# Is there a geometric object analagous to a spinor that encodes projections onto bivectors?

The most sensible geometric interpretation of spinors that I've come across is that they encode projections in the Clifford algebra. So if $\mathbf A$ is a vector with components $A_i$ and $\psi$ is a spinor, then $\psi^\dagger A_i \sigma^i \psi = \mathrm{Tr} (A_i\sigma^i \psi \psi^\dagger)$ gives the component of $\mathbf A$ along the direction encoded by $\psi$.

Is there a geometric object analagous to spinors, which projects onto a bivector orientation rather than a vector direction? So if $S_{ij}$ were the components of a bivector and $\chi$ were such an object, $\chi^\dagger S_{ij} \frac12[\sigma^i,\sigma^j] \chi$ might give the component of the bivector $\mathbf S$ with orientation encoded by $\chi$. As in the case of spinors, $\frac{1}{2}(\chi_1+\chi_2)(\chi_1+\chi_2)^\dagger$ would be another such projection.

Or can this already be accomplished using spinors?

## 1 Answer

The geometric object which corresponds to spinors is the external bundle, the bundle of differential forms. The natural equation on such a bundle is the Dirac-Kähler equation.

This bundle is essentially used in lattice computations under the name "staggered fermions".

A problem with this geometric interpretation is that it has too many components. So, the complexified bundle $\Lambda(\mathbb{R}^4,\mathbb{C})$ would describe four Dirac fermions. In arXiv:0908.0591 (see also http://ilja-schmelzer.de/matter/ ) a splitting spacetime into space and time allows to reduce this to two Dirac fermions, which, then, can be interpreted as an electroweak pair.

• Very interesting paper, though I'll have to take some more time with it. So the Dirac-Kähler equation describes a complex multivector field in 4d spacetime, with scalar, pseudoscalar, vector, pseudovector,... components. You've taken this equation for 3d Euclidean space and constructed two Dirac fermions from the 8 complex degrees of freedom that define the multivector? – rossng Sep 12 '16 at 1:09