Reflection positivity in general

In the Euclidean QFT obtained by "Wick-rotating" a unitary QFT, the correlation functions satisfy a property called reflection positivity, see e.g. this Wikipedia article for the case of a scalar field.

What's the correct formulation if you have chiral fermions and/or terms like the QCD theta angle? Could someone give the references?

This is one of the first hits for "reflection positivity" on google, so someone better answer it!

Reflection positivity is just positivity of the Hilbert space norm along with the fact that complex conjugation negates the imaginary time.

To begin the discussion, let's just consider spacetime of the form $$Y \times \mathbb{R}_t$$. We have a Hilbert space associated to $$Y \times 0$$ and some operators $$\phi(y,0)$$ which we're interested in computing correlation functions with. We define the usual time evolved operators by $$\phi(y,t) = e^{-itH} \phi(y,0) e^{itH}.$$ These have the nice property of being Hermitian if $$\phi(y,0)$$ is. We can also define the analytically continued operators by just replacing $$t$$ with $$z = t + i\tau$$. Now taking the adjoint we find (assuming $$\phi(y,0)$$ is Hermitian) $$\phi(y,z)^\dagger = \phi(y,z^*),$$ in particular $$\phi(y,i\tau)^\dagger = \phi(y,-i\tau).$$

Now let us consider the states $$\phi(y,z)|0\rangle$$ obtained from the vacuum $$|0\rangle$$. These are nonzero Hilbert space states, so they have positive norm, ie. $$\langle 0 | \phi(y,z)^\dagger \phi(y,z)|0\rangle = \langle 0 | \phi(y,z^*) \phi(y,z) | 0 \rangle > 0.$$ (Note that this is automatically properly imaginary-time-ordered as long as $$\tau>0$$, which we need anyway to have good states). In particular $$\langle 0 | \phi(y,-i\tau) \phi(y,i\tau) | 0 \rangle > 0.$$ We see the reflection principle at work here, $$i\tau \mapsto -i\tau$$, so if we imagined this was computed in a Euclidean path integral, it would be a reflection-symmetric configuration on $$Y \times \mathbb{R}_{\tau}$$.

The general statement of reflection positivity is that for all such reflection-symmetric configurations of Hermitian operators, even through other coordinates on other spacetime manifolds, the Euclidean path integral always computes something positive. In all these cases the proof is just to realize that what you're computing is the norm of some Hilbert space state.

It does not matter if there's chiral fermions or a theta angle, although these do Wick rotate in interesting ways, since (spacetime)parity-odd terms remain imaginary in the Euclidean action. Sorry I don't know any references, but I'm trying to write one!

• Sorry I don't know any references, but I'm trying to write one! – now that you got me excited, you must deliver or I will be left heartbroken. Feb 26, 2020 at 18:48
• @Prof.Legolasov Actually here is a nice reference that goes into some of the details: arxiv.org/abs/1602.07982 Feb 26, 2020 at 21:10
• CFT is nice, but I’m more interested in the axiomatic QFT. I’d kill for a good introduction to OS and Wightman with explicitly built examples such as $\phi^4_2$, $\phi^4_3$, $P(\phi)_2$, etc Feb 27, 2020 at 23:14