. Direct differentiation gives $(\log Z)''=\mathbb E_\nu[V^2]-\mathbb E_\nu[V]^2=\operatorname{Var}_\nu(V)\ge0$.

References

• Glimm, James, and Arthur Jaffe. “A $\lambda\phi^4_2$ Quantum Field Theory Without Cutoffs. II.” Annals of Mathematics 91 (1970): 362–401. DOI.
• Nelson, Edward. “A Quartic Interaction in Two Dimensions.” In Mathematical Theory of Elementary Particles, 69–73. MIT Press, 1966. Publisher record.
• Summers, Stephen J. “A Perspective on Constructive Quantum Field Theory.” arXiv:1203.3991, revised 2016. Open PDF.