In Theorem 1 of [this][1] paper Segal stablish a relation between states and generating functionals. 
He assert that in order to $\mu$ be a generating functional  must satisfy 
$$
\sum_{j,k\in F} \mu (z_j-z_k)e^{iB(z_j\cdot z_k)}\bar{\alpha}_k\alpha_j\ge 0
$$
Then, as an example he show the functional 
$$
\mu(z)= e^{-\frac{1}{4}|z|^2}
$$
is the *zero-interaction* vacuum generating functional.

**The question is**: Why this functional satisfies the desidered condition?


  [1]: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/foundations-of-the-theory-of-dynamical-systems-of-infinitely-many-degrees-of-freedom-ii/600D279603237D9BC69E458758DF682A
  [2]: https://i.sstatic.net/iLJY9.png
  [3]: https://i.sstatic.net/RZDcZ.png