Here we introduce reflection positive doubles, a general framework for
reflection positivity, covering a wide variety of systems in statistical
physics and quantum field theory. These systems may be bosonic, fermionic, or
parafermionic in nature. Within the framework of reflection positive doubles,
we give necessary and sufficient conditions for reflection positivity. We use a
reflection-invariant cone to implement our construction. Our characterization
allows for a direct interpretation in terms of coupling constants, making it
easy to check in concrete situations. We illustrate our methods with numerous
examples.Comment: 53 pages, 3 figure
