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Tensor[KroneckerDeltaSpinor] - create the Kronecker delta spinor

Calling Sequences

KroneckerDeltaSpinor(spinorType, fr)

Parameters

spinorType - a string, either "spinor" or "barspinor"

fr         - (optional) the name of a defined frame

Description

 • The Kronecker delta spinor is the type $\left(\genfrac{}{}{0}{}{1}{1}\right)$ spinor whose components in any coordinate system are given by the identity matrix.
 • The command KroneckerDeltaSpinor(spinorType) returns a Kronecker delta spinor of the type specified by 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 KroneckerDeltaSpinor(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KroneckerDeltaSpinor.

Examples

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

Example 1.

First create a vector bundle $M$ with base coordinates $\left(x,y,z,t\right)$ and fiber coordinates $\left(\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right)$.

 > $\mathrm{DGsetup}\left(\left[x,y,z,t\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Here are the 2 Kronecker delta spinors one can define:

 M > $\mathrm{K1}≔\mathrm{KroneckerDeltaSpinor}\left("spinor"\right)$
 ${\mathrm{K1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right)$ (2.2)
 M > $\mathrm{K2}≔\mathrm{KroneckerDeltaSpinor}\left("barspinor"\right)$
 ${\mathrm{K2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{7}{,}{7}\right]{,}{1}\right]{,}\left[\left[{8}{,}{8}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{7}{,}{7}\right]{,}{1}\right]{,}\left[\left[{8}{,}{8}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{7}{,}{7}\right]{,}{1}\right]{,}\left[\left[{8}{,}{8}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{7}{,}{7}\right]{,}{1}\right]{,}\left[\left[{8}{,}{8}\right]{,}{1}\right]\right]\right]\right)$ (2.3)

Define some other manifold $N$.

 M > $\mathrm{DGsetup}\left(\left[x,y,z,t\right],N\right)$
 ${\mathrm{frame name: N}}$ (2.4)

The current frame is $N$. Because there are no fiber variables, one cannot calculate a Kronecker delta spinor in this frame. To now re-calculate the Kronecker delta spinor $\mathrm{K1}$, either use the ChangeFrame command or pass KroneckerDeltaSpinor the frame name $M$ as a second argument.

 N > $\mathrm{KroneckerDeltaSpinor}\left("spinor",M\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right)$ (2.5)

Example 2.

The Kronecker delta spinor defines an identity mapping on spinors of the indicated type. The linear transformation associated to the Kronecker delta spinor $K$ is defined by contracting the covariant index of $K$ against the contravariant index of the spinor $\mathrm{S1}$. We see that the result is $\mathrm{S1}$ so that the linear transformation defined by $K$ is indeed the identity transformation.

 M > $\mathrm{DGsetup}\left(\left[x,y,z,t\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.6)
 M > $K≔\mathrm{KroneckerDeltaSpinor}\left("spinor"\right)$
 ${K}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right)$ (2.7)
 M > $\mathrm{S1}≔\mathrm{evalDG}\left(a\mathrm{D_z1}+b\mathrm{D_z2}\right)$
 ${\mathrm{S1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right)$ (2.8)
 M > $\mathrm{S2}≔\mathrm{ContractIndices}\left(\mathrm{S1},K,\left[\left[1,2\right]\right]\right)$
 ${\mathrm{S2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right)$ (2.9)