DifferentialGeometry/Tensor/KroneckerDeltaSpinor

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Tensor[KroneckerDeltaSpinor] - create the Kronecker delta spinor

Calling Sequences

KroneckerDeltaSpinor(spinorType, ,fr)

Parameters

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

fr - (optional) the name of a defined frame

Description

•	The Kronecker delta spinor is the type (1,1) spinor whose components in any coordinate system are given by the identity matrix.

•	The command KroneckerDeltaSpinor(spinorType) returns a Kronecker delta spinor of the type specified by 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 KroneckerDeltaSpinor(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KroneckerDeltaSpinor.

Examples

Example 1.

First create a vector bundle M with base coordinates [x, y, z, t] and fiber coordinates [z1, z2, w1, w2].

(2.1)

Here are the 2 Kronecker delta spinors one can define:

M >

(2.2)

M >

(2.3)

Define some other manifold N.

M >

(2.4)

The current frame is N. Because there are no fiber variables, one cannot calculate a Kronecker delta spinor in this frame. To now re-calculate the Kronecker delta spinor KK1, either use the ChangeFrame command or pass KroneckerDeltaSpinor the frame name M as a second argument.

N >

(2.5)

Example 2.

The Kronecker delta spinor defines an identity mapping on spinors of the indicated type. The linear transformation associated to the Kronecker delta spinor K is defined by contracting the covariant index of K against the contravariant index of the spinor S1. We see that the result is S2 = S1 so that the linear transformation defined by K is indeed the identity transformation.