The command ConjugateSpinor(S) calculates the complex conjugate of an arbitrary spinor S.

For spinors with real parameters, the assuming command of Maple can be used to properly calculate the complex conjugates.

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

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

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

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

$\mathrm{S1}≔\mathrm{D\_z1}$

$\mathrm{S1}:=\mathrm{D\_z1}$

$\mathrm{ConjugateSpinor}\left(\mathrm{S1}\right)$

$\mathrm{D\_w1}$

$\mathrm{S2}≔\mathrm{dw1}+3\mathrm{I}\mathrm{dw2}$

$\mathrm{S2}:=\mathrm{dw1}\+3\mathrm{I}\mathrm{dw2}$

$\mathrm{ConjugateSpinor}\left(\mathrm{S2}\right)$

$\mathrm{dz1}-3\mathrm{I}\mathrm{dz2}$

$\mathrm{S3}≔\mathrm{KroneckerDeltaSpinor}\left(''spinor''\right)$

$\mathrm{S3}:=\mathrm{D\_z1}\mathrm{dz1}+\mathrm{D\_z2}\mathrm{dz2}$

$\mathrm{S4}≔\mathrm{KroneckerDeltaSpinor}\left(''barspinor''\right)$

$\mathrm{S4}:=\mathrm{D\_w1}\mathrm{dw1}+\mathrm{D\_w2}\mathrm{dw2}$

$\mathrm{ConjugateSpinor}\left(\mathrm{S3}\right)\&minus\mathrm{S4}$

$0\mathrm{D\_z1}\mathrm{dz1}$

$F≔\left[\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\:$

$\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$

$\mathrm{\sigma }≔\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}+\frac{\sqrt{2}}{2}\mathrm{dz}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{\sqrt{2}}{2}\mathrm{dz}\mathrm{D\_z2}\mathrm{D\_w2}+\frac{\sqrt{2}}{2}\mathrm{dt}\mathrm{D\_z1}\mathrm{D\_w1}+\frac{\sqrt{2}}{2}\mathrm{dt}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{bar\_sigma}≔\mathrm{ConjugateSpinor}\left(\mathrm{\sigma }\right)$

$\mathrm{bar\_sigma}≔\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_w1}\mathrm{D\_z2}+\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_w2}\mathrm{D\_z1}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_w1}\mathrm{D\_z2}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_w2}\mathrm{D\_z1}+\frac{\sqrt{2}}{2}\mathrm{dz}\mathrm{D\_w1}\mathrm{D\_z1}-\frac{\sqrt{2}}{2}\mathrm{dz}\mathrm{D\_w2}\mathrm{D\_z2}+\frac{\sqrt{2}}{2}\mathrm{dt}\mathrm{D\_w1}\mathrm{D\_z1}+\frac{\sqrt{2}}{2}\mathrm{dt}\mathrm{D\_w2}\mathrm{D\_z2}$

$\mathrm{\sigma }\&minus\mathrm{RearrangeIndices}\left(\mathrm{bar\_sigma}\,\left[\left[2\,3\right]\right]\right)$

$0\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_z1}$

$\mathrm{S5}≔\mathrm{evalDG}\left(\mathrm{\alpha }\mathrm{D\_t}\&t\mathrm{D\_w1}+\frac{x}{\mathrm{\alpha }}\mathrm{D\_z}\&t\mathrm{D\_w2}\right)$

$\mathrm{S5}≔\frac{x}{\mathrm{\alpha }}\mathrm{D\_z}\mathrm{D\_w2}+\mathrm{\alpha }\mathrm{D\_t}\mathrm{D\_w1}$

$\mathrm{S6}≔\mathrm{ConjugateSpinor}\left(\mathrm{S5}\right)\mathbf{assuming}\mathrm{\alpha }::\mathrm{real}$

$\mathrm{S6}≔\frac{x}{\mathrm{\alpha }}\mathrm{D\_z}\mathrm{D\_z2}+\mathrm{\alpha }\mathrm{D\_t}\mathrm{D\_z1}$

$\mathrm{DGsetup}\left(\left[u\,v\,z\,t\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,N\right)$

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

$\mathrm{S7}≔\mathrm{evalDG}\left(v\mathrm{D\_z1}\&t\mathrm{dz2}+tu\mathrm{D\_z2}\&t\mathrm{dz1}\right)$

$\mathrm{S7}≔v\mathrm{D\_z1}\mathrm{dz2}+tu\mathrm{D\_z2}\mathrm{dz1}$

$\mathrm{ConjugateSpinor}\left(\mathrm{S7}\,\mathrm{conjugatecoordinates}=\left[\left[u\,v\right]\right]\right)$

$u\mathrm{D\_w1}\mathrm{dw2}+tv\mathrm{D\_w2}\mathrm{dw1}$

$\mathrm{S8}≔\mathrm{evalDG}\left(\mathrm{\alpha }\mathrm{D\_u}\&t\mathrm{dz2}\right)$

$\mathrm{S8}≔\mathrm{\alpha }\mathrm{D\_u}\mathrm{dz2}$

$\mathrm{ConjugateSpinor}\left(\mathrm{S8}\,\mathrm{conjugatecoordinates}=\left[\left[u\,v\right]\right]\right)\mathbf{assuming}\mathrm{\alpha }::\mathrm{real}$

$\mathrm{\alpha }\mathrm{D\_v}\mathrm{dw2}$