Theorem. Reflection positivity of a massive scalar lattice [fdip-007O]
✍️source

  • November 1, 2026

On the nearest-neighbor torus of Definition [fdip-007M], choose either reflection \(\theta \). Give every undirected edge a nonnegative weight \(w_e\) with \(w_{\theta e}=w_e\), and fix \(m>0\). The real scalar law \[\begin {aligned} H(\phi )&=\frac 12\sum _{e=\{u,v\}}w_e(\phi _u-\phi _v)^2 +\frac {m^2}{2}\sum _{v\in X}\phi _v^2,\\ d\mu (\phi )&=Z^{-1}e^{-H(\phi )} \prod _{v\in X}d\phi _v \end {aligned}\] is reflection invariant and reflection positive on \(\mathcal A_+\) and its positive-time \(L^2\) domain. Every polynomial in the positive-half field coordinates belongs to that domain. This is the massive scalar precision with zero-weight edges omitted. It uses ordinary differences, equivalently trivial transport in the connection energy.

Proof. 

  • November 1, 2026

Since \(H(\phi )\ge (m^2/2)\|\phi \|^2\), the normalizer is finite and positive, and every polynomial moment exists. Reflection is a permutation of the coordinates, preserves Lebesgue measure and preserves \(H\) by the weight hypothesis. This proves invariance separately from positivity.

For a site reflection, no nearest-neighbor edge joins \(X_+\setminus X_0\) directly to \(X_-\setminus X_0\). Partition the terms of \(H\) into \(H_+\), those supported in \(X_+\) but not wholly in \(X_0\); their reflected partners \(H_-=\Theta H_+\); and \(H_0\), those supported wholly in \(X_0\). Include onsite mass terms in the same partition. Because reflection fixes every boundary coordinate, \[-H=A+\Theta A,\qquad A=-H_+-\tfrac 12H_0.\] In particular the boundary terms are counted once.

For a link reflection the halves are disjoint. The only crossing edges are the time bonds at the two cuts, and each has the form \(e=\{v,\theta v\}\) with a unique \(v\in X_+\). Its contribution to \(-H\) is \[-\frac {w_e}{2}\phi _v^2 -\frac {w_e}{2}\phi _{\theta v}^2 +w_e\phi _v\phi _{\theta v}.\] Put its first term into \(A\), its second into \(\Theta A\), and write the last as \(C_e\Theta C_e\), where \(C_e=\sqrt {w_e}\phi _v\). All within-half edge energies and onsite mass terms are assigned to their own half, with a minus sign. Reflection invariance of the weights then gives the positive-half decomposition.

For \(R>0\), restrict every coordinate to \([-R,R]\) and use the uniform probability on this interval as the product reference. On that cube the functions \(A,C_e\) are bounded, so the lemma proves reflection positivity of the normalized cutoff law \(\mu _R\). The constant reference-density factor cancels from normalization; thus \(\mu _R\) is exactly \(\mu \) conditioned on \(\max _v|\phi _v|\le R\). For bounded \(f,g\), dominated convergence in the integrable Gaussian density gives \[B_{\mu _R}(f,g)\longrightarrow B_\mu (f,g).\] For polynomials the same argument uses a polynomial times the dominating massive Gaussian. In particular positivity survives the limit. Finally, bounded value truncations are dense in \(\mathcal H_+\); the norm bound in Definition [fdip-007M] extends the pairing and its positivity to all of that domain.

Compare [friedli2017statistical, pp. 443–444], Examples 10.9–10.10, for the half decompositions. The cutoff argument above supplies the noncompact Gaussian step. Arbitrary inhomogeneous weights need not respect the chosen reflection. Arbitrary fixed connections need not have the displayed nonnegative crossing terms. Neither case is asserted by this theorem.