## Archive for November 10th, 2022

### Proof Of NO LOCAL Hidden Variables From Magnets

November 10, 2022

Abstract: Looked at it the right way, the Stern Gerlach experiment with three consecutive magnets oriented various ways, show that there can’t be LOCAL hidden variables. No need, to exhibit nonlocality, for the precise, but obscure logic of the Bell Inequality. The argument here is less mathematically precise, but more intuitive.

***

Stern-Gerlach Magnets (SGM) reveal an aspect of the Quantum, namely the quantization of (some sort of) angular momentum. SGM launched in 1922 the saga of Quantum Spin (it turns out to be connected to deep pure geometry which had been developed independently by Élie Cartan, a decade earlier). Drive an electron, or (appropriate) atomic beam through a SGM, and one will get two dots, one up, one down. Whatever the axis of the SGM [1]. (The SGM is just a magnetic field with a specific linear direction.)

That means, at the emotional level that, at the smallest scale, spin, the electronic (sort of) angular momentum, or the orbital (sort of) angular momentum, reduce to UP and DOWN. First surprise. (This is actually the case of Spin 1/2, the simplest, such as for an electron; we are trying to learn the most from the simplest case.)

Say the first SGM is vertical (magnetic field along “z axis”) and a second SGM is horizontal (mag field along “x axis”). Call them respectively SGMV and SGMH. So SGMH produces LEFT-RIGHT beams. Put SGMH across the UP beam coming out of SGMV. One could call that beam SGMV (UP). Once goes through SGMH, one will get 50-50 LEFT-RIGHT. No surprise there.

Now say one selects the RIGHT beam coming out SGMH. Call that beam SGMH (UP; RIGHT)… because first the beam went up, then the beam went right.

Naively one would expect, from classical mechanics, that SGMH (UP; RIGHT) to have kept a memory of its initial source as SGMV(UP).

That would be to assume that beam SGMV (UP) and its descendant SGMH (UP;  RIGHT) to have kept some memory, in other words that some that the beams through the first SGMV and then the second SGM to harbor some LOCAL HIDDEN VARIABLES.

But that’s not the case.

Indeed, please run SGMH (UP; RIGHT) into a second SGMV (that is a Stern Gerlach Magnet parallel to the first magnet SGMV… Call that second vertical Stern Gerlach Magnet SGMV2 One gets fifty-fifty UP and DOWN coming out of SGMV2. It is as if the initial Stern Gerlach, SGMV, never happened. (This set-up is presented in Feynman Lectures on Physics III, Chapter 6, Spin 1/2)

So if there were local hidden variables carried after SGMV that is in the beam SGMV (UP), they got somehow completely eradicated by getting into the beam SGMH (RIGHT).

So these proposed local hidden variables do not stay hidden inside the “particle”: an outside agent can erase them…. Thus those putative local hidden variables aren’t really “local” anymore: the environment impacts them, outside of the particle, and drastically so, just as the potential impacts the phase of an electron in the Bohm-Aharonov experiment… non locally.

***

One can rerun the experiment, by using both beams SGMH (RIGHT) and SGMH (RIGHT), mixing them up. Then it turns out that SGMV2 deflects ONLY UP. So simply going through magnet SGMH, WITHOUT selecting a beam (either SGMH(LEFT) or SMGH (RIGHT)) doesn’t do anything: a collapsing of the Quantum space available to the Quantum wave, selecting either left or right space, is what does something.

Conventional Quantum Physics, newish, path integral version, phrases this by saying one can’t say which path has been followed [2] to keep the information SGV UP or SGV DOWN.  Copenhagen Interpretation of Quantum (CIQ) simply says that selecting beam SGMH (RIGHT) is a measurement thus collapses the wave function… SQPR says roughly the same thing.

In any case, this eradication of the influence of SGMH on the “particle” by just keeping open the OTHER beam, which the putative local hidden variable “particle” is by definition NOT taking, is itself a NONLOCAL effect, thus once again demolishing the “LOCAL Hidden Variable” concept. (One could say that one beam is entangled with the other…)

The advantage of this conceptual approach is that it exhibits directly the nonlocality… without hermetic complications [3]. It also shows the interest of a more philosophical rather than purely formalistic approach to physics.

Patrice Ayme
***

[1] Wolfgang Pauli in 1924 was the first to propose a doubling of the number of available electron states due to a two-valued non-classical “hidden rotation“. In 1925, George Uhlenbeck and Samuel Goudsmit suggested the simple physical interpretation for spin of a particle spinning around its own axis… But clearly that doesn’t fit what is observed. Pauli built a mathematical machinery which reflected the observed GSM behavior. It turned out to be a particular case of deep mathematical work from the French mathematician Élie Cartan who was born and initially educated in the small Alpine coal mining village of La Mure, and rose through merit in the republican educational system. It’s a bit like taking the square root of space. I don’t understand it, neither did the extremely famous mathematician Atiyah…

It is easy to be blinded by the math. But actually the math describes an observed physical behavior. Now this behavior may arise from deeper geometrical reason

***

[2] In SQPR, the “particles” are preceded by the linear guiding waves. Blocking some of them triggers “collapse”. By selecting SGMH (RIGHT) one clearly collapses the linear guidance.

***

[3] Stern Gerlach Magnets also directly illustrates Spin, as did in the first few lines above (magnetic field —> two dots!) The Pauli machinery is often how Spin is introduced in Quantum Physics courses, but that, philososophically is confusing the formalism derived from what is observed with the observation itself.

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