DifferentialGeometry/Tensor/BivectorSolderForm

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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 $\overline{S}$ which are defined in terms of the solder form s by

$S_{ij}^{AB} = σ_{i}^{AC'} σ_{j C'}^{B} - σ_{j}^{BC'} σ_{i C'}^{A}$

and

$\overline{S}_{ij}^{A' B'} = σ_{i}^{CA'} σ_{jC}^{B'} - σ_{j}^{CB'} σ_{iC}^{A'}$ .

•	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

>	$with (DifferentialGeometry) &colon;$ $with (Tensor) &colon;$

Example 1.

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

>	$DGsetup ([t, x, y, z], [z1, z2, w1, w2], M)$

$frame name: M$

(2.1)

Define a metric g on M. Note that our spinor conventions have the metric with signature $[+ 1, - 1, - 1, - 1] &period;$

>	$g ≔ evalDG (dt &t dt - dx &t dx - dy &t dy - dz &t dz)$

$g := dt dt - dx dx - dy dy - dz dz$

(2.2)

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

>	$F ≔ [D_t, D_x, D_y, D_z]$

$F := [D_t, D_x, D_y, D_z]$

(2.3)

Calculate the solder form sigma from the frame F.

>	$σ ≔ SolderForm (F)$

$σ := \frac{\sqrt{2}}{2} dt D_z1 D_w1 + \frac{\sqrt{2}}{2} dt D_z2 D_w2 + \frac{\sqrt{2}}{2} dx D_z1 D_w2 + \frac{\sqrt{2}}{2} dx D_z2 D_w1 - \frac{I}{2} \sqrt{2} dy D_z1 D_w2 + \frac{I}{2} \sqrt{2} dy D_z2 D_w1 + \frac{\sqrt{2}}{2} dz D_z1 D_w1 - \frac{\sqrt{2}}{2} dz D_z2 D_w2$

(2.4)

Calculate the bivector solder form S from sigma.

>	$S ≔ BivectorSolderForm (σ, spinor)$

$S := dt dx D_z1 D_z1 - dt dx D_z2 D_z2 - I dt dy D_z1 D_z1 - I dt dy D_z2 D_z2 - dt dz D_z1 D_z2 - dt dz D_z2 D_z1 - dx dt D_z1 D_z1 + dx dt D_z2 D_z2 - I dx dy D_z1 D_z2 - I dx dy D_z2 D_z1 - dx dz D_z1 D_z1 - dx dz D_z2 D_z2 + I dy dt D_z1 D_z1 + I dy dt D_z2 D_z2 + I dy dx D_z1 D_z2 + I dy dx D_z2 D_z1 + I dy dz D_z1 D_z1 - I dy dz D_z2 D_z2 + dz dt D_z1 D_z2 + dz dt D_z2 D_z1 + dz dx D_z1 D_z1 + dz dx D_z2 D_z2 - I dz dy D_z1 D_z1 + I dz dy D_z2 D_z2$

(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.

$S_{ij}^{AB} S_{CD}^{ij} = 4 (δ_{C}^{A} δ_{D}^{B} + δ_{C}^{B} δ_{D}^{A}) &period;$

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

>	$S1 ≔ BivectorSolderForm (σ, spinor)$

$S1 := dt dx D_z1 D_z1 - dt dx D_z2 D_z2 - I dt dy D_z1 D_z1 - I dt dy D_z2 D_z2 - dt dz D_z1 D_z2 - dt dz D_z2 D_z1 - dx dt D_z1 D_z1 + dx dt D_z2 D_z2 - I dx dy D_z1 D_z2 - I dx dy D_z2 D_z1 - dx dz D_z1 D_z1 - dx dz D_z2 D_z2 + I dy dt D_z1 D_z1 + I dy dt D_z2 D_z2 + I dy dx D_z1 D_z2 + I dy dx D_z2 D_z1 + I dy dz D_z1 D_z1 - I dy dz D_z2 D_z2 + dz dt D_z1 D_z2 + dz dt D_z2 D_z1 + dz dx D_z1 D_z1 + dz dx D_z2 D_z2 - I dz dy D_z1 D_z1 + I dz dy D_z2 D_z2$

(2.6)

>	$S2 ≔ BivectorSolderForm (σ, spinor, indextype = [con, con, cov, cov])$

$S2 := D_t D_x dz1 dz1 - D_t D_x dz2 dz2 + I D_t D_y dz1 dz1 + I D_t D_y dz2 dz2 - D_t D_z dz1 dz2 - D_t D_z dz2 dz1 - D_x D_t dz1 dz1 + D_x D_t dz2 dz2 + I D_x D_y dz1 dz2 + I D_x D_y dz2 dz1 - D_x D_z dz1 dz1 - D_x D_z dz2 dz2 - I D_y D_t dz1 dz1 - I D_y D_t dz2 dz2 - I D_y D_x dz1 dz2 - I D_y D_x dz2 dz1 - I D_y D_z dz1 dz1 + I D_y D_z dz2 dz2 + D_z D_t dz1 dz2 + D_z D_t dz2 dz1 + D_z D_x dz1 dz1 + D_z D_x dz2 dz2 + I D_z D_y dz1 dz1 - I D_z D_y dz2 dz2$

(2.7)

>	$LHS ≔ ContractIndices (S1, S2, [[1, 1], [2, 2]])$

$LHS := 8 D_z1 D_z1 dz1 dz1 + 4 D_z1 D_z2 dz1 dz2 + 4 D_z1 D_z2 dz2 dz1 + 4 D_z2 D_z1 dz1 dz2 + 4 D_z2 D_z1 dz2 dz1 + 8 D_z2 D_z2 dz2 dz2$

(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).

>	$δ ≔ KroneckerDeltaSpinor (spinor)$

$δ := D_z1 dz1 + D_z2 dz2$

(2.9)

>	$E ≔ RearrangeIndices (δ &t δ, [[2, 3]])$

$E := D_z1 D_z1 dz1 dz1 + D_z1 D_z2 dz1 dz2 + D_z2 D_z1 dz2 dz1 + D_z2 D_z2 dz2 dz2$

(2.10)

>	$RHS ≔ 8 &mult SymmetrizeIndices (E, [1, 2], Symmetric)$

$RHS := 8 D_z1 D_z1 dz1 dz1 + 4 D_z1 D_z2 dz1 dz2 + 4 D_z1 D_z2 dz2 dz1 + 4 D_z2 D_z1 dz1 dz2 + 4 D_z2 D_z1 dz2 dz1 + 8 D_z2 D_z2 dz2 dz2$

(2.11)

Check that the LHS and RHS are the same.

>	$LHS &minus RHS$

$0 D_z1 D_z1 dz1 dz1$

(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

$S_{ij}^{AB} S_{hkAB}^{} = 2 (g_{ih} g_{jk} - g_{jh} g_{ik} - i ε_{ijhk})$ .

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

>	$S3 ≔ BivectorSolderForm (σ, spinor, indextype = [cov, cov, cov, cov])$

$S3 := - dt dx dz1 dz1 + dt dx dz2 dz2 - I dt dy dz1 dz1 - I dt dy dz2 dz2 + dt dz dz1 dz2 + dt dz dz2 dz1 + dx dt dz1 dz1 - dx dt dz2 dz2 + I dx dy dz1 dz2 + I dx dy dz2 dz1 - dx dz dz1 dz1 - dx dz dz2 dz2 + I dy dt dz1 dz1 + I dy dt dz2 dz2 - I dy dx dz1 dz2 - I dy dx dz2 dz1 - I dy dz dz1 dz1 + I dy dz dz2 dz2 - dz dt dz1 dz2 - dz dt dz2 dz1 + dz dx dz1 dz1 + dz dx dz2 dz2 + I dz dy dz1 dz1 - I dz dy dz2 dz2$

(2.13)

>	$LHS ≔ ContractIndices (S1, S3, [[3, 3], [4, 4]])$

$LHS := - 2 dz dt dz dt - 2 I dt dy dz dx - 2 I dy dz dt dx + 2 I dt dz dy dx + 2 I dt dy dx dz - 2 dt dx dt dx + 2 dt dy dy dt + 2 dz dy dz dy + 2 I dy dz dx dt - 2 dt dy dt dy - 2 I dz dt dy dx + 2 I dx dt dy dz - 2 dx dt dx dt - 2 dz dy dy dz - 2 I dy dx dz dt - 2 I dx dy dt dz + 2 dx dy dx dy + 2 dz dx dz dx - 2 dy dz dz dy + 2 I dy dx dt dz + 2 I dy dt dz dx - 2 dy dt dy dt - 2 I dx dt dz dy + 2 I dz dx dy dt - 2 I dy dt dx dz + 2 dt dz dz dt - 2 dz dx dx dz + 2 dx dt dt dx + 2 dy dt dt dy + 2 dt dx dx dt + 2 I dz dt dx dy - 2 dx dz dz dx - 2 I dx dz dy dt + 2 dx dz dx dz + 2 dz dt dt dz - 2 dy dx dx dy + 2 dy dx dy dx + 2 I dz dy dt dx - 2 I dt dx dy dz + 2 dy dz dy dz - 2 I dz dy dx dt + 2 I dx dz dt dy - 2 dx dy dy dx + 2 I dt dx dz dy + 2 I dx dy dz dt - 2 I dz dx dt dy - 2 I dt dz dx dy - 2 dt dz dt dz$

(2.14)

To calculate the right-hand side we first construct the tensor product of the metric tensor with itself.

>	$G ≔ g &tensor g$

$G := dt dt dt dt - dt dt dx dx - dt dt dy dy - dt dt dz dz - dx dx dt dt + dx dx dx dx + dx dx dy dy + dx dx dz dz - dy dy dt dt + dy dy dx dx + dy dy dy dy + dy dy dz dz - dz dz dt dt + dz dz dx dx + dz dz dy dy + dz dz dz dz$

(2.15)

We re-arrange the indices of G to obtain the first two terms on the right-hand side.

>	$RHS1 ≔ RearrangeIndices (G, [[2, 3]])$

$RHS1 := dt dt dt dt - dt dx dt dx - dt dy dt dy - dt dz dt dz - dx dt dx dt + dx dx dx dx + dx dy dx dy + dx dz dx dz - dy dt dy dt + dy dx dy dx + dy dy dy dy + dy dz dy dz - dz dt dz dt + dz dx dz dx + dz dy dz dy + dz dz dz dz$

(2.16)

>	$RHS2 ≔ RearrangeIndices (RHS1, [[1, 2]])$

$RHS2 := dt dt dt dt - dx dt dt dx - dy dt dt dy - dz dt dt dz - dt dx dx dt + dx dx dx dx + dy dx dx dy + dz dx dx dz - dt dy dy dt + dx dy dy dx + dy dy dy dy + dz dy dy dz - dt dz dz dt + dx dz dz dx + dy dz dz dy + dz dz dz dz$

(2.17)

We construct the epsilon tensor using the commands MetricDensity and PermutationSymbol.

>	$E ≔ MetricDensity (g, 1, detmetric = - 1) &tensor PermutationSymbol (cov_bas)$

$E := dt dx dy dz - dt dx dz dy - dt dy dx dz + dt dy dz dx + dt dz dx dy - dt dz dy dx - dx dt dy dz + dx dt dz dy + dx dy dt dz - dx dy dz dt - dx dz dt dy + dx dz dy dt + dy dt dx dz - dy dt dz dx - dy dx dt dz + dy dx dz dt + dy dz dt dx - dy dz dx dt - dz dt dx dy + dz dt dy dx + dz dx dt dy - dz dx dy dt - dz dy dt dx + dz dy dx dt$

(2.18)

Evaluate the right-hand side of the identity and check that it agrees with the left-hand side.

>	$RHS ≔ evalDG (2 (RHS1 - RHS2 - I E))$

$RHS := 2 I dy dz dx dt + 2 dz dy dz dy + 2 I dt dx dz dy + 2 dx dz dx dz - 2 I dy dz dt dx + 2 I dx dy dz dt - 2 dt dy dt dy - 2 dx dz dz dx - 2 dx dy dy dx + 2 I dx dz dt dy - 2 dz dy dy dz - 2 dy dx dx dy - 2 I dz dy dx dt - 2 dy dt dy dt - 2 dx dt dx dt - 2 I dt dy dz dx + 2 dz dx dz dx + 2 I dz dx dy dt - 2 dz dx dx dz - 2 I dz dx dt dy - 2 dz dt dz dt + 2 dy dt dt dy - 2 I dz dt dy dx + 2 I dz dt dx dy + 2 dz dt dt dz + 2 dt dz dz dt + 2 dt dx dx dt - 2 I dy dx dz dt + 2 I dx dt dy dz - 2 I dy dt dx dz - 2 I dx dt dz dy - 2 I dx dy dt dz + 2 I dt dy dx dz - 2 dy dz dz dy - 2 I dt dz dx dy + 2 dy dx dy dx - 2 I dt dx dy dz - 2 I dx dz dy dt + 2 I dy dt dz dx + 2 dx dt dt dx + 2 dx dy dx dy - 2 dt dz dt dz + 2 dy dz dy dz + 2 I dz dy dt dx + 2 dt dy dy dt - 2 dt dx dt dx + 2 I dy dx dt dz + 2 I dt dz dy dx$

(2.19)

>	$LHS &minus RHS$

$0 dt dt dt dt$

(2.20)

Example 4.

The bivector solder form is anti-self-dual, that is,

$S_{ij}^{AB} = - \frac{i}{2} ε_{ijhk} S^{hkAB} &period;$

We check this identity using the solder form from Example 1. The left-hand side is just the solder form S1 from Example 1.

>	$LHS ≔ S1$

$LHS := dt dx D_z1 D_z1 - dt dx D_z2 D_z2 - I dt dy D_z1 D_z1 - I dt dy D_z2 D_z2 - dt dz D_z1 D_z2 - dt dz D_z2 D_z1 - dx dt D_z1 D_z1 + dx dt D_z2 D_z2 - I dx dy D_z1 D_z2 - I dx dy D_z2 D_z1 - dx dz D_z1 D_z1 - dx dz D_z2 D_z2 + I dy dt D_z1 D_z1 + I dy dt D_z2 D_z2 + I dy dx D_z1 D_z2 + I dy dx D_z2 D_z1 + I dy dz D_z1 D_z1 - I dy dz D_z2 D_z2 + dz dt D_z1 D_z2 + dz dt D_z2 D_z1 + dz dx D_z1 D_z1 + dz dx D_z2 D_z2 - I dz dy D_z1 D_z1 + I dz dy D_z2 D_z2$

(2.21)

To evaluate the right-hand side we begin with the contravariant form of the bivector solder form.

>	$S4 ≔ BivectorSolderForm (σ, spinor, indextype = [con, con, con, con])$

$S4 := - D_t D_x D_z1 D_z1 + D_t D_x D_z2 D_z2 + I D_t D_y D_z1 D_z1 + I D_t D_y D_z2 D_z2 + D_t D_z D_z1 D_z2 + D_t D_z D_z2 D_z1 + D_x D_t D_z1 D_z1 - D_x D_t D_z2 D_z2 - I D_x D_y D_z1 D_z2 - I D_x D_y D_z2 D_z1 - D_x D_z D_z1 D_z1 - D_x D_z D_z2 D_z2 - I D_y D_t D_z1 D_z1 - I D_y D_t D_z2 D_z2 + I D_y D_x D_z1 D_z2 + I D_y D_x D_z2 D_z1 + I D_y D_z D_z1 D_z1 - I D_y D_z D_z2 D_z2 - D_z D_t D_z1 D_z2 - D_z D_t D_z2 D_z1 + D_z D_x D_z1 D_z1 + D_z D_x D_z2 D_z2 - I D_z D_y D_z1 D_z1 + I D_z D_y D_z2 D_z2$

(2.22)

Construct the epsilon tensor and contract with S4 and to obtain the left-hand side.

>	$E ≔ MetricDensity (g, 1, detmetric = - 1) &tensor PermutationSymbol (cov_bas)$

(2.23)

>	$RHS ≔ (- \frac{I}{2}) &mult ContractIndices (E, S4, [[3, 1], [4, 2]])$

$RHS := dt dx D_z1 D_z1 - dt dx D_z2 D_z2 - I dt dy D_z1 D_z1 - I dt dy D_z2 D_z2 - dt dz D_z1 D_z2 - dt dz D_z2 D_z1 - dx dt D_z1 D_z1 + dx dt D_z2 D_z2 - I dx dy D_z1 D_z2 - I dx dy D_z2 D_z1 - dx dz D_z1 D_z1 - dx dz D_z2 D_z2 + I dy dt D_z1 D_z1 + I dy dt D_z2 D_z2 + I dy dx D_z1 D_z2 + I dy dx D_z2 D_z1 + I dy dz D_z1 D_z1 - I dy dz D_z2 D_z2 + dz dt D_z1 D_z2 + dz dt D_z2 D_z1 + dz dx D_z1 D_z1 + dz dx D_z2 D_z2 - I dz dy D_z1 D_z1 + I dz dy D_z2 D_z2$

(2.24)

>	$LHS &minus RHS$

$0 dt dt D_z1 D_z1$

(2.25)

	See Also
	DifferentialGeometry, Tensor, ContractIndices, KroneckerDeltaSpinor, MetricDensity, PermutationSymbol, SolderForm, SymmetrizeIndices, RearrangeIndices

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