Motivation behind reflection positivity

Question

I have taken a look at

• this Phys.SE question Reflection positivity for general fields,

• Wikipedia, and

• this paper Reflection Positivity Then and Now by Jaffe,

which go over reflection positivity. While they all nicely explain the definition behind reflection positivity and provide some intuition behind what it is doing, I would like to know more of the motivation behind it and why it is so important.

From what I understand, we start with a Euclidean quantum field theory and would like to answer when is there an associated Minkowski quantum field theory (i.e. when can we "undo" Wick rotation). To do this we need to construct a Hilbert space by defining an inner product on the desired Minkowski space, and this requires positivity of the norm induced from this inner product. Reflection positivity is the requirement that this induced norm is positive.

Is this the correct idea behind reflection positivity? If so, how does inverting the sign of the time in each field achieve this? If not, what is the correct idea?

Answer 1

1. Briefly speaking, reflection positivity is a property for an Euclidean QFT, such that the Wick rotated Minkowskian QFT is unitary.

2. E.g. if $\phi$ is a real scalar field in the Heisenberg picture, then reflection positivity $\phi_E({\bf x},t_E)^{\dagger}=\phi_E({\bf x},-t_E)$ in Euclidean signature corresponds under Wick rotation $t_E=it_M$ to unitary time evolution $\phi_M({\bf x},t_M)=U(t_M)^{-1}\phi_M({\bf x},0)U(t_M)$ in Minkowskian signature.

References:

1. D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, arXiv:1602.07982; Section 7. (Hat tip: Ryan Thorngren.)