The spinor inner product of two spinors $S$ and $T$ of the same type is calculated by contracting each pair of corresponding spinor indices (one from $S$ and one from $T$) with the appropriate epsilon spinor. For example, the inner product of two covariant rank 1 spinors with components $S_A$ and $T_B$ is ${\mathrm{\ε}}^{\mathrm{AB}}{S}_A{T}_B\.$  The inner product of two contravariant rank 1 spinors $S_{}^A$ and $T_{}^B$ is ${\mathrm{\ε}}_{\mathrm{AB}}^{}{S}_{}^A{T}_{}^B$. The inner product of two contravariant rank 2 spinors with components $S_{\mathrm{AB}}$and $T_{\mathrm{CD}}$ is ${\mathrm{\ε}}^{\mathrm{AC}}{\mathrm{\ε}}^{\mathrm{BD}}$$S_{\mathrm{AB}}$$T_{\mathrm{CD}}$ .

If $S$ and $T$ are odd rank spinors, then SpinorInnerProduct(S, T) = -SpinorInnerProduct(T, S) and therefore SpinorInnerProduct(S, S) = 0. (Strictly speaking, the spinor inner product is really just a bilinear pairing -- it is not a true inner product because it is not always symmetric in its arguments.)

If $S$ and $T$ are even rank spinors, then SpinorInnerProduct(S, T) = SpinorInnerProduct(T, S).

Unlike TensorInnerProduct, SpinorInnerProduct does not require specification of a metric tensor to perform the contractions.

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

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,t\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,M\right)$

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

$\mathrm{S1}≔\mathrm{evalDG}\left(a\mathrm{D\_z1}\+b\mathrm{D\_z2}\right)$

$\mathrm{S1}:=\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[5\right]\,a\right]\,\left[\left[6\right]\,b\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[5\right]\,a\right]\,\left[\left[6\right]\,b\right]\right]\right]\right)$

$\mathrm{T1}≔\mathrm{evalDG}\left(c\mathrm{D\_z1}\+d\mathrm{D\_z2}\right)$

$\mathrm{T1}:=\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[5\right]\,c\right]\,\left[\left[6\right]\,d\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[5\right]\,c\right]\,\left[\left[6\right]\,d\right]\right]\right]\right)$

$\mathrm{SpinorInnerProduct}\left(\mathrm{S1}\,\mathrm{T1}\right)$

$ad-bc$

$\mathrm{SpinorInnerProduct}\left(\mathrm{T1}\,\mathrm{S1}\right)$

$-ad\+bc$

$\mathrm{SpinorInnerProduct}\left(\mathrm{S1}\,\mathrm{S1}\right)$

$0$

$\mathrm{\ε}≔\mathrm{EpsilonSpinor}\left(''cov''\,''spinor''\right)$

$\mathrm{\ϵ}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{U1}≔\mathrm{ContractIndices}\left(\mathrm{\ε}\,\mathrm{S1}\,\left[\left[1\,1\right]\right]\right)$

$\mathrm{U1}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\right]\,-b\right]\,\left[\left[6\right]\,a\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\right]\,-b\right]\,\left[\left[6\right]\,a\right]\right]\right]\right)$

$\mathrm{ContractIndices}\left(\mathrm{U1}\,\mathrm{T1}\,\left[\left[1\,1\right]\right]\right)$

$ad-bc$

$\mathrm{S2}≔\mathrm{evalDG}\left(a\mathrm{D\_z1}\&t\mathrm{dw2}\+b\mathrm{D\_z1}\&t\mathrm{dw1}\right)$

$\mathrm{S2}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,7\right]\,b\right]\,\left[\left[5\,8\right]\,a\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,7\right]\,b\right]\,\left[\left[5\,8\right]\,a\right]\right]\right]\right)$

$\mathrm{T2}≔\mathrm{evalDG}\left(c\mathrm{D\_z1}\&t\mathrm{dw1}\+d\mathrm{D\_z2}\&t\mathrm{dw2}\right)$

$\mathrm{T2}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,7\right]\,c\right]\,\left[\left[6\,8\right]\,d\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,7\right]\,c\right]\,\left[\left[6\,8\right]\,d\right]\right]\right]\right)$

$\mathrm{SpinorInnerProduct}\left(\mathrm{S2}\,\mathrm{T2}\right)$

$bd$

$\mathrm{S3}≔\mathrm{evalDG}\left(\mathrm{D\_t}\&t\mathrm{dw1}\+\mathrm{D\_z}\&t\mathrm{dw2}\right)$

$\mathrm{S3}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\,8\right]\,1\right]\,\left[\left[4\,7\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\,8\right]\,1\right]\,\left[\left[4\,7\right]\,1\right]\right]\right]\right)$

$\mathrm{T3}≔\mathrm{evalDG}\left(\mathrm{D\_y}\&t\mathrm{dw1}\+\mathrm{D\_x}\&t\mathrm{dw2}\right)$

$\mathrm{T3}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,8\right]\,1\right]\,\left[\left[2\,7\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,8\right]\,1\right]\,\left[\left[2\,7\right]\,1\right]\right]\right]\right)$

$\mathrm{SpinorInnerProduct}\left(\mathrm{S3}\,\mathrm{T3}\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\,2\right]\,\mathrm{-1}\right]\,\left[\left[4\,1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\,2\right]\,\mathrm{-1}\right]\,\left[\left[4\,1\right]\,1\right]\right]\right]\right)$