DifferentialGeometry/Tensor/ConjugateSpinor - Maple Help
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Tensor[ConjugateSpinor] - calculate the complex conjugate of a spinor

Calling Sequences

ConjugateSpinor(S, ConjCoord)

Parameters

S         - a spinor

ConjCoord - (optional) keyword argument conjugatecoordinates = C, where C is a list of lists specifying conjugate coordinates

Description

 • The command ConjugateSpinor(S) calculates the complex conjugate of an arbitrary spinor S.
 • For spinors with real parameters, the assuming command of Maple can be used to properly calculate the complex conjugates.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form ConjugateSpinor(...) only after executing the commands with(DifferentialGeometry); with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-ConjugateSpinor.

Examples

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

Example 1.

First create a vector bundle $E\to M$ with base coordinates $\left(x,y,z,t\right)$ and fiber coordinates $\left(\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right)$. For spinor applications, it is tacitly assumed that are complex coordinates with complex conjugates $\left(\mathrm{w1},\mathrm{w2}\right)$.

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

Define spinors $\mathrm{S1}$ and $\mathrm{S2}$ and calculate their complex conjugates.

 E > $\mathrm{S1}≔\mathrm{D_z1}$
 ${\mathrm{S1}}{:=}{\mathrm{D_z1}}$ (2.2)
 E > $\mathrm{ConjugateSpinor}\left(\mathrm{S1}\right)$
 ${\mathrm{D_w1}}$ (2.3)
 E > $\mathrm{S2}≔\mathrm{dw1}+3I\mathrm{dw2}$
 ${\mathrm{S2}}{:=}{\mathrm{dw1}}{+}{3}{}{I}{}{\mathrm{dw2}}$ (2.4)
 E > $\mathrm{ConjugateSpinor}\left(\mathrm{S2}\right)$
 ${\mathrm{dz1}}{-}{3}{}{I}{}{\mathrm{dz2}}$ (2.5)

Example 2.

The two type $\left(1,1\right)$ Kronecker delta spinors are complex conjugates of each other.

 E > $\mathrm{S3}≔\mathrm{KroneckerDeltaSpinor}\left("spinor"\right)$
 ${\mathrm{S3}}{:=}{\mathrm{D_z1}}{}{\mathrm{dz1}}{+}{\mathrm{D_z2}}{}{\mathrm{dz2}}$ (2.6)
 E > $\mathrm{S4}≔\mathrm{KroneckerDeltaSpinor}\left("barspinor"\right)$
 ${\mathrm{S4}}{:=}{\mathrm{D_w1}}{}{\mathrm{dw1}}{+}{\mathrm{D_w2}}{}{\mathrm{dw2}}$ (2.7)
 E > $\mathrm{ConjugateSpinor}\left(\mathrm{S3}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{S4}$
 ${0}{}{\mathrm{D_z1}}{}{\mathrm{dz1}}$ (2.8)

Example 3.

The soldering form is always a Hermitian spinor.  To check this calculate, first define the solder form ${\mathrm{σ}}_{}$, then conjugate $\mathrm{\sigma }$ and interchange the 2nd and 3rd indices.  The result is the original solder form $\mathrm{\sigma }$.

 E > $F≔\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]:$
 E > $\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{\sigma }}{≔}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.9)
 E > $\mathrm{bar_sigma}≔\mathrm{ConjugateSpinor}\left(\mathrm{\sigma }\right)$
 ${\mathrm{bar_sigma}}{≔}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_w1}}{}{\mathrm{D_z2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_w2}}{}{\mathrm{D_z1}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_w1}}{}{\mathrm{D_z2}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_w2}}{}{\mathrm{D_z1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{D_w1}}{}{\mathrm{D_z1}}{-}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{D_w2}}{}{\mathrm{D_z2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_w1}}{}{\mathrm{D_z1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_w2}}{}{\mathrm{D_z2}}$ (2.10)
 E > $\mathrm{\sigma }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{RearrangeIndices}\left(\mathrm{bar_sigma},\left[\left[2,3\right]\right]\right)$
 ${0}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}$ (2.11)

Example 4.

Use the Maple assuming command to simplify the complex conjugate of a spinor-tensor containing a real parameter ${\mathrm{α}}_{}$.

 E > $\mathrm{S5}≔\mathrm{evalDG}\left(\mathrm{\alpha }\mathrm{D_t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w1}+\frac{x}{\mathrm{\alpha }}\mathrm{D_z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w2}\right)$
 ${\mathrm{S5}}{≔}\frac{{x}}{{\mathrm{\alpha }}}{}{\mathrm{D_z}}{}{\mathrm{D_w2}}{+}{\mathrm{\alpha }}{}{\mathrm{D_t}}{}{\mathrm{D_w1}}$ (2.12)
 E > $\mathrm{S6}≔\mathrm{ConjugateSpinor}\left(\mathrm{S5}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\alpha }::\mathrm{real}$
 ${\mathrm{S6}}{≔}\frac{{x}}{{\mathrm{\alpha }}}{}{\mathrm{D_z}}{}{\mathrm{D_z2}}{+}{\mathrm{\alpha }}{}{\mathrm{D_t}}{}{\mathrm{D_z1}}$ (2.13)

Example 6.

In some applications complex coordinates on the base space are used. Suppose, for example, that $z$ and $t$ are real coordinates and that $u$ is a complex coordinate with complex conjugate $v$.

 M > $\mathrm{DGsetup}\left(\left[u,v,z,t\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],N\right)$
 ${\mathrm{frame name: N}}$ (2.14)
 N > $\mathrm{S7}≔\mathrm{evalDG}\left(v\mathrm{D_z1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz2}+tu\mathrm{D_z2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz1}\right)$
 ${\mathrm{S7}}{≔}{v}{}{\mathrm{D_z1}}{}{\mathrm{dz2}}{+}{t}{}{u}{}{\mathrm{D_z2}}{}{\mathrm{dz1}}$ (2.15)

Use the keyword argument conjugatecoordinates to specify that the conjugate of $u$ is $v$ (and the conjugate of $v$ is $u$).

 N > $\mathrm{ConjugateSpinor}\left(\mathrm{S7},\mathrm{conjugatecoordinates}=\left[\left[u,v\right]\right]\right)$
 ${u}{}{\mathrm{D_w1}}{}{\mathrm{dw2}}{+}{t}{}{v}{}{\mathrm{D_w2}}{}{\mathrm{dw1}}$ (2.16)
 N > $\mathrm{S8}≔\mathrm{evalDG}\left(\mathrm{\alpha }\mathrm{D_u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz2}\right)$
 ${\mathrm{S8}}{≔}{\mathrm{\alpha }}{}{\mathrm{D_u}}{}{\mathrm{dz2}}$ (2.17)
 N > $\mathrm{ConjugateSpinor}\left(\mathrm{S8},\mathrm{conjugatecoordinates}=\left[\left[u,v\right]\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\alpha }::\mathrm{real}$
 ${\mathrm{\alpha }}{}{\mathrm{D_v}}{}{\mathrm{dw2}}$ (2.18)
 See Also