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