DifferentialGeometry/Tensor/EpsilonSpinor - Maple Help

Tensor[EpsilonSpinor] - create an epsilon spinor

Calling Sequences

EpsilonSpinor(indexType, spinorType, fr)

Parameters

indexType  - a string, either "cov" or "con"

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

fr         - (optional) the name of a defined frame

Description

 • The epsilon spinor is a rank 2 spinor which is fully skew-symmetric and whose component values are 1 or -1.
 • The command EpsilonSpinor(indexType, spinorType) returns the epsilon symbol of the type specified by indexType and 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 EpsilonSpinor(...) only after executing the commands with(DifferentialGeometry); with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-EpsilonSpinor.

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 4 epsilon spinors one can define:

 M > $\mathrm{P1}≔\mathrm{EpsilonSpinor}\left("cov","spinor"\right)$
 ${\mathrm{P1}}{:=}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{\mathrm{dz2}}{}{\mathrm{dz1}}$ (2.2)
 M > $\mathrm{P2}≔\mathrm{EpsilonSpinor}\left("con","spinor"\right)$
 ${\mathrm{P2}}{:=}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{\mathrm{D_z2}}{}{\mathrm{D_z1}}$ (2.3)
 M > $\mathrm{P3}≔\mathrm{EpsilonSpinor}\left("cov","barspinor"\right)$
 ${\mathrm{P3}}{:=}{\mathrm{dw1}}{}{\mathrm{dw2}}{-}{\mathrm{dw2}}{}{\mathrm{dw1}}$ (2.4)
 M > $\mathrm{P4}≔\mathrm{EpsilonSpinor}\left("con","spinor"\right)$
 ${\mathrm{P4}}{:=}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{\mathrm{D_z2}}{}{\mathrm{D_z1}}$ (2.5)

Define some other manifold $N$.

 M > $\mathrm{DGsetup}\left(\left[x,y,z,t\right],N\right)$
 ${\mathrm{frame name: N}}$ (2.6)

The current frame is $N$.  Because there are no fiber variables, one cannot calculate an epsilon spinor in this frame. To now re-calculate the epsilon spinor $\mathrm{P1}$, either use the ChangeFrame command or pass EpsilonSpinor the frame name $M$ as a third argument.

 N > $\mathrm{EpsilonSpinor}\left("cov","spinor",M\right)$
 ${\mathrm{dz1}}{}{\mathrm{dz2}}{-}{\mathrm{dz2}}{}{\mathrm{dz1}}$ (2.7)

Example 2.

The covariant and contravariant forms of the epsilon spinors are inverses of each other.

 M > $\mathrm{DGsetup}\left(\left[x,y,z,t\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.8)
 M > $\mathrm{P1}≔\mathrm{EpsilonSpinor}\left("cov","spinor"\right)$
 ${\mathrm{P1}}{:=}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{\mathrm{dz2}}{}{\mathrm{dz1}}$ (2.9)
 M > $\mathrm{P2}≔\mathrm{EpsilonSpinor}\left("con","spinor"\right)$
 ${\mathrm{P2}}{:=}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{\mathrm{D_z2}}{}{\mathrm{D_z1}}$ (2.10)

Contract the first index of $\mathrm{P1}$ with the first index of $\mathrm{P2}$.  The result is the Kronecker delta spinor.

 M > $\mathrm{P5}≔\mathrm{ContractIndices}\left(\mathrm{P2},\mathrm{P1},\left[\left[1,1\right]\right]\right)$
 ${\mathrm{P5}}{:=}{\mathrm{D_z1}}{}{\mathrm{dz1}}{+}{\mathrm{D_z2}}{}{\mathrm{dz2}}$ (2.11)
 M > $\mathrm{P5}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{KroneckerDeltaSpinor}\left("spinor"\right)$
 ${0}{}{\mathrm{D_z1}}{}{\mathrm{dz1}}$ (2.12)