See also :
With a metrics (+ - - -)
the Klein-Gordon equation is : ([] + m^2) PSI(x) = 0
with [] = d/dt^2 - d/dx^2 - d/dy^2 - d/dz^2
Let gamma^mu (i=0..3) be 4 matrices :
gamma^0 = ONE 0
0 -ONE
gamma^k = 0 sigma^k
-sigma^k 0
ONE = 1 0
0 1
sigma^1 = 0 1
1 0
sigma^2 = 0 -i
i 0
sigma^3 = 1 0
0 -1
Let A/ be (sum over mu = 0,1,2,3 of ) gamma^mu A_mu
The Dirac equations are :
(i d/ - m) PSI(x) = 0
and :
i (d_mu PSIbar) gamma^mu + m PSIbar = PSIbar (i <-d/ + m) = 0
With an electromagnetic field :
i d PSI/dt = (alpha (p - e A) + e PHI + beta m) PSI
i d/dt PSI(r,t) = (-i alpha nabla + beta + e aplha . A(r,t) - e A^0(r,t)) PSI(r,t)
(alpha p + beta (m + U(x)) PSI(x,t) = i d/dt PSI(x,t)
The lagrangian is :
L = PSIbar (i d/ - m) PSI
= - PSIbar (i <-d/ + m) PSI
= 1/2 (i PSIbar <-d/ PSI - i PSIbar d/ PSI) - m PSIbar PSI
= 1/2 (i PSIbar gamma^mu d_mu PSI - i d_mu PSIbar gamma^mu PSI) - m PSIbar PSI
L_DEM = PSIbar (i D/ - m) PSI = PSIbar (i d/ - e A/ - m) PSI
with D_mu = d_mu + i e A_mu
L_QED = -1/4 F_mu_nu F^mu^nu + PSIbar (i D/ - m) PSI