Gamma matrices Definition and 62 Threads

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 known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(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-½ particles.
In 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





=


(



0


0


0


−
i




0


0


i


0




0


i


0


0




−
i


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}&={\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&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, antihermitian matrices. More compactly,




γ

0


=

σ

3


⊗
I


{\displaystyle \gamma ^{0}=\sigma ^{3}\otimes I}
, 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.
The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3, 0). In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma-matrix to be presented below generate the Clifford algebra.

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