Gamma matrices
matrix representation of a Clifford algebra
In mathematical physics, the gamma matrices,
{
γ
0
,
γ
1
,
γ
2
,
γ
3
}
,
{\displaystyle \ \left\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\right\}\ ,}
also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra
C
l
1
,
3
(
R
)
.
{\displaystyle \ \mathrm {Cl} _{1,3}(\mathbb {R} )~.}
It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin
1
2
{\displaystyle {\tfrac {\ 1\ }{2}}}
particles. Gamma matrices were introduced by Paul Dirac in 1928.
In the Dirac basis of the Dirac representation, the four contravariant gamma matrices are
γ
0
=
(
1
0
0
0
0
1
0
0
0
0
−
1
0
0
0
0
−
1
)
,
γ
1
=
(
0
0
0
1
0
0
1
0
0
−
1
0
0
−
1
0
0
0
)
,
γ
2
=
i
(
0
0
0
−
1
0
0
1
0
0
1
0
0
−
1
0
0
0
)
,
γ
3
=
(
0
0
1
0
0
0
0
−
1
−
1
0
0
0
0
1
0
0
)
.
{\displaystyle {\begin{aligned}\gamma ^{0}\ &=~~{\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},&\gamma ^{1}&={\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{pmatrix}},\\\\\gamma ^{2}&=i\ {\begin{pmatrix}0&0&0&-1\\0&0&1&0\\0&1&0&0\\-1&0&0&0\end{pmatrix}},&\gamma ^{3}&={\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}}~.\end{aligned}}}
γ
0
{\displaystyle \gamma ^{0}}
is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices. More compactly,
γ
0
=
σ
3
⊗
I
2
,
{\displaystyle \ \gamma ^{0}=\sigma ^{3}\otimes I_{2}\ ,}
and
γ
j
=
i
σ
2
⊗
σ
j
,
{\displaystyle \ \gamma ^{j}=i\sigma ^{2}\otimes \sigma ^{j}\ ,}
where
⊗
{\displaystyle \ \otimes \ }
denotes the Kronecker product and the
σ
j
{\displaystyle \ \sigma ^{j}\ }
(for j = 1, 2, 3) denote the Pauli matrices.
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