Gauge invariance and the electron-electron interaction in a normal metal

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Abstract
The gauge invariance of the theory of an electron gas interacting via an electromagnetic field is examined. We show that the earlier results concerning unusual behavior due to the interaction of electrons with transverse photons are gauge invariant. To study the gauge invariance of the conventional Fermi-liquid results due to the interaction of electrons via the scalar potential, we consider the longitudinal gauge in which the scalar potential is absent (φ=0) and instead of the Coulomb interaction we have the interaction of electrons with the longitudinal photons. We show that the electron self-energy and the density of states are gauge dependent, while the thermodynamic potential and the energy-relaxation time are gauge invariant. In general, quantities that can be expressed as closed loops in the perturbation theory are gauge invariant. We study the effect of the electron-electron interaction in the longitudinal gauge on the electron density of states in an impure metal. It is shown that in the three-dimensional case the result coincides with the corresponding calculation for the Coulomb interaction, but in the two-dimensional case there is a difference in the logarithmic factor between the results of calculations in the different gauges.