DifferentialGeometry/Tensor/RaiseLowerSpinorIndices - Maple Help

Tensor[RaiseLowerSpinorIndices] - raise or lower a list of spinor indices using epsilon spinors

Calling Sequences

RaiseLowerSpinorIndices(S, Indices)

Parameters

S       - a spinor or spinor-tensor

Indices - a list of integers, referring to the arguments of S

Description

 • Spinor indices are raised and lowed using the epsilon spinor.
 • Indices are lowered by contraction with the first index of the covariant epsilon spinor and raised by contraction with the second index of the contravariant epsilon spinor. n terms of components:

 • The command RaiseLowerSpinorIndices(S, Indices) will raise or lower the indices of the spinor S given by the list Indices.
 • Unlike the command RaiseLowerIndices for raising and lowering tensor indices, no metric need be specified.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RaiseLowerSpinorIndices(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-RaiseLowerSpinorIndices.

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 .

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

Define a contravariant rank 1 spinor $\mathrm{S1}$ and lower its indices, that is, convert it to a covariant rank 1 spinor $\mathrm{T1}$.

 M > $\mathrm{S1}≔\mathrm{evalDG}\left(a\mathrm{D_z1}+b\mathrm{D_z2}\right)$
 ${\mathrm{S1}}{:=}{a}{}{\mathrm{D_z1}}{+}{b}{}{\mathrm{D_z2}}$ (2.2)
 M > $\mathrm{T1}≔\mathrm{RaiseLowerSpinorIndices}\left(\mathrm{S1},\left[1\right]\right)$
 ${\mathrm{T1}}{:=}{-}{b}{}{\mathrm{dz1}}{+}{a}{}{\mathrm{dz2}}$ (2.3)

Define the covariant epsilon spinor $\mathrm{ε1}$ and check that this result coincides with the contraction of $\mathrm{ε1}$ and $\mathrm{S1}$.

 M > $\mathrm{ε1}≔\mathrm{EpsilonSpinor}\left("cov","spinor"\right)$
 ${\mathrm{ϵ1}}{:=}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{\mathrm{dz2}}{}{\mathrm{dz1}}$ (2.4)
 M > $\mathrm{ContractIndices}\left(\mathrm{ε1},\mathrm{S1},\left[\left[1,1\right]\right]\right)$
 ${-}{b}{}{\mathrm{dz1}}{+}{a}{}{\mathrm{dz2}}$ (2.5)

Convert $\mathrm{T1}$ back to a contravariant rank 1 spinor, recovering $\mathrm{S1}$.

 M > $\mathrm{RaiseLowerSpinorIndices}\left(\mathrm{T1},\left[1\right]\right)$
 ${a}{}{\mathrm{D_z1}}{+}{b}{}{\mathrm{D_z2}}$ (2.6)

Example 2.

Define a rank 4 spinor-tensor $\mathrm{S2}$ and raise its 2nd index and lower its 4th index.

 M > $\mathrm{S2}≔\mathrm{evalDG}\left(a\left(\left(\mathrm{D_t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w1}+b\left(\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w2}\right)$
 ${\mathrm{S2}}{:=}{b}{}{\mathrm{D_x}}{}{\mathrm{dz2}}{}{\mathrm{D_w1}}{}{\mathrm{D_w2}}{+}{a}{}{\mathrm{D_t}}{}{\mathrm{dz1}}{}{\mathrm{D_w2}}{}{\mathrm{D_w1}}$ (2.7)
 M > $\mathrm{RaiseLowerSpinorIndices}\left(\mathrm{S2},\left[2,4\right]\right)$
 ${-}{b}{}{\mathrm{D_x}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{}{\mathrm{dw1}}{-}{a}{}{\mathrm{D_t}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{}{\mathrm{dw2}}$ (2.8)
 M >