Weyl Transform and the Magnetic Susceptibility of a Relativistic Dirac Electron Gas

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Abstract
The Weyl transform in relativistic quantum dynamics is formulated in terms of the coordinate eigenfunction and momentum eigenfunction that correspond to the ordinary and magnetic Wannier function and ordinary and magnetic Bloch function, respectively, in solid-state theory. Using this method, a rigorous but fairly simple dynamical derivation of the magnetic susceptibility of a relativistic Dirac electron gas (which includes the effect of an anomalous magnetic moment) is given. The terms coming from the Landau-Peierls formula and Pauli spin paramagnetism are cancelled by the terms arising from the second-order effect of spin and by an unfamiliar dynamical effect due to the inherent spread of the electron (approximately equal to its Compton wavelength). The very simple result reduces to the sum of Landau orbital diamagnetism and Pauli spin paramagnetism in the nonrelativistic limit. It is suggested that different physical processes dominate in the very-low-electron-density limit as compared to that in the very-high-electron-density limit. Striking similarities with the magnetic properties of the electrons in bismuth crystal are pointed out.

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