In Jurgen Jost's book on Riemannian Geometry, he describes the following equation as one of the Seiberg-Witten equations:
$$F^{+}_A = \frac{1}{4}\langle e_j \cdot e_k \cdot \varphi , \varphi \rangle e^j \wedge e^k$$
Here I think $\varphi$ is a section of the spinnor bundle on a 4-manifold with a $Spin^c$-structure, the dots stand for Clifford multiplication, $\{e_i\}$ is local tangent frame and $\{e^i\}$is its dual. $F_A^{+}$ is the self-dual part of the curvature of a $U(1)$-connection $A$. (This is equation 11.2.10 in Jost's book)
On the other hand, in Morgan's book about Seiberg-Witten equations and 4-manifolds, and also in most other places, this Seiberg-Witten equation is written something like this:
$$F_A^{+} = \varphi \otimes \varphi^{*} - \frac{|\varphi|^2}{2}Id$$
This is found in section 4.1 of Morgan's book. Are these two equations the same and how do I see that? When I try to calculate Morgan's equation from Jost's I get stuck (somewhat embarrassingly) because I don't really understand what the inner product looks like. I don't see how I get two terms. Perhaps I am missing some details about how the elements $e_i$ act on spinnors. Any help would be appreciated.
Edit: I believe the answer to this can be essentially found here: https://math.berkeley.edu/~hutching/pub/tn.pdf (in section 2.8) but I would still appreciate someone filling in the details.