  The concept of reflection positivity has its origins in the work of
Osterwalder--Schrader on constructive quantum field theory. It is a fundamental
tool to construct a relativistic quantum field theory as a unitary
representation of the Poincare group from a non-relativistic field theory as a
representation of the euclidean motion group. This is the second article in a
series on the mathematical foundations of reflection positivity. We develop the
theory of reflection positive one-parameter groups and the dual theory of
dilations of contractive hermitian semigroups. In particular, we connect
reflection positivity with the outgoing realization of unitary one-parameter
groups by Lax and Phillips. We further show that our results provide effective
tools to construct reflection positive representations of general symmetric Lie
groups, including the ax+b-group, the Heisenberg group, the euclidean motion
group and the euclidean conformal group.
