Working with Dirac Gamma Matrices using FeynCalc - A simple problem

Question

I need to obtain this using package FeynCalc: $$ \begin{align} [\gamma_{0},\gamma_{i}]=& 2 \gamma_{0}\gamma_{i}, \\ [\gamma_{i},\gamma_{0}]=& 2 \gamma_{i}\gamma_{0} , \\ [\gamma_{0},\gamma_{0}]=& 0, \\ \gamma_{0}.\gamma_{i}.\gamma_{0}.\gamma_{i} =& 1 , \\ \gamma_{0}.\gamma_{i}.\gamma_{i}.\gamma_{0} =& -1. \end{align}$$.

My gamma matrices are:

$\gamma_{0}= \lbrace \lbrace 0, \ \ \mathbb{I}_{2\times 2} \rbrace,\lbrace \mathbb{I}_{2\times 2}, \ \ 0 \rbrace \rbrace $

$\gamma_{i}= \lbrace \lbrace 0, \ \ -\sigma^{i} \rbrace,\lbrace \sigma^{i}, \ \ 0 \rbrace \rbrace $

and

$\sigma_{1}=\lbrace \lbrace 0, \ \ 1 \rbrace,\lbrace 1, \ \ 0 \rbrace \rbrace $

$\sigma_{2}=\lbrace \lbrace 0, \ \ -\mathrm i \rbrace,\lbrace \mathrm i, \ \ 0 \rbrace \rbrace $

$\sigma_{3}=\lbrace \lbrace 1, \ \ 0 \rbrace,\lbrace 0, \ \ -1 \rbrace \rbrace $

I'm working in D-4 space.

Answer 1

If you just want to verify some identities of Dirac gamma matrices in some certain representations. FeynCalc is not a necessity. Actually your choice of Dirac matrices is just the Weyl (chiral) representation:

$$ \gamma^\mu = \left( \begin{array}{ll} 0 & \sigma^\mu \\ \overline{\sigma}^\mu & 0 \end{array}\right), $$

which specifically are

$$ \begin{align} \gamma^0 &= \ \tau_1\otimes\sigma^0, \\ \gamma^i &= \mathrm i \tau_2\otimes\sigma^i. \end{align} $$

So according to the above equations, we can define the Dirac gamma matrices ourselves in the Wolfram language:

Clear[γ]
SetAttributes[γ, Listable]
γ[μ_] := If[μ == 0, KroneckerProduct @@ PauliMatrix[{1, μ}], 
                  I KroneckerProduct @@ PauliMatrix[{2, μ}]
           ]

Then that codes below return True means your first three identities are verified:

With[{commutator = #.#2 - #2.# &, γList = γ@Range[0, 3]},
     2 TensorProduct[DiagonalMatrix[{1, -1, -1, -1}], γ[0].γ[0]] + Outer[commutator, γList, γList, 1] == 
     2 Outer[Dot, γList, γList, 1]
    ]

True

As for the last two, it is easy to see:

Table[Dot @@ γ[{0, i, 0, i}], {i, 3}] == -Table[Dot @@ γ[{0, i, i, 0}], {i, 3}] == ConstantArray[IdentityMatrix[4], 3]

True

---

Bonus

To verify what speaks of the Clifford algebra, also the defining features of Dirac gamma matrices

$$ \{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}I_4, $$

one can compare the outcomes of the two pieces of codes:

With[{anticommutator = #.#2 + #2.# &, γList = γ[Range[0, 3]]},
     Outer[anticommutator, γList, γList, 1] // MatrixForm
    ]

2 DiagonalMatrix[{1, -1, -1, -1}]~TensorProduct~IdentityMatrix[4] // MatrixForm