DifferentialGeometry/Tensor/ConjugateSpinor

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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:-GeneralRelativity 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

Example 1.

First create a vector bundle M with base coordinates [x, y, z, t] and fiber coordinates [z1, z2, w1, w2]. For spinor applications, it is tacitly assumed that [z1, z2] are complex coordinates with complex conjugates [w1, w2].

(2.1)

Define spinors S1 and S2 and calculate their complex conjugates.

M >

(2.2)

M >

(2.3)

M >

(2.4)

M >

(2.5)

Example 2.

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

M >

(2.6)

M >

(2.7)

M >

(2.8)

Example 3.

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

M >

sigma := 2^(1/2)*dx*D_z1*D_w2/2+2^(1/2)*dx*D_z2*D_w1/2-((1/2)*I)*2^(1/2)*dy*D_z1*D_w2+((1/2)*I)*2^(1/2)*dy*D_z2*D_w1+2^(1/2)*dz*D_z1*D_w1/2-2^(1/2)*dz*D_z2*D_w2/2+2^(1/2)*dt*D_z1*D_w1/2+2^(1/2)*dt*D_z2*D_w2/2

(2.9)

M >

bar_sigma := 2^(1/2)*dx*D_w1*D_z2/2+2^(1/2)*dx*D_w2*D_z1/2+((1/2)*I)*2^(1/2)*dy*D_w1*D_z2-((1/2)*I)*2^(1/2)*dy*D_w2*D_z1+2^(1/2)*dz*D_w1*D_z1/2-2^(1/2)*dz*D_w2*D_z2/2+2^(1/2)*dt*D_w1*D_z1/2+2^(1/2)*dt*D_w2*D_z2/2

(2.10)

M >

(2.11)

Example 4.

Use the Maple assuming command to simplify the complex conjugate of a spinor-tensor containing a real parameter alpha.

M >

(2.12)

M >

(2.13)

Example 6.

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