We continue our investigations of the representation theoretic
side of reflection positivity by studying positive definite functions
$\psi $ on the
additive group
$\left(\mathbb{R},+\right)$
satisfying a suitably defined KMS condition. These functions take values in the space
$Bil\left(V\right)$ of bilinear forms on
a real vector space $V$.
As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic
continuation of
$\psi $
to the strip
$$\left\{z\in \u2102\phantom{\rule{2.22144pt}{0ex}}0\le Imz\le \beta \right\}$$
with a coupling condition
$\psi \left(i\beta +t\right)=\overline{\psi \left(t\right)}$
on the boundary. Our first main result consists of a characterization of these functions in terms of
modular objects
$\left(\Delta ,J\right)$
($J$ an antilinear
involution and
$\Delta >0$
selfadjoint with
$J\Delta J={\Delta}^{1}$)
and an integral representation.
Our second main result is the existence of a
$Bil\left(V\right)$valued positive
definite function
$f$
on the group
${\mathbb{R}}_{\tau}=\mathbb{R}\u22ca\left\{{id}_{\mathbb{R}},\tau \right\}$
with
$\tau \left(t\right)=t$
satisfying
$f\left(t,\tau \right)=\psi \left(it\right)$ for
$0\le t\le \beta $. We thus obtain
a
$2\beta $periodic
unitary oneparameter group on the GNS space
${\mathcal{\mathscr{H}}}_{f}$
for which the oneparameter group on the GNS space
${\mathcal{\mathscr{H}}}_{\psi}$ is
obtained by Osterwalder–Schrader quantization.
Finally, we show that the building blocks of these representations arise from
bundlevalued Sobolev spaces corresponding to the kernels
$${\left({\lambda}^{2}{d}^{2}\u2215d{t}^{2}\right)}^{1}$$
on the circle
$\mathbb{R}\u2215\beta \mathbb{Z}$
of length
$\beta $.
