Spinor indices are raised and lowed using the epsilon spinor.

Indices are lowered by contraction with the first index of the covariant epsilon spinor and raised by contraction with the second index of the contravariant epsilon spinor. n terms of components:

The command RaiseLowerSpinorIndices(S, Indices) will raise or lower the indices of the spinor S given by the list Indices.

Unlike the command RaiseLowerIndices for raising and lowering tensor indices, no metric need be specified.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RaiseLowerSpinorIndices(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-RaiseLowerSpinorIndices.

with(DifferentialGeometry): with(Tensor):

DGsetup([x, y, z, t], [z1, z2, w1, w2], M);

$\mathrm{frame\; name:\; M}$

S1 := evalDG(a*D_z1 + b*D_z2);

$\mathrm{S1}:=a\mathrm{D\_z1}\+b\mathrm{D\_z2}$

T1 := RaiseLowerSpinorIndices(S1, [1]);

$\mathrm{T1}:=-b\mathrm{dz1}\+a\mathrm{dz2}$

epsilon1 := EpsilonSpinor("cov", "spinor");

$\mathrm{\ϵ1}:=\mathrm{dz1}\mathrm{dz2}-\mathrm{dz2}\mathrm{dz1}$

ContractIndices(epsilon1, S1, [[1, 1]]);

$-b\mathrm{dz1}\+a\mathrm{dz2}$

RaiseLowerSpinorIndices(T1, [1]);

$a\mathrm{D\_z1}\+b\mathrm{D\_z2}$

S2 := evalDG(a*D_t &t dz1 &t D_w2 &t D_w1 + b*D_x &t dz2 &t D_w1 &t D_w2);

$\mathrm{S2}:=b\mathrm{D\_x}\mathrm{dz2}\mathrm{D\_w1}\mathrm{D\_w2}\+a\mathrm{D\_t}\mathrm{dz1}\mathrm{D\_w2}\mathrm{D\_w1}$

RaiseLowerSpinorIndices(S2, [2, 4]);

$-b\mathrm{D\_x}\mathrm{D\_z1}\mathrm{D\_w1}\mathrm{dw1}-a\mathrm{D\_t}\mathrm{D\_z2}\mathrm{D\_w2}\mathrm{dw2}$