The method of contractions in Clifford algebras

Dmitry Shirokov, Institute for Information Transmission Problems (Russian Academy of Sciences), Russia

We consider real and complex Clifford algebras ${\cal C}\!\ell(p,q)$ with fixed basis $\{e^A\}$ where $A$ is multi-index $A\in I=\{-, \, 1,\, \ldots,\, n,\, 12,\, 13,\, \ldots,\, 1\ldots n\}$. For any element $U\in{\cal C}\!\ell(p,q)$ we study expressions (contractions) $$\sum_{A\in S} e_A U e^A$$ for different $S\subseteq I.$ We present the relation between contractions and projection operators onto fixed subspaces of Clifford algebra. We present solutions of commutator system of equations $$e^A X +\epsilon X e^A=q^A,\qquad \forall A\in S,\qquad \epsilon\in{\Bbb C}$$ for unknown $X\in{\cal C}\!\ell(p,q)$. Also we consider generalized contractions $$\sum_{A\in S} \gamma_A U \beta^A,\qquad S\subseteq I$$ for two different bases $\{\gamma^A\}$, $\{\beta^A\}$. With the aid of these contractions we present formulas for element $T$ such that $$\gamma^a=J T^{-1}\beta^a T,\qquad a=1, \ldots ,n,\qquad J=\pm e, \pm e^{1\ldots n}, \pm i e^{1\ldots n}.$$ The method of contractions allow us to give solution of ''the simplest'' field equation in the pseudo-euclidian space and to present the new class of gauge-invariant solutions of the Yang-Mills equtions.