Reflection Positive One-Parameter Groups and Dilations
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 Poincaré 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 $$ax+b$$ax+b-group, the Heisenberg group, the euclidean motion group and the euclidean conformal group.