Lemma. A sufficient positive-half decomposition [fdip-007N]
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  • November 1, 2026

Let a finite set \(X\) have an involution \(\theta \) and halves \(X_-=\theta X_+\) with \(X_+\cup X_-=X\) and \(X_+\cap X_-=\operatorname {Fix}(\theta )=X_0\). Use the observable conventions of Definition [fdip-007M]; both torus reflections there have these properties. Let \(\nu \) be a probability measure supported on a compact interval of \(\mathbb R\), and put \(\mu _0=\bigotimes _{v\in X}\nu \). Suppose a real Hamiltonian has the form \[-H=A+\Theta A+\sum _{\alpha =1}^q C_\alpha \Theta C_\alpha ,\] where \(A,C_1,\ldots ,C_q\) are bounded real functions in \(\mathcal A_+\); the sum may be empty. Then the probability law \[d\mu =Z^{-1}e^{-H}\,d\mu _0,\qquad Z=\int e^{-H}\,d\mu _0\] is reflection invariant and reflection positive. This is the complex-observable version of [friedli2017statistical, Lemma 10.8, pp. 442–443].

Proof. 

  • November 1, 2026

First consider the product reference. Conditional on the field \(b=\phi |_{X_0}\) at the fixed sites, the two interiors are independent and have matching product laws under reflection. The boundary field is a single shared variable. For any bounded positive-half observable \(F\), therefore, \[\int (\Theta F)F\,d\mu _0 =\int \left |\int F(\phi _+,b)\,d\nu _+(\phi _+)\right |^2 \,d\nu _0(b)\ge 0.\] Here \(\nu _+\) is the product over \(X_+\setminus X_0\) and \(\nu _0\) the product over \(X_0\). When \(X_0\) is empty the outer integral is evaluation at the unique empty configuration. This also proves the product case of [friedli2017statistical, Lemma 10.7, p. 442].

The displayed decomposition is invariant under reflection because its functions are real and multiplication commutes. The reference law is invariant as well. Boundedness makes \(0<Z<\infty \) and permits uniform absolute expansion of the crossing exponential. For \(f\in \mathcal A_+\), this gives \[\begin {aligned} ZB_\mu (f,f) &=\sum _{n\in \mathbb N^q}\frac 1{n!} B_{\mu _0}\left (fe^A C^n,fe^A C^n\right ),\\ n!&=\prod _\alpha n_\alpha !,\qquad C^n=\prod _\alpha C_\alpha ^{n_\alpha }. \end {aligned}\] Every summand is nonnegative by the conditional product identity. The sum converges absolutely by the bounded exponential majorant, so the claimed positivity follows.