Hi everyone, this is my first StackExchange post, so all tips on how to improve the question are very welcome! The question I would like to ask comes from mathematical physics, so please also tell me if you think it may be better suited to a different Stackexchange community.
This question is about systematically finding the coefficients of polynomial functions on matrix representations of Clifford algebras. I would like to know if you have seen more systematic treatment of the following problem: Take the Clifford Algebra $\mathcal{Cl}_3$ of $\mathbb{R}^3$. Choose the Pauli matrices $\sigma^{i}$ as basis elements of its matrix representation. We are interested in functions of the form $$l: \mathbb{R}^{3n} \to \mathcal{Cl}_3,$$ where $l$
$$l(p_1,\cdots,p_n,m,\vec{\mu})= \sigma^{\mu_1} (- \vec{p}_1 \cdot \vec{\sigma}+m \mathbb{1}) \sigma^{\mu_2} (- \vec{p}_2 \cdot \vec{\sigma}+m \mathbb{1}) \cdots \sigma^{\mu_n} (- \vec{p}_n \cdot \vec{\sigma}+m \mathbb{1}),$$ where $\vec{p} \cdot \vec{\sigma} = \sum_i p_i \sigma^{i}$. Ultimately, I would like to know properties of this polynomials' trace, i.e. find an expression of the form
$$tr(l)= Q_n(\vec{p}_1,\cdots,\vec{p}_n)+m Q_{n-1} (\vec{p}_1,\cdots,\vec{p}_n)+\cdots $$ Expressions of these type naturally appear in the perturbative path integral formulation of Quantum Electrodynamics. It would be nice to know if a systematic treatment of these objects exist. Is there a structured approach on how to find the coefficients of a polynomial function like the one above on matrix representations on Clifford algebras?