Definition. Reflections, halves and the positive-time domain [fdip-007M]

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Let \(Y\) be a finite spatial torus with coordinate periods at least three, or a point. Put \(X=Y\times \mathbb Z/(2M)\), where \(M\ge 2\); the second coordinate is the chosen time direction. The nearest-neighbor graph joins sites differing by one coordinate step, with each undirected edge counted once. Time indices below are taken modulo \(2M\). Define the site reflection and link reflection by \[\theta _s(y,t)=(y,-t),\qquad \theta _\ell (y,t)=(y,1-t).\] Their positive halves and fixed sets are \[\begin {aligned} X_+^s&=Y\times \{0,\ldots ,M\},& X_0^s&=Y\times \{0,M\},\\ X_+^\ell &=Y\times \{1,\ldots ,M\},& X_0^\ell &=\varnothing . \end {aligned}\] In either case set \(X_-=\theta X_+\). Then \(X_+\cup X_-=X\) and \(X_+\cap X_-=X_0\); sites in \(X_0\) are fixed individually.

The diagram shows one time circle of length 8. A plus or minus marks an interior site of that half; a star marks a fixed site shared by both halves. Time \(7\) is adjacent to time \(0\). The boxes display membership in the halves, not separate field copies.

For a real scalar configuration \(\phi \in \mathbb R^X\), write \((\theta \phi )_v=\phi _{\theta v}\). On complex observables define \[(\Theta f)(\phi )=\overline {f(\theta \phi )}.\] Thus \(\Theta \) is an antilinear multiplicative involution. Let \(\mathcal A_+\) be the algebra of bounded complex measurable functions depending only on \(\phi |_{X_+}\). Let \(\mu \) be a probability measure on \(\mathbb R^X\). Reflection invariance means \(\theta _*\mu =\mu \). In addition, reflection positivity means \[B_\mu (f,f):=\int (\Theta f)f\,d\mu \ge 0 \qquad (f\in \mathcal A_+).\] The inequality includes the assertion that the integral is real. Under reflection invariance, \(B_\mu (f,g)=\int (\Theta f)g\,d\mu \) is Hermitian and conjugate-linear in its first argument. These are the complex-observable conventions corresponding to [friedli2017statistical, Definition 10.5, p. 441].

The larger positive-time domain is \(\mathcal H_+=L^2(\mathbb R^X,\mu ;\mathbb C)\) restricted to functions measurable in the coordinates of \(X_+\). It is a closed vector subspace, generally not an algebra under pointwise multiplication. Reflection invariance gives \[|B_\mu (f,g)|\le \|f\|_{L^2(\mu )}\|g\|_{L^2(\mu )}.\] Truncating the values of a function shows that \(\mathcal A_+\) is dense in \(\mathcal H_+\). Consequently positivity on \(\mathcal A_+\) extends to this domain. For a reflection-positive law, the quotient by \(\mathcal N=\{f:B_\mu (f,f)=0\}\), with the induced pairing, has a Hilbert-space completion. This definition supplies no translation operator on that quotient.