The $Phi^4$ Potential in Dimension One: Wick Ordering Is a Finite Mass Shift and Nelson's Estimate Is Not Needed

This remark concerns the construction of The Wick-Ordered Phi^4 Potential of the Free Field on the Torus of Dimension at Most Two, The Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every L^p, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff, Nelson's Exponential Integrability for the Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two and The Phi^4 Measure on the Torus of Dimension at Most Two as the Gibbs Measure of a Gibbs-Sobolev Potential, and Its Galerkin Approximation in Total Variation in the case $n=1$ of a one-dimensional torus, and explains which of the techniques used there are forced by renormalisation and which are not.

Wick ordering is a finite mass shift. For $n=1$ the Wick constants $C_{N}$, partial cube sums of positive terms, increase to the finite lattice sum $C_{\infty}=\sum_{k\in\mathbb{Z}}\mu_{k}^{-1}$ (The Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every L^p, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff §lower-bound-uniform). The Wick polynomial of The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §polynomial is $\frac{\varkappa}{4}s^{4}+\frac{\mathfrak{m}-3\varkappa C_{N}}{2}s^{2}$ plus a constant, so at each cutoff $V_{N}$ is the bare potential $\int(\frac{\varkappa}{4}\Phi_{N}^{4}+\frac{\mathfrak{m}_{N}}{2}\Phi_{N}^{2})$ of the truncated field with the mass $\mathfrak{m}_{N}=\mathfrak{m}-3\varkappa C_{N}$, which stays bounded in dimension one, plus a constant that cancels from every Gibbs measure. That the bare potential of the full field, with the limiting mass, gives the same Gibbs measure is classical but not proved in this database. Wick ordering is kept for $n=1$ only so that a single chain of items covers $n\le2$; it is not a renormalisation there.

Nelson's estimate is not needed. For $n=1$ the potential is bounded below by a constant independent of the cutoff (The Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every L^p, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff §lower-bound-uniform), so exponential integrability of $\exp(-tV)$ is immediate. The argument of Nelson's Exponential Integrability for the Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two, which balances the slow growth of the Wick constants (Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums §growth) against the decay of $V-V_{N}$ in the chaos up to order four, is needed only for $n=2$, where $C_{N}$ is unbounded (The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §divergent) and the cutoff potentials $V_{N}$ are not bounded below uniformly in $N$, their infima being of order $-C_{N}^{2}$ (as one sees on constant fields).

Other techniques would work for $n=1$. It is classical, though not proved in this database, that the free field on the circle is carried by continuous functions. A pathwise treatment is therefore available for $n=1$: $V$ is the integral of a polynomial of a function, its Hessian in the white-noise norm is formally multiplication by $3\varkappa\phi^{2}+\mathfrak{m}-3\varkappa C_{\infty}$ (at each cutoff this is The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §hessian), so $V$ is semiconvex with constant $(3\varkappa C_{\infty}-\mathfrak{m})_{+}$; and the one-variable quartic is a convex function plus a bounded one, so one expects the Bakry--'Emery and Holley--Stroock criteria, in an infinite-dimensional formulation not yet available here, to give a log-Sobolev inequality. These could replace the chaos expansions and hypercontractive moment bounds used here; the chaos route is taken because it is the one that survives at $n=2$.

What does not simplify. Even for $n=1$, in the geometry of square-integrable white noise the noise gradient of $V$ is $\varkappa\phi^{3}$ plus lower-order terms, whose square-integrability needs sixth moments of the field, and the trace terms $\sum_{j}a_{j}\partial_{j}\partial_{j}V_{N}$ of the white-noise Ornstein--Uhlenbeck operator applied to the cutoff potentials grow with the number of active modes. So the splitting of the Gibbs score into a Gaussian score and a potential gradient, available at each cutoff by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §splitting, does not pass to the full field, and the score relative to the Gibbs measure has to be defined intrinsically (The Intrinsic Fisher Information Relative to the Gibbs Measure of a Gibbs-Sobolev Potential) already for $n=1$.