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Free electron systems are ubiquitous in nature and have demonstrated intriguing effects in their collective interactions with weak electric and magnetic fields, especially in aqueous environments. Starting from the Dirac Hamiltonian, a fully relativistic expression is derived for the electron energy shift in the presence of a spatiotemporally constant, weak electromagnetic field. The expectation value of this energy shift is then computed explicitly using the Fourier transforms of the fermionic fields. To first order in the electromagnetic fields, the average relativistic energy shift is found to be completely independent of the electron spin-polarization coefficients. This effect is also considerably larger than that predicted in quantum mechanics by the analogous Zeeman shift. Finally, in the non-relativistic limit, it is shown how to discriminate between achiral and completely polarized states, which leads to a concluding discussion of possible mesoscopic and macroscopic manifestations of electron spin states across many orders of magnitude in the physical world, with stark implications for biological and other complex systems.

From Micro to Macro: A Relativistic Treatment of the Chiral Energy Shifts Caused by Static Electromagnetic Effects on Free Electrons