FrediFizzx wrote:gill1109 wrote:FrediFizzx wrote:Hi Folks,
Since
a,
b and
s can be all random, the Stern-Gerlach polarizer action should be random up or down also so let's try these functions,
.
I think we need to know your definition of "limit as +/-
s converges to
a". And why is there a "plus or minus" there?
I would say that
a and
b are chosen by the experimenter. Sometimes the experimenter will choose them by tossing coins. I think that in this formula,
a and
b are just any given directions. I would say that
s is what some mathematicians informally call a ``dummy variable''; it also takes values in the set of spatial directions.
What we really need to know is what is the limit of the pair
, B({\bf b},\lambda)))
. If it exists, it would have to be (-1,-1), or (-1, +1), or (+1, -1), or (+1, +1). And we need all four limits to be possible for different values of lambda.
Do you know how a Stern-Gerlach polarizer works? It splits a beam of randomly polarized spin 1/2 particles into beams of up and down spins. The up and down are the plus and minus on s.
In the model,
a and
b can be chosen at any random angle and even in 3D directions. This is a theory; not an experiment. And the theory still works just fine with completely random settings. Yes, the spin vector for
s can be and usually is, pointing in any 3D direction randomly also.
All four outcome possibilities come out the same as predicted for QM. You can easily calculate that for yourself.
.
Fred, to be honest, I don't know how a Stern-Gerlach device works, and I don't want to know. Or to be more precise: I want to obtain some understanding of how it works through gaining a mathematical understanding of various mathematical models which have been proposed for the Stern-Gerlach experiment.
I am a mathematician. I am not a physicist. I can't make any sense whatever of the symbols in your formulas. They look like mathematics but they break the standard rules and conventions, so I am lost.
Do you know the epsilon-delta definition of a limit? Could you please try to relate your concept of limit to the standard mathematical concept?
https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limitOf course, I can easily do the standard quantum mechanical computations myself. In the standard treatment, there is no limit being taken of a sum of products of two functions A(a, lambda) and B(b, lambda). Bell's theorem says that it is not possible to reproduce the quantum mechanical answers through a computation of the mean value, as lambda is chosen at random from some fixed probability distribution (fixed means: not depending on a or b), of a product A(a, lambda)B(b, lambda) where the two functions A and B take values in the set {-1, +1}. As we discussed elsewhere, Steve Gull sketched a direct proof of this "no-go" theorem using standard results from Fourier analysis which every STEM educated person knows by heart. I further elaborated on Steve's beautiful proof in a short paper on arXiv which is not yet submitted to a journal. I would love to find a co-author to help me finish it.
Incidentally, the epsilon-delta definition of limit was discovered by Augustin Cauchy as part of an attempt to repair the already highly successful Fourier theory, which unfortunately contained some embarrassing anomalies. Riemann's later work on integration theory was part of the same major research programme. Fourier theory had enormous engineering and military applications and mathematics had to be "fixed" so that people could use it with confidence. Later, Lebesgue's work was part of the same programme. Which finally led to Kolmogorovian probability theory as the solution of Hibert's sixth problem
https://en.wikipedia.org/wiki/Hilbert%27s_sixth_problem. It's a pity most physicists stayed stuck in the 19th century regarding their ignorance and even disdain for probability theory and for statistics. It's high time they caught up. Fortunately, modern technology means that most theoretical physics PhDs, and most algebraic geometry PhDs, end up getting jobs in big data, machine learning, AI, and data science ... i.e. in statistics and probability.
). If a limit of the pair exists, it would have to be (-1,-1), or (-1, +1), or (+1, -1), or (+1, +1). And we need all four limits to be possible for different values of lambda. Hence I think we need lambda to take on at least four different values.
