$\newcommand{\real}{\mathrm{real}}$I am having trouble with understanding the axiom (OS3) in [this book](https://link.springer.com/book/10.1007/978-1-4612-4728-9) by Glimm and Jaffe.
It defines
\begin{equation}
\mathcal{A} = \left \{ A(\phi) = \sum\_{j = 1}^N c\_j \exp \left( \phi(f\_j) \right) \; \Big \vert \; c\_j \in \mathbb{C}, f\_j \in \mathcal{D} \right \}
\end{equation}
where $\mathcal{D}$ is supposed to be the space of compactly supported, smooth functions on $\mathbb{R}^d$ and $\phi \in \mathcal{D}'(\mathbb{R}^d)$.
Now, this is notationally somewhat messed up (imho) but it is also stated earlier in the text, that $\mathcal{D}'(\mathbb{R}^d)$ should refer to *real* distributions hence, I assume that $\mathcal{D}$ should also refer to *real* functions - but I have not found this statement anywhere.


They then go on by defining the time-reflection operator $\theta : \mathcal{A} \to \mathcal{A}$ and consider a special supspace $\mathcal{A}\_+ \subset \mathcal{A}$ and give the axiom:
\begin{equation}
0 \le \int\_{\mathcal{D}'(\mathbb{R}^d)} \left( \theta A \right)^- A \mathrm{d} \mu
\end{equation}
where the superscript $-$ is supposed to mean complex conjugation.
This might sound okay but then they state that this is equivalent to the positive definiteness of matrices of the form
\begin{equation}
M\_{i j} = \int\_{\mathcal{D}'(\mathbb{R}^d)} \exp \left[ i \phi \left( f\_i - \theta f\_j \right) \right] \mathrm{d} \mu \left( \phi \right)
\end{equation}
with $f\_j$ in the new space $\mathcal{D}\_{\real}(\mathbb{R}^d)$.


I feel that it is completely unclear whether they are talking about $\mathcal{D}$ as a space of real or complex functions.
Furthermore, I cannot see the equivalence of the two statements unless $\mathcal{A}$ was defined with an additional $i$ in the exponent and with $\mathcal{D} \supset \mathcal{D}\_{\real}(\mathbb{R}^d)$ as a space of *real* functions.


[nlab](https://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem) also gives the almost identical definition as the book but without stating anything about real or complex spaces.



Understanding the Osterwalder-Schrader conditions as formulated by Glimm and Jaffe
Since this has confused me multiple times, I write this answer in the hope that it might help others.


First, recall that reflection positivity as formulated by [Osterwalder and Schrader](https://link.springer.com/article/10.1007/BF01645738) states that
\begin{equation}
\sum\_{n,m} G\_{n+m} \left( \theta f\_n^\* \otimes f\_m \right) \ge 0
\end{equation}
for all finite sequences $(f\_n)\_{n \in \mathbb{N}}$ of **complex** Schwartz functions on $(\mathbb{R}^d)^n$ with supports in $(\mathbb{R}\_{\ge 0} \times \mathbb{R}^{d-1})^n$.
Here,
\begin{equation}
\left( \theta f \right) \left( x^1\_1, \dots, x^1\_d, x^2\_1, \dots x^d\_d \right)
=
f \left( -x^1\_1, x^1\_2 \dots, x^1\_d, -x^2\_1, x^2\_2 \dots x^2\_d, \dots \right)
\end{equation}
for all $x^1, \dots x^d \in \mathbb{R}^d$.


Second, the whole idea of having a measure $\mu$ on a space of distributions is to be able to write the $G\_n$s as moments of that measure.
From here on $\mu$ will be a corresponding measure on the space of **real** distributions.
Also, let $Z$ denote the moment-generating function of $\mu$ (assuming it exists) and $\hat{\mu}$ its characteristic function.


Now, for any finite sequences $(\phi\_n, c\_n)\_{n \in \mathbb{N}}$ of **real** Schwartz functions on $\mathbb{R}^d$ with supports on $\mathbb{R}\_{\ge 0} \times \mathbb{R}^{d-1}$ and complex numbers, define
\begin{equation}
f\_a = \frac{1}{a!} \sum\_n c\_n \phi\_n^{\otimes a}
\end{equation}
for all $a \in \mathbb{N}$.
Then a simple calculation (assuming everything is well-defined) shows that
\begin{equation}
\sum\_{m, n} c\_n^\* c\_m Z \left( \theta \phi\_n + \phi\_m \right)
=
\sum\_{a, b = 0}^\infty G\_{a+b} \left( \theta f\_a^\* \otimes f\_b \right)
=
\lim\_{N \to \infty} \sum\_{a, b = 0}^N G\_{a+b} \left( \theta f\_a^\* \otimes f\_b \right)
\ge 0 \, .
\end{equation}


Similarly, we obtain a connection with the characteristic function by setting
\begin{equation}
g\_a = \frac{\left(-i\right)^a}{a!} \sum\_n c\_n \phi\_n^{\otimes a}
\end{equation}
for all $a \in \mathbb{N}$.
Then
\begin{equation}
\sum\_{m, n} c\_n^\* c\_m \hat{\mu} \left( \theta \phi\_n - \phi\_m \right)
=
\sum\_{a, b = 0}^\infty G\_{a+b} \left( \theta g\_a^\* \otimes g\_b \right)
=
\lim\_{N \to \infty} \sum\_{a, b = 0}^N G\_{a+b} \left( \theta g\_a^\* \otimes g\_b \right)
\ge 0 \, .
\end{equation}


Observe that we need either $Z( \theta \phi\_n + \phi\_m )$ or $\hat{\mu}( \theta \phi\_n - \phi\_m )$.


