This discussion centers on the mathematical foundations of Bell's theorem, particularly the derivation of inequalities related to quantum mechanics (QM) and local hidden-variable theories. Key references include Bell's original 1964 paper, specifically equations (1) and (14), which outline the spinorial representation of rotations and the mathematical structure underlying the theorem. The conversation emphasizes the importance of understanding the theorem's assumptions without philosophical biases, focusing on the algebraic and geometric aspects of the proof, including the role of probability spaces and random variables.
PREREQUISITESMathematicians, physicists, and students of quantum mechanics seeking a deeper understanding of Bell's theorem and its mathematical underpinnings, as well as those interested in the philosophical implications of quantum theory.
RockyMarciano said:Once again this thread is not about that, it discusses something previous to laying out the abstract probabilistic conditions that a theory fulfilling the EPR experiment must have. What the theorem says is not that all theories satisfying the inequality must be of certain type,
but rather that those that want to replicate all QM predictions (confirmed empirically) must violate them
Again, the goal of this thread is not directly concerned with the probabilities that lead to the Bell inequalities other than for the fact that all the physical theories with the minimal common mathematical description I'm trying to reach consensus about will certainly satisfy the inequalities when using classicall probabilities on them. The advantage is that by identifying the principal features of this physical theories spaces and mathematical objects in them that directly lead to satisfying the inequality(because that is a feature of EPR math models) it would be so much easier to discard from the beguinning a lot of mathematical models for physics that are not compatible with experiment.zonde said:Maybe it's trivial, but think it is good idea to put conditions of Bell theorem into mathematical form before doing the same thing with assumptions. So I propose these conditions:
@RockyMarciano, do these seem fine to you?
- We can produce series of paired events ##A_\alpha## and ##B_\beta## where ##\alpha## and ##\beta## are freely changeable parameters. ##A_\alpha## and ##B_\beta## each can be split into two subevents: ##A_\alpha=\pm 1## and ##B_\beta=\pm 1##.
- For every i-th pair of events ##A_{\alpha i}## and ##B_{\beta i}## in series, ##P(A_{\alpha i}=+1, B_{\beta i}=+1)=0## and ##P(A_{\alpha i}=-1, B_{\beta i}=-1)=0## whenever ##\alpha = \beta##.
RockyMarciano said:Again, the goal of this thread is not directly concerned with the probabilities that lead to the Bell inequalities
And therefore that those that don't satisfy them like the QM predictions or those appparently compatible with nature don't. This is what I'm saying. It is not backwards.stevendaryl said:No, that's backwards. It proves that all theories of a certain type satisfy the inequality.
Sure, look tho the possible source of confusion here above.He proved that all theories of a certain type satisfy a particular inequality. EPR experiments do not satisfy that inequality. Therefore, EPR experiments cannot be explained by a theory of that type.
Of course. Again there might be some trivial misunderstanding here, see the first commnet of this post.There is nothing in Bell's argument that relies on any property of quantum mechanics. The fact that EPR violates his inequality is an empirical matter.
Yes, please clarify these operational requirements of gedanken experiment in mathematical form as it seems you are not satisfied with my attempt in post #29.RockyMarciano said:Wait, first let's clarify what I mean by EPR experiment and EPR mode, I refer to Einstein's so called "local realist" model, and to the gedanken experiment operational requirements and assumptions not to the actual Aspect type experiments outcomes showing the violations.