DifferentialGeometry/Tensor/SpinorInnerProduct - Maple Help

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Tensor[SpinorInnerProduct] - calculate the inner product of 2 spinors

Calling Sequences

SpinorInnerProduct(S, T)

Parameters

S, T    - two spinors or spinor-tensors of the same spinor type

Description

 • The spinor inner product of two spinors $S$ and $T$ of the same type is calculated by contracting each pair of corresponding spinor indices (one from $S$ and one from $T$) with the appropriate epsilon spinor. For example, the inner product of two covariant rank 1 spinors with components ${S}_{A}$ and ${T}_{B}$ is ${\mathrm{ε}}^{\mathrm{AB}}{S}_{A}{T}_{B}.$  The inner product of two contravariant rank 1 spinors ${S}_{}^{A}$ and ${T}_{}^{B}$ is ${\mathrm{ε}}_{\mathrm{AB}}^{}{S}_{}^{A}{T}_{}^{B}$. The inner product of two contravariant rank 2 spinors with components and ${T}_{\mathrm{CD}}$ is ${S}_{\mathrm{AB}}$${T}_{\mathrm{CD}}$ .
 • If $S$ and $T$ are odd rank spinors, then SpinorInnerProduct(S, T) = -SpinorInnerProduct(T, S) and therefore SpinorInnerProduct(S, S) = 0. (Strictly speaking, the spinor inner product is really just a bilinear pairing -- it is not a true inner product because it is not always symmetric in its arguments.)
 • If $S$ and $T$ are even rank spinors, then SpinorInnerProduct(S, T) = SpinorInnerProduct(T, S).
 • Unlike TensorInnerProduct, SpinorInnerProduct does not require specification of a metric tensor to perform the contractions.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SpinorInnerProduct(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-SpinorInnerProduct.

Examples

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

Example 1.

First create a vector bundle $M$ with base coordinates  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 rank 1 spinors $\mathrm{S1}$ and $\mathrm{T1}$ and calculate their inner product.

 M > $\mathrm{S1}≔\mathrm{evalDG}\left(a\mathrm{D_z1}+b\mathrm{D_z2}\right)$
 ${\mathrm{S1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{a}\right]{,}\left[\left[{6}\right]{,}{b}\right]\right]\right]\right)$ (2.2)
 M > $\mathrm{T1}≔\mathrm{evalDG}\left(c\mathrm{D_z1}+d\mathrm{D_z2}\right)$
 ${\mathrm{T1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{c}\right]{,}\left[\left[{6}\right]{,}{d}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{c}\right]{,}\left[\left[{6}\right]{,}{d}\right]\right]\right]\right)$ (2.3)
 M > $\mathrm{SpinorInnerProduct}\left(\mathrm{S1},\mathrm{T1}\right)$
 ${a}{}{d}{-}{b}{}{c}$ (2.4)

Note the sign change:

 M > $\mathrm{SpinorInnerProduct}\left(\mathrm{T1},\mathrm{S1}\right)$
 ${-}{a}{}{d}{+}{b}{}{c}$ (2.5)

The inner product of a rank 1 spinor with itself vanishes

 M > $\mathrm{SpinorInnerProduct}\left(\mathrm{S1},\mathrm{S1}\right)$
 ${0}$ (2.6)

Calculate the inner product of $\mathrm{S1}$ and $\mathrm{T1}$ from the definition.

 M > $\mathrm{\epsilon }≔\mathrm{EpsilonSpinor}\left("cov","spinor"\right)$
 ${\mathrm{ϵ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{6}{,}{5}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{6}{,}{5}\right]{,}{-1}\right]\right]\right]\right)$ (2.7)
 M > $\mathrm{U1}≔\mathrm{ContractIndices}\left(\mathrm{\epsilon },\mathrm{S1},\left[\left[1,1\right]\right]\right)$
 ${\mathrm{U1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}{b}\right]{,}\left[\left[{6}\right]{,}{a}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}{b}\right]{,}\left[\left[{6}\right]{,}{a}\right]\right]\right]\right)$ (2.8)
 M > $\mathrm{ContractIndices}\left(\mathrm{U1},\mathrm{T1},\left[\left[1,1\right]\right]\right)$
 ${a}{}{d}{-}{b}{}{c}$ (2.9)

Example 2.

Calculate the inner product of two rank 2 spinors $\mathrm{S2}$ and $\mathrm{T2}$.

 M > $\mathrm{S2}≔\mathrm{evalDG}\left(a\mathrm{D_z1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}+b\mathrm{D_z1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}\right)$
 ${\mathrm{S2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{7}\right]{,}{b}\right]{,}\left[\left[{5}{,}{8}\right]{,}{a}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{7}\right]{,}{b}\right]{,}\left[\left[{5}{,}{8}\right]{,}{a}\right]\right]\right]\right)$ (2.10)
 M > $\mathrm{T2}≔\mathrm{evalDG}\left(c\mathrm{D_z1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}+d\mathrm{D_z2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)$
 ${\mathrm{T2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{7}\right]{,}{c}\right]{,}\left[\left[{6}{,}{8}\right]{,}{d}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{7}\right]{,}{c}\right]{,}\left[\left[{6}{,}{8}\right]{,}{d}\right]\right]\right]\right)$ (2.11)
 M > $\mathrm{SpinorInnerProduct}\left(\mathrm{S2},\mathrm{T2}\right)$
 ${b}{}{d}$ (2.12)

Example 3.

Calculate the inner product of two rank 2 spinor-tensors $\mathrm{S3}$ and $\mathrm{T3}$. Note that in this example the result is a rank 2 tensor.

 M > $\mathrm{S3}≔\mathrm{evalDG}\left(\mathrm{D_t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}+\mathrm{D_z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)$
 ${\mathrm{S3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{3}{,}{8}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{3}{,}{8}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}\right]{,}{1}\right]\right]\right]\right)$ (2.13)
 M > $\mathrm{T3}≔\mathrm{evalDG}\left(\mathrm{D_y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}+\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)$
 ${\mathrm{T3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{8}\right]{,}{1}\right]{,}\left[\left[{2}{,}{7}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{8}\right]{,}{1}\right]{,}\left[\left[{2}{,}{7}\right]{,}{1}\right]\right]\right]\right)$ (2.14)
 M > $\mathrm{SpinorInnerProduct}\left(\mathrm{S3},\mathrm{T3}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{3}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}{1}\right]\right]\right]\right)$ (2.15)