Spinors in dimensions other than 4

[spaghetti3451]

The Dirac equation describes the behaviour of non-interacting spin-$1/2$ fermions in a quantum-field-theoretic framework and is given by

$i\gamma^{\mu}\partial_{\mu}\psi=-m\psi,$

where $\gamma^{\mu}$ are the so-called gamma matrices which obey the Clifford algebra $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$ and the spinor $\psi$ is the vector space on which the gamma matrices act. Therefore, the dimension of the gamma matrices fixes the dimension of the spinor.

The spinor $\psi$ that describes spin-$1/2$ fermions in this quantum-field-theoretic framework is a $4$-dimensional vector and the gamma matrices are $4$-dimensional matrices.The smallest number of dimensions of the gamma matrices that satisfy the Clifford algebra is $4$. Can we not consider higher-dimensional gamma matrices and corresponding spinors? Are these higher-dimensional spinors at all physical?

What determines the dimensions of the gamma matrices and the spinors?

What are the possible generalisations of the Dirac equation in higher dimensions? Does this involve an increase in the number of gamma matrices?

 

[Spinnor]

https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions

Also see, http://motls.blogspot.com/2012/04/why-are-there-spinors.html

"Why are there spinors?

Spinors are competitors of vectors and tensors. In other words, they are representations of the orthogonal (rotational) group or the pseudoorthogonal (Lorentz) group, a space of possible objects whose defining property is the very characteristic behavior of their components under these transformations."

 

[PeterDonis]

The smallest number of dimensions of the gamma matrices that satisfy the Clifford algebra is 4.

No, it isn't. It's 2. See here:

https://en.wikipedia.org/wiki/Higher-dimensional_gamma_matrices#d_.3D_2

Can we not consider higher-dimensional gamma matrices and corresponding spinors?

Yes. The article I linked to discusses this as well.

 

[haushofer]

See e.g. Van Proeyen's Tools for Supersymmetry :)