<P> where γ μ (\ displaystyle \ scriptstyle \ gamma ^ (\ mu)) are the gamma matrices (known as Dirac matrices) and i is the imaginary unit . A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by i, have the same squaring and commutation properties as the Pauli matrices . What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles . This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness . The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner . First the equation is written in the form of coupled equations for 2 - spinors with the units restored: </P> <Dl> <Dd> ((m c 2 − E + e φ) c σ ⋅ (p − e c A) − c σ ⋅ (p − e c A) (m c 2 + E − e φ)) (ψ + ψ −) = (0 0). (\ displaystyle (\ begin (pmatrix) (mc ^ (2) - E + e \ phi) &c \ sigma \ cdot \ left (\ mathbf (p) - (\ frac (e) (c)) \ mathbf (A) \ right) \ \ - c (\ boldsymbol (\ sigma)) \ cdot \ left (\ mathbf (p) - (\ frac (e) (c)) \ mathbf (A) \ right) & \ left (mc ^ (2) + E-e \ phi \ right) \ end (pmatrix)) (\ begin (pmatrix) \ psi _ (+) \ \ \ psi _ (-) \ end (pmatrix)) = (\ begin (pmatrix) 0 \ \ 0 \ end (pmatrix)).) </Dd> </Dl> <Dd> ((m c 2 − E + e φ) c σ ⋅ (p − e c A) − c σ ⋅ (p − e c A) (m c 2 + E − e φ)) (ψ + ψ −) = (0 0). (\ displaystyle (\ begin (pmatrix) (mc ^ (2) - E + e \ phi) &c \ sigma \ cdot \ left (\ mathbf (p) - (\ frac (e) (c)) \ mathbf (A) \ right) \ \ - c (\ boldsymbol (\ sigma)) \ cdot \ left (\ mathbf (p) - (\ frac (e) (c)) \ mathbf (A) \ right) & \ left (mc ^ (2) + E-e \ phi \ right) \ end (pmatrix)) (\ begin (pmatrix) \ psi _ (+) \ \ \ psi _ (-) \ end (pmatrix)) = (\ begin (pmatrix) 0 \ \ 0 \ end (pmatrix)).) </Dd> <Dl> <Dd> (E − e φ) ψ + − c σ ⋅ (p − e c A) ψ − = m c 2 ψ + − (E − e φ) ψ − + c σ ⋅ (p − e c A) ψ + = m c 2 ψ − (\ displaystyle (\ begin (aligned) (E-e \ phi) \ psi _ (+) - c (\ boldsymbol (\ sigma)) \ cdot \ left (\ mathbf (p) - (\ frac (e) (c)) \ mathbf (A) \ right) \ psi _ (-) & = mc ^ (2) \ psi _ (+) \ \ - (E-e \ phi) \ psi _ (-) + c (\ boldsymbol (\ sigma)) \ cdot \ left (\ mathbf (p) - (\ frac (e) (c)) \ mathbf (A) \ right) \ psi _ (+) & = mc ^ (2) \ psi _ (-) \ end (aligned))) </Dd> </Dl>

Magnetic dipole moments due to orbital and spin motions of an electron