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Tensor[EpsilonSpinor] - create an epsilon spinor
Calling Sequences
EpsilonSpinor(indexType, spinorType, fr)
Parameters
indexType - a string, either "cov" or "con"
spinorType - a string, either "spinor" or "barspinor"
fr - (optional) the name of a defined frame
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Description
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The epsilon spinor is a rank 2 spinor which is fully skew-symmetric and whose component values are 1 or -1.
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The command EpsilonSpinor(indexType, spinorType) returns the epsilon symbol of the type specified by indexType and spinorType in the current frame unless the frame is explicitly specified.
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form EpsilonSpinor(...) only after executing the commands with(DifferentialGeometry); with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-EpsilonSpinor.
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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=5644/file05870/math72.png)
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| (2.1) |
Here are the 4 epsilon spinors one can define:
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M >
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M >
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| (2.3) |
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M >
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| (2.4) |
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M >
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| (2.5) |
Define some other manifold N.
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M >
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![DGsetup([x, y, z, t], N)](/support/helpjp/helpview.aspx?si=5644/file05870/math115.png)
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| (2.6) |
The current frame is N. Because there are no fiber variables, one cannot calculate an epsilon spinor in this frame. To now re-calculate the epsilon spinor P1, either use the ChangeFrame command or pass EpsilonSpinor the frame name M as a third argument.
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N >
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| (2.7) |
Example 2.
The covariant and contravariant forms of the epsilon spinors are inverses of each other.
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![DGsetup([x, y, z, t], [z1, z2, w1, w2], M)](/support/helpjp/helpview.aspx?si=5644/file05870/math150.png)
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| (2.8) |
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M >
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| (2.9) |
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M >
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| (2.10) |
Contract the first index of P1 with the first index of P2. The result is the Kronecker delta spinor.
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M >
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![P5 := ContractIndices(P2, P1, [[1, 1]])](/support/helpjp/helpview.aspx?si=5644/file05870/math185.png)
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| (2.11) |
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M >
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| (2.12) |
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See Also
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DifferentialGeometry, Tensor, BivectorSolderForm, CanonicalTensors, ChangeFrame, ContractIndices, KroneckerDelta, Physics[KroneckerDelta], KroneckerDeltaSpinor, PermutationSymbol, Physics[LeviCivita], SolderForm
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