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

Examples

See Also

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 SA and TB is εABSATB.  The inner product of two contravariant rank 1 spinors SA and TB is εABSATB. The inner product of two contravariant rank 2 spinors with components SAB and TCD is εACεBD SABTCD .

• 

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

withDifferentialGeometry:withTensor:

 

Example 1.

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

DGsetupx,y,z,t,z1,z2,w1,w2,M

frame name: M

(2.1)

 

Define rank 1 spinors S1 and T1 and calculate their inner product.

M > 

S1evalDGaD_z1+bD_z2

S1:=aD_z1+bD_z2

(2.2)
M > 

T1evalDGcD_z1+dD_z2

T1:=cD_z1+dD_z2

(2.3)
M > 

SpinorInnerProductS1,T1

adbc

(2.4)

 

Note the sign change:

M > 

SpinorInnerProductT1,S1

ad+bc

(2.5)

 

The inner product of a rank 1 spinor with itself vanishes

M > 

SpinorInnerProductS1,S1

0

(2.6)

 

Calculate the inner product of S1 and T1 from the definition.

M > 

εEpsilonSpinorcov,spinor

ϵ:=dz1dz2dz2dz1

(2.7)
M > 

U1ContractIndicesε,S1,1,1

U1:=bdz1+adz2

(2.8)
M > 

ContractIndicesU1,T1,1,1

adbc

(2.9)

 

Example 2.

Calculate the inner product of two rank 2 spinors S2 and T2.

M > 

S2evalDGaD_z1&tdw2+bD_z1&tdw1

S2:=bD_z1dw1+aD_z1dw2

(2.10)
M > 

T2evalDGcD_z1&tdw1+dD_z2&tdw2

T2:=cD_z1dw1+dD_z2dw2

(2.11)
M > 

SpinorInnerProductS2,T2

bd

(2.12)

 

Example 3.

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

M > 

S3evalDGD_t&tdw1+D_z&tdw2

S3:=D_zdw2+D_tdw1

(2.13)
M > 

T3evalDGD_y&tdw1+D_x&tdw2

T3:=D_xdw2+D_ydw1

(2.14)
M > 

SpinorInnerProductS3,T3

D_zD_y+D_tD_x

(2.15)

See Also

DifferentialGeometry, Tensor, ContractIndices, EpsilonSpinor, RaiseLowerSpinorIndices, TensorInnerProduct