Trivial reflection-positive Schwinger hierarchy

The zero functional is complex-linear and continuous on every Schwartz space. Its pullback by every Euclidean isometry or block permutation is still zero, so the normalization in [F1] and covariance and symmetry equalities in [F2] hold; the chosen degree-zero value is the required normalization.

In [F3], every summand with $j+k\ge 1$ vanishes by [A1]. The only possible surviving term is $j=k=0$, and it equals $\mathrm{\mid }{a}_0{\mathrm{\mid }}^2\mathrm{\mid }{f}_0{\mathrm{\mid }}^2{S}_0=\mathrm{\mid }{a}_0{f}_0{\mathrm{\mid }}^2$. Hence $Q_S(F)=\mathrm{\mid }{a}_0{f}_0{\mathrm{\mid }}^2\ge 0$ is real for every finite family, proving reflection positivity.

Let $\eta (s)=\exp (-1\mathrm{/}(1-s^2))$ for $\mathrm{\mid }s\mathrm{\mid }<1$ and $\eta (s)=0$ for $\mathrm{\mid }s\mathrm{\mid }\ge 1$, and choose $f_1(t,x)=\eta (t-2)\eta (x)$, which is a nonzero smooth compactly supported test function in $O_{+}$; set $a_1=1$ and $a_0=0$, with every other coefficient zero. Then the polynomial is nonzero while step 2.1 gives $Q_S(F)=0$, so the positive form is degenerate. The example asserts neither a nontrivial field nor OS reconstruction.