Dirac algebra

In <a href="/facts/Mathematical_physics/sLKg8T1G">mathematical physics</a>, the Dirac algebra is the <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a> 
 
 
 
 
 
 Cl
 
 
 1
 ,
 3
 
 
 (
 
 C
 
 )
 
 
 {\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )}
 
. This was introduced by the mathematical physicist <a href="/facts/P._A._M._Dirac/6bt5yRdk">P. A. M. Dirac</a> in 1928 in developing the <a href="/facts/Dirac_equation/DzFaXS4p">Dirac equation</a> for <a href="/facts/Spin-1/2/3I929M4X">spin-⁠1/2⁠</a> particles with a matrix representation of the <a href="/facts/Gamma_matrices/C2y9Aumy">gamma matrices</a>, which represent the generators of the algebra.
The gamma matrices are a set of four 
 
 
 
 4
 ×
 4
 
 
 {\displaystyle 4\times 4}
 
 matrices 
 
 
 
 {
 
 γ
 
 μ
 
 
 }
 =
 {
 
 γ
 
 0
 
 
 ,
 
 γ
 
 1
 
 
 ,
 
 γ
 
 2
 
 
 ,
 
 γ
 
 3
 
 
 }
 
 
 {\displaystyle \{\gamma ^{\mu }\}=\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\}}
 
 with entries in 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, that is, elements of 
 
 
 
 
 
 Mat
 
 
 4
 ×
 4
 
 
 (
 
 C
 
 )
 
 
 {\displaystyle {\text{Mat}}_{4\times 4}(\mathbb {C} )}
 
 that satisfy

{
          
            γ
            
              μ
            
          
          ,
          
            γ
            
              ν
            
          
          }
          =
          
            γ
            
              μ
            
          
          
            γ
            
              ν
            
          
          +
          
            γ
            
              ν
            
          
          
            γ
            
              μ
            
          
          =
          2
          
            η
            
              μ
              ν
            
          
          ,
        
      
    
    {\displaystyle \displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu },}

where by convention, an identity matrix has been suppressed on the right-hand side. The numbers 
 
 
 
 
 η
 
 μ
 ν
 
 
 
 
 
 {\displaystyle \eta ^{\mu \nu }\,}
 
 are the components of the <a href="/facts/Minkowski_metric/vDkfkQJn">Minkowski metric</a>. 
For this article we fix the signature to be mostly minus, that is, 
 
 
 
 (
 +
 ,
 −
 ,
 −
 ,
 −
 )
 
 
 {\displaystyle (+,-,-,-)}
 
. 
The Dirac algebra is then the linear span of the identity, the gamma matrices 
 
 
 
 
 γ
 
 μ
 
 
 
 
 {\displaystyle \gamma ^{\mu }}
 
 as well as any linearly independent products of the gamma matrices. This forms a finite-dimensional <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> over the field 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 or 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, with dimension 
 
 
 
 16
 =
 
 2
 
 4
 
 
 
 
 {\displaystyle 16=2^{4}}
 
.

Dirac algebra open-in-new

Dirac algebra