Example 1.

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

Here are the 4 epsilon spinors one can define:

Define some other manifold N.

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.

Example 2.

The covariant and contravariant forms of the epsilon spinors are inverses of each other.

Contract the first index of P1 with the first index of P2.  The result is the Kronecker delta spinor.

DifferentialGeometry, Tensor, BivectorSolderForm, CanonicalTensors, ChangeFrame,  ContractIndices, KroneckerDelta, Physics[KroneckerDelta], KroneckerDeltaSpinor, PermutationSymbol, Physics[LeviCivita], SolderForm