Gauge invariance of the Dirac equation

The Dirac equation in its most usual form can be derived by the minimum action principle from the Lagrangian density where γ k ( k =1,2,3,4) are the Dirac 4 × 4 matrices satisfying relations; ψ( x ) is a column matrix with 4 complex elements being functions of x 1 , x 2 , x 3 , x 4 , coordinates of the pseudo-Euclidean space with metric g ik .(