The Exact Theory of the Stern–Gerlach Experiment and Why It Does Not Imply That a Fermion Can Only Have Its Spin Up or Down

The Stern&ndash
Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern&ndash
Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles &theta
between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2&plusmn
&mu
B. The values &plusmn
B thus do not correspond to the continuum of values &minus
&middot
B Einstein and Ehrenfest had conjectured. The energy term V=&minus
B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies &plusmn
B, and as such not valid for individual fermions. The two fermion states with energy &plusmn
B are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.