Dirac matrices
Four Hermitian matrices 
, 
, and 
of dimension 
which satisfy the following conditions
 |
 |
where 
is the unit matrix of dimension 
. The matrices 
may also be replaced by the Hermitian matrices 
, 
and by the anti-Hermitian matrix 
, which satisfy the condition
 |
where 
, 
; 
if 
, 
, which makes it possible to write the Dirac equation in a form which is covariant with respect to the Lorentz group of transformations. The matrices 
, 
and 
are defined up to an arbitrary unitary transformation, and may be represented in various ways. One such representation is
 |
where 
are 
Pauli matrices while 
and 
are the 
unit and zero matrix respectively. Dirac matrices may be used to factorize the Klein–Gordon equation:
 |
 |
where 
is the d'Alembert operator.
Introduced by P. Dirac in 1928 in the derivation of the Dirac equation.
Comments
For references see – of Dirac equation.
Dirac matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirac_matrices&oldid=40015