The spinor inner product of two spinors S and T of the same type is calculated by contracting each pair of corresponding spinor indices (one from S and one from T) with the appropriate epsilon spinor.

For example, the inner product of two covariant rank 1 spinors with components S_A and T_B is epsilon^{AB} S_A T_B.  The inner product of two contravariant rank 1 spinors with components S^A and T^B is epsilon_{AB} S^A T^B.  The inner product of two contravariant rank 2 spinors with components S_{AX} and T_{BY} is epsilon^{AB} epsilon^{XY} S_{AX} T_{BY}.

If S and T are odd rank spinors, then SpinorInnerProduct(S, T) = -SpinorInnerProduct(T, S) and therefore SpinorInnerProduct(S, S) = 0.  (Strictly speaking, the spinor inner product is really just a bilinear pairing -- it is not a true inner product because it is not always symmetric in its arguments.)

If S and T are even rank spinors, then SpinorInnerProduct(S, T) = SpinorInnerProduct(T, S).

Unlike TensorInnerProduct, SpinorInnerProduct does not require specification of a metric tensor to perform the contractions.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SpinorInnerProduct(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-SpinorInnerProduct.