Hard thermal loops in a magnetic field and the chiral anomaly

The fermionic dispersion relation in the presence of a background magnetic field and a high temperature QED plasma is calculated exactly in the external field, using the Hard Thermal Loop effective action. As the field strength increases there is a smooth transition from the weak-field (qB much less than q(2)T(2)) thermal dispersion relations to the vacuum Landau levels when the background field is much stronger than any thermal effects (qB much greater than q(2)T(2)). The self-energy at finite field strength acquires an imaginary part. The spectral width becomes important for critical field strength (qB similar to q(2)T(2)), necessitating the use of the full spectral function. It is shown that the spectral function satisfies the usual condition of normalization and causality. Using the exact spectral function I also show that the production of chirality in an external electromagnetic field at high temperature is unaffected by the presence of the thermal masses of the fermions.