Local Realistic Hidden Variables Quantum Mechanics (LRHVQM)

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Local Realistic Hidden Variables Quantum Mechanics (LRHVQM)

Dear Friends:

In the past, I have alluded to my belief that quantum mechanics itself, based on its use of the Pauli matrices and Pauli linear algebra, can in fact be made local and realistic, and that the uncertainty principle when properly-understood is actually a "reality hiding" principle which gives rise to a hidden variable which is central to this restoring of locality and realism. I have also said that I will not make any claims until I am ready to make a claim. Now I am ready.

In a 7-page document linked at https://jayryablon.files.wordpress.com/ ... lrhvqm.pdf, for the first time, I am comfortable enough with my own analysis of whether QM can be made local and realistic, on its own terms, to put it out in public.

My conclusion: Indeed it can!

And I emphasize that the attached is based entirely on Pauli linear algebra with many are familiar, with no use of geometric algebra.

I will provide my assessment of how this may or may not relate to Joy Christian's work in a separate post in the near future. But for the moment, I want to stay centered in "Pauli linear algebra land" and get your feedback whether you find the attached to be a convincing case in favor of local, realistic, hidden variable quantum mechanics (LRHVQM). If there are aspects of this for which you wish further elaboration or detail, please let me know and I will be pleased to try to do so. I know I usually go into too much detail. Here, I have really endeavored to be brief, and was able to because I finally have a very clear view of how all the pieces fit together. Enjoy!

Jay
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

Interesting work Jay! I had a quick read but will take a second more careful read to make sure I understand the math.
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

So here you have it folks. There can be no further doubt that QM is in fact local for the EPR-Bohm scenario!

Using eigenvalues, Jay's manifestly local measurement functions are essentially equivalent to the following upon implementing the polarizer functions.

$A({\bf a},\, h) := +\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf a}\cdot{\bf s}){\bf a}}\Big[\lambda_{a, s\, h} \Big]=+\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf a}\cdot{\bf s}){\bf a}}\Big[{\bf a}\cdot {\bf s}+i\,h\,||{\bf a} \times {\bf s}|| \Big]$ $= +\, \text{sgn}({\bf a}\cdot{\bf s}) =\pm 1$

$B({\bf b},\, h) :=-\lim_{{\bf s} \rightarrow \, \text{sgn}({\bf b}\cdot{\bf s}){\bf b}}\Big[\lambda_{s, b\, h}\Big]=-\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf b}\cdot{\bf s}){\bf b}}\Big[{\bf s}\cdot {\bf b}+i\,h\,||{\bf s} \times {\bf b}|| \Big]$ $= -\, \text{sgn}({\bf s}\cdot{\bf b})= \mp 1$

where $h=\pm 1$ is the hidden variable. This is more of the beginning of "The New Quantum Mechanics".
.
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

FrediFizzx wrote:So here you have it folks. There can be no further doubt that QM is in fact local for the EPR-Bohm scenario!

Using eigenvalues, Jay's manifestly local measurement functions are essentially equivalent to the following upon implementing the polarizer functions.

$A({\bf a},\, h) := +\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf a}\cdot{\bf s}){\bf a}}\Big[\lambda_{a, s\, h} \Big]=+\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf a}\cdot{\bf s}){\bf a}}\Big[{\bf a}\cdot {\bf s}+i\,h\,||{\bf a} \times {\bf s}|| \Big]$ $= +\, \text{sgn}({\bf a}\cdot{\bf s}) =\pm 1$

$B({\bf b},\, h) :=-\lim_{{\bf s} \rightarrow \, \text{sgn}({\bf b}\cdot{\bf s}){\bf b}}\Big[\lambda_{s, b\, h}\Big]=-\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf b}\cdot{\bf s}){\bf b}}\Big[{\bf s}\cdot {\bf b}+i\,h\,||{\bf s} \times {\bf b}|| \Big]$ $= -\, \text{sgn}({\bf s}\cdot{\bf b})= \mp 1$

where $h=\pm 1$ is the hidden variable. This is more of the beginning of "The New Quantum Mechanics".
.

Can you confirm my reading of this, that sequence of equalities here allows us to deduce

$A({\bf a},\, h) = +\, \text{sgn}({\bf a}\cdot{\bf s}),$

$B({\bf b},\, h) = -\, \text{sgn}({\bf b}\cdot{\bf s}),$

rather similar to formulas which I saw before in the writings of J.S. Bell?

Is it true that the right-hand sides do not depend on the hidden variable $h=\pm 1$, but do depend on another (hidden?) variable $\bf s$? And that ${\bf b}\cdot{\bf s} = {\bf s}\cdot{\bf b}$ ?
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

Did you read Jay's new paper?
.
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

FrediFizzx wrote:Did you read Jay's new paper?
.

Not yet in detail.

He can also answer my questions, if you can't. And I will read the paper, soon...

I don't see the formulas you quote in the paper.

He does say "so, the strong correlation is now connected to binary-valued measurement functions, on a local and realistic footing" but I don't see his definitions of "local" and "realistic". Over the years, there have been tens or even hundreds of papers redefining the words "local" and "realistic" in order to enable the authors to say something like he says in his new paper. The many worlds people have been saying this for years. The most recent and most serious one (many pages of heavy-duty mathematical logic and mathematical philosophy) being that of Gilles Brassard and Paul Raymond-Robichaud.

https://arxiv.org/abs/1710.01380

I would also be interested to know Jay's opinion as to whether his new re-writing of familiar derivations (ie, the derivation of the singlet correlations) can be used to win my 64 thousand dollar computer programming challenge. If he can do that, he will certainly get the Nobel prize, and all doubt about Bell's theorem would be erased, for ever. I hope he will demonstrate his computer programs at our symposium
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

gill1109 wrote:
FrediFizzx wrote:Did you read Jay's new paper?
.

Not yet in detail.

He can also answer my questions, if you can't. And I will read the paper, soon...

I don't see the formulas you quote in the paper.

Eqs. (16) and (17). The answers to your questions are really obvious. You basically answered them yourself.
.
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

FrediFizzx wrote:
gill1109 wrote:
FrediFizzx wrote:Did you read Jay's new paper?
.

Not yet in detail.

He can also answer my questions, if you can't. And I will read the paper, soon...

I don't see the formulas you quote in the paper.

Eqs. (16) and (17). The answers to your questions are really obvious. You basically answered them yourself.
.

Thank you!

Yes, but I had already seen (16) and (17). But these equations don't make any sense. A term which clearly depends on a, s and h is stated to be equal to a term which only depends on a and h. These equations must be wrong. Or only true in special circumstances.

And a limit in which "s" occurs as a dummy variable is supposed to be equal to an expression in which "s" is still present. This is just gobbeldly-gook. Jay has got to get back to the drawing board.

[No disrepect intended! Just my opinion. it's forty-five years since I did calculus 101]
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

gill1109 wrote:
FrediFizzx wrote:So here you have it folks. There can be no further doubt that QM is in fact local for the EPR-Bohm scenario!

Using eigenvalues, Jay's manifestly local measurement functions are essentially equivalent to the following upon implementing the polarizer functions.

$A({\bf a},\, h) := +\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf a}\cdot{\bf s}){\bf a}}\Big[\lambda_{a, s\, h} \Big]=+\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf a}\cdot{\bf s}){\bf a}}\Big[{\bf a}\cdot {\bf s}+i\,h\,||{\bf a} \times {\bf s}|| \Big]$ $= +\, \text{sgn}({\bf a}\cdot{\bf s}) =\pm 1$

$B({\bf b},\, h) :=-\lim_{{\bf s} \rightarrow \, \text{sgn}({\bf b}\cdot{\bf s}){\bf b}}\Big[\lambda_{s, b\, h}\Big]=-\lim_{{\bf s} \rightarrow\, \text{sgn}({\bf b}\cdot{\bf s}){\bf b}}\Big[{\bf s}\cdot {\bf b}+i\,h\,||{\bf s} \times {\bf b}|| \Big]$ $= -\, \text{sgn}({\bf s}\cdot{\bf b})= \mp 1$

where $h=\pm 1$ is the hidden variable. This is more of the beginning of "The New Quantum Mechanics".
.

Can you confirm my reading of this, that sequence of equalities here allows us to deduce

$A({\bf a},\, h) = +\, \text{sgn}({\bf a}\cdot{\bf s}),$

$B({\bf b},\, h) = -\, \text{sgn}({\bf b}\cdot{\bf s}),$
rather similar to formulas which I saw before in the writings of J.S. Bell?

Yes.
gill1109 wrote:Is it true that the right-hand sides do not depend on the hidden variable $h=\pm 1$,. . .

Sort of, but the better way to think about it is this. What Fred writes as a "limit" I prefer to write as an "observation transformation," because limits have a calculus meaning and I do not see any connection to calculus here. Prior to observation there is a dependency on h. But at and following observation, because $\bf s\rightarrow \pm \bf a$ and $-\bf s\rightarrow \pm \bf b$, the cross products become zero, so the term with the hidden variable becomes $h \cdot 0 = 0$. So any remnant of the hidden variables is "destroyed" upon observation.
gill1109 wrote:. . . but do depend on another (hidden?) variable $\bf s$?

Yes, I would say that s is also hidden.
gill1109 wrote:And that ${\bf b}\cdot{\bf s} = {\bf s}\cdot{\bf b}$ ?

Yes

I see some new comments posted the past half hour. I will try to reply later today.

Jay
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

Yablon wrote:
gill1109 wrote:Can you confirm my reading of this, that sequence of equalities here allows us to deduce

$A({\bf a},\, h) = +\, \text{sgn}({\bf a}\cdot{\bf s}),$

$B({\bf b},\, h) = -\, \text{sgn}({\bf b}\cdot{\bf s}),$
rather similar to formulas which I saw before in the writings of J.S. Bell?

Yes.
[...]
Jay

But this is just Bell's example (9) in his original paper. And we know that this does not reproduce the quantum correlations; in fact the correlation in this case can be shown to be $-1 + \frac{2}{\pi}\theta$, where $\theta$ is the angle between the detector settings.
Heinera

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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

Heinera wrote:
Yablon wrote:
gill1109 wrote:Can you confirm my reading of this, that sequence of equalities here allows us to deduce

$A({\bf a},\, h) = +\, \text{sgn}({\bf a}\cdot{\bf s}),$

$B({\bf b},\, h) = -\, \text{sgn}({\bf b}\cdot{\bf s}),$
rather similar to formulas which I saw before in the writings of J.S. Bell?

Yes.
[...]
Jay

But this is just Bell's example (9) in his original paper. And we know that this does not reproduce the quantum correlations; in fact the correlation in this case can be shown to be $-1 + \frac{2}{\pi}\theta$, where $\theta$ is the angle between the detector settings.

Well, if you do the product calculation, A*B, you have (+/-1)(+/-1) = (+/-1). Doesn't really tell us a lot. However, if you do Jay's product calculation you get -a.b. So you probably should read the paper.
.
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

FrediFizzx wrote:Well, if you do the product calculation, A*B, you have (+/-1)(+/-1) = (+/-1).
.

Well, that was an extremely rudimentary calculation. It is of course possible to do much better than that, by evaluating the integral
$\int p({\bf s}) \, \text{sgn}({\bf a\cdot s}) \text{sgn}({\bf b\cdot s})\, d {\bf s} = -1 + \frac{2}{\pi}\theta$.
Heinera

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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

Heinera wrote:
FrediFizzx wrote:Well, if you do the product calculation, A*B, you have (+/-1)(+/-1) = (+/-1).
.

Well, that was an extremely rudimentary calculation. It is of course possible to do much better than that, by evaluating the integral
$\int p({\bf s}) \, \text{sgn}({\bf a\cdot s}) \text{sgn}({\bf b\cdot s})\, d {\bf s} = -1 + \frac{2}{\pi}\theta$.

Well, I don't think you are doing something right because I get,
$\int p({\bf s}) \, (\pm 1)(\pm 1)\, d {\bf s} = \pm {\bf s}$
.
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

gill1109 wrote:. . . I had already seen (16) and (17). But these equations don't make any sense. A term which clearly depends on a, s and h is stated to be equal to a term which only depends on a and h. These equations must be wrong. Or only true in special circumstances.

And a limit in which "s" occurs as a dummy variable is supposed to be equal to an expression in which "s" is still present. This is just gobbeldly-gook. Jay has got to get back to the drawing board.

[No disrepect intended! Just my opinion. it's forty-five years since I did calculus 101]

Hi Richard,

I do not believe I have to go back to the drawing board. But, I will agree that there is a lot to be discussed, and the questions you raised are fair questions.

So, let's take this all a step at a time. Below I have reproduced my equation (1) and a like-equation (1) for b. These are the Pauli identities.

Q1: Do you agree that these are correct equations when the vectors a and s are in the same Hilbert space for (1), and when s and b are in the same Hilbert space for (1) for b?

Next, I have written (16) and (17) to only show the part before the right arrow, and to show that (16) is a function of a, s and h, and that (17) is a function of s, b and h.

Q2: Do you agree that these contain correct calculations of the eigenvalues for the two equations (1), and that in general they are properly written down, including that h=+1 and h=-1 represent the two possible eigenvalues based on the Pauli matrices being 2x2 operators?

Q3: Though I did not show the calculation because it is a standard calculation, do you have any reason to disbelieve, when the two related eigenstates are normalized to sum to 1, and using the probabilistic interpretations customarily applied to 2x2 operators in quantum mechanics, that there is a 50% probability for h=+1 and 50% for h=-1?

Let me also ask on more thing:

Q4: Expressions such as sgn (a dot s) = +/-1 appear regularly in the EPRB literature. When you see such an expression, what does it mean to you, physically?

Let's go from there.

Jay
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

Yablon wrote:
gill1109 wrote:. . . I had already seen (16) and (17). But these equations don't make any sense. A term which clearly depends on a, s and h is stated to be equal to a term which only depends on a and h. These equations must be wrong. Or only true in special circumstances.

And a limit in which "s" occurs as a dummy variable is supposed to be equal to an expression in which "s" is still present. This is just gobbeldly-gook. Jay has got to get back to the drawing board.

[No disrepect intended! Just my opinion. it's forty-five years since I did calculus 101]

Hi Richard,

I do not believe I have to go back to the drawing board. But, I will agree that there is a lot to be discussed, and the questions you raised are fair questions.

So, let's take this all a step at a time. Below I have reproduced my equation (1) and a like-equation (1) for b. These are the Pauli identities.

Q1: Do you agree that these are correct equations when the vectors a and s are in the same Hilbert space for (1), and when s and b are in the same Hilbert space for (1) for b?

Next, I have written (16) and (17) to only show the part before the right arrow, and to show that (16) is a function of a, s and h, and that (17) is a function of s, b and h.

Q2: Do you agree that these contain correct calculations of the eigenvalues for the two equations (1), and that in general they are properly written down, including that h=+1 and h=-1 represent the two possible eigenvalues based on the Pauli matrices being 2x2 operators?

Q3: Though I did not show the calculation because it is a standard calculation, do you have any reason to disbelieve, when the two related eigenstates are normalized to sum to 1, and using the probabilistic interpretations customarily applied to 2x2 operators in quantum mechanics, that there is a 50% probability for h=+1 and 50% for h=-1?

Let me also ask on more thing:

Q4: Expressions such as sgn (a dot s) = +/-1 appear regularly in the EPRB literature. When you see such an expression, what does it mean to you, physically?

Let's go from there.

Jay

Dear Jay, dear friends

I respond to the matters of notation.

Note that Jay’s bold a, b and s are vectors in R^3. They are not operators on the same Hilbert space.
His bold sigma is a vector of three 2x2 matrices
His “Id” is the 2x2 identity matrix
All his 2x2 matrices can be thought of as operators on the same complex Hilbert space of dimension 2
There is (up to isomorphism) only one 2-dimensional Hilbert space and we may represent it with C^2

I have no problems with his equation (1)

There is a bold a in the middle expression of its unnumbered friend which can’t be right. But the outside equality (first expression equals third) is just rewriting (1) with roles of some vectors replaced and multiplying throughout by minus one.

Equations (1) and its friend (corrected) are therefore just two times the same familiar identity, just changing names of things in a consistent way on left and right hand sides; please fix the middle expression of the second identity

Equations (16) and (17) are, I take it, definitions. The notation is obscure. Why does lambda have two subscripts *and* three arguments? If on the right hand sides one merges h with s, one could take (16) as definition of something which has just two arguments, or just two subscripts (but not both) (depending on which notation one prefers). (17) is superfluous since it is just a different way to say the same thing, similar to Jay’s rewriting of (1) as its companion unnumbered equation.

One could also define something which has which has just three arguments, or just three subscripts (but not both).

Jay’s paper, his notation!
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

FrediFizzx wrote:
Heinera wrote:
FrediFizzx wrote:Well, if you do the product calculation, A*B, you have (+/-1)(+/-1) = (+/-1).
.

Well, that was an extremely rudimentary calculation. It is of course possible to do much better than that, by evaluating the integral
$\int p({\bf s}) \, \text{sgn}({\bf a\cdot s}) \text{sgn}({\bf b\cdot s})\, d {\bf s} = -1 + \frac{2}{\pi}\theta$.

Well, I don't think you are doing something right because I get,
$\int p({\bf s}) \, (\pm 1)(\pm 1)\, d {\bf s} = \pm {\bf s}$
.

This makes absolutely no sense. There is not much point in discussing Bell's theorem if you don't understand integrals. For a correct evaluation of the integral, see Peres eqn. (6.21) on page 161.
Heinera

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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

Heinera wrote:
FrediFizzx wrote:
Heinera wrote:
FrediFizzx wrote:Well, if you do the product calculation, A*B, you have (+/-1)(+/-1) = (+/-1).
.

Well, that was an extremely rudimentary calculation. It is of course possible to do much better than that, by evaluating the integral
$\int p({\bf s}) \, \text{sgn}({\bf a\cdot s}) \text{sgn}({\bf b\cdot s})\, d {\bf s} = -1 + \frac{2}{\pi}\theta$.

Well, I don't think you are doing something right because I get,
$\int p({\bf s}) \, (\pm 1)(\pm 1)\, d {\bf s} = \pm {\bf s}$
.

This makes absolutely no sense. There is not much point in discussing Bell's theorem if you don't understand integrals. For a correct evaluation of the integral, see Peres eqn. (6.21) on page 161.

There is absolutely nothing wrong with what I did. Is not the sgn(n.s) = +/-1? Sure they are thus you should be able to substitute in the integral and that is the result Mathematica gives. So of course it doesn't make much sense to worry about the sgn(n.s) = +/-1. It's the wrong product calculation!
.
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

Heinera wrote:
FrediFizzx wrote:Well, if you do the product calculation, A*B, you have (+/-1)(+/-1) = (+/-1).
.

Well, that was an extremely rudimentary calculation. It is of course possible to do much better than that, by evaluating the integral
$\int p({\bf s}) \, \text{sgn}({\bf a\cdot s}) \text{sgn}({\bf b\cdot s})\, d {\bf s} = -1 + \frac{2}{\pi}\theta$.

Your notation is not very good. If it were true, I would write something like
$\int \text{sgn}({\bf a\cdot s}) \text{sgn}({\bf b\cdot s})\, \mu(\text{d} {\bf s}) = -1 + \frac{2}{\pi}\theta$ where $\mu$ is the uniform Haar measure on the sphere $S^2$. But maybe you should first restrict attention to directions and a hidden variable on the circle $S^1$.

One certainly does have to worry about the sign. The devil is in the details. One also has to worry about Mathematica's precise rules for evaluating mathematical expressions, and how Mathematica interprets multiple occurrences of +/- and -/+
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

gill1109 wrote:
Heinera wrote:
FrediFizzx wrote:Well, if you do the product calculation, A*B, you have (+/-1)(+/-1) = (+/-1).
.

Well, that was an extremely rudimentary calculation. It is of course possible to do much better than that, by evaluating the integral
$-\int p({\bf s}) \, \text{sgn}({\bf a\cdot s}) \text{sgn}({\bf b\cdot s})\, d {\bf s} = -1 + \frac{2}{\pi}\theta$.

Your notation is not very good. If it were true, I would write something like
$\int -\text{sgn}({\bf a\cdot s}) \text{sgn}({\bf b\cdot s})\, \mu(\text{d} {\bf s}) = -1 + \frac{2}{\pi}\theta$ where $\mu$ is the uniform Haar measure on the sphere $S^2$.

It's the same, I was just using Bell's notation to avoid unnecessary confusion. Here $p({\bf s})$ is the probability density for the uniform (rotationally invariant) distribution on the sphere. (But I forgot a minus in the integrand, corrected now in the quotes above).

There is no doubt the result is correct; see page 161 in Peres for the derivation. It's just the familiar piecewise linear zigzag correlation.

gill1109 wrote:One certainly does have to worry about the sign. The devil is in the details. One also has to worry about Mathematica's precise rules for evaluating mathematical expressions, and how Mathematica interprets multiple occurrences of +/- and -/+

I doubt Mathematica can do this integral without a lot of tweaking, but I haven't checked.
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Re: Local Realistic Hidden Variables Quantum Mechanics (LRHV

gill1109 wrote:Dear Jay, dear friends

I respond to the matters of notation.

Note that Jay’s bold a, b and s are vectors in R^3. They are not operators on the same Hilbert space.
His bold sigma is a vector of three 2x2 matrices
His “Id” is the 2x2 identity matrix
All his 2x2 matrices can be thought of as operators on the same complex Hilbert space of dimension 2
There is (up to isomorphism) only one 2-dimensional Hilbert space and we may represent it with C^2

I have no problems with his equation (1)

There is a bold a in the middle expression of its unnumbered friend which can’t be right. But the outside equality (first expression equals third) is just rewriting (1) with roles of some vectors replaced and multiplying throughout by minus one.

Equations (1) and its friend (corrected) are therefore just two times the same familiar identity, just changing names of things in a consistent way on left and right hand sides; please fix the middle expression of the second identity

Equations (16) and (17) are, I take it, definitions. The notation is obscure. Why does lambda have two subscripts *and* three arguments? If on the right hand sides one merges h with s, one could take (16) as definition of something which has just two arguments, or just two subscripts (but not both) (depending on which notation one prefers). (17) is superfluous since it is just a different way to say the same thing, similar to Jay’s rewriting of (1) as its companion unnumbered equation.

One could also define something which has which has just three arguments, or just three subscripts (but not both).

Jay’s paper, his notation!

Dear Richard and other friends,

As to notation: Yes, the friend of equation (1) has a typo. The cross product should be sxb.

Equations (16) and (17) are NOT definitions. They are the respective eigenvalues calculated for (1) and (1) friend. The calculation is in equations (2) through (4) of https://jayryablon.files.wordpress.com/ ... lrhvqm.pdf.

My Q2 below is whether there is agreement that these are in fact the correct eigenvalues, with h=+/-1. Lambda does have three subscripts and I also showed these same subscripts as arguments. The subscripts are not clear from the JPG file so it may look like there are two subscripts. My notation, yes. I just want to know if anybody disagrees with (16) and (17), left side of the right arrow, being the correct eigenvalues of (1) and (1) friend.

Then, I'd like to hear in response to my Q3 and Q4.

Jay
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