Racah Algebra

Abstract

It is convenient in many vector problems to express the vectors in terms of spherical bases given by

$$\begin{gathered} {\mathop e\limits^ \wedge _{ \pm 1}}\; = \; \mp \left( {{{\mathop e\limits^ \wedge }_{x\;}} \pm \;i{{\mathop e\limits^ \wedge }_y}} \right)/\sqrt 2 \hfill \\ {\mathop e\limits^ \wedge _{0\; = }}\;{\mathop e\limits^ \wedge _z} \hfill \\ \end{gathered} $$

((6.1))

Then

$$\begin{array}{*{20}{c}} {\hat e_\mu ^* = {{\left( { - 1} \right)}^\mu }{{\hat e}_{ - \mu }},} \\ {{{\hat e}_\mu } \times {{\hat e}_\upsilon } = - i\sqrt 2 \left\langle {1\left( \upsilon \right)1\left( {\mu + \upsilon } \right)} \right\rangle {{\hat e}_{\mu + \upsilon }},} \\ {\hat e_\mu ^* \cdot {{\hat e}_\upsilon } = {\delta _{\mu \upsilon }}.} \end{array}$$

((6.2))