DifferentialGeometry/Tensor/BivectorSolderForm - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : DifferentialGeometry/Tensor/BivectorSolderForm

Tensor[BivectorSolderForm] - construct the bivector solder form defined by a solder form

Calling Sequences

     BivectorSolderForm(sigma, spinorType, indexlist)

Parameters

   sigma      - a solder form

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

   indexlist  - (optional) the keyword argument indexlist = ind, where ind is a list of 4 index types "con" or "cov"

 

Description

Examples

See Also

Description

• 

A bivector is a skew-symmetric, rank 2 contravariant tensor. On a 4-dimensional manifold with solder form σ there is a 1-1 correspondence between bivectors and symmetric rank 2 spinors.  This correspondence is explicitly furnished by the bivector solder forms S and S which are defined in terms of the solder form s by

SijAB=σiAC 'σj C 'BσjBC 'σi C 'A

    and

SijA'B'=σiCA 'σjC     B'σjCB 'σiC    A'.

    

• 

The tensor indices of the bivector solder forms are raised and lowered with the metric g defined by σ

• 

The keyword argument indexlist = ind allows the user to specify the index structure for the bivector solder form. For example, with indexlist = ["con", "con", "con", "con"], the contravariant form S ijAB is returned.

• 

The bivector soldering forms satisfy a large number of identities, some of which are illustrated in Examples 2 - 4.

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form BivectorSolderForm(...) only after executing the commands with(DifferentialGeometry; with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-BivectorSolderForm.

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

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

DGsetupt,x,y,z,z1,z2,w1,w2,M

frame name: M

(2.1)

 

Define a metric g on M. Note that our spinor conventions have the metric with signature +1, 1, 1, 1.

gevalDGdt &t dtdx &t dxdy &t dydz &t dz

g:=_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,−1,3,3,−1,4,4,−1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,−1,3,3,−1,4,4,−1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,−1,3,3,−1,4,4,−1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,−1,3,3,−1,4,4,−1

(2.2)

 

Define an orthonormal frame on M with respect to the metric g.

FD_t,D_x,D_y,D_z

F:=_DGvector,M,,1,1,_DGvector,M,,1,1,_DGvector,M,,1,1,_DGvector,M,,1,1,_DGvector,M,,2,1,_DGvector,M,,2,1,_DGvector,M,,2,1,_DGvector,M,,2,1,_DGvector,M,,3,1,_DGvector,M,,3,1,_DGvector,M,,3,1,_DGvector,M,,3,1,_DGvector,M,,4,1,_DGvector,M,,4,1,_DGvector,M,,4,1,_DGvector,M,,4,1

(2.3)

 

Calculate the solder form sigma from the frame F.

σSolderFormF

σ:=_DGtensor,M,cov_bas,con_vrt,con_vrt,,1,5,7,22,1,6,8,22,2,5,8,22,2,6,7,22,3,5,8,I22,3,6,7,I22,4,5,7,22,4,6,8,22,_DGtensor,M,cov_bas,con_vrt,con_vrt,,1,5,7,22,1,6,8,22,2,5,8,22,2,6,7,22,3,5,8,I22,3,6,7,I22,4,5,7,22,4,6,8,22,_DGtensor,M,cov_bas,con_vrt,con_vrt,,1,5,7,22,1,6,8,22,2,5,8,22,2,6,7,22,3,5,8,I22,3,6,7,I22,4,5,7,22,4,6,8,22,_DGtensor,M,cov_bas,con_vrt,con_vrt,,1,5,7,22,1,6,8,22,2,5,8,22,2,6,7,22,3,5,8,I22,3,6,7,I22,4,5,7,22,4,6,8,22

(2.4)

 

Calculate the bivector solder form S from sigma.

SBivectorSolderFormσ,spinor

S:=_DGtensor,M,cov_bas,cov_bas,con_vrt,con_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,−I,2,3,6,5,−I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,I,3,2,6,5,I,3,4,5,5,I,3,4,6,6,−I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,−I,4,3,6,6,I,_DGtensor,M,cov_bas,cov_bas,con_vrt,con_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,−I,2,3,6,5,−I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,I,3,2,6,5,I,3,4,5,5,I,3,4,6,6,−I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,−I,4,3,6,6,I,_DGtensor,M,cov_bas,cov_bas,con_vrt,con_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,−I,2,3,6,5,−I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,I,3,2,6,5,I,3,4,5,5,I,3,4,6,6,−I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,−I,4,3,6,6,I,_DGtensor,M,cov_bas,cov_bas,con_vrt,con_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,−I,2,3,6,5,−I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,I,3,2,6,5,I,3,4,5,5,I,3,4,6,6,−I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,−I,4,3,6,6,I

(2.5)

 

Example 2.

The contraction of two bivector solder forms on their tensor indices can be expressed in terms of the Kronecker delta spinor.

 

SijABSCDij=4 δCAδDB+δCBδDA.

 

We check this identity using the solder form from Example 1.  First we calculate the left-hand side.

S1BivectorSolderFormσ,spinor

S1:=_DGtensor,M,cov_bas,cov_bas,con_vrt,con_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,−I,2,3,6,5,−I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,I,3,2,6,5,I,3,4,5,5,I,3,4,6,6,−I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,−I,4,3,6,6,I,_DGtensor,M,cov_bas,cov_bas,con_vrt,con_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,−I,2,3,6,5,−I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,I,3,2,6,5,I,3,4,5,5,I,3,4,6,6,−I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,−I,4,3,6,6,I,_DGtensor,M,cov_bas,cov_bas,con_vrt,con_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,−I,2,3,6,5,−I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,I,3,2,6,5,I,3,4,5,5,I,3,4,6,6,−I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,−I,4,3,6,6,I,_DGtensor,M,cov_bas,cov_bas,con_vrt,con_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,−I,2,3,6,5,−I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,I,3,2,6,5,I,3,4,5,5,I,3,4,6,6,−I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,−I,4,3,6,6,I

(2.6)

S2BivectorSolderFormσ,spinor,indextype=con,con,cov,cov

S2:=_DGtensor,M,con_bas,con_bas,cov_vrt,cov_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,I,1,3,6,6,I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,I,2,3,6,5,I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,−I,3,1,6,6,−I,3,2,5,6,−I,3,2,6,5,−I,3,4,5,5,−I,3,4,6,6,I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,I,4,3,6,6,−I,_DGtensor,M,con_bas,con_bas,cov_vrt,cov_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,I,1,3,6,6,I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,I,2,3,6,5,I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,−I,3,1,6,6,−I,3,2,5,6,−I,3,2,6,5,−I,3,4,5,5,−I,3,4,6,6,I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,I,4,3,6,6,−I,_DGtensor,M,con_bas,con_bas,cov_vrt,cov_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,I,1,3,6,6,I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,I,2,3,6,5,I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,−I,3,1,6,6,−I,3,2,5,6,−I,3,2,6,5,−I,3,4,5,5,−I,3,4,6,6,I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,I,4,3,6,6,−I,_DGtensor,M,con_bas,con_bas,cov_vrt,cov_vrt,,1,2,5,5,1,1,2,6,6,−1,1,3,5,5,I,1,3,6,6,I,1,4,5,6,−1,1,4,6,5,−1,2,1,5,5,−1,2,1,6,6,1,2,3,5,6,I,2,3,6,5,I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,−I,3,1,6,6,−I,3,2,5,6,−I,3,2,6,5,−I,3,4,5,5,−I,3,4,6,6,I,4,1,5,6,1,4,1,6,5,1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,I,4,3,6,6,−I

(2.7)

LHSContractIndicesS1,S2,1,1,2,2

LHS:=_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,8,5,6,5,6,4,5,6,6,5,4,6,5,5,6,4,6,5,6,5,4,6,6,6,6,8,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,8,5,6,5,6,4,5,6,6,5,4,6,5,5,6,4,6,5,6,5,4,6,6,6,6,8,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,8,5,6,5,6,4,5,6,6,5,4,6,5,5,6,4,6,5,6,5,4,6,6,6,6,8,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,8,5,6,5,6,4,5,6,6,5,4,6,5,5,6,4,6,5,6,5,4,6,6,6,6,8

(2.8)

 

To calculate the right-hand side we construct the symmetrized tensor product of 2 Kronecker delta spinors and multiply by 8 (because SymmetrizeIndices will include a factor of 1/2).

δKroneckerDeltaSpinorspinor

δ:=_DGtensor,M,con_vrt,cov_vrt,,5,5,1,6,6,1,_DGtensor,M,con_vrt,cov_vrt,,5,5,1,6,6,1,_DGtensor,M,con_vrt,cov_vrt,,5,5,1,6,6,1,_DGtensor,M,con_vrt,cov_vrt,,5,5,1,6,6,1

(2.9)

ERearrangeIndicesδ &t δ,2,3

E:=_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,1,5,6,5,6,1,6,5,6,5,1,6,6,6,6,1,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,1,5,6,5,6,1,6,5,6,5,1,6,6,6,6,1,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,1,5,6,5,6,1,6,5,6,5,1,6,6,6,6,1,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,1,5,6,5,6,1,6,5,6,5,1,6,6,6,6,1

(2.10)

RHS8 &mult SymmetrizeIndicesE,1,2,Symmetric

RHS:=_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,8,5,6,5,6,4,5,6,6,5,4,6,5,5,6,4,6,5,6,5,4,6,6,6,6,8,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,8,5,6,5,6,4,5,6,6,5,4,6,5,5,6,4,6,5,6,5,4,6,6,6,6,8,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,8,5,6,5,6,4,5,6,6,5,4,6,5,5,6,4,6,5,6,5,4,6,6,6,6,8,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,8,5,6,5,6,4,5,6,6,5,4,6,5,5,6,4,6,5,6,5,4,6,6,6,6,8

(2.11)

 

Check that the LHS and RHS are the same.

LHS &minus RHS

_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,0,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,0,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,0,_DGtensor,M,con_vrt,con_vrt,cov_vrt,cov_vrt,,5,5,5,5,0

(2.12)

 

Example 3.

The contraction of two bivector soldering forms on their tensor indices can be expressed in terms of the metric and the permutation tensor

 

SijABShkAB=2gih gjkgjhgiki εijhk.

 

We check this identity using the solder form from Example 1.  First we calculate the left-hand side.

S3BivectorSolderFormσ,spinor,indextype=cov,cov,cov,cov

S3:=_DGtensor,M,cov_bas,cov_bas,cov_vrt,cov_vrt,,1,2,5,5,−1,1,2,6,6,1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,1,1,4,6,5,1,2,1,5,5,1,2,1,6,6,−1,2,3,5,6,I,2,3,6,5,I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,−I,3,2,6,5,−I,3,4,5,5,−I,3,4,6,6,I,4,1,5,6,−1,4,1,6,5,−1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,I,4,3,6,6,−I,_DGtensor,M,cov_bas,cov_bas,cov_vrt,cov_vrt,,1,2,5,5,−1,1,2,6,6,1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,1,1,4,6,5,1,2,1,5,5,1,2,1,6,6,−1,2,3,5,6,I,2,3,6,5,I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,−I,3,2,6,5,−I,3,4,5,5,−I,3,4,6,6,I,4,1,5,6,−1,4,1,6,5,−1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,I,4,3,6,6,−I,_DGtensor,M,cov_bas,cov_bas,cov_vrt,cov_vrt,,1,2,5,5,−1,1,2,6,6,1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,1,1,4,6,5,1,2,1,5,5,1,2,1,6,6,−1,2,3,5,6,I,2,3,6,5,I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,−I,3,2,6,5,−I,3,4,5,5,−I,3,4,6,6,I,4,1,5,6,−1,4,1,6,5,−1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,I,4,3,6,6,−I,_DGtensor,M,cov_bas,cov_bas,cov_vrt,cov_vrt,,1,2,5,5,−1,1,2,6,6,1,1,3,5,5,−I,1,3,6,6,−I,1,4,5,6,1,1,4,6,5,1,2,1,5,5,1,2,1,6,6,−1,2,3,5,6,I,2,3,6,5,I,2,4,5,5,−1,2,4,6,6,−1,3,1,5,5,I,3,1,6,6,I,3,2,5,6,−I,3,2,6,5,−I,3,4,5,5,−I,3,4,6,6,I,4,1,5,6,−1,4,1,6,5,−1,4,2,5,5,1,4,2,6,6,1,4,3,5,5,I,4,3,6,6,−I

(2.13)

LHSContractIndicesS1,S3,3,3,4,4

LHS:=_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,2,1,2,−2,1,2,2,1,2,1,2,3,4,2I,1,2,4,3,2I,1,3,1,3,−2,1,3,2,4,2I,1,3,3,1,2,1,3,4,2,2I,1,4,1,4,−2,1,4,2,3,2I,1,4,3,2,2I,1,4,4,1,2,2,1,1,2,2,2,1,2,1,−2,2,1,3,4,2I,2,1,4,3,2I,2,3,1,4,2I,2,3,2,3,2,2,3,3,2,−2,2,3,4,1,2I,2,4,1,3,2I,2,4,2,4,2,2,4,3,1,2I,2,4,4,2,−2,3,1,1,3,2,3,1,2,4,2I,3,1,3,1,−2,3,1,4,2,2I,3,2,1,4,2I,3,2,2,3,−2,3,2,3,2,2,3,2,4,1,2I,3,4,1,2,2I,3,4,2,1,2I,3,4,3,4,2,3,4,4,3,−2,4,1,1,4,2,4,1,2,3,2I,4,1,3,2,2I,4,1,4,1,−2,4,2,1,3,2I,4,2,2,4,−2,4,2,3,1,2I,4,2,4,2,2,4,3,1,2,2I,4,3,2,1,2I,4,3,3,4,−2,4,3,4,3,2,_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,2,1,2,−2,1,2,2,1,2,1,2,3,4,2I,1,2,4,3,2I,1,3,1,3,−2,1,3,2,4,2I,1,3,3,1,2,1,3,4,2,2I,1,4,1,4,−2,1,4,2,3,2I,1,4,3,2,2I,1,4,4,1,2,2,1,1,2,2,2,1,2,1,−2,2,1,3,4,2I,2,1,4,3,2I,2,3,1,4,2I,2,3,2,3,2,2,3,3,2,−2,2,3,4,1,2I,2,4,1