Quantum Mechanics with HV is Local!

[FrediFizzx]

...
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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So this example I gave is basically for when s_A is at zero degrees relative to a. But for when b = a the relative angle doesn't matter since s_A = -s_B always. The relative angle can be random and it still works like for the zero angle case.
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[Joy Christian]

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.

I don't need to brag about my "traditional formal mathematics training." It is a matter of documentary evidence, granted to me by at least three highly respected, world-class universities (ask Sir Roger Penrose about it, for example).

But this is not a forum about ad hominem attacks, however subtly executed. I will address your attack on my "mathematics training" with the precise mathematical question at hand:

Have a look at equations (72) and (73) of this paper. Note that I have used the standard calculus rule of “product of limits equal to limits of the product.” The validity of this rule in the context can be verified at once by recognizing that the same quaternion −D(a)L(a, λ)L(b, λ)D(b) results from both the limits in Eq. (72) and the limits in Eq. (73).

Sorry for shattering your dream of having found a flaw in my mathematics, or deficiency in my mathematics training.

Nice try anyway!

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[Heinera]

So this example I gave is basically for when s_A is at zero degrees relative to a. But for when b = a the relative angle doesn't matter since s_A = -s_B always. The relative angle can be random and it still works like for the zero angle case.
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To me it looks like the you mean that the variables s_A and s_B are random variables. Then the lim notation is mathematically superfluous; you could rather just explain that in the paper. The question is now: Are they part of the state of the entangled particles, i.e., are they part of the hidden variable? If not, how do you ensure that " s_A = -s_B always"?

[gill1109]

I will address your attack on my "mathematics training" with the precise mathematical question at hand:

Have a look at equations (72) and (73) of this paper. Note that I have used the standard calculus rule of “product of limits equal to limits of the product.” The validity of this rule in the context can be verified at once by recognizing that the same quaternion −D(a)L(a, λ)L(b, λ)D(b) results from both the limits in Eq. (72) and the limits in Eq. (73).

Sorry if I hurt anyone's feeling, but I'm not attacking anyone's mathematics training.

I have mathematical comments on a mathematical derivation in this paper

The mistake I believe you are making in that paper is not in the step from equation (72) to (73) but the very big step from equation (73) to (74). Here you, in fact, compress three steps into one. As you yourself say, (74) follows "(i) using the relation (60) [thus setting all bivectors in the spin bases], (ii) the associativity of the geometric product, and (iii) the conservation of spin angular momentum specified in Eq. (70)". That's a whole lot. If you write out the three intermediate steps in full, we will be able to discuss what went wrong, if anything.

But why do the routine calculations a hard way when there are fast track alternatives? One could have done the computation of the correlation by substituting the findings of (58) and (59) directly into (71). Then we would arrive at a different answer (the wrong answer). Well, my analysis is well documented in various places and I certainly don't want to go on repeating my arguments. I'm also not asking you to repeat yours.

[FrediFizzx]

So this example I gave is basically for when s_A is at zero degrees relative to a. But for when b = a the relative angle doesn't matter since s_A = -s_B always. The relative angle can be random and it still works like for the zero angle case.
.

To me it looks like the you mean that the variables s_A and s_B are random variables. Then the lim notation is mathematically superfluous; you could rather just explain that in the paper. The question is now: Are they part of the state of the entangled particles, i.e., are they part of the hidden variable? If not, how do you ensure that " s_A = -s_B always"?

Because this is about EPR-Bohm. We defined the singlet state at the beginning of the paper. Don't take the example I presented out of context. Thanks.

[Joy Christian]

The mistake I believe you are making in that paper is not in the step from equation (72) to (73) but the very big step from equation (73) to (74).

There is no mistake. Eq. (74) follows trivially from Eq. (73) within the Geometric Algebra framework presented in the paper I have linked above. It is not a "big step."

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[Heinera]

So this example I gave is basically for when s_A is at zero degrees relative to a. But for when b = a the relative angle doesn't matter since s_A = -s_B always. The relative angle can be random and it still works like for the zero angle case.
.

To me it looks like the you mean that the variables s_A and s_B are random variables. Then the lim notation is mathematically superfluous; you could rather just explain that in the paper. The question is now: Are they part of the state of the entangled particles, i.e., are they part of the hidden variable? If not, how do you ensure that " s_A = -s_B always"?

Because this is about EPR-Bohm. We defined the singlet state at the beginning of the paper. Don't take the example I presented out of context. Thanks.

But that didn't answer my question: If s_A and s_B are not part of the state of the entangled particles, how do you ensure that " s_A = -s_B always"?

[Joy Christian]

But that didn't answer my question: If s_A and s_B are not part of the state of the entangled particles, how do you ensure that " s_A = -s_B always"?

The constraint s_A = -s_B follows from the conservation of zero spin angular momentum: s_A + s_B = 0. It remains valid for the free motion of the two spins all the way up to the detectors.

***

[Heinera]

But that didn't answer my question: If s_A and s_B are not part of the state of the entangled particles, how do you ensure that " s_A = -s_B always"?

The constraint s_A = -s_B follows from the conservation of zero spin angular momentum: s_A + s_B = 0. It remains valid for the free motion of the two spins all the way up to the detectors.

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Ok, then it is part of the initial state of the pair. In other words, part of the hidden variable. "Hidden variable" is a synonym for "state" in Bell's terminology.

[Joy Christian]

But that didn't answer my question: If s_A and s_B are not part of the state of the entangled particles, how do you ensure that " s_A = -s_B always"?

The constraint s_A = -s_B follows from the conservation of zero spin angular momentum: s_A + s_B = 0. It remains valid for the free motion of the two spins all the way up to the detectors.

***

Ok, then it is part of the initial state of the pair. In other words, part of the hidden variable. "Hidden variable" is a synonym for "state" in Bell's terminology.

Only partly true. It does not require summing over, like traditional hidden variables lambda of Bell.

***

[Heinera]

Only partly true. It does not require summing over, like traditional hidden variables lambda of Bell.

***

But it does require summing over, if you want to compute correlations, which are averages (empirical correlations) or integrals (theoretical correlations á la Kolmogorov).

[Joy Christian]

Only partly true. It does not require summing over, like traditional hidden variables lambda of Bell.

***

But it does require summing over, if you want to compute correlations, which are averages (empirical correlations) or integrals (theoretical correlations á la Kolmogorov).

No, it does not require summing over, either in my GA model or in the current attempt to have a quantum mechanical version of it. The devil is in the details.

***

[Heinera]

No, it does not require summing over, either in my GA model or in the current attempt to have a quantum mechanical version of it. The devil is in the details.

***

But that is only because you somehow think that correlations have nothing to do with statistics or probabilities. That is wrong; a correlation is simply nothing more than a normalized average of a function of two random variables (thus, this function being itself a random variable). We can speak of empirical correlation from finite experiments (a random variable), or theoretically derived correlation (expectation of that random variable). But the correlation is in principle no different from other random variables.

[Joy Christian]

No, it does not require summing over, either in my GA model or in the current attempt to have a quantum mechanical version of it. The devil is in the details.

***

But that is only because you somehow think that correlations have nothing to do with statistics or probabilities. That is wrong; a correlation is simply nothing more than a normalized average of a function of two random variables (thus, this function being itself a random variable). We can speak of empirical correlation from finite experiments (a random variable), or theoretically derived correlation (expectation of that random variable). But the correlation is in principle no different from other random variables.

This is all empty talk. Read my GA-based papers or the QM paper under discussion. I have no interest in the traditional Bell dogma, which relies on statistical and probabilistic obfuscations.

**

[FrediFizzx]

The conclusion from this Stern-Gerlach polarizer simulation,

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

Is that when a = b you will not get AB = -1 always if the angle between the particle spin vector and a is random in the actual experiment. So these functions are correct in that case.



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[minkwe]

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.

***

A rotation of -90 degrees and a rotation 270 degrees are mathematically the same but physically different.

[FrediFizzx]

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.

***

A rotation of -90 degrees and a rotation 270 degrees are mathematically the same but physically different.

Yep, and you would write it like you did to convey the actual physics. But we are past that now as it doesn't matter. The limits s --> +/- n is how the experiment works with Stern-Gerlach polarizers for EPR-Bohm.
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[Yablon]

I prefer not to think of what was being discussed here using the expression “limit” because that has a connotation with calculus that really is not applicable here. I prefer to think of it as “observational effect on spin” or the like.

[Joy Christian]

I prefer not to think of what was being discussed here using the expression “limit” because that has a connotation with calculus that really is not applicable here. I prefer to think of it as “observational effect on spin” or the like.

That preference may be justified in the quantum mechanical version of the model being discussed here, but it is not justified in my Geometric Algebra model. In fact, limits, as used in my GA model, are the most natural way to represent what is happening geometrically within the topology of the 3-sphere. My GA model is a purely geometrical model, with no quantum stuff.

Have a look at equations (72) and (73) of this paper. Note that I have used the standard calculus rule of “product of limits equal to limits of the product.” The validity of this rule in the context can be verified at once by recognizing that the same quaternion −D(a)L(a, λ)L(b, λ)D(b) results from both the limits in Eq. (72) and the limits in Eq. (73).

I think it is important to appreciate just how radical my GA approach to quantum correlations is. In it, absolutely everything concerning the experiment is represented purely geometrically.

***

[Yablon]

I prefer not to think of what was being discussed here using the expression “limit” because that has a connotation with calculus that really is not applicable here. I prefer to think of it as “observational effect on spin” or the like.

That preference may be justified in the quantum mechanical version of the model being discussed here, but it is not justified in my Geometric Algebra model. In fact, limits, as used in my GA model, are the most natural way to represent what is happening geometrically within the topology of the 3-sphere. My GA model is a purely geometrical model, with no quantum stuff.

Have a look at equations (72) and (73) of this paper. Note that I have used the standard calculus rule of “product of limits equal to limits of the product.” The validity of this rule in the context can be verified at once by recognizing that the same quaternion −D(a)L(a, λ)L(b, λ)D(b) results from both the limits in Eq. (72) and the limits in Eq. (73).

I think it is important to appreciate just how radical my GA approach to quantum correlations is. In it, absolutely everything concerning the experiment is represented purely geometrically.

***

However, as you know, I believe that there are isomorphisms between QM and GA, and that eventually what you are doing needs to be understood as quantum mechanics in a different mathematical language. There are insights which QM can gain from GA, and vice versa. I think it unwise to sell short, one or the other. And I think the primary problems with QM are interpretive. Jay