Joy Christian, Disproof of Bell's Theorem

[FlorinM]

Dear Delta Kilo,

the multiplication in:
A(a,λ)={−ajβj}{akβk(λ)}

is not the usual multiplication, but the "geometric algebra" multiplication as the elements beying multiplied are bivectors. Consequently your following math is wrong.

 

[FlorinM]

A quick note:

On FQXi's website Joy Christian and I are arguing for and against his "disproof"

http://www.fqxi.org/community/blogs

Please join in the discussion there. Let the best argument win.

 

[Delta Kilo]

the multiplication in:
A(a,λ)={−ajβj}{akβk(λ)}

is not the usual multiplication, but the "geometric algebra" multiplication as the elements beying multiplied are bivectors. Consequently your following math is wrong.

Please tell me which line is wrong. I am aware that these are elements of Clifford algebra, they follow their fancy rules for multiplication. But they can still be multiplied by ordinary (complex) numbers and follow associativity and distributivity laws (but not commutativity of course). I believe I handled them correctly. If there is an error, please point it to me.

PS: Had a quick look at the paper again and just noticed that it actually says at the very beginning in eq (1):
$A(a,\lambda)= \cdots = \begin{cases} +1, & \text{if } \lambda=+1 \\ -1, & \text{if } \lambda=-1 \end{cases}$
Which means (a) my math is correct, (b) I shouldn't have bothered and (c) WTF all these $\beta_j$ are there for in the first place?

Regards,
DK

 

[FlorinM]

DK,

The multiplication between the sigmas
A(a,λ)=−λΣj[ajβj]Σk[akβk]
is not the regular multiplication.

And indeed, Eq.1 looks like is agreeing with your calculation, but it is not. The variables A and B Alice and Bob are equipped are not scalars (as resulting from your math), but bivectors representing the handedness of a shared sense of rotation.

Your kind of approach for proving Joy Christian wrong was tried 2 years ago, but his math still stands. However, I am not agreeing with him and I think I have a solid argument against his position in my achive preprint. I am challenging him on FQXi's website and I will attempt to make my position easier to understand. Please join the discussion there. I am preparing a massive rebuttal of his arguments.

 

[Delta Kilo]

The multiplication between the sigmas
A(a,λ)=−λΣj[ajβj]Σk[akβk]
is not the regular multiplication.

Indeed it is not. But it is nevertheless associative and distributive is it not? As in $a \beta_i(b \beta_j+c \beta_k) = a b \beta_i \beta_j + ac \beta_i \beta_k$
And I believe I have been careful about that. I'm sorry for not numbering my equations. I've copied them here with numbers. Please specify exactly which steps (from-to) you believe to be in error:
(1) A(a,λ)={−ajβj}{akβk(λ)}
(2) A(a,λ)=Σj[−ajβj]Σk[akβk(λ)]
(3) βj(λ)=λβj:
(4) A(a,λ)=Σj[−ajβj]Σk[ak(λβk)]
(5) A(a,λ)=−λΣj[ajβj]Σk[akβk]
(6) A(a,λ)=−λΣj,k(ajakβjβk)
(7) A(a,λ)=−λ[Σj(a2jβjβj)+Σj≠kajak(βjβk+βkβj)]
(8) βjβj=−1 and βjβk=−βkβj,j≠k:
(9) A(a,λ)=λΣja2j
(10) |a|=1:
(11) A(a,λ)=λ, and similarly B(b,λ)=−λ

And indeed, Eq.1 looks like is agreeing with your calculation, but it is not.

How is it so? It agrees for every possible value of $\lambda$, that is for -1 and 1 and for every possible $a$, that is for every possible real $a_j$ provided that $\sum{a_j^2}=1$

The variables A and B Alice and Bob are equipped are not scalars (as resulting from your math), but bivectors representing the handedness of a shared sense of rotation.

There are no variables A and B. There are functions A(a,λ) and B(b,λ). These functions were introduced by Bell in his paper as possible outcomes of the experiment. Their range was explicitly given as a set {-1, 1}. This agrees with my previous post and with eq (1) of the paper in question.

 

[DrChinese]

Your kind of approach for proving Joy Christian wrong was tried 2 years ago, but his math still stands. However, I am not agreeing with him and I think I have a solid argument against his position in my achive preprint. I am challenging him on FQXi's website and I will attempt to make my position easier to understand. Please join the discussion there. I am preparing a massive rebuttal of his arguments.

Welcome to PhysicsForums, Florin!

I am very interested in learning more about this. Any comments you can share, including background on the subject, is very welcome.

-DrC

 

[Delta Kilo]

A quick note:

On FQXi's website Joy Christian and I are arguing for and against his "disproof"

http://www.fqxi.org/community/blogs

Please join in the discussion there. Let the best argument win.

Thanks for the invitation but no thanks. I went there, pointed out some issues and received a sermon back.

All right, I'll try one last time.

This time I draw your attention to http://arxiv.org/abs/1106.0748 by the same author.

Equation (16) says

$\mathcal{A} (\alpha,\boldsymbol{ \mu})=(-I \cdot \tilde{a})(+\boldsymbol{ \mu} \cdot \tilde{a})= \begin{cases} +1 & \text{if } \mu = +I \\ -1 & \text{if } \mu = -I \end{cases}$

Here $I$ is a unit trivector, $\boldsymbol{ \mu}= \pm I$ is two-valued random parameter with equal probability of outcomes, $\tilde{a}$ is a vector derived from scalar parameter $\alpha$. Incidentally the values in brackets are bivectors and the multiplication between the brackets is geometric product, all that in grassman algebra. The result is $\pm 1$ as it should be. So far so good.

Now the author wants to calculate correlation. And for that he needs standard deviation which appears in the denominator.

Well, since the only values of $\mathcal{A}$ are -1 and +1 and they are equally probable, it is immediately obvious that the expectation $E[\mathcal{A}] =0$ and the standard deviation $\sigma(\mathcal{A})=1$.
I'll do it again real slow just in case. We have 2 equiprobable outcomes, $n=2, p_1=p_2=\frac{1}{2}, \mathcal{A}_1 = \mathcal{A}(\alpha,\mu_1) = -1, \mathcal{A}_2 = \mathcal{A}(\alpha,\mu_2) = +1$,
$E[\mathcal{A}] =\displaystyle \sum_{i=1}^n p_i \mathcal{A}(\alpha,\mu_i) = \frac{1}{2}(-1) + \frac{1}{2}(+1) = 0$,
$\sigma(\mathcal{A})=\sqrt{\displaystyle \sum_{i=1}^n p_i [\mathcal{A}(\alpha,\mu_i) - E[\mathcal{A}]]^2 } = \sqrt{\frac{1}{2}(-1)^2 + \frac{1}{2}(+1)^2} = 1$

But apparently it's not good enough for the author for he knows better. Allow me to quote:

These deviations can be calculated easily. Since errors in linear relations such as (16) and (17) propagate linearly, the standard deviation of $\mathcal{A} (\alpha,\boldsymbol{ \mu})$ is equal to $(−I \cdot \tilde{a})$ times the standard deviation of $(+\boldsymbol{ \mu} \cdot \tilde{a})$ (which we write as $\sigma(A)$)

Basically, the author just claimed that standard deviation is linear with respect to geometric product of grassman bivectors. And the words are quickly followed by deeds, eq (23):

$\sigma(\mathcal{A})=(−I \cdot \tilde{a})\sigma(A)$

Note that while $\mathcal{A}$ as defined by eq (16) has a value range $\pm 1$ and $\sigma(\mathcal{A})$ is quite ok , $A=(+\boldsymbol{ \mu} \cdot \tilde{a})$ is a grassman bivector and $\sigma(A)$ simply does not compute. So what, the author just quietly replaces the bivector with its norm in eq (24):

$\sigma(A)=\sqrt{\frac{1}{n} \displaystyle \sum_{i=1}^n \left| \left|A(\alpha, \boldsymbol{ \mu}^i) - \overline{A(\alpha, \boldsymbol{ \mu}^i) } \right|\right| ^2 }$

As a result, $\sigma(A)$ comes out as 1 (a scalar). What was geometric product in eq (16) now becomes multiplication by a scalar 1 in (23) so now $\sigma(\mathcal{A})$ comes out as a bivector!
The author now uses this strange quantity $\sigma(\mathcal{A})$ to "normalize" $\mathcal{A} (\alpha,\boldsymbol{ \mu})$, eq (25):

$A(\alpha,\boldsymbol{ \mu}) = \frac{\mathcal{A}(\alpha, \boldsymbol{ \mu}) - \overline{\mathcal{A}(\alpha, \boldsymbol{ \mu}) }}{\sigma(\mathcal{A})} = (+\boldsymbol{ \mu} \cdot \tilde{a})$

Note that $A(\alpha,\boldsymbol{ \mu})$ again comes out as a bivector. And as a final touch the author plugs these grassman whatsises instead of outcomes into the formula for covariance eq (30):

$E(\alpha,\beta)=\displaystyle \lim_{n \gg 1}[\frac{1}{n}\displaystyle \sum_{i=1}^n A(\alpha,\boldsymbol{\mu}^i)B(\beta, \boldsymbol{\mu}^i)]$

Now the trick finally pays off, things get canceled out and the value comes out which was supposed to violate Bell's inequality. And it does not matter that a direct application of a standard textbook formula gives different answer (which happen to agree with Bell).

I pointed all these issues to the author and received the following reply:

Neither Bell’s, nor your calculations agree with what is observed in the experiments. This is because neither Bell, nor you are calculating the correlations correctly. Your calculation, as I pointed out to you more than once, produces statistical nonsense, because it is based on elementary errors. My calculation, on the other hand, agrees with the experiment, event-by-event, number-by-number, because it is based on a conceptually superior framework, and is entirely free of error. It is based on the correct model of the physical space introduced by Grassmann some 160 years ago, and further developed by many people, including Clifford and Hestenes. It is a pity that you do not have the proper background to see this.

I'll be blunt but I'm going to call it a bluff. I do not believe the author has any answers at all.

DK
PS: Can I too get a mini-grant please?

 

[DrChinese]

All right, I'll try one last time.

This time I draw your attention to http://arxiv.org/abs/1106.0748 by the same author.
...

DK
PS: Can I too get a mini-grant please?

Delta Kilo,

Riddle me this: if someone (i.e. Christian) has a model which is local non-contextual, why won't they simply supply a set of values for 3 simultaneous angle settings (you know the kind I mean) for a set of data points and be done with it? I can't get past this simple requirement. It seems as if the focus is on presenting a complex model which will emulate the predictions of QM (for Alice and Bob, 2 values) but FAILS the EPR test (i.e. multiple simultaneous elements of reality independent of the act of observation). By presenting a complicated mathematical derivation, it just pulls things away for what I think are the real issues.

I guess I am just dumb on this point. Maybe you can enlighten me... The de Raedt team is the only one who has even attempted to address this with their simulations (which present values for any simultaneously desired angles).

-DrC

 

[FlorinM]

DK,

Thanks for participating on FQXi's web site. Indeed, Joy is not the easiest guy to challenge and he even got criticised for it on the achive for the lack of a collegial tone. I had some doubts about challenging him myself for the same very reason, but his results were too interesting and his interpretation too wrong to pass the opportunity.

I answered one of your questions on FQXi's blog, and I read your comments above. I did not find any mathematical mistakes in his approach and after I'll be done rebutting his reply I may come back here and show in detail why he is correct. In the meantime, I recommend you to read the geometric algebra book by David Hestenes.

 

[Delta Kilo]

Floring,

Please try answering the following quiz:

* Do you agree that standard deviation of a random variable $A$ is computed according to $\sigma(A)=\sqrt{E[(A-E[A])^2]}$? (if not, please post alternative definition)

* Do you agree that if random variable $A$ takes the value of either -1 or +1, each with probability of 1/2, then its standard deviation $\sigma(A)=1$?

* Do you agree that functions $A(\alpha,\lambda)$ and $B(\beta,\lambda)$ representing individual outcomes in Bell's experiment satisfy the above criteria ant therefore have standard variation of 1?

* Do you agree that standard deviation is not a linear function, that $\sigma(aA)=a\sigma(A)$ is incorrect in general, and in particular it is violated for a=-1, not to mention complex numbers, vectors, bivectors, quaternions etc.?

* Do you agree that standard deviation is a non-negative real number (fer crying out loud)?

* In the view of the above, do you agree that eq (23) from the paper which I cited in my previous post is incorrect?

* Finally, do you agree that if two mathematical derivations starting from the same premise, arrive at different results, then at least one of them must be in error?

* Did you point out the error in my (or better yet, Bell's) derivation, indicating which particular equation is not correct? (Please quote)

* Did I point out the error in the paper in question? (I can answer that: yes I did. See above)

DK
[rant]I'm sick of people on high horses telling me to go read some books. All right, it's a deal: I'll go read to refresh my memory on Grassman algebra, and you guys go read up some basics on statistics 101, starting with the definition of standard deviation. Wake me up when you are ready to point which one of my equations is incorrect.[/rant]

 

[FlorinM]

DK,

Against my better judgement not to get sidetracked, here are the answers:

* Do you agree that standard deviation of a random variable A is computed according to σ(A)=E[(A−E[A])2]−−−−−−−−−−−−√? (if not, please post alternative definition)
Yes, it's valid

* Do you agree that if random variable A takes the value of either -1 or +1, each with probability of 1/2, then its standard deviation σ(A)=1?
Yes

* Do you agree that functions A(α,λ) and B(β,λ) representing individual outcomes in Bell's experiment satisfy the above criteria ant therefore have standard variation of 1?
Yes

* Do you agree that standard deviation is not a linear function, that σ(aA)=aσ(A) is incorrect in general, and in particular it is violated for a=-1, not to mention complex numbers, vectors, bivectors, quaternions etc.?
yes, the correct formula is σ(aA)=norm(a)σ(A) when a is a constant (because the expectation value can be redefined with norms). Alternatively σ(aA)=aσ(A) when σ(A) is (re)defined correctly.

* Do you agree that standard deviation is a non-negative real number (fer crying out loud)?
Not necessarily. In geometric algebra it is not. It is a "number" in that formalism. Joy makes this distiction between "raw" and "standard" scores. For the standard score you are correct, but not for the raw ones.

* In the view of the above, do you agree that eq (23) from the paper which I cited in my previous post is incorrect?
Eq. 23 is correct. This may sound paradoxical especially since I agreed that σ(aA)=aσ(A) is not correct in general, but there is no contradiction. σ(aA)=aσ(A) is right in geometric algebra only for raw intermediate calculations, but not in the end for standard results where we deal only with pure scalars as outcomes of experiments. Eq. 23 is an intermediate "raw" geometric algebra step.

* Finally, do you agree that if two mathematical derivations starting from the same premise, arrive at different results, then at least one of them must be in error?
yes (but Joy's computation is not the one in error - I wish it were, and in that case it would make my challenge of his results that much easier)

* Did you point out the error in my (or better yet, Bell's) derivation, indicating which particular equation is not correct? (Please quote)
In your case you make geometric algebra mistakes when analysing Joy's computations. Bell does not make any mistakes, and Joy is incorrect in asserting that. Joy states that Bell makes a "topological error" and I am after Joy proving him wrong on that.

* Did I point out the error in the paper in question? (I can answer that: yes I did. See above)
See my answers

Florin

 

[SpectraCat]

* In the view of the above, do you agree that eq (23) from the paper which I cited in my previous post is incorrect?
Eq. 23 is correct. This may sound paradoxical especially since I agreed that σ(aA)=aσ(A) is not correct in general, but there is no contradiction. σ(aA)=aσ(A) is right in geometric algebra only for raw intermediate calculations, but not in the end for standard results where we deal only with pure scalars as outcomes of experiments. Eq. 23 is an intermediate "raw" geometric algebra step.

Sorry, but that claim is utterly opaque to a non-expert. Can you please provide a deeper explanation, or at least an example where what you say is true? Specifically, which properties of a and A cause the simple linear relationship that you claim holds true? Is this general or coincidental (and thus true for this specific "raw geometric algebra step")? What is the distinction you are using to define a "raw" geometric algebra step?

 

[FlorinM]

SpectraCat

You say: "Sorry, but that claim is utterly opaque to a non-expert. Can you please provide a deeper explanation, or at least an example where what you say is true? Specifically, which properties of a and A cause the simple linear relationship that you claim holds true? Is this general or coincidental (and thus true for this specific "raw geometric algebra step")? What is the distinction you are using to define a "raw" geometric algebra step? "

Let me try to explain it by an analogy. The results of experiments are numbers. To an experimentalist standard statistical methods do apply. However, in standard QM formalism, a theoretician uses complex numbers. There are stranger "raw" rules which work there and you have this Born rule which acts as a translation layer between raw "complex probabilities" or the complex wavefunction and standard probabilities. In a similar way, Joy Christian is using a different formalism (the geometric algebra formalism) and in the end he converts the "raw" calculations into "standard" ones. When checking his computation you need to watch 2 things: 1. is the raw (or internal, or geometric algebra) computation correct? and 2. does he apply the correct translation mechanism at the end to recover standard probabilities?

DK's mistake in geometric algebra was to impose the rules of standard statistics in the middle of computation. The corresponding mistake in standard QM formalism would be to add probabilities and not amplitudes in the middle of computation.

 

[DrChinese]

Let me try to explain it by an analogy. The results of experiments are numbers. To an experimentalist standard statistical methods do apply. However, in standard QM formalism, a theoretician uses complex numbers. There are stranger "raw" rules which work there and you have this Born rule which acts as a translation layer between raw "complex probabilities" or the complex wavefunction and standard probabilities. In a similar way, Joy Christian is using a different formalism (the geometric algebra formalism) and in the end he converts the "raw" calculations into "standard" ones. When checking his computation you need to watch 2 things: 1. is the raw (or internal, or geometric algebra) computation correct? and 2. does he apply the correct translation mechanism at the end to recover standard probabilities?

If Christian's technique were correct, he could provide answers for any group of angle settings I choose REGARDLESS of whether they could be tested experimentally or not. What else does it mean to be realistic if you cannot do that? In other words: For Alice and Bob, I want to see a dataset in which the "answer" for polarization for 0, 120 and 240 degrees is presented for every photon. Then for each of the 6 theta=120 pairing permutations, I want them to average to the QM value of .25 [.75]. For each of the 3 theta=0 pairing permutations, I want them to average to the QM value of 1.00 [0.00]. Hopefully, you understand the intent of the challenge - a data point by data point result set from the candidate formula.

Unless he can provide that, I fail to see the significance of anything being done here other than an exercise in hyperbole. On the other hand, there is a local realistic simulation from the group of de Raedt et al which provides answers to the above challenge (and exploits the so-called fair sampling assumption to operate). Of course, it suffers from other issues but at least addresses what I consider to be the acid test.

If the formula works, where is the example data? Why not generate 30 or 40 data points and be done with it? I realize you do not speak for Christian, I am simply asking why you do not demand the same of any candidate model.

 

[FlorinM]

Dr Chinese,

I don't quite get your challenge, but let me make a critical point for spin 1/2. Joy's method is completely equivalent with the standard QM formalism in this case. The state space in this case is SU(2) which is isomorphic with SO(3) where geometric algebra can be naturally used. It can be actually proven mathematically that what he is doing in those cases are a 100% faithful translation to the standard complex QM formalism into a geometric algebra formalism. (If he does not recover all QM predictions completely it means that he a mathematical mistake in his computation.) QM can be done in many formalisms: complex numbers, real numbers, quaternions, Bohm. Joy simply found another equivalent formalism (for spin 1/2 only).

For SU(2)~SO(3) Joy is using the double cover property to introduce his "hidden variables" which are basically the disambiguation on which one-to-two map you are located (similar with Riemann's sheets in complex analasys).

My challenge to his method is using spin 1 where there is no such kind of isomorphism and this clearly illuminates his interpretation mistakes.

Florin

 

[Delta Kilo]

here are the answers:

Thank you very much. I appreciate that we are back from wooly vague words and into the realm of verifiable math. Please bear with me, this might take a while.

We are still talking about http://arxiv.org/abs/1106.0748 as it appears to be far more detailed than the original paper that started this topic.

Start with eq(1). Here the author gives the results predicted by QM and observed in experiments:

$\mathcal{A}(\alpha)=\pm 1, \mathcal{B}(\beta)=\pm 1$
$E(\alpha)=0, E(\beta)=0$
$E(\alpha,\beta)=-\cos^2(\alpha-\beta)$
(eq 1)

Here $E(\alpha,\beta)$ represents the expected value of simultaneously observing remote measurement results $\mathcal{A}(\alpha)$ and $\mathcal{B}(\beta)$ along the polarization angles $\alpha$ and $\beta$, respectively.

First a small clarification, the text should read: "$E(\alpha,\beta)$ represents the expected value of the product $\mathcal{A}(\alpha)\mathcal{B}(\beta)$ of simultaneously observing...". I added the words in bold because it is important exactly what kind of product we are dealing with here.

Now, the range of $\mathcal{A}(\alpha)$ and $\mathcal{B}(\beta)$ is a set of {-1, +1}. I stress that these are normal ordinary everyday integer +1 and -1, not some fancy Grassman +1 and -1 and the multiplication between $\mathcal{A}(\alpha)$ and $\mathcal{B}(\beta)$ for the purposes of computing $E(\alpha,\beta)$ is normal everyday multiplication, not an inner product, not an outer product, not an wedge product, not a geometric product, not any other fancy kind of product.

Why is that so? Because Bell chose it to be so. The experiment itself can produce any kind of indication of the outcome, it could be 0 or 1, 'X' or 'O', up or down, red LED or green LED. Bell chose to associate these outcomes with numbers +1 and -1 for the purposes of deriving his inequality. And this is how the data is presented in real Bell-type experiments.

Obviously these numbers, be it theoretical results or real experimental data, are computed using normal everyday arithmetic, normal everyday definitions of expectation value, standard deviation, correlation etc, taken from the statistics 101.

Therefore if the author claims to disprove Bell and to demonstrate $\cos^2$ rule arising from locally realistic $\mathcal{A}(\alpha)$ and $\mathcal{B}(\beta)$, then he has to play by the rules. This means, internally $\mathcal{A}(\alpha)$ and $\mathcal{B}(\beta)$ can use whatever fancy math you want, but their outcomes should be counted the same way the outcomes of real experiments are counted.

To summarize: for the results to be relevant to Bell's theorem and to real-life experiments, functions $\mathcal{A}(\alpha)$ and $\mathcal{B}(\beta)$ should return either -1 or +1 which are to be treated as normal integer numbers using normal arithmetic and statistics. Do you agree with this statement?

Why do I have to explain is so painstakingly? Because I'm sick of people saying "this is not an ordinary multiplication/You won't understand/Go read a book" when in fact it is (should have been) ordinary multiplication.

Now, fast-forward to eq (16).

To this end, we have assumed that the complete state of the photons is given by $\mu = \pm I$, where $I$ is the fundamental trivector defined in Eq. (2). The detections of photon polarizations observed by Alice and Bob along their respective axes $\alpha$ and $\beta$, with the bivector basis fixed by the trivector $\mu$, can then be represented intrinsically as points of the physical space $S^3$, by the following two local variables:
$S^3 \ni \mathcal{A} (\alpha,\mu)=(-I \cdot \tilde{a})(+\mu \cdot \tilde{a})= \begin{cases} +1 & \text{if } \mu = +I \\ -1 & \text{if } \mu = -I \end{cases}$
(eq 16)
and
$S^3 \ni \mathcal{B} (\beta,\mu)=(+I \cdot \tilde{b})(+\mu \cdot \tilde{b})= \begin{cases} -1 & \text{if } \mu = +I \\ +1 & \text{if } \mu = -I \end{cases}$
(eq 17)
with equal probabilities for $\mu$ being either $+I$ or $-I$, and the rotating vectors $\tilde{a}$ and $\tilde{b}$ defined as

$\tilde{a} = e_x \cos 2\alpha + e_y \sin 2\alpha, \tilde{b} = e_x \cos 2\beta + e_y \sin 2\beta$
(eq 18)
[not sure it is meant to be $\cos 2\alpha$ or $\cos^2\alpha$ it won't matter much though]

and further down:

Putting these two results together, we arrive at the following standard scores corresponding to the raw scores (16) and (17):
$A(\alpha,\mu) = \frac {\mathcal{A} (\alpha,\mu) - \overline{\mathcal{A} (\alpha,\mu)}} {\sigma(\mathcal{A})} = \frac {\mathcal{A} (\alpha,\mu) - 0} {(-I \cdot \tilde{a})} = (+\mu \cdot \tilde{a})$
(eq 25)
$B(\beta,\mu) = \frac {\mathcal{B} (\beta,\mu) - \overline{\mathcal{B} (\beta,\mu)}} {\sigma(\mathcal{B})} = \frac {\mathcal{B} (\beta,\mu) - 0} {(+I \cdot \tilde{b})} = (+\mu \cdot \tilde{b})$
(eq 26)

The question is: which one of these should be identified with $A(a,\lambda)$ and $B(b,\lambda)$ from Bell's paper and with the outcomes collected in the actual experiments to compute E(a,b)? Should it be $\mathcal{A} (\alpha,\mu)$ and $\mathcal{B} (\beta,\mu)$ from eq 16-17, or "normalized" $A(\alpha,\mu)$ and $B(\beta,\mu)$ from eq 25-26? Please answer.

Case 1: the answer is the former ($\mathcal{A} (\alpha,\mu)$ and $\mathcal{B} (\beta,\mu)$):

We agreed (I hope) that individual outcomes of measurements are represented by (mapped onto) normal integer numbers -1 and 1. So the first order of business is to drop the notion of $\mathcal{A} \in S^3$ and replace it with simple $\mathcal{A} \in \{-1, +1\}$ (by establishing 1:1 map if you wish).
The next thing we do is define $\mu_+ = +I, \mu_- = -I$. Once this is done we can rewrite eq 16-17, removing all traces of Grassman algebra from them:

$\mathcal{A} (\alpha,\mu)=\begin{cases} +1 & \text{if } \mu = \mu_+ \\ -1 & \text{if } \mu = \mu_- \end{cases}, \mathcal{B} (\beta,\mu)=\begin{cases} -1 & \text{if } \mu = \mu_+ \\ +1 & \text{if } \mu = \mu_- \end{cases}$

where $\mu \in \{ \mu_+, \mu_- \}$ is some opaque random parameter taking up one of the two opaque values with equal probability.

From here we can immediately obtain:

$\mathcal{A} (\alpha,\mu)\mathcal{B} (\beta,\mu)=-1, \forall \mu \in \{ \mu_+, \mu_- \}$

and therefore

$E(a,b)=-1, \forall a,b$

and therefore

$|E(a,b) + E(a',b) + E(a,b') -E(a',b')|= 2, \forall a,b,a',b'$

So far the results agree with Bell and do not exhibit $\cos^2$ rule, which is exactly the opposite of what the author claimed.

Case 2: The answer is $A(\alpha,\mu)$ and $B(\beta,\mu)$ from eq 25-26. That appears to be author's intention because that's what he uses in eq 30 to calculate E(a,b). But what is the value of $A(\alpha,\mu)$? It is a whatsis bivector in whatever space.

Since the goal is to provide a working model explaining experimental results of $\cos^2$ rule (and thus disproves Bell) , we need to identify $A(\alpha,\mu)$ unambiguously with the outcome of a measurement, such as either detector D+ or D- clicking in a typical two-channel Bell type experiment by mapping it into { -1, +1 }. The answer is that we cannot because $A(\alpha,\mu)$ is not a two-valued function. It's value, whatever is it, cannot be obtained in the experiment, therefore it cannot be used to calculate E(a,b) (since E(a,b) is calculated from experimental data and we wish to provide a model for it).

As it is, $A(\alpha,\mu)$ might refer to some internal state of the system, but an extra step is required to obtain the actual outcome of a measurement. This extra step ( which can be achieved by some sort of map $M: A(\alpha,\mu) \mapsto \{-1, +1\}$ will encapsulate in itself the process of measurement. And to maintain connection with actual physical experiments, we would have to use the value of this $M(\alpha,\mu)$ and not the unobservable $A(\alpha,\mu)$. Well, guess what, doing this will bring us back to agreement with Bell and disagreement with reality.

So where it all went wrong? Well, when calculating standard deviation.

To begin with, the whole issue of standard deviation and "normalizing" is a red herring. If you bother to read Bell's original paper, you will see that there is no reference to mean or standard deviation. What's more, Bell's derivation works just fine for any $A(a,\lambda)$ as long as $A(a,\lambda) \in \{-1, +1 \}$ and $A(a,\lambda)=-B(a,\lambda)$. The mean does not have to be 0 and sigma does not have to be 1 and there is no need to "normalize" anything.

Having said that, everyone knows that standard deviation of individual measurements in Bell type experiment is 1 (assuming ideal 100% efficient detector) . It is so bleedingly obvious that no-one needs to explain that. Still, there is nothing wrong with actually calculating one, as long as one's math is correct. The sigma would come out as 1, eq 25-26 would be exactly the same as 16-17 and we would be back to where we started.

But the math is not correct. Instead of directly calculating σ from the definition, which would be far easier but would not produce the desired effect, the author [STRIKE]averts his eyes and carefully walks along the wall pretending there is no elephant in the room[/STRIKE] starts mucking around with it with no clear purpose.

As I already pointed out, eq (23) is wrong. I said and you agreed that σ(aA)=aσ(A) is in general incorrect. You said:

yes, the correct formula is σ(aA)=norm(a)σ(A) when a is a constant (because the expectation value can be redefined with norms).

Well, I have news for you: σ(aA)=norm(a)σ(A) does not work either. I gave you the example already:

$\sigma( \vec{a} \cdot \vec{b} ) \ne \vec{a} \cdot \sigma( \vec{b} ) \ne ||\vec{a}||\sigma( \vec{b} ) \ne \vec{a}\sigma( ||\vec{b})|| ) \ne ||\vec{a}||\sigma( ||\vec{b}|| )$

in fact, 2d ad 3rd terms simply do not compute and 4th term gives a value of a vector where the original was a scalar. This is, by the way, exactly the case with eq. 23-24.

Alternatively σ(aA)=aσ(A) when σ(A) is (re)defined correctly.

Please enlighten us, what is the correct redefinition of σ(A) that allows σ(aA)=aσ(A). All I can see in eq (24) is the same old σ with the argument A (which is a vector) quietly replaced with its norm |A|, which bring us back to my previous point.

This is all so wrong and so crude I'm surprised anyone can fall for this trick. The whole thing reminds me of http://en.wikipedia.org/wiki/Technology_in_The_Hitchhiker%27s_Guide_to_the_Galaxy#Bistromathic_drive"

DK

 

[ppnl]

Does everyone agree that anyone who claims to have developed a local realistic model for QM should be able to meet Sascha's Quantum Crackpot Randi Challenge?

http://www.science20.com/alpha_meme...icist_joy_christian_collect_nobel_prize-79614

Bell's theorem seems to me to say nothing more than that such programs cannot exist.

 

[DrChinese]

Dr Chinese,

I don't quite get your challenge, but let me make a critical point for spin 1/2. Joy's method is completely equivalent with the standard QM formalism in this case. ...

I follow the assertion that Joy's method is completely equivalent with the QM expectation value for electrons. I say (following Bell) that won't ever provide a realistic dataset to be produced for election angle settings A=-22.5, B=0, C=22.5. Here is a very small sample to illustrate:

Alice / Bob
A B C/A B C
+ + +/- - -
+ + +/- - -
+ + +/- - -
+ + -/- - +

The AC expectation value for correlation is .25 (.5*sin(theta)^2) which matches the dataset (AC: 1 of 4). However, the AB and BC expectation values, being equal, should average .073. However, they actually come out as .125 above (AB:0 of 4 and BC:1 of 4). In fact, there is no dataset possible which will be counterfactually realistic AND match QM. (This is basic Bell/Sakurai, right?)

So my point is that Christian's method is actually incapable of making counterfactual predictions even if the math yields the QM expectation for 2 angles. So it seems at best he has a non-realistic local model, in accordance with Bell's Theorem.

 

[DrChinese]

...

http://www.science20.com/alpha_meme...icist_joy_christian_collect_nobel_prize-79614

Bell's theorem seems to me to say nothing more than that such programs cannot exist.

Why, this is almost exactly the same as the DrChinese challenge! Awesome!



I am glad SOMEBODY sees my point. I was starting to feel lonely.

 

[FlorinM]

Dear Dr. Chinese and DK,

Thank you for your messages, there were really helpful.

Let me start with Dr. Chinese.
I enjoyed the link "http://www.science20.com/alpha_meme/...el_prize-79614" a lot, I was not aware of it. No, the classical computer model is not possible in this case. And this can be established rigurously mathematically by a theorem by Clifton arXiv:quant-ph/9711009v1 which I cite in my preprint: http://arxiv.org/abs/1107.1007 Clifton proved under what conditions Bell's beables must be commutative and Joy's are not. The reason why Joy's theory fails to be modeled on a computer is because his hidden variable theory is contextual. (and contextual hidden variables' ontology is basically junk). Joy's interpretations are all wrong and misleading. What he calls realistic is actually factorizable.

DK,

You are 100% right from the beginning until "So where it all went wrong? Well, when calculating standard deviation." The right approach is your step 2.

Let me quote you: "This extra step ( which can be achieved by some sort of map M:A(α,μ)↦{−1,+1} will encapsulate in itself the process of measurement. And to maintain connection with actual physical experiments, we would have to use the value of this M(α,μ) and not the unobservable A(α,μ). Well, guess what, doing this will bring us back to agreement with Bell and disagreement with reality."

So here is the deal: consider the map M ("which can be achieved by some sort of map M:A(α,μ)↦{−1,+1} will encapsulate in itself the process of measurement"). Such a map is illegal in his formalism and computations should be caried all the way in geometric algebra formalism until you reach the answer. If you say at this point: "but this is not a realistic local model" you are right. The pollitically correct description for his model is "contextual hidden variable theory", and the pollitically incorrect description is "BS".

I was pointing earlir to Dr. Chinese that what Joy uses is the SU(2)~SO(3) isomorphism and his hidden variable is the extra degree of freedom resulting from the double cover property. As such his formalism is actually only a rewrite of QM standard formalism from the spin1/2 SU(2) state space in the fancy geometric algebra on SO(3). What he gets is a factorization between Alice and Bob in the new formalism which he illegally calls "realism". Applying the map M calls his realistic bluff because the ontological meaning of his hidden variables is not fixed. Joy's is protected by appying M by the Hestenes' formalism and he will always argue that appying M at any stage violates geometric algebra (go directly to jail, do not pass go, do not collect 200, and read a geometric algebra book). What is needed is another way of proving him wrong.

Please see my preprint and my FQXi post to see how I prove that his model is only a contextual hidden variable theory with the help of a spin 1 state and a nice decomposition trick into 2 spin 1/2's where I can use Joy's model. This bypasses all geometric algebra defence from Joy. Right now I am preparring a massive rebuttal of his answer to my FQXi post which I hope will clearly show his interpretation mistakes.

Florin

 

[Delta Kilo]

Florin,

There are 2 main things wrong.

First, is the conceptual BS, his insistence on extending the use of his fancy geometric formalism beyond the boundaries of the model and into the statistical processing of the outcomes of measurements. I spent the first 1/3 of my post debunking that, I'm not going to repeat myself again.

The second is the fact that equation 23 is plain WRONG. It violates basic rules of arithmetic.

DK

 

[FlorinM]

Dr. Chinese and DK,

I believe I made a wrong statement earlier when I said that Joy Christian's work contained no mathematical mistakes. I unfortunately got blinded by high level arguments and did not see the trees from the forest. However, I am here to set the record straight and point you to my latest preprint: http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.0535v1.pdf which hopefully will close this debate once and for all. (see also my FQXi blog post: http://www.fqxi.org/community/forum/topic/983)

By the way, I am still disagreeing with DK on σ(aA)=aσ(A). Suppose “a” is the unit of measurement (temperature, meters, kilograms, etc). Then the equation is actually correct, and therefore it is not incorrect in general and cannot be used as a decisive argument against Joy's math. Other more blatant mistakes can be used however. It is embarrassing to admit for me I never bothered to check Joy's math up close before, but now that I did I hope this would absolve me for at least part of the blame.

 

[DrChinese]

Dr. Chinese and DK,

I believe I made a wrong statement earlier when I said that Joy Christian's work contained no mathematical mistakes. I unfortunately got blinded by high level arguments and did not see the trees from the forest. However, I am here to set the record straight and point you to my latest preprint: http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.0535v1.pdf which hopefully will close this debate once and for all. (see also my FQXi blog post: http://www.fqxi.org/community/forum/topic/983)

Nice paper. I didn't follow all of it, but it is well written. No question that Christian is wrong in my opinion anyway, hardly surprising.

 

[FlorinM]

Thanks. There are some heated exchanges between Joy and me now on FQXi blog right now. The icing on the cake was when Joy called Hodge duality his own: "It is a Christian duality not Hodge duality, with a very specific Christian meaning attached to it." :) Now this is precious.

 

[billschnieder]

Florin,
After reading your paper, I doubt that you understand Joy's model at all. It does not appear you have recognized the difference between averaging over a series of events each of which can only be one of two possibilities, and picking a convention for a series of equations.

Your complex number example is humorous. If 3 + 2i and 3 - 2i are equal alternate posibilities for z, <z> is 3. But if you select a convention for your equations where only one is possible, then <z> = 3 is wrong. This is what you are missing.

 

[FlorinM]

Bill,

I don't quite get your criticism and I don't want to give an answer which may not be what you are looking for. Can you please specify the context a bit more? What do you mean by: "averaging over a series of events"? Let's frame the discussion around http://arxiv.org/PS_cache/quant-ph/pdf/0703/0703179v3.pdf to be specific. Are you talking about Eqs. 18, 19 of that paper, or are you talking about what Bell does in his theorem?

Thanks,

Florin

 

[billschnieder]

Bill,

I don't quite get your criticism and I don't want to give an answer which may not be what you are looking for. Can you please specify the context a bit more? What do you mean by: "averaging over a series of events"? Let's frame the discussion around http://arxiv.org/PS_cache/quant-ph/pdf/0703/0703179v3.pdf to be specific. Are you talking about Eqs. 18, 19 of that paper, or are you talking about what Bell does in his theorem?

Thanks,

Florin

Not sure what is not clear as I'm responding directly to what you have written in your paper for which you gave the complex number analogy. You are using the orientation of the 3-sphere as a convention in your equations, whereas Joy is using it as a hidden variable.

Remember Alice is making multiple measurements of different particles and averaging over them not repeated measurements of the same particle. But each particle has a different hidden variable or in other words, there is an ambiguity in the orientation for the different particles arriving at Alice. Finally remember that the hidden variables are not the outcomes of the experiments. The *different* hidden variables must interact with Alice's device in Alice's frame and only after that can you average and obtain Alice's result.

Until you understand this simple fact, you will not understand his model. Your rebuttal is flawed because of this.

 

[FlorinM]

Dear billschnieder,

Or should I say Joy Christian?

First naming equations after yourself, and now sockpuppetry?

I have exchaged way too many messages with you already not to recognize your writing style. I guess it is time for a new pen name, this one was already exposed.



Florin

 

[Gordon Watson]

Dear billschnieder,

Or should I say Joy Christian?

First naming equations after yourself, and now sockpuppetry?

I have exchaged way too many messages with you already not to recognize your writing style. I guess it is time for a new pen name, this one was already exposed.



Florin

Are you sure? The styles are not nearly the same, imho!

 

[billschnieder]

Dear billschnieder,

Or should I say Joy Christian?

First naming equations after yourself, and now sockpuppetry?

I have exchaged way too many messages with you already not to recognize your writing style. I guess it is time for a new pen name, this one was already exposed.



Florin

Now this is funny. Your judgement is so obviously clouded for you to think that everyone challenging your rebuttal must somehow be Joy Christian.

In your paper you say on page 1:

"Even without spelling in detail the error, it is easy
to see that the exterior product term should not vanish
on any handedness average because handedness is just
a paper convention on how to consistently make compu-
tations."

All I have done is point out to you that you are missing the point because Joy Christian is not using handedness as a convention but as the hidden variable itself.

My criticism is very clear and instead of addressing it, you decide to accuse Joy Christian of acts for which you have no proof. Very disappointing.

 

[FlorinM]

Dear billschnieder,

Let me start by saying that on the very remote possibility that you are indeed not Joy Christian, I am apologizing to you.

I replied earlier, but my post did not appear and unfortunaley I did not saved it.

Let me list the reasons why Joy's model is wrong:

Physical reasons:

- Never in his model he is using the fact that the original state in in the Bell state. Start with any other Psi and you will still get -a.b if you believe his math.
- The model does not respect the detector swapping symmetry: Swap Alice and Bob's detectors and you get the same results. Joy is using DIFFERENT analyzers for Alice and Bob to recover the minus sign on -a.b. Restoring the symmetry results in + a.b
- Holman's argument: Once MU is set, perform the EPR-B experiment on z axis and do a subsequent measurement on one arm of the experiment on the x axis. You get 2 choices: MU does not change between measurement, or MU changes between measurement. MU does not change: this means the x measurement outcome is always the same as the z outcome. Experiments show you get 50% the same answer and 50% the opposite answer. MU does change: than you have problems explaining 3 1/2 spin particle experimental results.

Mathematical reasons:

-incorrect Hodge duality between pseudo-vectors and bivectors in a left handed basis. In a right handed bases a^b = I (axb) (Joys agrees with it). In a left handed basis Joy claims incorrectly a^b = -I (axb). This is wrong, it is still with +. Easy way of seeing this: changing handedness comes from a mirror reflection. In a mirror reflection I = e1^e2^e3 changes signs because it is a PSEUDO-scalar (Joy does this correctly). However (axb) changes signs as well (Joy forgets that axb is a PSEUDO-vectors and treats it like a vector)
-On FQXi website Joy now claims a different thing: he is using left and right algebras instead of left and right handedness. To debunk this I spelled out all 4 combinations: left algebra-left handedness, left algebra-right handedness, right algebra-left handedness, right algebra-right handedness. In each algebra Hodge duality preserves the sign, and mixing algebras is inconsistent (it is like adding kets with bras, row and column vectors: "go direcly to jail, do not pass go do not collect 200"). All associative algebras have left and right implementations (and the name comes from the matrix formalism). Only in 3D there is handedness-a property of the cross product. Handedness is the sign of the pseudo-scalar I = e1^e2^e3 = e1e2e3 and not of the bivector product: B1B2B3. The sign of the bivector product gives you the left or right algebra.
-Any generalization of Joy's model in the Clifford algebra formalism breaks either -a.b correlation, or the zero average in each arm of the experiment
-Joy takes a 0/0 limit: sin(epsilon)/sin(epsilon) and claims it equals zero because the nominator goes to zero.
-Joy computes incorrectly a rotation with a bad rotor in geometric algebra. (the last 2 errors are used to fight Holman's analysis)

Computer simulation arguments:
-By now there are 2 independent simulations of Joy's model both recovering the classical limit. One of the simulation was validated by obtaining -cos correlation on other models

Sociological factors:
-I have never ever got any mathematical arguments from Joy. Instead he used only lies, insults, fallacious arguments, and obfuscation of simple mathematical facts.
-naming the Hodge duality after himself – a major score on Baez’s crackpot index.
-His archive replies are using a bullying tone which scared away critics. You want proof? Sure. The +1=-1 mistake from the wrong sign of Hodge duality was almost found by the very first critic and the tone of Joy’s reply: “rectify this pedagogical error”-like the first critic was an idiot, scared other people from checking his math.
Frankly, I have no explanation for his behavior and obstinate denial of obvious elementary mistakes except that he is doing a cover-up. But a coverup is worse than the offense, and if he can now say: look, I made a sign mistake and I did not treat axb as a pseudo-vector – I am only human, publishing anything else on the archive denying the obvious mistakes can only be achieved by doing other mistakes. And after that he will lose all his mathematical credibility. I plead with him to see reason and stop this self-destruction madness.

 

[DrChinese]

Simple refutation of Joy Christian's simple refutation of Bell's simple theorem

Posted today by Richard Gill, of the Mathematical Institute:

http://arxiv.org/abs/1203.1504

Abstract:

"I point out a simple algebraic error in Joy Christian's refutation of Bell's theorem. In substituting the result of multiplying some derived bivectors with one another by consultation of their multiplication table, he confuses the generic vectors which he used to define the table, with other specific vectors having a special role in the paper, which had been introduced earlier. The result should be expressed in terms of the derived bivectors which indeed do follow this multiplication table. When correcting this calculation, the result is not the singlet correlation any more. Moreover, curiously, his normalized correlations are independent of the number of measurements and certainly do not require letting n converge to infinity. On the other hand his unnormalized or raw correlations are identically equal to -1, independently of the number of measurements too. Correctly computed, his standardized correlations are the bivectors - a . b - a x b, and they find their origin entirely in his normalization or standardization factors; the raw product moment correlations are all -1. I conclude that his research program has been set up around an elaborately hidden but trivial mistake. "

--------------------------------------------

It is interesting to add this note, addressed to those who suggest Jaynes is the only person who properly understands how probability applies to Bell's Theorem, entanglement, etc: Gill is also an expert in statistical theory, and has done extensive research in this area (including the application of Bayes). He apparently does not see the issue Jaynes does. Gill frequently collaborates with the top scientists in the study of entanglement, so I think it is safe to say this area has been well considered and has not been overlooked somehow.

 

[kith]

I conclude that his research program has been set up around an elaborately hidden but trivial mistake.

Puh, this is definitely not something you want to read in a serious paper addressing your work. ;-)

 

[DrChinese]

Puh, this is definitely not something you want to read in a serious paper addressing your work. ;-)

That would sting. I would say that Gill addressing this shows that top teams take challenges to Bell quite seriously. Gill has previously brought down at least one of the Hess-Philipp stochastic models.

 

[Delta Kilo]

Told you so!

... and no-one actually bothered to look at the half-a-page of math to see the elephants lurking therein.

Well, let's look at eq (5). ...