Quantum Theory of an Electron Gas in Intense Magnetic Fields

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 ∼${10}^{13}$ G) are believed to exist during gravitational collapse and in collapsed astronomical objects. Our results show that the stress tensor is very anisotropic when $\frac{〈{p_z}^2〉}{m^2{c}^2}$ [$p_z=\left(z \mathrm{momentum}\right)=\left(\mathrm{momentum}\mathrm{parallel}\mathrm{to}\mathrm{the}\mathrm{field}\right)$] is smaller than ${\left(\frac{2H}{H_c}\right)}^{\frac{1}{2}}$, where $H$ is the field and $H_c=\frac{m^2{c}^3}{\hslash e}=4.414\mathrm{}\times {10}^{13}$ G. At a temperature of ${10}^9$ °K or a density of ${10}^5$ g/${\mathrm{cm}}^3$, the theory developed here must be used if $H\gtrsim {10}^{12}$ G.