Fermi algebra

Here are collected some results related with the operator $\sigma$ which commutation with its adjoint $\ud{\sigma}$ reads:

$$\{\sigma,\ud{\sigma}\}=\mathbb{1}$$

(cf. Bose algebra).

A fairly general result is obtained with $\mu,\nu,\eta,\theta$ boolean variables (0 or 1), in which case the following general commutator reads in normal order (see here):

$$[{\sigma}^{\dagger\mu}\sigma^\nu,{\sigma}^{\dagger\eta}\sigma^\theta] = \nu\theta(\eta-\mu)\sigma +\mu\eta(\nu-\theta){\sigma}^\dagger +[(1-\mu)\nu\eta(1-\theta)-\mu(1-\nu)(1-\eta)\theta](1-2{\sigma}^\dagger\sigma).$$

This is a list of relations to normal-order the ladder operators of a two-level system, with $\mu, \nu\in\{0,1\}$:

1. $[\ud{\sigma}\sigma,\ud{\sigma}^\mu\sigma^\nu]=(\mu-\nu)\ud{\sigma}^\mu\sigma^\nu$
2. $[\sigma,\ud{\sigma}^\mu]=\mu(\mathbb{1}-2\ud{\sigma}^\mu\sigma)$
3. $[\ud{\sigma},\sigma^\nu]=\nu(2\ud{\sigma}\sigma^\nu-\mathbb{1})$
4. $[\sigma,\ud{\sigma}^\mu]\sigma^\nu=\mu\ud{\sigma}^{1-\mu}\sigma^\nu-2\mu(1-\nu)\ud{\sigma}^\mu\sigma^{1-\nu}$
5. $\ud{\sigma}^\mu[\ud{\sigma},\sigma^\nu]=2(1-\mu)\nu\ud{\sigma}^{1-\mu}\sigma^\nu-\nu\ud{\sigma}^\mu\sigma^{1-\nu}$
6. $\sigma^{\nu+1}=(1-\nu)\sigma^{1-\nu}$
7. $\ud{\sigma}^{\mu+1}=(1-\mu)\ud{\sigma}^{1-\mu}$
8. $\ud{\sigma}^{\mu+1}\sigma^{\nu+1}=(1-\mu)(1-\nu)\ud{\sigma}^{1-\mu}\sigma^{1-\nu}$
9. $\sigma^\nu\ud{\sigma}=\nu\mathbb{1}+(1-2\nu)\ud{\sigma}\sigma^\nu$
10. $\sigma\ud{\sigma}^\mu=\mu\mathbb{1}+(1-2\mu)\ud{\sigma}^{\mu}\sigma$
11. $\ud{\sigma}^\mu\sigma^\nu\ud{\sigma}=\nu\ud{\sigma}^\mu\sigma^{1-\nu}+(1-2\nu)(1-\mu)\ud{\sigma}^{1-\mu}\sigma^\nu$
12. $\ud{\sigma}^\mu\sigma^\nu\ud{\sigma}=\nu\ud{\sigma}^\mu\sigma^{1-\nu}+(1-2\nu)(1-\mu)\ud{\sigma}^{1-\mu}\sigma^\nu$,
13. $\ud{\sigma}\sigma\ud{\sigma}^\mu\sigma^\nu=\mu\ud{\sigma}^\mu\sigma^\nu+(1-\mu)(1-\nu)\ud{\sigma}^{1-\mu}\sigma^{1-\nu}$,
14. $\ud{\sigma}^\mu\sigma^\nu\ud{\sigma}\sigma=\nu\ud{\sigma}^\mu\sigma^\nu+(1-\mu)(1-\nu)\ud{\sigma}^{1-\mu}\sigma^{1-\nu}$,
15. $\sigma\ud{\sigma}^{1+\mu}\sigma^\nu=(1-\mu)\ud{\sigma}^\mu\sigma^\nu-(1-\mu)(1-\nu)\ud{\sigma}^{1-\mu}\sigma^{1-\nu}$,
16. $\ud{\sigma}^\mu{\sigma}^{1+\nu}\ud{\sigma}=(1-\nu)\ud{\sigma}^\mu\sigma^\nu-(1-\mu)(1-\nu)\ud{\sigma}^{1-\mu}\sigma^{1-\nu}$,
17. $\sigma\ud{\sigma}^\mu\sigma^\nu\ud{\sigma}=(1-\mu-\nu)(\ud{\sigma}^\mu\sigma^\nu-\ud{\sigma}^{1-\mu}\sigma^{1-\nu})$,

A condensed version appears in a footnote of my Electrodynamic's Wolverhampton Lectures on Physics.

I have written a piece of Mathematica code to check these results (and variations, if needed) semi-symbolically. This is detailed in this blog post.