Orthonormality of Volkov Solutions and the Suffcient Condition

We present a simple demonstration on the orthonormality of Volkov solutions with emphasizing on the suffcient condition to the orthonormality. Properly aligning the external electromagnetic wave along the third-axis, the Volkov solutions are eigenfunctions of the hermitian momentum ■1, ■2 and the light-cone hamiltonian operators with real eigenvalues, which can lead to a verification of the orthonormality in the context of quantum mechanics when the x3-integration of the external potential is not singularity as severe as δ(0). The hermiticity of the fermion field fourmomentum operators validates the application of the demonstration to the intense field quantum electrodynamic. The proof based on a direct calculation to the inner products of the solutions is recapitulated as well in a general manner without dependence on explicit representation of the Dirac matrices and spinors, which can be conducive to understand the suffcient condition and to the study of the polarized electron production where a convenient representation is selected elaborately to project out the spin-polarization.