4.3 Clifford Algebras

In 1106.3197 conventions: In Euclidean signature, we can generate $\gamma$ matrices via the Mathematica code:

ClearAll[cliffordGenerators];
cliffordGenerators[1] = {PauliMatrix[1]};
cliffordGenerators[2] = {PauliMatrix[1], PauliMatrix[2]};
cliffordGenerators[3] = {PauliMatrix[1], PauliMatrix[2], PauliMatrix[3]};
cliffordGenerators[4] = Join[
TensorProduct[PauliMatrix[1], #] & /@ cliffordGenerators[3],
{TensorProduct[PauliMatrix[2], IdentityMatrix[2]]},
{TensorProduct[PauliMatrix[3], IdentityMatrix[2]]}
];

cliffordGenerators[n_?EvenQ] /; n > 3 := cliffordGenerators[n] = Join[
(TensorProduct[PauliMatrix[1], #] & /@ cliffordGenerators[n - 2]),
{TensorProduct[PauliMatrix[2], Nest[TensorProduct[IdentityMatrix[2], #] &,
IdentityMatrix[2], n/2 - 2]]}, {TensorProduct[PauliMatrix[3],
Nest[TensorProduct[IdentityMatrix[2], #] &, IdentityMatrix[2], n/2 - 2]]}
];

In the conventions of “Park, Jeong - Hyuck.(2016).Lecture Note on Clifford Algebra : Expanded version” In this convention, gamma matrices can be generated as

ClearAll[\[Gamma]];
\[Gamma][2, 1] = PauliMatrix[1]; \[Gamma][2, 2] = PauliMatrix[2];
\[Gamma][d_?EvenQ, i_] /; i <= d - 2 := ArrayFlatten[TensorProduct[
\[Gamma][d - 2, i], PauliMatrix[1]]];
\[Gamma][d_?EvenQ, i_] /; d - i == 1 := ArrayFlatten[TensorProduct[
IdentityMatrix[2^((d - 2)/2)], PauliMatrix[2]]];
\[Gamma][d_?EvenQ, i_] /; d - i == 0 := ArrayFlatten[TensorProduct[
IdentityMatrix[2^((d - 2)/2)], PauliMatrix[3]]];

ClearAll[\[CapitalGamma]];
\[CapitalGamma][d_?EvenQ] := IdentityMatrix[2^(d/2)];
\[CapitalGamma][d_?EvenQ, j__] /; Apply[And, 1 <= # <= d & /@ {j}] := Block[
{permutations, signatures},{permutations, signatures} = Transpose[{#,
Signature[#]} & /@ Permutations[{j}]];
Signature[{j}]/Length[signatures] signatures.(Apply[Dot,
Table[\[Gamma][d, i], {i, #}]] & /@ permutations)];