We now allow ourselves to formally add differential forms of different ranks, and introduce a new product on the resulting space. For any function $f$ and 1-forms $\alpha$, $\beta$, define \begin{align} f \vee \alpha &= f\alpha \\ \alpha \vee \beta &= \alpha \wedge \beta + g(\alpha,\beta) \end{align} The operation $\vee$ is called the Clifford product, and is read as “vee”. We extend the Clifford product to (formal sums of) differential forms of all ranks by requiring associativity. It is then a straightforward exercise to work out that, for example, \begin{equation} \alpha\vee\beta\vee\gamma = \alpha\wedge\beta\wedge\gamma + g(\alpha,\beta)\gamma - g(\alpha,\gamma)\beta + g(\beta,\gamma)\alpha \end{equation} for any 1-forms $\alpha$, $\beta$, $\gamma$. The space of formal sums of differential forms under the Clifford product is a Clifford algebra.

It turns out that the algebra we have just constructed is precisely the algebra of gamma matrices used in quantum field theory! All one needs to do is make the association \begin{equation} \gamma^i \longleftrightarrow \sigma^i \end{equation} and then identify matrix multiplication of gamma matrices with the Clifford product of differential forms. So our orthonormal basis $\{\sigma^i\}$ plays the role of the gamma matrices, and this construction is particularly simple in rectangular coordinates, where $\sigma^i=dx^i$. Spinors can be thought of as the objects acted upon by gamma matrices, which in turn can be reinterpreted in terms of the gamma matrices themselves. That's not quite the whole story, as spinors actually live in certain subspaces of Clifford algebras. Nonetheless, the study of spinors is therefore the study of Clifford algebras, for which we can use differential forms.

An example of the power of this approach is obtained by considering the Kähler operator $d-\delta$, where \begin{equation} \delta = (-1)^p\,{*}^{-1}d{*} \end{equation} where ${*}^{-1}$ is of course the same as $*$ up to sign (and where this $\delta$ has nothing to do with the variational $\delta$ introduced in the previous section). Since $\delta^2=0$, we also have \begin{equation} (d-\delta)^2 = -d\delta-\delta d \end{equation} Applying this operator to a function $f$ yields \begin{equation} (d-\delta)^2 f = -d\delta f-\delta df = 0 + {*}^{-1}d{*}df = \Delta f \end{equation} so that the Kähler operator is a sort of square root of the Laplacian — or, in Minkowski space, the square root of the wave equation. But that is precisely the role of the Dirac equation, the fundamental equation for spinor fields: it serves as a square root of the wave equation, also known as the (massless) Klein-Gordon equation. The Dirac equation, which describes both electrons and quarks, can therefore be rewritten in terms of differential forms.