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# Blog: Notes

This is a collection of technical and research notes. Browse only if curious and/or undemanding.

## Commutation of Pauli matrices

$\sigma$ being the annihilation operator of a two-level system (one of the Pauli matrices), $\sigma^\dagger$ its conjugate, $\mu,\nu,\eta,\theta$ boolean variables (0 or 1), the following general commutator reads in normal order:

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

I sometimes need this formula but always have to derive it again, which is very annoying (see this). (Unrelatedly, I also hope it is the last time I typeset something with texvc, as I plan to move to MathJax).

## NormalOrder examples

In this note I give some particular cases of the module NormalOrder (see here for this module itself).

## NormalOrder module

Something that comes recurrently when you work with quantum fields is, given any operator that consists of products of powers of annihilation $$a$$ and creation $$a^\dagger$$ Bose operators, such as, e.g.,

$a^{\dagger3}a^2a^{\dagger3}a^2a^{\dagger}a$

compute its normal order[1]:

$:a^{\dagger3}a^2a^{\dagger3}a^2a^{\dagger}a:$

## Associate module

I am writing a code that computes arbitrary commutation relations. As part of this code is the following module that performs the associative part of the algebra: