Reflection-positive Euclidean field data

Use the two-dimensional hierarchy and bilinear distribution-pairing convention defined earlier on this page. Write Euclidean coordinates as $(t,x)\in {\mathbb{R}}^2$, set $O_{+}=\{(t,x):t>0\}$, and let $\vartheta (t,x)=(-t,x)$. For $n\ge 1$, let ${\mathcal{D}}_{\mathrm{s}\mathrm{y}\mathrm{m}}(O_{+}^n)$ be the smooth compactly supported complex test functions on $O_{+}^n$ invariant under permutations of their $n$ two-dimensional argument blocks; set the degree-zero test space to $\mathbb{C}$. The reflection of a degree-$m$ test function is the antilinear map $(\mathrm{\Theta }{f}_m)(z_1,\dots ,z_m)=\overline{f_m(\vartheta z_1,\dots ,\vartheta z_m)},\mathrm{\Theta }{f}_0=\overline{f_0}\mathrm{.}$

For a finite family $F=(a_0,f_0;\dots ;a_N,f_N)$ with $a_j\in \mathbb{C}$ and $f_j\in {\mathcal{D}}_{\mathrm{s}\mathrm{y}\mathrm{m}}(O_{+}^j)$ (using $f_0\in \mathbb{C}$), define $Q_S(F)={\sum }_{j,k=0}^N\overline{a_j}{a}_k{S}_{j+k}(\mathrm{\Theta }{f}_j\otimes f_k)\mathrm{.}$ The tensor $\mathrm{\Theta }{f}_j\otimes f_k$ is a compactly supported smooth test function on $({\mathbb{R}}^2{)}^{j+k}$; for $j+k=0$ the term means $S_0\overline{f_0}{f}_0$. The hierarchy is reflection-positive on the positive-time test algebra when $Q_S(F)\in [0,\mathrm{\infty })$ for every such finite family. This scalar form is conjugate-linear in its first polynomial variable and linear in its second; when expressed using the imported convention that an inner product is linear in its first variable, the corresponding form is obtained by reversing the two arguments. No quotient or Hilbert-space completion is asserted here.

Reflection positivity is only this positivity condition. Euclidean covariance, block symmetry, clustering, regularity, and any linear-growth hypothesis remain distinct clauses; this definition alone is not an Osterwalder–Schrader reconstruction theorem.

Euclidean Schwinger hierarchy in two dimensions

Real and complex inner-product spaces and their induced length