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The one-loop Fermi spectra (one-particle and collective ones) are found forall momenta in the $T^2$-approximation and their gauge invariance in hot anddense Abelian and non-Abelian theories is studied. It is shown that theone-particle spectrum, if the calculation accuracy is kept strictly, is gaugeinvariant for all momenta and has two branches as the bare one. The collectivespectrum always has four branches which are gauge dependent including alsotheir $|\q|=0$ limit. The exception is the case $m,\mu=0$ for which thisspectrum is gauge invariant for all momenta as well.Comment: 16 pages, latex, no figure

Fermi Spectra and their Gauge Invariance in Hot and Dense Abelian and Non-Abelian theories