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Tensor[RaiseLowerSpinorIndices] - raise or lower a list of spinor indices using epsilon spinors
Calling Sequences
RaiseLowerSpinorIndices(S, Indices)
Parameters
S - a spinor or spinor-tensor
Indices - a list of integers, referring to the arguments of S
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Description
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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.
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The command RaiseLowerSpinorIndices(S, Indices) will raise or lower the indices of the spinor S given by the list Indices.
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Unlike the command RaiseLowerIndices for raising and lowering tensor indices, no metric need be specified.
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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.
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Examples
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Example 1.
First create a vector bundle M with base coordinates [x, y, z, t] and fiber coordinates [z1, z2, w1, w2].
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![DGsetup([x, y, z, t], [z1, z2, w1, w2], M)](/support/helpjp/helpview.aspx?si=5647/file05911/math70.png)
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| (2.1) |
Define a contravariant rank 1 spinor S1 and lower its indices, that is, convert it to a covariant rank 1 spinor.
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| (2.2) |
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![T1 := RaiseLowerSpinorIndices(S1, [1])](/support/helpjp/helpview.aspx?si=5647/file05911/math90.png)
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| (2.3) |
Define the covariant epsilon spinor epsilon1 and check that this result coincides with the contraction of epsilon1 and S1.
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| (2.4) |
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![ContractIndices(epsilon1, S1, [[1, 1]])](/support/helpjp/helpview.aspx?si=5647/file05911/math114.png)
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| (2.5) |
Convert T1 back to a contravariant rank 1 spinor.
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![RaiseLowerSpinorIndices(T1, [1])](/support/helpjp/helpview.aspx?si=5647/file05911/math127.png)
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| (2.6) |
Example 2.
Define a rank 4 spinor-tensor S2 and raise its 2nd index and lower its 4th index.
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| (2.7) |
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![RaiseLowerSpinorIndices(S2, [2, 4])](/support/helpjp/helpview.aspx?si=5647/file05911/math149.png)
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| (2.8) |
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