Quantum Theory of an Electron Gas in Intense Magnetic Fields

Abstract

In this paper we have developed the quantum theory of a relativistic electron gas in magnetic fields of arbitrary strength, and we have obtained the equation of state for such a gas. Intense magnetic fields (of field strengths \ensuremath{\sim}${10}^{13}$ G) are believed to exist during gravitational collapse and in collapsed astronomical objects. Our results show that the stress tensor is very anisotropic when $\frac{〈{{p}_{z}}^{2}〉}{{m}^{2}{c}^{2}}$ [${p}_{z}=(z \mathrm{momentum})=(\mathrm{momentum}\mathrm{parallel}\mathrm{to}\mathrm{the}\mathrm{field})$] is smaller than ${(\frac{2H}{{H}_{c}})}^{\frac{1}{2}}$, where $H$ is the field and ${H}_{c}=\frac{{m}^{2}{c}^{3}}{\ensuremath{\hbar}e}=4.414\ifmmode\times\else\texttimes\fi{}{10}^{13}$ G. At a temperature of ${10}^{9}$ \ifmmode^\circ\else\textdegree\fi{}K or a density of ${10}^{5}$ g/${\mathrm{cm}}^{3}$, the theory developed here must be used if $H\ensuremath{\gtrsim}{10}^{12}$ G.