Infeld–Van der Waerden symbols

The Infeld–Van der Waerden symbols, sometimes called simply Van der Waerden symbols, are an invariant symbol associated to the Lorentz group used in quantum field theory. They are named after Leopold Infeld and Bartel Leendert van der Waerden.

The Infeld–Van der Waerden symbols are index notation for Clifford multiplication of covectors on left handed spinors giving a right-handed spinors or vice versa, i.e. they are off diagonal blocks of gamma matrices. The symbols are typically denoted in Van der Waerden notation as

\sigma ^{m}{}_{\alpha {\dot {\beta }}}\quad {\text{and}}\quad {\bar {\sigma }}^{m\,{\dot {\alpha }}\beta }.

and so have one Lorentz index (m), one left-handed (undotted Greek), and one right-handed (dotted Greek) Weyl spinor index. They satisfy

{\begin{aligned}\sigma ^{m}{}_{\alpha {\dot {\beta }}}{\bar {\sigma }}^{n\,{\dot {\beta }}\gamma }+\sigma ^{n}{}_{\alpha {\dot {\beta }}}{\bar {\sigma }}^{m\,{\dot {\beta }}\gamma }&=2\delta _{\alpha }^{\gamma }g^{mn},\quad \\\sigma ^{m}{}_{\alpha {\dot {\beta }}}{\bar {\sigma }}^{n\,{\dot {\gamma }}\alpha }+\sigma ^{n}{}_{\alpha {\dot {\beta }}}{\bar {\sigma }}^{m\,{\dot {\gamma }}\alpha }&=2\delta _{\dot {\beta }}^{\dot {\gamma }}g^{mn}.\end{aligned}}

They need not be constant, however, and can therefore be formulated on curved space time.

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