Tensor[RaiseLowerSpinorIndices] - raise or lower a list of spinor indices using epsilon spinors
S - a spinor or spinor-tensor
Indices - a list of integers, referring to the arguments of S
Spinor indices are raised and lowed using the epsilon spinor.
Indices are lowered by contraction with the first index of the covariant epsilon spinor and raised by contraction with the second index of the contravariant epsilon spinor. n terms of components:
The command RaiseLowerSpinorIndices(S, Indices) will raise or lower the indices of the spinor S given by the list Indices.
Unlike the command RaiseLowerIndices for raising and lowering tensor indices, no metric need be specified.
This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RaiseLowerSpinorIndices(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-RaiseLowerSpinorIndices.
First create a vector bundle M with base coordinates x,y,z,t and fiber coordinates z1, z2, w1, w2.
frame name: M
Define a contravariant rank 1 spinor S1 and lower its indices, that is, convert it to a covariant rank 1 spinor T1.
S1 ≔ evalDG⁡a⁢D_z1+b⁢D_z2
T1 ≔ RaiseLowerSpinorIndices⁡S1,1
Define the covariant epsilon spinor ε1 and check that this result coincides with the contraction of ε1 and S1.
ε1 ≔ EpsilonSpinor⁡cov,spinor
Convert T1 back to a contravariant rank 1 spinor, recovering S1.
Define a rank 4 spinor-tensor S2 and raise its 2nd index and lower its 4th index.
S2 ≔ evalDG⁡a⁢D_t &t dz1 &t D_w2 &t D_w1+b⁢D_x &t dz2 &t D_w1 &t D_w2
DifferentialGeometry, Tensor, ContractIndices, EpsilonSpinor, RaiseLowerIndices
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