FCSetPauliSigmaScheme[scheme] allows you to specify how
Pauli matrices will be handled in D-1
dimensions.
This is mainly related to the commutator of two Pauli matrices, which involves a Levi-Civita tensor. The latter is not a well-defined quantity in D-1 dimensions. Following schemes are supported:
"None" - This is the default value. The
anticommutator relation is not applied to D-1 dimensional Pauli matrices.
"Naive" - Naively apply the commutator relation in
D-1-dimensions, i.e. \{\sigma^i, \sigma^j \} = 2 i \varepsilon^{ijk}
\sigma^k. The Levi-Civita tensor lives in D-1-dimensions, so that a contraction of two
such tensors which have all indices in common yields (D-3) (D-2) (D-1).
Overview, PauliSigma, FCGetPauliSigmaScheme.
FCGetPauliSigmaScheme[]\text{None}
CSID[i, j, k]
PauliSimplify[%, PauliReduce -> True]\sigma ^i.\sigma ^j.\sigma ^k
\sigma ^i.\sigma ^j.\sigma ^k
FCSetPauliSigmaScheme["Naive"];FCGetPauliSigmaScheme[]\text{Naive}
ex = PauliSimplify[CSID[i, j, k], PauliReduce -> True]i \overset{\text{}}{\epsilon }^{ijk}+D \sigma ^i \delta ^{jk}-D \sigma ^j \delta ^{ik}-3 \sigma ^i \delta ^{jk}+3 \sigma ^j \delta ^{ik}+\sigma ^k \delta ^{ij}
ex // FCE // StandardForm
(*I CLCD[i, j, k] + CSID[k] KDD[i, j] + 3 CSID[j] KDD[i, k] - D CSID[j] KDD[i, k] - 3 CSID[i] KDD[j, k] + D CSID[i] KDD[j, k]*)FCSetPauliSigmaScheme["None"];