The epsilon spinor is a rank 2 spinor which is fully skew-symmetric and whose component values are 1 or -1.

The command EpsilonSpinor(indexType, spinorType) returns the epsilon symbol of the type specified by indexType and spinorType in the current frame unless the frame is explicitly specified.

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

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

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

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

$\mathrm{P1}≔\mathrm{EpsilonSpinor}\left(''cov''\,''spinor''\right)$

$\mathrm{P1}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{P2}≔\mathrm{EpsilonSpinor}\left(''con''\,''spinor''\right)$

$\mathrm{P2}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{P3}≔\mathrm{EpsilonSpinor}\left(''cov''\,''barspinor''\right)$

$\mathrm{P3}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[7\,8\right]\,1\right]\,\left[\left[8\,7\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[7\,8\right]\,1\right]\,\left[\left[8\,7\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{P4}≔\mathrm{EpsilonSpinor}\left(''con''\,''spinor''\right)$

$\mathrm{P4}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,t\right]\,N\right)$

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

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

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

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

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

$\mathrm{P1}≔\mathrm{EpsilonSpinor}\left(''cov''\,''spinor''\right)$

$\mathrm{P1}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{P2}≔\mathrm{EpsilonSpinor}\left(''con''\,''spinor''\right)$

$\mathrm{P2}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,6\right]\,1\right]\,\left[\left[6\,5\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{P5}≔\mathrm{ContractIndices}\left(\mathrm{P2}\,\mathrm{P1}\,\left[\left[1\,1\right]\right]\right)$

$\mathrm{P5}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\right]\,1\right]\,\left[\left[6\,6\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\right]\,1\right]\,\left[\left[6\,6\right]\,1\right]\right]\right]\right)$

$\mathrm{P5}\&minus\left(\mathrm{KroneckerDeltaSpinor}\left(''spinor''\right)\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\right]\,0\right]\right]\right]\right)$