Suppose we have a differential operator of order $n$ that maps sections of a bundle $E \rightarrow M$ to sections of a bundle $F \rightarrow M$:
$$ D(\sigma) = \sum_{I} \alpha_{I} \frac{\partial^{I}}{\partial x^{I}} +\ \text{lower order terms}$$
Where $I$ ranges over multi-indices of length $n$.
Then I read that the symbol is defined to be:
$$Symb(D)(\xi) = i^{n} \sum_{I} \alpha_{I} \xi^{I}$$
Where $\xi \in T^*_xM$. The symbol is supposed to be a map between the pullbacks of the bundles $E$ and $F$ over the bundle $T^*M$.
My question is: why is there a power of $i = \sqrt{-1}$ in the expression for the symbol? If the bundles are just real vector bundles, where would the $i$ come from?
(FYI I am reading this in Morgan's book on Seiberg Witten theory)