The solder form $\mathrm{\σ}$ is a rank 3 spin-tensor which defines an isomorphism between vectors and Hermitian rank 2 spinors. The first index type is a covariant tensor index, the second index type is a contravariant spinor index, and the third index is a contravariant barred (primed) spinor index. Denote the components of the solder form by ${\mathrm{\σ}}_i^{\mathrm{AA}\'}\.$ (The components of the solder form are often referred to as the Infeld-van der Waerden symbols.) To define the solder form, first recall the definition of an orthonormal frame. Let $M$ be a 4-dimensional manifold and let $g$ be a metric on $M$ with signature $\left[1\,-1\,-1\,-1\right]$. A tetrad of vectors $\left(X_1\,X_2\,X_3\,X_4\right)$ is an orthonormal frame with respect to the metric if $g\left(X_a\,X_b\right)\=0$ for $a\ne b$, $g\left(X_1\,X_1\right)\=1$, $g\(X_a\, X_a$) = -1, $a\=2\,3\,4$. The command DGGramSchmidt can be used to create an orthonormal frame. The command GRQuery or TensorInnerProduct can used to check that a list of vectors constitutes an orthonormal frame for a given metric. Recall also the definition of the 4 Pauli spin matrices $\left(P_1\,P_2\,P_3\,P_4\right)$, given below in Example 1. The matrix elements of the $P_a$ can be viewed as components of a Hermitian spinor, $P_a^{\mathrm{AA}\'}\.$ Let $\left(X_1\,X_2\,X_3\,X_4\right)$ be an orthonormal frame with respect to a metric $g$and let $\left({\mathrm{\ω}}^1\,{\mathrm{\ω}}^2\,{\mathrm{\ω}}^3\,{\mathrm{\ω}}^4\right)$ be the dual co-frame (see DualBasis). Then the associated solder form is

The command SolderForm(OrthFr) calculates the solder form from the orthonormal frame OrthFr.

The keyword argument indexlist = ind allows the user to specify the index structure for the solder form. For example, withindexlist = ["con", "con", "con"], the contravariant form ${\mathrm{\σ}}^{\mathrm{iAA}\'}$ is returned.

The solder form satisfies a number of important identities.  These are given in Example 2.

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

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

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

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

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

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

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

$\mathrm{GRQuery}\left(F\,g\,''OrthonormalTetrad''\right)$

$\mathrm{true}$

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

$\mathrm{\σ}:=\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{D\_z1}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{D\_z2}\mathrm{D\_w2}\+\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}\+\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}\+\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{dz}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{1}{2}\sqrt{2}\mathrm{dz}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{P1},\mathrm{P2},\mathrm{P3},\mathrm{P4}≔\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[1\,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0\,-\mathrm{I}\right]\,\left[\mathrm{I}\,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,-1\right]\right]\right)$

$S\left[1\right]≔\mathrm{evalDG}\left(\mathrm{D\_z1}\&t\mathrm{D\_w1}+\mathrm{D\_z2}\&t\mathrm{D\_w2}\right)$

$S_1:=\mathrm{D\_z1}\mathrm{D\_w1}\+\mathrm{D\_z2}\mathrm{D\_w2}$

$S\left[2\right]≔\mathrm{evalDG}\left(\mathrm{D\_z1}\&t\mathrm{D\_w2}+\mathrm{D\_z2}\&t\mathrm{D\_w1}\right)$

$S_2:=\mathrm{D\_z1}\mathrm{D\_w2}\+\mathrm{D\_z2}\mathrm{D\_w1}$

$S\left[3\right]≔\mathrm{evalDG}\left(-\mathrm{I}\mathrm{D\_z1}\&t\mathrm{D\_w2}+\mathrm{I}\mathrm{D\_z2}\&t\mathrm{D\_w1}\right)$

$S_3:=-\mathrm{I}\mathrm{D\_z1}\mathrm{D\_w2}\+\mathrm{I}\mathrm{D\_z2}\mathrm{D\_w1}$

$S\left[4\right]≔\mathrm{evalDG}\left(\mathrm{D\_z1}\&t\mathrm{D\_w1}-\mathrm{D\_z2}\&t\mathrm{D\_w2}\right)$

$S_4:=\mathrm{D\_z1}\mathrm{D\_w1}-\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{\omega }≔\left[\mathrm{dt}\,\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]$

$\mathrm{\ω}:=\left[\mathrm{dt}\,\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]$

$\mathrm{\σ0}≔\mathrm{evalDG}\left(\frac{\mathrm{sqrt}\left(2\right)}{2}\mathrm{add}\left(\mathrm{\omega }\left[i\right]\&tensorS\left[i\right]\,i=1..4\right)\right)$

$\mathrm{\σ0}:=\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{D\_z1}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{D\_z2}\mathrm{D\_w2}\+\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}\+\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}\+\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{dz}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{1}{2}\sqrt{2}\mathrm{dz}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{evalDG}\left(\mathrm{\sigma }-\mathrm{\σ0}\right)$

$0$

$\mathrm{sigmaCov}≔\mathrm{SolderForm}\left(F\,\mathrm{indextype}=\left[''cov''\,''cov''\,''cov''\right]\right)$

$\mathrm{sigmaCov}:=\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{dz1}\mathrm{dw1}\+\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{dz2}\mathrm{dw2}-\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{dz1}\mathrm{dw2}-\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{dz2}\mathrm{dw1}-\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{dz1}\mathrm{dw2}\+\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{dz2}\mathrm{dw1}-\frac{1}{2}\sqrt{2}\mathrm{dz}\mathrm{dz1}\mathrm{dw1}\+\frac{1}{2}\sqrt{2}\mathrm{dz}\mathrm{dz2}\mathrm{dw2}$

$\mathrm{sigmaCov}\&minus\mathrm{RaiseLowerSpinorIndices}\left(\mathrm{\sigma }\,\left[2\,3\right]\right)$

$0\mathrm{dt}\mathrm{dz1}\mathrm{dz1}$

$\mathrm{ContractIndices}\left(\mathrm{\sigma }\,\mathrm{sigmaCov}\,\left[\left[2\,2\right]\,\left[3\,3\right]\right]\right)$

$\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

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

$\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{sigmaCon}≔\mathrm{SolderForm}\left(F\,\mathrm{indextype}=\left[''con''\,''con''\,''con''\right]\right)$

$\mathrm{sigmaCon}:=\frac{1}{2}\sqrt{2}\mathrm{D\_t}\mathrm{D\_z1}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{D\_t}\mathrm{D\_z2}\mathrm{D\_w2}-\frac{1}{2}\sqrt{2}\mathrm{D\_x}\mathrm{D\_z1}\mathrm{D\_w2}-\frac{1}{2}\sqrt{2}\mathrm{D\_x}\mathrm{D\_z2}\mathrm{D\_w1}\+\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{D\_y}\mathrm{D\_z1}\mathrm{D\_w2}-\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{D\_y}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{1}{2}\sqrt{2}\mathrm{D\_z}\mathrm{D\_z1}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{D\_z}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{sigmaCon}\&minus\mathrm{RaiseLowerIndices}\left(\mathrm{InverseMetric}\left(g\right)\,\mathrm{\sigma }\,\left[1\right]\right)$

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

$\mathrm{E1}≔\mathrm{ContractIndices}\left(\mathrm{\sigma }\,\mathrm{sigmaCon}\,\left[\left[1\,1\right]\right]\right)$

$\mathrm{E1}:=\mathrm{D\_z1}\mathrm{D\_w1}\mathrm{D\_z2}\mathrm{D\_w2}-\mathrm{D\_z1}\mathrm{D\_w2}\mathrm{D\_z2}\mathrm{D\_w1}-\mathrm{D\_z2}\mathrm{D\_w1}\mathrm{D\_z1}\mathrm{D\_w2}\+\mathrm{D\_z2}\mathrm{D\_w2}\mathrm{D\_z1}\mathrm{D\_w1}$

$\mathrm{E2}≔\mathrm{RearrangeIndices}\left(\mathrm{E1}\,\left[\left[2\,3\right]\right]\right)$

$\mathrm{E2}:=\mathrm{D\_z1}\mathrm{D\_z2}\mathrm{D\_w1}\mathrm{D\_w2}-\mathrm{D\_z1}\mathrm{D\_z2}\mathrm{D\_w2}\mathrm{D\_w1}-\mathrm{D\_z2}\mathrm{D\_z1}\mathrm{D\_w1}\mathrm{D\_w2}\+\mathrm{D\_z2}\mathrm{D\_z1}\mathrm{D\_w2}\mathrm{D\_w1}$

$\mathrm{evalDG}\left(\mathrm{E2}-\mathrm{EpsilonSpinor}\left(''con''\,''spinor''\right)\&t\mathrm{EpsilonSpinor}\left(''con''\,''barspinor''\right)\right)$

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

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

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

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}+\exp \left(x\right)\mathrm{dt}\&s\mathrm{dz}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}+\frac{1}{2}\exp \left(2x\right)\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\mathrm{dt}\mathrm{dt}\+\frac{1}{2}{\ⅇ}^x\mathrm{dt}\mathrm{dz}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}\+\frac{1}{2}{\ⅇ}^x\mathrm{dz}\mathrm{dt}\+\frac{1}{2}{\ⅇ}^{2x}\mathrm{dz}\mathrm{dz}$

$F≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\,g\,\mathrm{signature}=\left[1\,-1\,-1\,-1\right]\right)\mathbf{assuming}x::\mathrm{real}$

$F:=\left[\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{I}\mathrm{D\_t}-2\mathrm{I}{\ⅇ}^{-x}\mathrm{D\_z}\right]$

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

$\mathrm{\σ}:=\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{D\_z1}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{D\_z2}\mathrm{D\_w2}\+\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}\+\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}\+\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}\+\left(\frac{1}{4}\mathrm{I}{\ⅇ}^x\sqrt{2}\+\frac{1}{4}{\ⅇ}^x\sqrt{2}\right)\mathrm{dz}\mathrm{D\_z1}\mathrm{D\_w1}\+\left(-\frac{1}{4}\mathrm{I}{\ⅇ}^x\sqrt{2}\+\frac{1}{4}{\ⅇ}^x\sqrt{2}\right)\mathrm{dz}\mathrm{D\_z2}\mathrm{D\_w2}$

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

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

$\mathrm{g3}≔\mathrm{evalDG}\left(x^4\mathrm{du}\&s\mathrm{dv}-y^2\mathrm{dx}\&t\mathrm{dx}-v^2\mathrm{dy}\&t\mathrm{dy}\right)$

$\mathrm{g3}:=\frac{1}{2}{x}^4\mathrm{du}\mathrm{dv}\+\frac{1}{2}{x}^4\mathrm{dv}\mathrm{du}-y^2\mathrm{dx}\mathrm{dx}-v^2\mathrm{dy}\mathrm{dy}$

$\mathrm{F3}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_u}\&comma;\mathrm{D\_v}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\right]\&comma;\mathrm{g3}\&comma;\mathrm{signature}=\left[\left[1\&comma;-1\right]\&comma;-1\&comma;-1\right]\right)\mathbf{assuming}0<y,0<v,x::\mathrm{real}$

$\mathrm{F3}:=\left[\frac{\mathrm{D\_u}}{x^2}\&plus;\frac{\mathrm{D\_v}}{x^2}\&comma;\frac{\mathrm{D\_u}}{x^2}-\frac{\mathrm{D\_v}}{x^2}\&comma;\frac{\mathrm{D\_x}}{y}\&comma;\frac{\mathrm{D\_y}}{v}\right]$

$\mathrm{\&sigma;3}≔\mathrm{SolderForm}\left(\mathrm{F3}\right)$

$\mathrm{\&sigma;3}:=\frac{1}{4}{x}^2\sqrt{2}\mathrm{du}\mathrm{D\_z1}\mathrm{D\_w1}\&plus;\frac{1}{4}{x}^2\sqrt{2}\mathrm{du}\mathrm{D\_z1}\mathrm{D\_w2}\&plus;\frac{1}{4}{x}^2\sqrt{2}\mathrm{du}\mathrm{D\_z2}\mathrm{D\_w1}\&plus;\frac{1}{4}{x}^2\sqrt{2}\mathrm{du}\mathrm{D\_z2}\mathrm{D\_w2}\&plus;\frac{1}{4}{x}^2\sqrt{2}\mathrm{dv}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{1}{4}{x}^2\sqrt{2}\mathrm{dv}\mathrm{D\_z1}\mathrm{D\_w2}-\frac{1}{4}{x}^2\sqrt{2}\mathrm{dv}\mathrm{D\_z2}\mathrm{D\_w1}\&plus;\frac{1}{4}{x}^2\sqrt{2}\mathrm{dv}\mathrm{D\_z2}\mathrm{D\_w2}-\frac{1}{2}\mathrm{I}y\sqrt{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}\&plus;\frac{1}{2}\mathrm{I}y\sqrt{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}\&plus;\frac{1}{2}v\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{1}{2}v\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{SpinorInnerProduct}\left(\mathrm{\&sigma;3}\&comma;\mathrm{\&sigma;3}\right)$

$\frac{1}{2}{x}^4\mathrm{du}\mathrm{dv}\&plus;\frac{1}{2}{x}^4\mathrm{dv}\mathrm{du}-y^2\mathrm{dx}\mathrm{dx}-v^2\mathrm{dy}\mathrm{dy}$