Quantum Mechanics with HV is Local!

[FrediFizzx]

is the same as . Think of it that way if it helps you to connect the math to the way the polarizer works.

But this was the thing you dropped because then you can't ensure that the outcomes are always opposite for equal detector settings on both sides:

You were right that extra randomness. We took that out because it can give AB = +1 instead of AB = -1 when a = b.

Sorry about that. I was wrong again. Do you agree that when a = b that there is no randomness between A and B?
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[Heinera]

Sorry about that. I was wrong again. Do you agree that when a = b that there is no randomness between A and B?
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When a = b there can be no individual randomness in the functions A and B since that would ruin the observation that we should always get opposite outcomes.

[FrediFizzx]

Sorry about that. I was wrong again. Do you agree that when a = b that there is no randomness between A and B?
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When a = b there can be no individual randomness in the functions A and B since that would ruin the observation that we should always get opposite outcomes.

Not exactly. There still can be randomness in A or B but not between A and B. That is because of the two particles always having their spins anti-correlated. If they are not anti-correlated then we don't have a singlet to start with. IOW, if when a = b and if you get +1 for A you have to get -1 for B. And if you get -1 for A you have to get +1 for B. So A and B are locked into the anti-correlation when a = b. No randomness between them. IOW, s_A = -s_B always! Now you can see why I wrote the limits the way I did. And the minus sign in front of the A function was actually pulled out from s_A.
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[FrediFizzx]

Sorry about that. I was wrong again. Do you agree that when a = b that there is no randomness between A and B?
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When a = b there can be no individual randomness in the functions A and B since that would ruin the observation that we should always get opposite outcomes.

Not exactly. There still can be randomness in A or B but not between A and B. That is because of the two particles always having their spins anti-correlated. If they are not anti-correlated then we don't have a singlet to start with. IOW, if when a = b and if you get +1 for A you have to get -1 for B. And if you get -1 for A you have to get +1 for B. So A and B are locked into the anti-correlation when a = b. No randomness between them. IOW, s_A = -s_B always! Now you can see why I wrote the limits the way I did. And the minus sign in front of the A function was actually pulled out from s_A.
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Let me expand on that a bit. So we have the functions,




And we can see that s_A = - s_B as required for anti-correlation. So now let's take the situation for b = a and lambda = +1. Then we have,







So we are cool. AB = -1. Then we can have,







So we are still cool. AB = -1 again. But that is all we are allow to do otherwise we don't have s_A = -s_B and no singlet. And changing lambda to equal -1 just flips the signs on the outcomes.

Granted, we probably need a better way to express that the limit functions are locked to s_A = -s_B. I'm still thinking about how to do that. And of course what is presented here is a typical example. It really depends on the relationship of the spin vector direction of s to the direction of a. But since b = a and s_A is locked to -s_B, any changes in one will change the other one.
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[FrediFizzx]

Here is a pretty cool interactive animation for the Stern-Gerlach polarizer.

https://phet.colorado.edu/sims/stern-ge ... ch_en.html

This verifies my previous post.

[FrediFizzx]

Here is a pretty cool interactive animation for the Stern-Gerlach polarizer.

https://phet.colorado.edu/sims/stern-ge ... ch_en.html

This verifies my previous post.

After playing with that interactive animation a bit I think the limits are better expressed as,




With the caveat that plus on the LHS of the arrow goes to plus on the RHS and minus on LHS goes to minus on the RHS.
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[Joy Christian]

...plus on the LHS of the arrow goes to plus on the RHS and minus on LHS goes to minus on the RHS.

That is the same as the vector s_A goes to the vector a and vector S_B goes to the vector b, without needing the +/- on any of them.

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[FrediFizzx]

...plus on the LHS of the arrow goes to plus on the RHS and minus on LHS goes to minus on the RHS.

That is the same as the vector s_A goes to the vector a and vector S_B goes to the vector b, without needing the +/- on any of them.

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You didn't play with the interactive animation, did you?
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[Joy Christian]

...plus on the LHS of the arrow goes to plus on the RHS and minus on LHS goes to minus on the RHS.

That is the same as the vector s_A goes to the vector a and vector S_B goes to the vector b, without needing the +/- on any of them.

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You didn't play with the interactive animation, did you?
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I did. As you have them, the +/- does not make any difference for vector quantities like s_A and a.

***

[FrediFizzx]

...plus on the LHS of the arrow goes to plus on the RHS and minus on LHS goes to minus on the RHS.

That is the same as the vector s_A goes to the vector a and vector S_B goes to the vector b, without needing the +/- on any of them.

***

You didn't play with the interactive animation, did you?
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I did. As you have them, the +/- does not make any difference for vector quantities like s_A and a.

***

Then what don't you understand about +Z --> +a is not the same as -Z --> -a?
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[Joy Christian]

Then what don't you understand about +Z --> +a is not the same as -Z --> -a?

If both Z and a are vectors, then +Z --> +a is the same as -Z --> -a, and both of them are the same as Z --> a.

***

[FrediFizzx]

Then what don't you understand about +Z --> +a is not the same as -Z --> -a?

If both Z and a are vectors, then +Z --> +a is the same as -Z --> -a, and both of them are the same as Z --> a.

***

What exactly do you mean by "the same". The interactive animation says they are not the same physically. Keeping a fixed at zero degrees, a +Z up goes to the up + detector and a -Z down goes to the down - detector. Maybe we should be using up and down arrows instead of plus and minus?

[Joy Christian]

Then what don't you understand about +Z --> +a is not the same as -Z --> -a?

If both Z and a are vectors, then +Z --> +a is the same as -Z --> -a, and both of them are the same as Z --> a.

***

What exactly do you mean by "the same". The interactive animation says they are not the same physically. Keeping a fixed at zero degrees, a +Z up goes to the up + detector and a -Z down goes to the down - detector. Maybe we should be using up and down arrows instead of plus and minus?

I mean that, mathematically, if both Z and a are vector quantities, then the limit +Z --> +a is the same as the limit -Z --> -a, and both of them are the same as the limit Z --> a.

Mathematically, the +/- signs, as you have them, do not make any difference for the limits. Mathematically, changing +/- signs to up and down arrows will not make any difference either.

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[FrediFizzx]

Then what don't you understand about +Z --> +a is not the same as -Z --> -a?

If both Z and a are vectors, then +Z --> +a is the same as -Z --> -a, and both of them are the same as Z --> a.

***

What exactly do you mean by "the same". The interactive animation says they are not the same physically. Keeping a fixed at zero degrees, a +Z up goes to the up + detector and a -Z down goes to the down - detector. Maybe we should be using up and down arrows instead of plus and minus?

I mean that, mathematically, if both Z and a are vector quantities, then the limit +Z --> +a is the same as the limit -Z --> -a, and both of them are the same as the limit Z --> a.

Mathematically, the +/- signs, as you have them, do not make any difference for the limits. Mathematically, changing +/- signs to up and down arrows will not make any difference either.

***

Well, something is not correct with that because the interactive animation says different; not the same. Up goes to up detector and down goes to down detector. It has to be that a is not really a vector it is just an alignment direction. An angle relative to the lab frame. At least in the limit process. So how do we express that?
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[gill1109]

Then what don't you understand about +Z --> +a is not the same as -Z --> -a?

If both Z and a are vectors, then +Z --> +a is the same as -Z --> -a, and both of them are the same as Z --> a.

Guys, maybe one of you would like to define what you mean by saying +Z converges to +a? In mathematics, we talk about the limit as Z converges to a, of a function of Z. One can replace the "target" a by any free variable you like. Or by an expression defining a function of any number of free variables (including none - then it's a constant). But the bound variable "Z" is just a place-holder, a dummy variable. It appears in the expression whose limit we are taking. In usual mathematics syntax, the limit as x tends to 0 of f(x) is the same as the limit as y tends to 0 of f(y). You might be elsewhere using x and y to stand for different things, but in the notation for a limit, it doesn't matter what symbol you use as long as it is free for use, ie not defined elsewhere as something particular.

What we mathematicians (and computer scientists, too) mean, precisely, is written up nicely on Wikipedia https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit.

I'm not aware of different meanings in physics or in engineering, but I'm eager to learn if you guys have something different in mind.. The water has somewhat been muddied by computer science having typed languages and not typed languages, and there being clever ways with non-typed languages of using the same name for a family of different functions, which one you mean being determined by the types or even the *names* of the arguments.

[Joy Christian]

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There nothing mysterious, problematic, or deep about a vector such as s mapped to a vector such as a during a physical process of detection, represented by the limit s --> a.

Imagine a spin vector s in some direction in a 3D space. Now imagine a different vector a, freely chosen by Alice. During the detection of the spin by Alice, the vector s is rotated from its original direction to the direction a, which can be represented by the limit s --> a. It is our model. We choose to represent the detection process by such a limit. That is all there is to it.

Let us also not forget that a vector has only two properties. A magnitude and a direction. A vector has no other properties in the 3D space. And we are working mostly with unit vectors.

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[FrediFizzx]

What exactly do you mean by "the same". The interactive animation says they are not the same physically. Keeping a fixed at zero degrees, a +Z up goes to the up + detector and a -Z down goes to the down - detector. Maybe we should be using up and down arrows instead of plus and minus?

I mean that, mathematically, if both Z and a are vector quantities, then the limit +Z --> +a is the same as the limit -Z --> -a, and both of them are the same as the limit Z --> a.

Mathematically, the +/- signs, as you have them, do not make any difference for the limits. Mathematically, changing +/- signs to up and down arrows will not make any difference either.

***

Well, something is not correct with that because the interactive animation says different; not the same. Up goes to up detector and down goes to down detector. It has to be that a is not really a vector it is just an alignment direction. An angle relative to the lab frame. At least in the limit process. So how do we express that?
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I am wrong about that. a has to be a vector for the replacement in the function. And I think Joy is right that -Z --> -a is mathematically the same as Z --> a. But to show the correct physics, it simply should be shown as -Z --> -a. However, this is for Z fixed to a at zero degrees. In the interactive animation simulation,

https://phet.colorado.edu/sims/stern-ge ... ch_en.html

you can see how the up-down probability changes with different relative angles between Z and a. At 45 degrees up Z's are about 85 percent up and 15 percent down from the polarizer action. We probably should have something more sophisticated for the limit description of the polarizer action. Not sure that it matters though since Z_A = -Z_B always.
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[gill1109]

There nothing mysterious, problematic, or deep about a vector such as s mapped to a vector such as a during a physical process of detection, represented by the limit s --> a.

Imagine a spin vector s in some direction in a 3D space. Now imagine a different vector a, freely chosen by Alice. During the detection of the spin by Alice, the vector s is rotated from its original direction to the direction a, which can be represented by the limit s --> a. It is our model. We choose to represent the detection process by such a limit. That is all there is to it.

Let us also not forget that a vector has only two properties. A magnitude and a direction. A vector has no other properties in the 3D space. And we are working mostly with unit vectors.

So when you write s converges to a in what looks like more or less conventional mathematics notation you actually mean that a particle bearing the spin s moves to the detector which has been set in the direction a. And you actually imagine the spin of the particle rotating so as to be aligned with the detector.

This is not the same as what a mathematician means when they write "take the limit as the bound variable s converges in the tedious mathematical epsilon-delta sense to the limit a".

No problem. But it does mean that you cannot blindly use familiar calculus results about limits, since your notation does not have the familiar calculus meaning! I think we have reached the nub of the matter. I have been thinking that this was it, for quite a while now. You have deep physics insights (Lucien Hardy confirmed this in our conversation in Växjö) but you apparently are missing some parts of a traditional formal mathematics training.

[Joy Christian]

***
“When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean — neither more nor less.”

“The question is,” said Alice, “whether you can make words mean so many different things.”

“The question is,” said Humpty Dumpty, “which is to be master – – that's all.”

***

[FrediFizzx]

...
So when you write s converges to a in what looks like more or less conventional mathematics notation you actually mean that a particle bearing the spin s moves to the detector which has been set in the direction a. And you actually imagine the spin of the particle rotating so as to be aligned with the detector.

The term is polarizer. The detector just records a click. There is no problem using the limit function as we have done for the polarizer action. I can't think of any other mathematical function that could be used to describe the physics of the polarizer.
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