If you are aware of an interesting new academic paper (that has been published in a peer-reviewed journal or has appeared on the arXiv), a conference talk (at an official professional scientific meeting), an external blog post (by a professional scientist) or a news item (in the mainstream news media), which you think might make an interesting topic for an FQXi blog post, then please contact us at forums@fqxi.org with a link to the original source and a sentence about why you think that the work is worthy of discussion. Please note that we receive many such suggestions and while we endeavour to respond to them, we may not be able to reply to all suggestions.

Please also note that we do not accept unsolicited posts and we cannot review, or open new threads for, unsolicited articles or papers. Requests to review or post such materials will not be answered. If you have your own novel physics theory or model, which you would like to post for further discussion among then FQXi community, then please add them directly to the "Alternative Models of Reality" thread, or to the "Alternative Models of Cosmology" thread. Thank you.

Please also note that we do not accept unsolicited posts and we cannot review, or open new threads for, unsolicited articles or papers. Requests to review or post such materials will not be answered. If you have your own novel physics theory or model, which you would like to post for further discussion among then FQXi community, then please add them directly to the "Alternative Models of Reality" thread, or to the "Alternative Models of Cosmology" thread. Thank you.

Forum Home

Introduction

Terms of Use

RSS feed | RSS help

Introduction

Terms of Use

*Posts by the blogger are highlighted in orange; posts by FQXi Members are highlighted in blue.*

RSS feed | RSS help

RECENT POSTS IN THIS TOPIC

**songjoong df**: *on* 12/27/17 at 7:27am UTC, wrote KLIK DISINI KLIK DISINI KLIK DISINI KLIK DISINI KLIK DISINI KLIK...

**Darren**: *on* 5/7/12 at 13:30pm UTC, wrote I am glad seeing this continuation, it is very much informative where I can...

**Joy Christian**: *on* 4/2/12 at 11:15am UTC, wrote Richard Gill, In my opinion you are an incompetent mathematician who is...

**Lawrence B. Crowell**: *on* 3/29/12 at 23:17pm UTC, wrote The statement by Bohr indicates the difficulty, initially pointed out by...

**Jason Wolfe**: *on* 3/29/12 at 20:32pm UTC, wrote Pencho, You said, "Is it true? Clever Einsteinians know that there is no...

**Pentcho Valev**: *on* 3/29/12 at 20:14pm UTC, wrote Jason Wolfe wrote: "Some may argue that the relationship between gravity...

**Jason Wolfe**: *on* 3/29/12 at 19:49pm UTC, wrote Some may argue that the relationship between gravity and light is already...

**Jason Wolfe**: *on* 3/29/12 at 19:25pm UTC, wrote We take measurements so that we can understand how nature works; we define...

RECENT FORUM POSTS

**Jason Wolfe**: "Hi Georgina, Steve, What is reality? The humorous answer, almost at the..."
*in* Schrödinger’s Zombie:...

**Jason Wolfe**: "Joe, What you are saying sounds like mathematics. But mathematics doesn't..."
*in* First Things First: The...

**Joe Fisher**: "Jason, You can only unnaturally make an infinite number of finite written..."
*in* First Things First: The...

**Jason Wolfe**: "As for religious fundamentalists, I would rather deal with them, then with..."
*in* More on agency from the...

**Jason Wolfe**: "The best we can do with the environment is to plant more trees and..."
*in* More on agency from the...

**gmail login**: "Thanks a lot for the post. It has helped me get some nice ideas. I hope I..."
*in* Bonus Koan: A Lake of...

**Georgina Woodward**: "Steve, I don't think the quantum representation of the hydrogen atom is an..."
*in* Schrödinger’s Zombie:...

**SAJ Real Estate**: "Nice one. Real Estate Sales St Kitts"
*in* A Close Encounter

RECENT ARTICLES

*click titles to read articles*

**First Things First: The Physics of Causality**

Why do we remember the past and not the future? Untangling the connections between cause and effect, choice, and entropy.

**Can Time Be Saved From Physics?**

Philosophers, physicists and neuroscientists discuss how our sense of time’s flow might arise through our interactions with external stimuli—despite suggestions from Einstein's relativity that our perception of the passage of time is an illusion.

**Thermo-Demonics**

A devilish new framework of thermodynamics that focuses on how we observe information could help illuminate our understanding of probability and rewrite quantum theory.

**Gravity's Residue**

An unusual approach to unifying the laws of physics could solve Hawking's black-hole information paradox—and its predicted gravitational "memory effect" could be picked up by LIGO.

**Could Mind Forge the Universe?**

Objective reality, and the laws of physics themselves, emerge from our observations, according to a new framework that turns what we think of as fundamental on its head.

RECENT FORUM POSTS

RECENT ARTICLES

Why do we remember the past and not the future? Untangling the connections between cause and effect, choice, and entropy.

Philosophers, physicists and neuroscientists discuss how our sense of time’s flow might arise through our interactions with external stimuli—despite suggestions from Einstein's relativity that our perception of the passage of time is an illusion.

A devilish new framework of thermodynamics that focuses on how we observe information could help illuminate our understanding of probability and rewrite quantum theory.

An unusual approach to unifying the laws of physics could solve Hawking's black-hole information paradox—and its predicted gravitational "memory effect" could be picked up by LIGO.

Objective reality, and the laws of physics themselves, emerge from our observations, according to a new framework that turns what we think of as fundamental on its head.

FQXi BLOGS

November 17, 2019

This follows on from my post last week, “To Be or Not To Be (a Local Realist),” which discusses Joy Christian’s “Disproof of Bell’s Theorem.” Joy’s response to my post is here.

As I was stating at the end of part one, I am grateful to Joy Christian for give such a detailed answer in “Quantum Music from a Classical Sphere.” I was puzzled by his arguments and I was preparing several short replies to various points, when I realized this was getting out of hand and it was better to have a single coherent reply in a new post.

Let me start by saying that the topics we are presenting here are hard and there is no absolute agreement even between experts on the very definitions of what one means by locality, realism or contextuality, or a no-go theorem for hidden variable theories. Some people consider quantum mechanics as local and non-realistic, while others insist on non-locality. Contextual hidden variable theories encompass a wide and ill-defined domain that is probably best defined by their opposition to the much sharper-defined non-contextual hidden variable theories. Some people think that non-contextual hidden variable theories were rejected by the theorems of Gleason and Kochen- Specker and Bell’s theorem is important only to reject some additional contextual hidden variable theories. Other people point out that the Kochen- Specker theorem cannot be put to any experimental test, and the importance of Bell’s theorem is to reject non-contextual hidden variable theories. Finally, some people think contextual hidden variable theories are unphysical, while others think they are the future of physics.

To cut through this confusing mess, the best approach is to rely on accurate historical narratives. This would provide a stable setting for the meanings of the terms used, which would allow a common discussion ground where we can compare apples to apples. Also it is best to seek experimental results to settle disputes instead of debates, however interesting. Last, speculation about future results is best not to be discussed at all. If those tenets are followed, charges of “alarming confusions” and a fruitless dialogue can be avoided.

With this preparation, let me attempt to explain what contextuality for hidden variable is, why Joy’s result is contextual, and what it all means.

First some preliminaries. Earlier I discussed the EPR paper and then I jumped to Bell’s contribution. However, historically Bohr also made an early critical observation in his attempt to refute EPR’s argument. Bohr was using the good old fashioned complementarity principle: in some experiments the particles behaves like waves, while in others like point particles. However, one cannot see them both in the same experiment. Therefore this invalidates EPR’s argument where on one arm of the experiment one behavior is measured, while in the other arm the complementary behavior is measured. To justify this experimentally, Bohr noted that observations depend not only on the state of the system but also on the disposition of the apparatus. And indeed, no physicist is ever going to deny this. Joy mentioned general relativity, and I can also point out for example electrostatics, and any other theories requiring boundary value problems. Joy considers this “contextuality,” but I would argue that this is clearly an overreaching usage of the term, which in hidden variable theories has a very different meaning which I will define later in its historical context. For the sake of the clarity, let’s call this “Bohr’s contextuality”.

As a supporting argument for his position, Joy mentioned Bell’s classical paper “On the problem of hidden variables in quantum mechanics.” However, Bell’s paper has a slightly different intention than Joy presented. Bell did not have an abstract for this paper and please allow me to explain how I understand this famous paper.

Einstein once said: “God does not play dice” and Bohr replied: “Do not tell God what to do.” In a similar vein, I would say Bell’s main idea of the paper is: “Do not tell me how to construct my hidden variable theories.” Requirements which are obeyed by quantum mechanics are not necessarily obeyed by hidden variable theories, which should only reconstruct quantum mechanics’ predictions.

For the interest of time and space let me only discuss Bell’s criticism of von Neumann and Gleason in this paper. Von Neumann had enormous influence in quantum mechanics due to his seminal work on axiomatizing it. He produced a “no-go” theorem on hidden variable theories which Bell found unjustified in its assumptions. In particular, von Neumann required that the linear combination of expectation values is the expectation value of the combination. While true in quantum mechanics, this is too strong to be demanded for any particular value of the hidden variable, as an elementary example can show (from Reflections on Relativity by Kevin Brown):

Suppose we have two variables X and Y and two hidden variables 1 and 2. Suppose for hidden variable 1: X=2, Y=5 and X+Y=5 while for hidden variable 2: X=4, Y=3 and X+Y=9. Averaging for hidden variables 1 and 2, avg(x) = 3, avg(Y) = 4, and avg(X+Y)=7 and the sum of the averages is the average of the sums. On each hidden variable however, this property is not valid. (If you are worried about the funny math X+Y = 5 when X = 2 and Y = 5, “X”, “Y”, and “X+Y” are three separate experiments requiring separate experimental setups and the number appearing to the right hand side of the equal sign is the experimental outcome. The “+” in “X+Y” is only part of the label for an experimental setup and not a genuine plus operator.)

Gleason’s theorem improves on von Neumann’s result because it makes it a mathematical necessity to have a particular form of the average (the trace formalism) which does obey von Neumann’s condition for compatible (commuting) observables. Bell’s criticism on Gleason’s result is much more subtle and only here “Bohr’s contextuality” becomes a mandatory part of the argument. One can argue that von Neumann’s condition from above can be dismissed even without the toy example (which is not part of the paper, but I use it for the clarity of the argument), by pointing out that it is nonsensical for non-commutative variables which requires a different experimental setup. However, for compatible commutative variables it is a natural requirement (and this is what is used by Gleason). Here Bell’s analysis becomes subtle. To rule out hidden variables Gleason uses this requirement for commuting observables on spaces of dimensionality higher than two, meaning it has to be applied at least twice: on variables X and Y as well as X and Z, and while X and Y are compatible (commute), and X and Z are compatible as well, X and Z may not be (and the funny math from above can happen). It is only at this point where Bell says: “the result of an observation may reasonably depend not only on the state of the system (including hidden variables) but also on the disposition of the apparatus.”

So, Bell’s main argument was a defense of the freedom to construct a hidden variable theory. Unjustified requirements were also proven unjustified by showing they violate Bohr’s contextuality (but this is only a secondary supporting idea in the paper).

But why bother splitting hairs over the meaning of contextuality? Because it critically appears in many proposals of hidden variable theories aimed at recovering the predictions of quantum mechanics. Everyone agrees that quantum mechanics accurately describes nature, but not everyone is satisfied with quantum mechanics’ interpretation. Why is this so? In the traditional quantum world, there are no safe harbors for quantum objects (pure states are not immune to “collapse”) and you get epistemology (a statistical description) without ontology (objective reality) which generates a sense that something is fundamentally missing in this scheme. Hidden variables are supposed to fill this missing ontological gap and restore the “ignorance interpretation“ for the statistical nature of quantum mechanical predictions. First and foremost, in my opinion, hidden variables have to have a rock solid ontological value (ontological definiteness – typically known as non-contextuality). (Ontological definiteness is a stronger property than mere objective definiteness, which can be claimed for example by some parts of Bohm’s theory. Still, all hidden variable theories corresponding to Hilbert spaces of a dimension higher than two suffer in one form or another of ontological indefiniteness. In the case of Bohm’s theory--a contextual algebraic hidden variable theory--momentum is ill defined but still has objective definiteness. However spin is treated just like in standard quantum mechanics with ontological indefiniteness. I would challenge anyone to name a single hidden variable theory with dimensions higher than two which does not have ANY ontological problems.)

For measurement results, a dependency on the experimental context is completely acceptable. However, if hidden variables or their meaning changes with the experimental context, we are back to square one in terms of the strangeness of quantum mechanics. Not only do hidden variable theories suffer from strangeness, but all interpretations of quantum mechanics aimed at making it palatable to our classical intuition employ a bag of tricks: transactional interpretation is wonderful if you can swallow messages from the future; Bohm’s theory restores classicality at the expense of action at a distance; Everrett’s approach transforms “OR” into “AND” and deprives even classical physics of counterfactual definiteness, etc, etc.

So what do we call this rock solid (ontological) foundation? (Ontological) non-contextuality. If you are unhappy with standard quantum mechanics and its wave/particle complementarity, then you should not be happy with any other ontological flip-flopping explanation regardless of the packaging and other genuinely intuitive but ultimately non-essential features tied to the particular mathematical formalism used (lack of superposition, lack of entanglement, etc). The big pink elephant of epistemology without ontology remains in the middle of the room.

Ontological non-contextuality represents the golden standard for any physical theory aimed at restoring classical intuition. (As a side remark, general relativity and all other classical physical theories obey ontological non-contextuality.) Local realism obeys ontological non-contextuality too. We can now drop the ontological qualifier and simply state either contextual or non-contextual in the remainder of the text. This is the standard meaning of contextuality or non-contextuality when discussing hidden variables (and in this meaning general relativity is non-contextual). All non-contextual hidden variable theories are ruled out by many no-go theorems, and the majority of physicists consider contextual hidden variables unphysical with no experimental evidence ever backing their existence. There are also many flavors of contextual hidden variable theories and we already encountered the algebraic type in Bohm’s theory.

Now we are in a position to put everything together.

What does Bell’s theorem assert (regardless of any fine mathematical print)? Any locally realistic theory obeys certain inequalities which are violated by quantum mechanics and ultimately by nature in experiments. What is Joy’s challenge to Bell’s theorem? Non-commutative beables allows him to go over Bell’s inequality precisely like quantum mechanics. Is Joy’s theory noncontextual as he claims? No. Non-commutative beables are ruled out by Clifton’s theorem for non-contextual theories, meaning Joy’s theory cannot be non-contextual (and I clarified some more what kind of contextual theory it is in my preprint: http://arxiv.org/abs/1107.1007).

But let’s assume that both Clifton and I are mistaken. There is a simple way to put this to the test: if Joy’s claim of non-contextuality is correct, then one can write a computer program modeling his theory and obtain Tsirelson’s bounds. (This is not my idea; I got it from researching the web for reactions to Joy’s results.) I claim this cannot be done. If anyone manages to actually write such a program which goes above Bell’s limit then I admit I am completely wrong and Joy is right in his non-contextuality claim. And if he is right on his theory’s non-contextuality claim, then he did manage to disprove Bell’s theorem as usually understood by the physics community.

In the meantime, let me explain why I think Joy’s theory cannot be modeled on a computer and why it is not a strictly locally realistic theory.

Recently Joy published http://arxiv.org/abs/1106.0748. It is relevant to note that the state space (encoding the complete information) for spin 1/2 is SU(2) which is isomorphic with SO(3) where geometric algebra can naturally be used. There is an additional degree of freedom coming from the double cover property of the isomorphism and this allows for the possibility of a hidden variable which encodes the information on which one-to-two map you are located. It is not hard to see that Joy’s formalism when done consistently in the geometric algebra formalism is an equivalent representation of the standard quantum mechanics and his agreement with quantum mechanics comes as no surprise. However, Joy’s formalism lacks a “Born rule” which translates his formalism into the actual experimental outcomes (real numbers). If in http://arxiv.org/abs/1106.0748 one applies at any intermediate steps a map from geometric algebra objects to real numbers, it is easy to see that one only recovers Bell’s limit. (In standard complex quantum mechanics formalism this would correspond to applying Born’s rule too early and adding probabilities instead of amplitudes resulting in incorrect classical behavior predictions). But not applying any translation mechanism until the end in Joy’s formulation means that this step is delayed until the interaction taking place during the measurement process (which has to somehow implement/model the geometric algebra operations).Tung Ten Yong (http://arxiv.org/abs/0712.1637) has criticized Joy’s model because the final results are expressed as geometric algebra objects. I do not think this is a decisive criticism of Joy’s theory as Yong claims because if nature is quantum mechanical at heart, and if Joy’s theory reproduces all quantum mechanics results, this means that geometric algebra should be the proper ontology of nature. And then it is very natural to expect interaction during measurement to realize geometric algebra operations.

But geometric algebra objects are non-local and while Joy can call his model local due to factorizability, it is non-local due to the critical usage of a geometric algebra product (during measurement). Locality does not mean only separability between Alice and Bob, but must include a notion of physical distance as well. This additional non-locality would prevent any faithful simulation on a computer program because one cannot program a way to compute the experimental outcome for Alice at one end without accessing all information encoded in SO(3), meaning the inclusion of the information at Bob’s end as well (which is unavailable due to physical separation).

Again, if you disagree with the analysis above, Clifton’s theorems, or my preprint http://arxiv.org/abs/1107.1007, write a computer program modeling Joy’s theory for EPR-Bohm and show you get Tsirelson’s bound. If you succeed, I am wrong, this post is essentially incorrect, Joy’s theory is non-contextual, and Joy did disprove Bell’s theorem by counterexample.

Joy’s theory can be considered both local due to factorizability, and non-local due to geometric algebra. It can also be considered both realistic as it employs classical three dimensional geometrical objects, and non-realistic as it is (ontologically) contextual (the local beables critically depend on the experimental setting).

Finally, the same conclusions from my prior post and preprint stand: Joy does not disprove Bell’s theorem mathematically because he starts with a different assumption, that of a non-commutative beable algebra. I really did not understand Joy’s remark: “In particular, contrary to what Florin asserts, I do not start with mathematical assumptions different from those of Bell,” Quoting Joy (http://arxiv.org/abs/quant-ph/0703244v12): “In fact, apart from the multiplicative non-commutativity, they satisfy exactly the same algebra as do the local beables of Bell. […] Bell’s algebra, by contrast, is a commutative and associative normed division algebra over the field of real numbers.” Also: “Finally, Grangier expresses his unhappiness with the choice of my running title: “Disproof of Bell’s Theorem.” He rightly points out that as a mathematical theorem Bell’s theorem cannot be disproved, since its conclusions follow from well defined premises in a mathematically impeccable manner.”

Contrary to his claims and paper titles, Joy does not disprove the importance of Bell’s theorem as the key result validated by experiments rejecting local realism. If Clifton’s theorem is right or if my spin one analysis is correct, Joy’s theory is a contextual theory not classifiable as locally realistic. This is a scientific falsifiable statement, as any computer simulation of Joy’s theory exceeding Bell’s limit would render both Clifton’s theorems and my preprint wrong.

Last, Joy’s theory does expose a weakness in Bell’s theorem. Clifton’s theorems fill this weakness and Bell and Clifton’s theorem combined do rule out all hidden variable theories obeying local realism because Bell requires commutativity of beable algebra and Clifton shows that beable algebra must be commutative for all non-contextual theories and for any relativistic quantum field theories with bounded energies. As a counterexample, Joy’s theory does limit Bell’s importance in Shimony’s interpretation for some contextual hidden variable theories. However, other physicists including myself disagree with Shimony and place Bell’s theorem’s importance solely for rejecting non-contextual hidden variable theories in an experimentally verifiable fashion (in my preprint’s abstract I presented both points of view).

Scientific conclusions aside, I did enjoy learning about Joy’s result, and I do think they represent very valuable contributions in our understanding of quantum mechanics and hidden variable theories. More interesting work remains to be done and I would be very interested to see any proofs for his “Theorema Egregium” or (Eq. 10 in my preprint). I wish Joy success in further developing his ideas.

this post has been edited by the author since its original submission

report post as inappropriate

As I was stating at the end of part one, I am grateful to Joy Christian for give such a detailed answer in “Quantum Music from a Classical Sphere.” I was puzzled by his arguments and I was preparing several short replies to various points, when I realized this was getting out of hand and it was better to have a single coherent reply in a new post.

Let me start by saying that the topics we are presenting here are hard and there is no absolute agreement even between experts on the very definitions of what one means by locality, realism or contextuality, or a no-go theorem for hidden variable theories. Some people consider quantum mechanics as local and non-realistic, while others insist on non-locality. Contextual hidden variable theories encompass a wide and ill-defined domain that is probably best defined by their opposition to the much sharper-defined non-contextual hidden variable theories. Some people think that non-contextual hidden variable theories were rejected by the theorems of Gleason and Kochen- Specker and Bell’s theorem is important only to reject some additional contextual hidden variable theories. Other people point out that the Kochen- Specker theorem cannot be put to any experimental test, and the importance of Bell’s theorem is to reject non-contextual hidden variable theories. Finally, some people think contextual hidden variable theories are unphysical, while others think they are the future of physics.

To cut through this confusing mess, the best approach is to rely on accurate historical narratives. This would provide a stable setting for the meanings of the terms used, which would allow a common discussion ground where we can compare apples to apples. Also it is best to seek experimental results to settle disputes instead of debates, however interesting. Last, speculation about future results is best not to be discussed at all. If those tenets are followed, charges of “alarming confusions” and a fruitless dialogue can be avoided.

With this preparation, let me attempt to explain what contextuality for hidden variable is, why Joy’s result is contextual, and what it all means.

First some preliminaries. Earlier I discussed the EPR paper and then I jumped to Bell’s contribution. However, historically Bohr also made an early critical observation in his attempt to refute EPR’s argument. Bohr was using the good old fashioned complementarity principle: in some experiments the particles behaves like waves, while in others like point particles. However, one cannot see them both in the same experiment. Therefore this invalidates EPR’s argument where on one arm of the experiment one behavior is measured, while in the other arm the complementary behavior is measured. To justify this experimentally, Bohr noted that observations depend not only on the state of the system but also on the disposition of the apparatus. And indeed, no physicist is ever going to deny this. Joy mentioned general relativity, and I can also point out for example electrostatics, and any other theories requiring boundary value problems. Joy considers this “contextuality,” but I would argue that this is clearly an overreaching usage of the term, which in hidden variable theories has a very different meaning which I will define later in its historical context. For the sake of the clarity, let’s call this “Bohr’s contextuality”.

As a supporting argument for his position, Joy mentioned Bell’s classical paper “On the problem of hidden variables in quantum mechanics.” However, Bell’s paper has a slightly different intention than Joy presented. Bell did not have an abstract for this paper and please allow me to explain how I understand this famous paper.

Einstein once said: “God does not play dice” and Bohr replied: “Do not tell God what to do.” In a similar vein, I would say Bell’s main idea of the paper is: “Do not tell me how to construct my hidden variable theories.” Requirements which are obeyed by quantum mechanics are not necessarily obeyed by hidden variable theories, which should only reconstruct quantum mechanics’ predictions.

For the interest of time and space let me only discuss Bell’s criticism of von Neumann and Gleason in this paper. Von Neumann had enormous influence in quantum mechanics due to his seminal work on axiomatizing it. He produced a “no-go” theorem on hidden variable theories which Bell found unjustified in its assumptions. In particular, von Neumann required that the linear combination of expectation values is the expectation value of the combination. While true in quantum mechanics, this is too strong to be demanded for any particular value of the hidden variable, as an elementary example can show (from Reflections on Relativity by Kevin Brown):

Suppose we have two variables X and Y and two hidden variables 1 and 2. Suppose for hidden variable 1: X=2, Y=5 and X+Y=5 while for hidden variable 2: X=4, Y=3 and X+Y=9. Averaging for hidden variables 1 and 2, avg(x) = 3, avg(Y) = 4, and avg(X+Y)=7 and the sum of the averages is the average of the sums. On each hidden variable however, this property is not valid. (If you are worried about the funny math X+Y = 5 when X = 2 and Y = 5, “X”, “Y”, and “X+Y” are three separate experiments requiring separate experimental setups and the number appearing to the right hand side of the equal sign is the experimental outcome. The “+” in “X+Y” is only part of the label for an experimental setup and not a genuine plus operator.)

Gleason’s theorem improves on von Neumann’s result because it makes it a mathematical necessity to have a particular form of the average (the trace formalism) which does obey von Neumann’s condition for compatible (commuting) observables. Bell’s criticism on Gleason’s result is much more subtle and only here “Bohr’s contextuality” becomes a mandatory part of the argument. One can argue that von Neumann’s condition from above can be dismissed even without the toy example (which is not part of the paper, but I use it for the clarity of the argument), by pointing out that it is nonsensical for non-commutative variables which requires a different experimental setup. However, for compatible commutative variables it is a natural requirement (and this is what is used by Gleason). Here Bell’s analysis becomes subtle. To rule out hidden variables Gleason uses this requirement for commuting observables on spaces of dimensionality higher than two, meaning it has to be applied at least twice: on variables X and Y as well as X and Z, and while X and Y are compatible (commute), and X and Z are compatible as well, X and Z may not be (and the funny math from above can happen). It is only at this point where Bell says: “the result of an observation may reasonably depend not only on the state of the system (including hidden variables) but also on the disposition of the apparatus.”

So, Bell’s main argument was a defense of the freedom to construct a hidden variable theory. Unjustified requirements were also proven unjustified by showing they violate Bohr’s contextuality (but this is only a secondary supporting idea in the paper).

But why bother splitting hairs over the meaning of contextuality? Because it critically appears in many proposals of hidden variable theories aimed at recovering the predictions of quantum mechanics. Everyone agrees that quantum mechanics accurately describes nature, but not everyone is satisfied with quantum mechanics’ interpretation. Why is this so? In the traditional quantum world, there are no safe harbors for quantum objects (pure states are not immune to “collapse”) and you get epistemology (a statistical description) without ontology (objective reality) which generates a sense that something is fundamentally missing in this scheme. Hidden variables are supposed to fill this missing ontological gap and restore the “ignorance interpretation“ for the statistical nature of quantum mechanical predictions. First and foremost, in my opinion, hidden variables have to have a rock solid ontological value (ontological definiteness – typically known as non-contextuality). (Ontological definiteness is a stronger property than mere objective definiteness, which can be claimed for example by some parts of Bohm’s theory. Still, all hidden variable theories corresponding to Hilbert spaces of a dimension higher than two suffer in one form or another of ontological indefiniteness. In the case of Bohm’s theory--a contextual algebraic hidden variable theory--momentum is ill defined but still has objective definiteness. However spin is treated just like in standard quantum mechanics with ontological indefiniteness. I would challenge anyone to name a single hidden variable theory with dimensions higher than two which does not have ANY ontological problems.)

For measurement results, a dependency on the experimental context is completely acceptable. However, if hidden variables or their meaning changes with the experimental context, we are back to square one in terms of the strangeness of quantum mechanics. Not only do hidden variable theories suffer from strangeness, but all interpretations of quantum mechanics aimed at making it palatable to our classical intuition employ a bag of tricks: transactional interpretation is wonderful if you can swallow messages from the future; Bohm’s theory restores classicality at the expense of action at a distance; Everrett’s approach transforms “OR” into “AND” and deprives even classical physics of counterfactual definiteness, etc, etc.

So what do we call this rock solid (ontological) foundation? (Ontological) non-contextuality. If you are unhappy with standard quantum mechanics and its wave/particle complementarity, then you should not be happy with any other ontological flip-flopping explanation regardless of the packaging and other genuinely intuitive but ultimately non-essential features tied to the particular mathematical formalism used (lack of superposition, lack of entanglement, etc). The big pink elephant of epistemology without ontology remains in the middle of the room.

Ontological non-contextuality represents the golden standard for any physical theory aimed at restoring classical intuition. (As a side remark, general relativity and all other classical physical theories obey ontological non-contextuality.) Local realism obeys ontological non-contextuality too. We can now drop the ontological qualifier and simply state either contextual or non-contextual in the remainder of the text. This is the standard meaning of contextuality or non-contextuality when discussing hidden variables (and in this meaning general relativity is non-contextual). All non-contextual hidden variable theories are ruled out by many no-go theorems, and the majority of physicists consider contextual hidden variables unphysical with no experimental evidence ever backing their existence. There are also many flavors of contextual hidden variable theories and we already encountered the algebraic type in Bohm’s theory.

Now we are in a position to put everything together.

What does Bell’s theorem assert (regardless of any fine mathematical print)? Any locally realistic theory obeys certain inequalities which are violated by quantum mechanics and ultimately by nature in experiments. What is Joy’s challenge to Bell’s theorem? Non-commutative beables allows him to go over Bell’s inequality precisely like quantum mechanics. Is Joy’s theory noncontextual as he claims? No. Non-commutative beables are ruled out by Clifton’s theorem for non-contextual theories, meaning Joy’s theory cannot be non-contextual (and I clarified some more what kind of contextual theory it is in my preprint: http://arxiv.org/abs/1107.1007).

But let’s assume that both Clifton and I are mistaken. There is a simple way to put this to the test: if Joy’s claim of non-contextuality is correct, then one can write a computer program modeling his theory and obtain Tsirelson’s bounds. (This is not my idea; I got it from researching the web for reactions to Joy’s results.) I claim this cannot be done. If anyone manages to actually write such a program which goes above Bell’s limit then I admit I am completely wrong and Joy is right in his non-contextuality claim. And if he is right on his theory’s non-contextuality claim, then he did manage to disprove Bell’s theorem as usually understood by the physics community.

In the meantime, let me explain why I think Joy’s theory cannot be modeled on a computer and why it is not a strictly locally realistic theory.

Recently Joy published http://arxiv.org/abs/1106.0748. It is relevant to note that the state space (encoding the complete information) for spin 1/2 is SU(2) which is isomorphic with SO(3) where geometric algebra can naturally be used. There is an additional degree of freedom coming from the double cover property of the isomorphism and this allows for the possibility of a hidden variable which encodes the information on which one-to-two map you are located. It is not hard to see that Joy’s formalism when done consistently in the geometric algebra formalism is an equivalent representation of the standard quantum mechanics and his agreement with quantum mechanics comes as no surprise. However, Joy’s formalism lacks a “Born rule” which translates his formalism into the actual experimental outcomes (real numbers). If in http://arxiv.org/abs/1106.0748 one applies at any intermediate steps a map from geometric algebra objects to real numbers, it is easy to see that one only recovers Bell’s limit. (In standard complex quantum mechanics formalism this would correspond to applying Born’s rule too early and adding probabilities instead of amplitudes resulting in incorrect classical behavior predictions). But not applying any translation mechanism until the end in Joy’s formulation means that this step is delayed until the interaction taking place during the measurement process (which has to somehow implement/model the geometric algebra operations).Tung Ten Yong (http://arxiv.org/abs/0712.1637) has criticized Joy’s model because the final results are expressed as geometric algebra objects. I do not think this is a decisive criticism of Joy’s theory as Yong claims because if nature is quantum mechanical at heart, and if Joy’s theory reproduces all quantum mechanics results, this means that geometric algebra should be the proper ontology of nature. And then it is very natural to expect interaction during measurement to realize geometric algebra operations.

But geometric algebra objects are non-local and while Joy can call his model local due to factorizability, it is non-local due to the critical usage of a geometric algebra product (during measurement). Locality does not mean only separability between Alice and Bob, but must include a notion of physical distance as well. This additional non-locality would prevent any faithful simulation on a computer program because one cannot program a way to compute the experimental outcome for Alice at one end without accessing all information encoded in SO(3), meaning the inclusion of the information at Bob’s end as well (which is unavailable due to physical separation).

Again, if you disagree with the analysis above, Clifton’s theorems, or my preprint http://arxiv.org/abs/1107.1007, write a computer program modeling Joy’s theory for EPR-Bohm and show you get Tsirelson’s bound. If you succeed, I am wrong, this post is essentially incorrect, Joy’s theory is non-contextual, and Joy did disprove Bell’s theorem by counterexample.

Joy’s theory can be considered both local due to factorizability, and non-local due to geometric algebra. It can also be considered both realistic as it employs classical three dimensional geometrical objects, and non-realistic as it is (ontologically) contextual (the local beables critically depend on the experimental setting).

Finally, the same conclusions from my prior post and preprint stand: Joy does not disprove Bell’s theorem mathematically because he starts with a different assumption, that of a non-commutative beable algebra. I really did not understand Joy’s remark: “In particular, contrary to what Florin asserts, I do not start with mathematical assumptions different from those of Bell,” Quoting Joy (http://arxiv.org/abs/quant-ph/0703244v12): “In fact, apart from the multiplicative non-commutativity, they satisfy exactly the same algebra as do the local beables of Bell. […] Bell’s algebra, by contrast, is a commutative and associative normed division algebra over the field of real numbers.” Also: “Finally, Grangier expresses his unhappiness with the choice of my running title: “Disproof of Bell’s Theorem.” He rightly points out that as a mathematical theorem Bell’s theorem cannot be disproved, since its conclusions follow from well defined premises in a mathematically impeccable manner.”

Contrary to his claims and paper titles, Joy does not disprove the importance of Bell’s theorem as the key result validated by experiments rejecting local realism. If Clifton’s theorem is right or if my spin one analysis is correct, Joy’s theory is a contextual theory not classifiable as locally realistic. This is a scientific falsifiable statement, as any computer simulation of Joy’s theory exceeding Bell’s limit would render both Clifton’s theorems and my preprint wrong.

Last, Joy’s theory does expose a weakness in Bell’s theorem. Clifton’s theorems fill this weakness and Bell and Clifton’s theorem combined do rule out all hidden variable theories obeying local realism because Bell requires commutativity of beable algebra and Clifton shows that beable algebra must be commutative for all non-contextual theories and for any relativistic quantum field theories with bounded energies. As a counterexample, Joy’s theory does limit Bell’s importance in Shimony’s interpretation for some contextual hidden variable theories. However, other physicists including myself disagree with Shimony and place Bell’s theorem’s importance solely for rejecting non-contextual hidden variable theories in an experimentally verifiable fashion (in my preprint’s abstract I presented both points of view).

Scientific conclusions aside, I did enjoy learning about Joy’s result, and I do think they represent very valuable contributions in our understanding of quantum mechanics and hidden variable theories. More interesting work remains to be done and I would be very interested to see any proofs for his “Theorema Egregium” or (Eq. 10 in my preprint). I wish Joy success in further developing his ideas.

this post has been edited by the author since its original submission

report post as inappropriate

Dear Florin,

Thank you very much for your detailed reply. I will need to write down some equations to properly address the points you are making. So instead of posting my reply here I will post it in this collection of my ongoing replies. I hope that is alright with you.

All the Best,

Joy

report post as inappropriate

Thank you very much for your detailed reply. I will need to write down some equations to properly address the points you are making. So instead of posting my reply here I will post it in this collection of my ongoing replies. I hope that is alright with you.

All the Best,

Joy

report post as inappropriate

Dear Joy,

I am looking forward to your reply. I hope by now I made clear the motivation and background info for my preprint.

By the way, I have Bell's Speakable and Unspeakable book and in there Bell's papers have no abstracts. I was quite surprised when FQXi found an on-line version with an abstract and I was very eager/nervous to see how Bell himself summarized his paper.

Also I want to let you know what I think about the author of the computer modeling idea. I felt a bit awkward to put this idea in the reply because I think the author engaged in what I consider unethical behavior. However I consider the idea valid and my way of distancing from its author was not cite his name to reward/advertise his dishonest approach.

Best regards,

Florin

report post as inappropriate

I am looking forward to your reply. I hope by now I made clear the motivation and background info for my preprint.

By the way, I have Bell's Speakable and Unspeakable book and in there Bell's papers have no abstracts. I was quite surprised when FQXi found an on-line version with an abstract and I was very eager/nervous to see how Bell himself summarized his paper.

Also I want to let you know what I think about the author of the computer modeling idea. I felt a bit awkward to put this idea in the reply because I think the author engaged in what I consider unethical behavior. However I consider the idea valid and my way of distancing from its author was not cite his name to reward/advertise his dishonest approach.

Best regards,

Florin

report post as inappropriate

Dear Florin,

I appreciate your honesty and courtesy. There is a world of difference between you and the nameless author. Just for the record, his behaviour was far worse than unethical. He has in fact violated several criminal laws.

In any case, I do not buy the simulation argument, as you will find out from my reply.

More later,

Joy

report post as inappropriate

I appreciate your honesty and courtesy. There is a world of difference between you and the nameless author. Just for the record, his behaviour was far worse than unethical. He has in fact violated several criminal laws.

In any case, I do not buy the simulation argument, as you will find out from my reply.

More later,

Joy

report post as inappropriate

Dear Florin,

About a computer simulation of Joy's approach. We can always ask S.L. for the source code, to check it. (I don't think it is a good practice to hide someone's result, even if we suspect he is dishonest. People can verify his results, and decide whether his results are correct or not. [Not whether he is dishonest or not.])

Or we can implement ourselves something like this. Joy can provide the specs and can personally verify each function, to remove any suspicion. Anyway, I predict that if we take the scalar part of the Clifford product of Pauli bivectors, we will always obtain the cos correlation (both for singlet states and for independent states).

Best regards,

Cristi

report post as inappropriate

About a computer simulation of Joy's approach. We can always ask S.L. for the source code, to check it. (I don't think it is a good practice to hide someone's result, even if we suspect he is dishonest. People can verify his results, and decide whether his results are correct or not. [Not whether he is dishonest or not.])

Or we can implement ourselves something like this. Joy can provide the specs and can personally verify each function, to remove any suspicion. Anyway, I predict that if we take the scalar part of the Clifford product of Pauli bivectors, we will always obtain the cos correlation (both for singlet states and for independent states).

Best regards,

Cristi

report post as inappropriate

Dear Joy and Florin,

If I am missing something, I would be grateful if you would explain me. I will say briefly what I think so far about Joy's theory. To make the things clear, I split the theory in two layers.

1. (a simplified version of Joy's theory, as I see it). We want to obtain the The EPR correlation for two particles in a singlet state, which is the cos of the angle between the orientations of the measurement devices. The obvious way to obtain cos is to take the dot product between the two orientations.

2. The second layer - the usage of Pauli's algebra - seems to provide a justification for the 1st layer, but I think that in fact it is just a mean to obtain the dot product, hence the cos, and it obfuscates the idea.

Being it dot or Clifford product, I fail to see other justification for it than to get the desired cos.

But let's assume that Joy's papers present a good reason for this. I don't see why this is considered local, and a disproof of Bell's theorem. The particles are "at a distance", so this operation would be non-local. Also, we can take the dot/Clifford product for any two particles, being from a singlet or not, and get the same cos, which this time is not the correct correlation. So, we would have to say that if they are entangled, we can dot them, if not, we should do something else. So we couldn't get rid of entanglement.

Briefly, I think that the "exact sequence" is this:

justify "space algebra" -> take Clifford product -> keep dot product -> get cos correlations

and at this moment I don't see other reason for this approach than to get the cos. And the approach itself sounds to me like a "spooky Clifford product at a distance" :-), so I wouldn't consider it a disproof of Bell's theorem.

Are the things really that simple as I see them, or I am missing something?

Cristi

report post as inappropriate

If I am missing something, I would be grateful if you would explain me. I will say briefly what I think so far about Joy's theory. To make the things clear, I split the theory in two layers.

1. (a simplified version of Joy's theory, as I see it). We want to obtain the The EPR correlation for two particles in a singlet state, which is the cos of the angle between the orientations of the measurement devices. The obvious way to obtain cos is to take the dot product between the two orientations.

2. The second layer - the usage of Pauli's algebra - seems to provide a justification for the 1st layer, but I think that in fact it is just a mean to obtain the dot product, hence the cos, and it obfuscates the idea.

Being it dot or Clifford product, I fail to see other justification for it than to get the desired cos.

But let's assume that Joy's papers present a good reason for this. I don't see why this is considered local, and a disproof of Bell's theorem. The particles are "at a distance", so this operation would be non-local. Also, we can take the dot/Clifford product for any two particles, being from a singlet or not, and get the same cos, which this time is not the correct correlation. So, we would have to say that if they are entangled, we can dot them, if not, we should do something else. So we couldn't get rid of entanglement.

Briefly, I think that the "exact sequence" is this:

justify "space algebra" -> take Clifford product -> keep dot product -> get cos correlations

and at this moment I don't see other reason for this approach than to get the cos. And the approach itself sounds to me like a "spooky Clifford product at a distance" :-), so I wouldn't consider it a disproof of Bell's theorem.

Are the things really that simple as I see them, or I am missing something?

Cristi

report post as inappropriate

Dear Florin,

Both your posts, as well as your paper, are very well-written, and as usual you did a great job conveying highly abstract ideas so that they can be followed by a wide audience. Joy's response deserves equal praises, and he answered with patience the questions addressed to him.

You said: "Tung Ten Yong (http://arxiv.org/abs/0712.1637) has criticized Joy's model because the final results are expressed as geometric algebra objects. I do not think this is a decisive criticism of Joy's theory as Yong claims because if nature is quantum mechanical at heart, and if Joy's theory reproduces all quantum mechanics results, this means that geometric algebra should be the proper ontology of nature."

I must say that the very moment when I read your remark, I was convinced that Yong was in fact criticizing the use of geometric algebra objects AS EXPECTATION VALUES. This was one of the first features that stroke me when reading Joy's work.

The expectation value is a real number. The measurement device indicates real numbers. Quantum mechanics is based on complex numbers, Hilbert spaces, matrices, yet the results are real numbers, ALWAYS. Because the expectation value is a probabilistic thing. So, in this particular point, I must agree with Yong.

Joy's expectation value is an even element of the Pauli algebra. While it looks structurally similar to the expectation value for real numbers, there is no obvious meaning of it. What is the meaning of a a Clifford-valued expectation value?

But I don't want to reject this novel type of expectation value just for not being real numbers. For example, the expectation values of momentum, along the three axes, are real numbers, but you can if you really want make statistics with the vector which has them as components. So, one may be able to write "vector valued expectation values".

So, in my opinion, when introducing a novel kind of expectation value, it should come with some explanations of why is this an expectation value, and what actually is a Clifford algebra valued expectation value. I couldn't find this explanation in Joy's papers.

Of course, if we take the limit for large number of observations, the bivector part cancels and we remain with a real number - the cosine. But what if we make only one observation? Would the result have a bivector part? If so, what's its meaning?

Best regards,

Cristi

report post as inappropriate

Both your posts, as well as your paper, are very well-written, and as usual you did a great job conveying highly abstract ideas so that they can be followed by a wide audience. Joy's response deserves equal praises, and he answered with patience the questions addressed to him.

You said: "Tung Ten Yong (http://arxiv.org/abs/0712.1637) has criticized Joy's model because the final results are expressed as geometric algebra objects. I do not think this is a decisive criticism of Joy's theory as Yong claims because if nature is quantum mechanical at heart, and if Joy's theory reproduces all quantum mechanics results, this means that geometric algebra should be the proper ontology of nature."

I must say that the very moment when I read your remark, I was convinced that Yong was in fact criticizing the use of geometric algebra objects AS EXPECTATION VALUES. This was one of the first features that stroke me when reading Joy's work.

The expectation value is a real number. The measurement device indicates real numbers. Quantum mechanics is based on complex numbers, Hilbert spaces, matrices, yet the results are real numbers, ALWAYS. Because the expectation value is a probabilistic thing. So, in this particular point, I must agree with Yong.

Joy's expectation value is an even element of the Pauli algebra. While it looks structurally similar to the expectation value for real numbers, there is no obvious meaning of it. What is the meaning of a a Clifford-valued expectation value?

But I don't want to reject this novel type of expectation value just for not being real numbers. For example, the expectation values of momentum, along the three axes, are real numbers, but you can if you really want make statistics with the vector which has them as components. So, one may be able to write "vector valued expectation values".

So, in my opinion, when introducing a novel kind of expectation value, it should come with some explanations of why is this an expectation value, and what actually is a Clifford algebra valued expectation value. I couldn't find this explanation in Joy's papers.

Of course, if we take the limit for large number of observations, the bivector part cancels and we remain with a real number - the cosine. But what if we make only one observation? Would the result have a bivector part? If so, what's its meaning?

Best regards,

Cristi

report post as inappropriate

Dear Cristi,

You state: "So, in my opinion, when introducing a novel kind of expectation value, it should come with some explanations of why is this an expectation value, and what actually is a Clifford algebra valued expectation value. I couldn't find this explanation in Joy's papers."

In Joy's defense I'll say his theory is a work in progress and he chose to direct his energy to other issues first. I can say Yong is right to point out strange things, but that cannot be used as grounds for discarding a new theory.

To your question: "If so, what's its meaning?" my simple answer is: I don't know. I could speculate and have an educate guess, but I don't want to do that because I could be wrong. However, I am not concerned about this at this stage. All I would say is that at some point some consistent translation mechanism has to be found from Joy's formulation to standard statistics. I disagree with Joy not on his math, but on his interpretation. And if his math is right I am confident such a mechanism does exist and will be eventually discovered. In other words, if Joy found an equivalent way of formulating QM, Born rule's translation into the new formalism will follow. It is just not done at this point.

report post as inappropriate

You state: "So, in my opinion, when introducing a novel kind of expectation value, it should come with some explanations of why is this an expectation value, and what actually is a Clifford algebra valued expectation value. I couldn't find this explanation in Joy's papers."

In Joy's defense I'll say his theory is a work in progress and he chose to direct his energy to other issues first. I can say Yong is right to point out strange things, but that cannot be used as grounds for discarding a new theory.

To your question: "If so, what's its meaning?" my simple answer is: I don't know. I could speculate and have an educate guess, but I don't want to do that because I could be wrong. However, I am not concerned about this at this stage. All I would say is that at some point some consistent translation mechanism has to be found from Joy's formulation to standard statistics. I disagree with Joy not on his math, but on his interpretation. And if his math is right I am confident such a mechanism does exist and will be eventually discovered. In other words, if Joy found an equivalent way of formulating QM, Born rule's translation into the new formalism will follow. It is just not done at this point.

report post as inappropriate

Dear Florin,

> "In Joy's defense I'll say his theory is a work in progress".

Isn't the entire Physics a work in progress? :-)

> "I can say Yong is right to point out strange things, but that cannot be used as grounds for discarding a new theory"

My opinion is that we point problems when we find them to advance the work in progress, not to discard it. The software tester is not the enemy of the software developer. My remark was intended to clarify that particular observation of Yong, which seemed to be misunderstood. It happens that this problem occurred to me independently of Yong, but my reaction was to speculate on Joy's page that his formula may be a gate toward a new kind of probability theory.

I agree we should not rush to discard a theory in progress. No theory is born already mature. We can only try to offer our help.

Of course, some could say that this means we should not discard Bell's theorem as well. It is hard to see it as a work in progress, but nevertheless, I would vote to give it the same immunity.

I am not a fan of no-go theorems. I think they are often (usually?) abused to reject whatever theories are a menace for our own views. Especially if we invest our entire life in a particular direction of research, we would like an insurance that we are on the right track, and the no-go theorems plus some metaphysical principles can provide just this. Sometimes, they even proclaim the alternative research ways as pseudoscience, as it is too often done with the hidden variables theories, by claiming that Bell eliminated them altogether.

This is why I admire Joy, for questioning Bell's well-established theorem. And I admire you, for questioning Joy's work. And both of you, for allowing others to question your views, and for taking their observations seriously.

Although I am not a fan of the abuse of no-go theorems, I see no reason to proclaim Bell's theorem disproved, not even in a tiny part.

report post as inappropriate

> "In Joy's defense I'll say his theory is a work in progress".

Isn't the entire Physics a work in progress? :-)

> "I can say Yong is right to point out strange things, but that cannot be used as grounds for discarding a new theory"

My opinion is that we point problems when we find them to advance the work in progress, not to discard it. The software tester is not the enemy of the software developer. My remark was intended to clarify that particular observation of Yong, which seemed to be misunderstood. It happens that this problem occurred to me independently of Yong, but my reaction was to speculate on Joy's page that his formula may be a gate toward a new kind of probability theory.

I agree we should not rush to discard a theory in progress. No theory is born already mature. We can only try to offer our help.

Of course, some could say that this means we should not discard Bell's theorem as well. It is hard to see it as a work in progress, but nevertheless, I would vote to give it the same immunity.

I am not a fan of no-go theorems. I think they are often (usually?) abused to reject whatever theories are a menace for our own views. Especially if we invest our entire life in a particular direction of research, we would like an insurance that we are on the right track, and the no-go theorems plus some metaphysical principles can provide just this. Sometimes, they even proclaim the alternative research ways as pseudoscience, as it is too often done with the hidden variables theories, by claiming that Bell eliminated them altogether.

This is why I admire Joy, for questioning Bell's well-established theorem. And I admire you, for questioning Joy's work. And both of you, for allowing others to question your views, and for taking their observations seriously.

Although I am not a fan of the abuse of no-go theorems, I see no reason to proclaim Bell's theorem disproved, not even in a tiny part.

report post as inappropriate

Dear Florin,

I don't understand your remark: "geometric algebra objects are non-local". Could you please explain it more?

For example, we can rewrite Maxwell's equations using geometric algebra. Does this make the electromagnetic field non-local?

What if, instead of the geometric product in Joy's theory, we take just the dot product (which is commutative)? We obtain directly the cos, without worrying about the bivector part of the expectation value, and we avoid using geometric algebra. Does this make Joy's theorem more local?

Best regards,

Cristi

report post as inappropriate

I don't understand your remark: "geometric algebra objects are non-local". Could you please explain it more?

For example, we can rewrite Maxwell's equations using geometric algebra. Does this make the electromagnetic field non-local?

What if, instead of the geometric product in Joy's theory, we take just the dot product (which is commutative)? We obtain directly the cos, without worrying about the bivector part of the expectation value, and we avoid using geometric algebra. Does this make Joy's theorem more local?

Best regards,

Cristi

report post as inappropriate

"I don't understand your remark: "geometric algebra objects are non-local". Could you please explain it more?"

Sure. In GA one puts all geometric objects on equal footing: scalars, vectors, bi-vectors, etc and one does add and multiply apples and oranges. It is legal in GA. The flip side is that a GA object is not always a scalar and can encode information about spatially extended geometric objects. In other words, it is non-local.

"For example, we can rewrite Maxwell's equations using geometric algebra. Does this make the electromagnetic field non-local?"

One can rewrite all Lie groups in GA and almost anything is expressible in this language. The devil is in details about locality. That is why I did not make a blanket statement about nonlocality in computer modeling for Joy's theory and I tried to explain where I think one will encounter a problem.

"What if, instead of the geometric product in Joy's theory, we take just the dot product (which is commutative)"

Then you get nonsensical or trivial results. The GA product combines the dot product and the exterior product into one. One reduces the dimesionality, the other increases it and you get a rich algebra. Using only dots is equivalent to using only projector operators and applying several times results very quickly is the uninteresting empty set.

report post as inappropriate

Sure. In GA one puts all geometric objects on equal footing: scalars, vectors, bi-vectors, etc and one does add and multiply apples and oranges. It is legal in GA. The flip side is that a GA object is not always a scalar and can encode information about spatially extended geometric objects. In other words, it is non-local.

"For example, we can rewrite Maxwell's equations using geometric algebra. Does this make the electromagnetic field non-local?"

One can rewrite all Lie groups in GA and almost anything is expressible in this language. The devil is in details about locality. That is why I did not make a blanket statement about nonlocality in computer modeling for Joy's theory and I tried to explain where I think one will encounter a problem.

"What if, instead of the geometric product in Joy's theory, we take just the dot product (which is commutative)"

Then you get nonsensical or trivial results. The GA product combines the dot product and the exterior product into one. One reduces the dimesionality, the other increases it and you get a rich algebra. Using only dots is equivalent to using only projector operators and applying several times results very quickly is the uninteresting empty set.

report post as inappropriate

Dear Florin,

Given that it is often stated on these pages that people don't understand geometric algebra well enough, I have to say something.

Yes, with geometric algebra you can model all finite dimensional Lie groups. Take the spin group Spin(n,n), and you can find GL(n, R)'s double cover as subgroup. The rest is easy. Many equations become simpler in this formalism. But they do not add any magical feature which cannot be obtained, often in a much non-elegant way, with other structures like tensors.

Using Clifford algebras concentrates sometimes more equations, apparently very different, in only one equation. It takes some work to unwind such equations. So yes, they may be a barrier in understanding someone's theory - both of the good points, and of the mistakes.

You say "The flip side is that a GA object is not always a scalar and can encode information about spatially extended geometric objects. In other words, it is non-local."

A Clifford algebra is associated to a vector space (endowed with a dot product). A Clifford-algebra valued field is a section of a Clifford bundle: each point of space has its own Clifford algebra, associated to the tangent space at that point. This is local.

Yes, if you want you can encode non-local information in it, but this is artificial. It is not because of their intrinsic nature. Take any section of any Clifford bundle. It has nothing non-local in it, except if you put it there. You can encode non-local information in scalars as well, if you want.

I consider Joy's theory non-local, but not just for the reason of using Clifford algebras, and certainly not just because violates Bell's inequality.

Best regards,

Cristi

report post as inappropriate

Given that it is often stated on these pages that people don't understand geometric algebra well enough, I have to say something.

Yes, with geometric algebra you can model all finite dimensional Lie groups. Take the spin group Spin(n,n), and you can find GL(n, R)'s double cover as subgroup. The rest is easy. Many equations become simpler in this formalism. But they do not add any magical feature which cannot be obtained, often in a much non-elegant way, with other structures like tensors.

Using Clifford algebras concentrates sometimes more equations, apparently very different, in only one equation. It takes some work to unwind such equations. So yes, they may be a barrier in understanding someone's theory - both of the good points, and of the mistakes.

You say "The flip side is that a GA object is not always a scalar and can encode information about spatially extended geometric objects. In other words, it is non-local."

A Clifford algebra is associated to a vector space (endowed with a dot product). A Clifford-algebra valued field is a section of a Clifford bundle: each point of space has its own Clifford algebra, associated to the tangent space at that point. This is local.

Yes, if you want you can encode non-local information in it, but this is artificial. It is not because of their intrinsic nature. Take any section of any Clifford bundle. It has nothing non-local in it, except if you put it there. You can encode non-local information in scalars as well, if you want.

I consider Joy's theory non-local, but not just for the reason of using Clifford algebras, and certainly not just because violates Bell's inequality.

Best regards,

Cristi

report post as inappropriate

Computer Simulation of the Physical Reality:

Much has been made out of a possible demonstration of the validity of my arguments by means of a computer simulation. It has been suggested that if my ideas are correct, then one can write a computer program modelling them and obtain the Tsirelson bounds. While such a demonstration would indeed be sociologically important (especially...

view entire post

Much has been made out of a possible demonstration of the validity of my arguments by means of a computer simulation. It has been suggested that if my ideas are correct, then one can write a computer program modelling them and obtain the Tsirelson bounds. While such a demonstration would indeed be sociologically important (especially...

view entire post

report post as inappropriate

Dear Joy,

I activate as computer programmer for almost 13 years, mostly in computational geometry, and I used quaternions and Clifford algebras in computations. Below I explain why I agree with what Florin said about computer simulation. But I agree with you that such a simulation will not add something qualitatively relevant to the mathematical description. Except that it forces us to unwind the mathematical solution step-by-step to see what it actually does.

A physical theory can be viewed as describing the possible states of the world, and how the world evolves from one state to another. If there is a mathematical description of the states, and a mathematical description of the evolution from one state to another, I don't see any reason which can prevent us simulate this description on a computer. There are only technical limitations: the simulation may require more memory and computational power than the computer can provide. I don't think this is the case.

To simulate EPR from the standard QM view, is easy. The state is the singlet state, which can be simulated as an element of the tensor product of the Hilbert space of the particle, and the orientations of the measurement devices can be simulated as unit vectors, and the correlation will occur because the superposition from the singlet state is destroyed. All steps can very well be simulated. Of course, projecting the singlet state into a separable state is a non-local operation.

For a local theory, the things should go very straight. To simulate a mathematical description of reality, we can use a "dead" computer, for the same reason why a dead computer can hold a pdf containing that mathematical description.

A hidden variable theory can provide a list of elements of reality, and the prescriptions to move from one state to another. For example, there can be a function which receives as input the element of reality describing the complete state of a particle, and the orientation of the measurement device, and output the observed value of the spin. The question is: can we call this function independently for each particle - without passing the information regarding the other particle (its elements of reality or the orientation of the measurement device) - and still get the correlations?

Best regards,

Cristi

report post as inappropriate

I activate as computer programmer for almost 13 years, mostly in computational geometry, and I used quaternions and Clifford algebras in computations. Below I explain why I agree with what Florin said about computer simulation. But I agree with you that such a simulation will not add something qualitatively relevant to the mathematical description. Except that it forces us to unwind the mathematical solution step-by-step to see what it actually does.

A physical theory can be viewed as describing the possible states of the world, and how the world evolves from one state to another. If there is a mathematical description of the states, and a mathematical description of the evolution from one state to another, I don't see any reason which can prevent us simulate this description on a computer. There are only technical limitations: the simulation may require more memory and computational power than the computer can provide. I don't think this is the case.

To simulate EPR from the standard QM view, is easy. The state is the singlet state, which can be simulated as an element of the tensor product of the Hilbert space of the particle, and the orientations of the measurement devices can be simulated as unit vectors, and the correlation will occur because the superposition from the singlet state is destroyed. All steps can very well be simulated. Of course, projecting the singlet state into a separable state is a non-local operation.

For a local theory, the things should go very straight. To simulate a mathematical description of reality, we can use a "dead" computer, for the same reason why a dead computer can hold a pdf containing that mathematical description.

A hidden variable theory can provide a list of elements of reality, and the prescriptions to move from one state to another. For example, there can be a function which receives as input the element of reality describing the complete state of a particle, and the orientation of the measurement device, and output the observed value of the spin. The question is: can we call this function independently for each particle - without passing the information regarding the other particle (its elements of reality or the orientation of the measurement device) - and still get the correlations?

Best regards,

Cristi

report post as inappropriate

I have been puzzled the last couple of days by references to a computer simulation challenge that I could not find in these forums. I finally looked looked it up on the web.

No need to worry about those criticisms. It's a load of simple minded crap.

Tom

report post as inappropriate

No need to worry about those criticisms. It's a load of simple minded crap.

Tom

report post as inappropriate

Dear Tom,

Your objections to my conclusions have never bothered me. However, there is something that has always bothered me and it is: For someone who identifies themselves as an independent researcher, your breadth of knowledge continues to amaze me. On previous occasions, I have thought about writing this message but didn't. You are deserving of respect. You are someone to listen to even during disagreements. Moreover, whether you intended your last message to be humorous or not, I found it amusing. Thanks for being here.

James

report post as inappropriate

Your objections to my conclusions have never bothered me. However, there is something that has always bothered me and it is: For someone who identifies themselves as an independent researcher, your breadth of knowledge continues to amaze me. On previous occasions, I have thought about writing this message but didn't. You are deserving of respect. You are someone to listen to even during disagreements. Moreover, whether you intended your last message to be humorous or not, I found it amusing. Thanks for being here.

James

report post as inappropriate

Florin,

I am intrigued by this matter of computation. First off it sounds as if there is some scuttlebutt about this. Yet of greater interest, if one really wanted to address this question it seems to come down to an issue of algorithmic structure and computability. If you are right this means the Tsirelson bound is simply not computable by JC’s system. Unfortunately I am not familiar with Clifton’s theorem and other matters so I do not see clearly what the problem is. This issue of hidden variables and contextuality is pretty tangential to my main focus. Yet this seems to suggest that either this is some problem involving a contradiction, or it is a problem involving NP completeness, where the space is somehow to large to ever compute this, or maybe (though I doubt it) some sort of Turing machine halting problem or Godel’s theorem.

Cheers LC

report post as inappropriate

I am intrigued by this matter of computation. First off it sounds as if there is some scuttlebutt about this. Yet of greater interest, if one really wanted to address this question it seems to come down to an issue of algorithmic structure and computability. If you are right this means the Tsirelson bound is simply not computable by JC’s system. Unfortunately I am not familiar with Clifton’s theorem and other matters so I do not see clearly what the problem is. This issue of hidden variables and contextuality is pretty tangential to my main focus. Yet this seems to suggest that either this is some problem involving a contradiction, or it is a problem involving NP completeness, where the space is somehow to large to ever compute this, or maybe (though I doubt it) some sort of Turing machine halting problem or Godel’s theorem.

Cheers LC

report post as inappropriate

Lawrence,

This is all much easier, it has nothing to do with Godel or NP complete directly. With quantum resources one can achieve many things impossible to do with local operation and classical communications: teleportation, speed-up of algorithms, etc. Joy claims his theory is local and realistic, and if true, we should be able to do all those things using only local resources because Joy's theory do reproduce QM result exactly. If it is only a local and realistic model then it should be able to be simulated on a classic Turing machine. In other words, a classical computer could have all the computation power of a quantum computer.

But I disagree that Joy's theory is local, and hence the challenge. Alternatively, prove my spin one analysis wrong, or prove Clifton's theorems wrong. Any of those 3 will do the trick. However, geometric algebra software packages are freely available on the web, and this is the easy way to prove me wrong if I am wrong indeed (which I think I am not).

And yes, the person originating the idea acted as a jerk towards Joy and I had some doubts about putting this in the post or not. I hope Joy is not mad on me about it. But an idea is an idea and I was judging it on its merits alone and not on its history.

report post as inappropriate

This is all much easier, it has nothing to do with Godel or NP complete directly. With quantum resources one can achieve many things impossible to do with local operation and classical communications: teleportation, speed-up of algorithms, etc. Joy claims his theory is local and realistic, and if true, we should be able to do all those things using only local resources because Joy's theory do reproduce QM result exactly. If it is only a local and realistic model then it should be able to be simulated on a classic Turing machine. In other words, a classical computer could have all the computation power of a quantum computer.

But I disagree that Joy's theory is local, and hence the challenge. Alternatively, prove my spin one analysis wrong, or prove Clifton's theorems wrong. Any of those 3 will do the trick. However, geometric algebra software packages are freely available on the web, and this is the easy way to prove me wrong if I am wrong indeed (which I think I am not).

And yes, the person originating the idea acted as a jerk towards Joy and I had some doubts about putting this in the post or not. I hope Joy is not mad on me about it. But an idea is an idea and I was judging it on its merits alone and not on its history.

report post as inappropriate

Dear Florin,

One does not need any of the stuff you are demanding to see that my framework is both local and non-contextual. No elaborate theorems or computer programs are needed to see this. Bell has given us rigorous and foolproof criteria for both locality and non-contextuality. My models fully comply with these criteria, as can be easily checked. Let A(a, L) and B(b, L) be the results of Alice and Bob in the usual notations. Then the correlation between A and B can be non-local *if and only if* either (1) the result A depends on the context b, or (2) the result B depends on the context a, or (3) the result A depends on the result B, or (4) the result B depends on the result A. There is absolutely no other way that the correlation can be non-local. This is Bell’s own criterion, and it is universally accepted. Now you can easily check that this criterion of locality is rigorously satisfied by all of my models. Thus my framework is clearly and manifestly local. Next, here is Bell’s criterion of contextuality: The model is contextual *if and only if* the result, such as A(a, L), changes its value (say from +1 to -1) when the context “a” is changed from, say, a to a’. You can easily check that this does not ever happen in any of my models, because the local results are entirely determined by the hidden variable L alone. The local results do not change when the context is changed, because the correlations are purely topological effects, not contextual effects. In other words, all of the locality and non-contextuality demands made by Bell are rigorously met by my models. So, in my opinion, all the other elaborate stuff you are raising is simply “grasping for straws” in order to hold on to orthodoxy, which has been squarely defeated.

Best,

Joy

report post as inappropriate

One does not need any of the stuff you are demanding to see that my framework is both local and non-contextual. No elaborate theorems or computer programs are needed to see this. Bell has given us rigorous and foolproof criteria for both locality and non-contextuality. My models fully comply with these criteria, as can be easily checked. Let A(a, L) and B(b, L) be the results of Alice and Bob in the usual notations. Then the correlation between A and B can be non-local *if and only if* either (1) the result A depends on the context b, or (2) the result B depends on the context a, or (3) the result A depends on the result B, or (4) the result B depends on the result A. There is absolutely no other way that the correlation can be non-local. This is Bell’s own criterion, and it is universally accepted. Now you can easily check that this criterion of locality is rigorously satisfied by all of my models. Thus my framework is clearly and manifestly local. Next, here is Bell’s criterion of contextuality: The model is contextual *if and only if* the result, such as A(a, L), changes its value (say from +1 to -1) when the context “a” is changed from, say, a to a’. You can easily check that this does not ever happen in any of my models, because the local results are entirely determined by the hidden variable L alone. The local results do not change when the context is changed, because the correlations are purely topological effects, not contextual effects. In other words, all of the locality and non-contextuality demands made by Bell are rigorously met by my models. So, in my opinion, all the other elaborate stuff you are raising is simply “grasping for straws” in order to hold on to orthodoxy, which has been squarely defeated.

Best,

Joy

report post as inappropriate

Dear Joy,

I agree with you that the result obtained by Bob is not determined by the orientation chosen by Alice. It is the probability to obtain the result which depends on both the orientation a chosen by her, and on the result A(a, L), obtained by her. The probability to get B=1 is not equally distributed for all possible choices of b, but also depends on A and a.

Can we obtain the probability for Bob's result from a formula which doesn't depend on the orientation chosen by Alice and on her result?

In your approach, the two bivectors representing the two orientations are used to obtain this correlation. This correlation is postulated to be the scalar part of their Clifford algebra product (or their dot product). Thus, the cos correlation is obtained. But the formula involves both orientations. The probability of the result obtained by Bob depends on A and a of Alice, "at a distance", because the formula involves both directions simultaneously, although they are at different locations.

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

What I find interesting about how you get this cos is that you simply generalize the hidden variables correlations as they are stated by Bell. On the one hand, that formula (19) from 0703179 is equivalent to (4), the quantum one for the case of a singlet state. On the other hand it looks like Bell's.

While Bell's formula refers to scalars only, it is not because he was "far too sloppy". It is because the correlations and probabilities happen to be scalars, and this is how probabilities work. Your correlation equation (4) cannot be directly given a probabilistic meaning. I don't know how to take it seriously as an equation involving probabilities. I look forward to see if you can provide an interpretation for it. Is this important? I think it is, because this may help us to see exactly how your hidden variables act to predict the correlations of the outcomes (this is, IMHO, the ontology). But more important, this part is necessary if you want to show that there is no non-local correlation between the two particles. Otherwise, to some may look just like a trick to obtain cos correlation for something which looks only formally like Bell's correlations for hidden variables, and actually masks non-locality.

Best regards,

Cristi

report post as inappropriate

I agree with you that the result obtained by Bob is not determined by the orientation chosen by Alice. It is the probability to obtain the result which depends on both the orientation a chosen by her, and on the result A(a, L), obtained by her. The probability to get B=1 is not equally distributed for all possible choices of b, but also depends on A and a.

Can we obtain the probability for Bob's result from a formula which doesn't depend on the orientation chosen by Alice and on her result?

In your approach, the two bivectors representing the two orientations are used to obtain this correlation. This correlation is postulated to be the scalar part of their Clifford algebra product (or their dot product). Thus, the cos correlation is obtained. But the formula involves both orientations. The probability of the result obtained by Bob depends on A and a of Alice, "at a distance", because the formula involves both directions simultaneously, although they are at different locations.

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

What I find interesting about how you get this cos is that you simply generalize the hidden variables correlations as they are stated by Bell. On the one hand, that formula (19) from 0703179 is equivalent to (4), the quantum one for the case of a singlet state. On the other hand it looks like Bell's.

While Bell's formula refers to scalars only, it is not because he was "far too sloppy". It is because the correlations and probabilities happen to be scalars, and this is how probabilities work. Your correlation equation (4) cannot be directly given a probabilistic meaning. I don't know how to take it seriously as an equation involving probabilities. I look forward to see if you can provide an interpretation for it. Is this important? I think it is, because this may help us to see exactly how your hidden variables act to predict the correlations of the outcomes (this is, IMHO, the ontology). But more important, this part is necessary if you want to show that there is no non-local correlation between the two particles. Otherwise, to some may look just like a trick to obtain cos correlation for something which looks only formally like Bell's correlations for hidden variables, and actually masks non-locality.

Best regards,

Cristi

report post as inappropriate

Florin,

This almost seems to boil down to asking whether a Rubik’s cube has nonlocal properties. The 4x10^{19} configurations of the cube could be a Hilbert space. If I wanted to teleport a state (Rubik's cube configuration) to Alice I would need to perform a set of Hadamard operations on an ancillary state that she and I have access to. This would demolish some entanglement. The question might amount to asking whether or not the teleportation could be completely accomplished with just the Rubik’s cube operations and nothing else. Hence the contextuality of states in a quantum setting would be defined entirely by the cube operations, without a Hadamard matrix operation for a SLOCC operation.

This might be an interesting thing to chew over for a while.

Cheers LC

report post as inappropriate

This almost seems to boil down to asking whether a Rubik’s cube has nonlocal properties. The 4x10^{19} configurations of the cube could be a Hilbert space. If I wanted to teleport a state (Rubik's cube configuration) to Alice I would need to perform a set of Hadamard operations on an ancillary state that she and I have access to. This would demolish some entanglement. The question might amount to asking whether or not the teleportation could be completely accomplished with just the Rubik’s cube operations and nothing else. Hence the contextuality of states in a quantum setting would be defined entirely by the cube operations, without a Hadamard matrix operation for a SLOCC operation.

This might be an interesting thing to chew over for a while.

Cheers LC

report post as inappropriate

That is indeed interesting, Lawrence. However, I don't think the discrete states of a Rubik's cube on S^2, which are commutative and associative, correspond to the continuous noncommutative and nonassociative correlations on S^7 in which Joy is working. In fact, I think he obviates "configuration space" and "discrete state" as we normally speak of them in quantum mechanics.

I'm trying to get used to his language.

Tom

report post as inappropriate

I'm trying to get used to his language.

Tom

report post as inappropriate

Take a Rubik’s cube that is in the “new state” and rotate the red face and then the yellow face. Record the configuration of the cube. Now restore it and rotate the yellow and the red face. The two operations are not commutative.

Cheers LC

report post as inappropriate

Cheers LC

report post as inappropriate

Also, 3-D vector cross-products are non-commutative, a x b = -b x a, such that these cross-products look like the spatial component of a Quaternion.

report post as inappropriate

report post as inappropriate

Dear Florin,

I wish to start by saying I have enjoyed the discussion between yourself and Joy and I look forward to the continuation. My own studies of Bell's work have led me to the following position. Bell's theorem can be generalized as follows:

--

IF there exists a single probability distribution p(a,b,c) from which three distributions p(a,b), p(b,c), and p(a,c) can be extracted, then the expectation values from these distributions must necessarily obey the following inequality:

|E(a,b) - E(a,c)|

report post as inappropriate

report post as inappropriate

I wish to start by saying I have enjoyed the discussion between yourself and Joy and I look forward to the continuation. My own studies of Bell's work have led me to the following position. Bell's theorem can be generalized as follows:

--

IF there exists a single probability distribution p(a,b,c) from which three distributions p(a,b), p(b,c), and p(a,c) can be extracted, then the expectation values from these distributions must necessarily obey the following inequality:

|E(a,b) - E(a,c)|

report post as inappropriate

For some reason my previous post was cut so here is another try:

Dear Florin,

I wish to start by saying I have enjoyed the discussion between yourself and Joy and I look forward to the continuation. My own studies of Bell's work have led me to the following position. Bell's theorem can be generalized as follows:

--

IF there exists a single probability distribution...

view entire post

Dear Florin,

I wish to start by saying I have enjoyed the discussion between yourself and Joy and I look forward to the continuation. My own studies of Bell's work have led me to the following position. Bell's theorem can be generalized as follows:

--

IF there exists a single probability distribution...

view entire post

report post as inappropriate

Dear Nkwenti,

Sorry for answering late, I was realy busy at work and I have only a very limited bandwith to devote to FQXi besides answering to Joy. I would be interested to read some more of your ideas, do you have anything published?

I am not a supporter of counterfactuals, but I am making the same assumptions Joy is making to see where they may lead.

You state: "My view is that violation of inequalities by experiments points to the lack of a single p(a,b,c) distribution, owing to the fact that no two particles can be measured simultaneously at 3 detector settings."

I am not quite agreeing with this because the lack of p(a,b,c) can also mean no inequality at all: you may go either above and below the limit. But going below the limit makes irrelevant the fact that in some experiment the limit is obeyed. In other words, one cannot reach any conclusion regardless of the violation or not of the inequality. Maybe I am missing something.

report post as inappropriate

Sorry for answering late, I was realy busy at work and I have only a very limited bandwith to devote to FQXi besides answering to Joy. I would be interested to read some more of your ideas, do you have anything published?

I am not a supporter of counterfactuals, but I am making the same assumptions Joy is making to see where they may lead.

You state: "My view is that violation of inequalities by experiments points to the lack of a single p(a,b,c) distribution, owing to the fact that no two particles can be measured simultaneously at 3 detector settings."

I am not quite agreeing with this because the lack of p(a,b,c) can also mean no inequality at all: you may go either above and below the limit. But going below the limit makes irrelevant the fact that in some experiment the limit is obeyed. In other words, one cannot reach any conclusion regardless of the violation or not of the inequality. Maybe I am missing something.

report post as inappropriate

Dear Florin,

I am afraid the main message I am trying to convey is getting lost in the erudition (as Einstein once put it). The correct reading of my results is this: quantum correlations are nothing but correlations among the points of a parallelized 7-sphere; and as such they are purely classical, topological effects. There is no such thing as non-locality, non-reality, or contextuality...

view entire post

I am afraid the main message I am trying to convey is getting lost in the erudition (as Einstein once put it). The correct reading of my results is this: quantum correlations are nothing but correlations among the points of a parallelized 7-sphere; and as such they are purely classical, topological effects. There is no such thing as non-locality, non-reality, or contextuality...

view entire post

report post as inappropriate

Here's how I see a software simulation of a local hidden variable theory applyed to the EPR experiment:

1. Make a computer program which generates randomly the hidden variables corresponding to each of the two particles corresponding to a singlet state (stored in a data structure).

2. Make a computer program which receives as input a data structure containing the hidden variables corresponding to one particle, and the orientation of the measurement device, and returns either +1, or -1. Give a copy of this program to Alice and another to Bob.

3. Prepare two states with the algorithm #1.

4. Send one of the states, by email, to Alice, and the other to Bob, which are at distant locations from one another.

5. Alice disconnects from the Internet, then runs the algorithm #2, with the first parameter being the hidden variables she got by email, and the second parameter the orientation of the measurement device, randomly chosen. Then she saves the orientation and the result (+1 or -1).

In the same time, Bob does the same as Alice, but for his particle.

7. Alice and Bob connect simultaneously to the Internet and send the orientations and the results to a server, to be recorded.

8. Repeat the steps 3 to 7 a large number of times.

9. Calculate the correlations between the data recorded on the server.

10. Compare the results with those predicted by Quantum Mechanics (and verified by experiments).

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

I think that it is a good exercise to think at possible ways to make the program output the two results correlated as they are in QM. The ardent practitioner of this sadhana receives the insight into the meaning of Bell's theorem :-)

report post as inappropriate

1. Make a computer program which generates randomly the hidden variables corresponding to each of the two particles corresponding to a singlet state (stored in a data structure).

2. Make a computer program which receives as input a data structure containing the hidden variables corresponding to one particle, and the orientation of the measurement device, and returns either +1, or -1. Give a copy of this program to Alice and another to Bob.

3. Prepare two states with the algorithm #1.

4. Send one of the states, by email, to Alice, and the other to Bob, which are at distant locations from one another.

5. Alice disconnects from the Internet, then runs the algorithm #2, with the first parameter being the hidden variables she got by email, and the second parameter the orientation of the measurement device, randomly chosen. Then she saves the orientation and the result (+1 or -1).

In the same time, Bob does the same as Alice, but for his particle.

7. Alice and Bob connect simultaneously to the Internet and send the orientations and the results to a server, to be recorded.

8. Repeat the steps 3 to 7 a large number of times.

9. Calculate the correlations between the data recorded on the server.

10. Compare the results with those predicted by Quantum Mechanics (and verified by experiments).

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

I think that it is a good exercise to think at possible ways to make the program output the two results correlated as they are in QM. The ardent practitioner of this sadhana receives the insight into the meaning of Bell's theorem :-)

report post as inappropriate

I think this won't work, for the reasons I posted in the "Quantum Music ..." forum.

A time dependent network of continuous functions input and outpout (aka the Real World) is never disconnected from information bounded by time connectedness and in which arbitrary outputs are correlated by the time scale of observation. What you suggest is 2-node network of discontinuous functions, while what Joy proposes is a many node network of continuous functions. Not the same thing.

Tom

report post as inappropriate

A time dependent network of continuous functions input and outpout (aka the Real World) is never disconnected from information bounded by time connectedness and in which arbitrary outputs are correlated by the time scale of observation. What you suggest is 2-node network of discontinuous functions, while what Joy proposes is a many node network of continuous functions. Not the same thing.

Tom

report post as inappropriate

Tom,

Thank you for your support earlier. I agree with your point here. Cristi has not understood the real problem yet. As a simple illustration of the problem, consider S^3, which has a non-trivial twist in its fibration. Now remove just one single point from it. That reduces S^3 to R^3, which has a trivial topology compared to S^3. As I have shown, Bell's theorem goes through for R^3 but not for S^3.

Joy

report post as inappropriate

Thank you for your support earlier. I agree with your point here. Cristi has not understood the real problem yet. As a simple illustration of the problem, consider S^3, which has a non-trivial twist in its fibration. Now remove just one single point from it. That reduces S^3 to R^3, which has a trivial topology compared to S^3. As I have shown, Bell's theorem goes through for R^3 but not for S^3.

Joy

report post as inappropriate

Dear Tom,

There are the following types of processes involved: the disintegration leading to the two particles, and the spin measurement. Just these. It doesn't matter where the planet Jupiter is, or if it is raining. I only ask the hidden variable theory to pass the simplest test, forget about the rest of the Universe.

If you have a hidden variable theory, it means that you have a realistic description of these two processes, and this realistic description can be used to predict the outcomes of the measurements, from the local elements of reality. You can put in each of the two programs whatever complicated networks or nonlinear differential equations you want. At the end, you will have two programs, more or less complicated.

The point is that any local hidden variables theory (not only Joy's) has to face the separation in location of the two particles.

You provided there some arguments about the impossibility to simulate. They are very general, and seem to apply as well to any physical phenomenon. It seems to imply that none of them should be simulable. Or, our experience shows that most physical process to which we know the detailed description could be simulated, even some highly non-linear, within a good approximation. I don't see from your arguments why this particular theory cannot be simulated at all, at least to give the difference between a linear function and cosine.

Anyway, the purpose of this post was to provide a more pragmatic way to think about Bell's theorem. I can make it even more pragmatic: instead of computer software, make mechanical devices. One contraption for the particles which generates the electrons, one contraption for each electron. Make the two particles be two contraptions, constructed so that after you select a direction on their dials, they show +1 or -1.

Best regards,

Cristi

report post as inappropriate

There are the following types of processes involved: the disintegration leading to the two particles, and the spin measurement. Just these. It doesn't matter where the planet Jupiter is, or if it is raining. I only ask the hidden variable theory to pass the simplest test, forget about the rest of the Universe.

If you have a hidden variable theory, it means that you have a realistic description of these two processes, and this realistic description can be used to predict the outcomes of the measurements, from the local elements of reality. You can put in each of the two programs whatever complicated networks or nonlinear differential equations you want. At the end, you will have two programs, more or less complicated.

The point is that any local hidden variables theory (not only Joy's) has to face the separation in location of the two particles.

You provided there some arguments about the impossibility to simulate. They are very general, and seem to apply as well to any physical phenomenon. It seems to imply that none of them should be simulable. Or, our experience shows that most physical process to which we know the detailed description could be simulated, even some highly non-linear, within a good approximation. I don't see from your arguments why this particular theory cannot be simulated at all, at least to give the difference between a linear function and cosine.

Anyway, the purpose of this post was to provide a more pragmatic way to think about Bell's theorem. I can make it even more pragmatic: instead of computer software, make mechanical devices. One contraption for the particles which generates the electrons, one contraption for each electron. Make the two particles be two contraptions, constructed so that after you select a direction on their dials, they show +1 or -1.

Best regards,

Cristi

report post as inappropriate

AN EXPLICIT LOCAL VARIABLES TOPOLOGICAL MECHANISM FOR THE EPR CORRELATIONS

It is based on a non-trivial topology (wormholes).

Cut two spheres out of our space, and glue the two boundaries of the space together. This wormhole can be traversed by a source free electric field, and used to model a pair of electrically charged particles of opposite charges as its mouths (Einstein-Risen 1935, Misner-Wheeler's charge-without-charge 1957, Rainich 1925).

For EPR we need a wormhole which connects two electrons instead of an electron-positron pair. A wormhole having as mouths two equal charges can be obtained as follows: instead of just gluing together the two spherical boundaries, we first flip the orientation of one of them. Since the electric field is a bivector, the change in orientation changes the sign of the electric field, and the two topological charges have the same sign.

Now associate to the two electrons your favorite local classical description. The communication required to obtain the correlation can be done through the wormhole.

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

This may be the basis of a mathematically correct local hidden variable theory. Also, it seems to disprove, or rather circumvent, Bell's theorem. For Bohm's hidden variable theory, it provides a mechanism to get the correlation without faster than light signals. I proposed it here for theoretical purposes only, as an example. My favorite interpretation is another one.

Cristi

report post as inappropriate

It is based on a non-trivial topology (wormholes).

Cut two spheres out of our space, and glue the two boundaries of the space together. This wormhole can be traversed by a source free electric field, and used to model a pair of electrically charged particles of opposite charges as its mouths (Einstein-Risen 1935, Misner-Wheeler's charge-without-charge 1957, Rainich 1925).

For EPR we need a wormhole which connects two electrons instead of an electron-positron pair. A wormhole having as mouths two equal charges can be obtained as follows: instead of just gluing together the two spherical boundaries, we first flip the orientation of one of them. Since the electric field is a bivector, the change in orientation changes the sign of the electric field, and the two topological charges have the same sign.

Now associate to the two electrons your favorite local classical description. The communication required to obtain the correlation can be done through the wormhole.

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

This may be the basis of a mathematically correct local hidden variable theory. Also, it seems to disprove, or rather circumvent, Bell's theorem. For Bohm's hidden variable theory, it provides a mechanism to get the correlation without faster than light signals. I proposed it here for theoretical purposes only, as an example. My favorite interpretation is another one.

Cristi

report post as inappropriate

Of course, the construction of topological pairs of electrons requires that in the same universe there are also at least two positive topological charges.

report post as inappropriate

report post as inappropriate

Dear Florin,

I've seen your nice arsenal against Joy's theory.

I would like to invite you to test your weapons on the theory of local hidden variables I sketched above :-)

I know that it is a work in progress, but the major elements are already developed: topological charges and Bohmian mechanics. Can you see any major difficulties in applying it to make Bohmian mechanics local?

Best regards,

Cristi

report post as inappropriate

I've seen your nice arsenal against Joy's theory.

I would like to invite you to test your weapons on the theory of local hidden variables I sketched above :-)

I know that it is a work in progress, but the major elements are already developed: topological charges and Bohmian mechanics. Can you see any major difficulties in applying it to make Bohmian mechanics local?

Best regards,

Cristi

report post as inappropriate

It is because being and experience and thought/ideas are integrated and interactive that there are limits to the understanding. Where do you think that ideas come from? The requirements, limits, and extensiveness of truth center upon the extent to which thought is similar to sensory experience. We all come from the center of the human body.

AND, the purpose of vision is to advise of the consequences of touch in time. Another great physical truth -- (by Berkeley).

We do not move (naturally) with regard to the Sun and photons because there are no consequences of touch in time. That is, no gravity, black space, death, destruction of vision, inability to discern distance in outer space.

Sorry folks, physics ultimately involves natural experience that gives us life.

FUNDAMENTALLY, THE ENTIRE APPROACH OF MODERN PHYSICS IS INCORRECT, AND ULTIMATELY DESTRUCTIVE.

report post as inappropriate

AND, the purpose of vision is to advise of the consequences of touch in time. Another great physical truth -- (by Berkeley).

We do not move (naturally) with regard to the Sun and photons because there are no consequences of touch in time. That is, no gravity, black space, death, destruction of vision, inability to discern distance in outer space.

Sorry folks, physics ultimately involves natural experience that gives us life.

FUNDAMENTALLY, THE ENTIRE APPROACH OF MODERN PHYSICS IS INCORRECT, AND ULTIMATELY DESTRUCTIVE.

report post as inappropriate

lET'S START FOCUSING ON CONSTRUCTIVE PHYSICS -- THE PHYSICS OF NATURAL EXPERIENCE, GROWTH, AND LIFE.

report post as inappropriate

report post as inappropriate

Florin,

When I pointed out Joy's work to you I was hoping for just the type of result you've produced above. I'd like to thank you for this effort. In particular I enjoyed your clear discussion of contextual and non-contextual. One cannot even evaluate Joy's statements about the relevance of 'contextuality' without knowing what is meant by the terms. Many people are interested in this topic and you have done all of us a favor by clarifying these definitions.

I am also pleased that you are more open on this subject. You earlier claimed that I brought disgrace on my theory by even referring to de Broglie-Bohm or 'hidden variable'. Now you note that "there is no absolute agreement even between experts on the very definitions of what one means by locality, realism, or contextuality, or a no-go theorem for hidden variable theories."

In addition to Joy Christian's analysis challenging the status quo, I've also pointed out recently that weak quantum measurements resulted in the following observation:

"Single-particle trajectories measured in this fashion reproduce those predicted by the Bohm-deBroglie interpretation of quantum mechanics."

In my mind, both of these results, theoretical and experimental, challenge John Bell's test. In addition you have noted that, unlike Bell, Kochen-Specker cannot be put to any experimental test. It is from this perspective that I ask the following:

Is violation of Bell's inequality *the* basis for rejection of local realism in physics?

That is, if Bell had derived Tsirelson's bound, would most physicists today reject local realism in quantum mechanics?

Thanks again for analyzing Joy's work and sharing your analysis with us. I look forward to your answer.

Edwin Eugene Klingman

report post as inappropriate

When I pointed out Joy's work to you I was hoping for just the type of result you've produced above. I'd like to thank you for this effort. In particular I enjoyed your clear discussion of contextual and non-contextual. One cannot even evaluate Joy's statements about the relevance of 'contextuality' without knowing what is meant by the terms. Many people are interested in this topic and you have done all of us a favor by clarifying these definitions.

I am also pleased that you are more open on this subject. You earlier claimed that I brought disgrace on my theory by even referring to de Broglie-Bohm or 'hidden variable'. Now you note that "there is no absolute agreement even between experts on the very definitions of what one means by locality, realism, or contextuality, or a no-go theorem for hidden variable theories."

In addition to Joy Christian's analysis challenging the status quo, I've also pointed out recently that weak quantum measurements resulted in the following observation:

"Single-particle trajectories measured in this fashion reproduce those predicted by the Bohm-deBroglie interpretation of quantum mechanics."

In my mind, both of these results, theoretical and experimental, challenge John Bell's test. In addition you have noted that, unlike Bell, Kochen-Specker cannot be put to any experimental test. It is from this perspective that I ask the following:

Is violation of Bell's inequality *the* basis for rejection of local realism in physics?

That is, if Bell had derived Tsirelson's bound, would most physicists today reject local realism in quantum mechanics?

Thanks again for analyzing Joy's work and sharing your analysis with us. I look forward to your answer.

Edwin Eugene Klingman

report post as inappropriate

Dear Edwin,

Thank you for your kind words.

You say: "Single-particle trajectories measured in this fashion reproduce those predicted by the Bohm-deBroglie interpretation of quantum mechanics."

I was not aware of that, however, I do not put much importance to it as regular evolution disagrees strongly with the trajectories of Bohm.

Also about: "That is, if Bell had derived Tsirelson's bound, would most physicists today reject local realism in quantum mechanics?"

The name of the bounds and who discovered them is irrelevant, they may equally be called Edwin and Florin's bounds. What counts is their meaning. Bell's bound are derived from local realism assumptions, and Tsirelson's bounds are derived from QM. There is a third bound: Popescu-Rohrlich which is derived from non-signaling.

And by the way, hidden variables are still a dead end. Joy's is trying to revive the area but he is getting no traction. Still, I consider his model worthy of investigation as his results do increase our knowledge, regardless of the meaning he ascribes them.

report post as inappropriate

Thank you for your kind words.

You say: "Single-particle trajectories measured in this fashion reproduce those predicted by the Bohm-deBroglie interpretation of quantum mechanics."

I was not aware of that, however, I do not put much importance to it as regular evolution disagrees strongly with the trajectories of Bohm.

Also about: "That is, if Bell had derived Tsirelson's bound, would most physicists today reject local realism in quantum mechanics?"

The name of the bounds and who discovered them is irrelevant, they may equally be called Edwin and Florin's bounds. What counts is their meaning. Bell's bound are derived from local realism assumptions, and Tsirelson's bounds are derived from QM. There is a third bound: Popescu-Rohrlich which is derived from non-signaling.

And by the way, hidden variables are still a dead end. Joy's is trying to revive the area but he is getting no traction. Still, I consider his model worthy of investigation as his results do increase our knowledge, regardless of the meaning he ascribes them.

report post as inappropriate

As Joy said, one of the obstacles encountered by the readers of his papers is insufficient understanding of topology. I'll try to help a little bit here.

The particularity of the hypersphere S^{3} which is supposed to make the difference between linear correlations and cosine correlation is its topology.

Other properties remain almost unchanged by removing one point. For...

view entire post

The particularity of the hypersphere S

Other properties remain almost unchanged by removing one point. For...

view entire post

report post as inappropriate

This image shows very clearly that the circles remain entangled after teh stereographic projection. The red line is the circle which was "destroyed" by removing one point.

report post as inappropriate

report post as inappropriate

I argued last week that the removal of this point was the assignement of this point as the pole under a stereogrphic projection to infinity. The Hopf fibration is a form of the Desargue projection, or a set of rays which form the heavenly sphere. This is a conformal map amd I argue that the relevant geometry for this physics are those circles which remain conformal, which excludes the points at infinity. The conformal theory of QM SL(2,R) will then necessarily exclude this point removed by the stereographic projection.

Cheers LC

report post as inappropriate

Cheers LC

report post as inappropriate

Dear Cristi,

"... covering S3 is lost by removing one point." If you did read my essay, you might have rejected the result of my heretical nearly intuitionistic reasoning.

I maintain that for really real numbers, y=|sign(x)|=1 is valid for any x without an excepted point y=0 at x=0. In other words, plain logics prevents me to remove any point from the line of real numbers because a point does not have any extension. Euclid defined a point something that does not have parts. On the other hand in good old mathematics a continuum is something every part of which has parts. I know, this is at odds with the mathematical thinking of those who proclaimed the freedom to arbitrarily define real numbers as the union of rational and irrational numbers and to ascribe trichotomy to them.

Of course, it requires the ability for consequent thinking if one is willing to accept that not a single number that exists in an continuum as a potentiality can be addressed or removed if one demands its actually infinite precision. My essay suggested using a limit from the lower side instead. My reasoning allows degeneration into a double root but not into a singularity. To me, EPR were not far from Buridan's ass.

By the way, I would like you to read my last comment concerning "Evolving Time's Arrow".

Regards,

Eckard

report post as inappropriate

"... covering S3 is lost by removing one point." If you did read my essay, you might have rejected the result of my heretical nearly intuitionistic reasoning.

I maintain that for really real numbers, y=|sign(x)|=1 is valid for any x without an excepted point y=0 at x=0. In other words, plain logics prevents me to remove any point from the line of real numbers because a point does not have any extension. Euclid defined a point something that does not have parts. On the other hand in good old mathematics a continuum is something every part of which has parts. I know, this is at odds with the mathematical thinking of those who proclaimed the freedom to arbitrarily define real numbers as the union of rational and irrational numbers and to ascribe trichotomy to them.

Of course, it requires the ability for consequent thinking if one is willing to accept that not a single number that exists in an continuum as a potentiality can be addressed or removed if one demands its actually infinite precision. My essay suggested using a limit from the lower side instead. My reasoning allows degeneration into a double root but not into a singularity. To me, EPR were not far from Buridan's ass.

By the way, I would like you to read my last comment concerning "Evolving Time's Arrow".

Regards,

Eckard

report post as inappropriate

KEEP AN EYE ON PHYSICS TODAY MAGAZINE FOR MY UNIFIED AND FUNDAMENTAL EXPLANATION OF PHYSICS (AND OUR ORIGIN).

report post as inappropriate

report post as inappropriate

Hi Florin,

I regret that I just got around to your arXiv paper ("Comments on 'disproof of Bell's theorem')but things have been flying pretty thick and fast.

I stumbled on this point:

"Joy Christian then attempts a generalization and makes an extraordinary claim without proof [4]: for any Hilbert space H of arbitrary dimension there exist a topological space corresponding to EPR counterfactual elements of reality and a morphism m : H -> Omega such that the following relation holds:

m(|psi_a> + |psi_b>) = (m|psi_a>) + (m|psi_b>) = A_aA_b members of Omega (10)

with A_a and A_b two points in the topological space. From this it is not hard to prove that exact local-realistic completion of any arbitrary entangled state is always possible. At this point, it is not clear if Eq. 10

can always be achieved, and this is a very interesting open problem."

I wasn't aware it was an open problem; however, it is simply proved by consequence of Joy's condition of 4pi (vice 2pi) rotation, q.e.d. Details from my results in the "Quantum music ..." blog, 16 August, text and attached figure.

His whole framework is, as he has claimed, is complete in its generalization.

Tom

report post as inappropriate

I regret that I just got around to your arXiv paper ("Comments on 'disproof of Bell's theorem')but things have been flying pretty thick and fast.

I stumbled on this point:

"Joy Christian then attempts a generalization and makes an extraordinary claim without proof [4]: for any Hilbert space H of arbitrary dimension there exist a topological space corresponding to EPR counterfactual elements of reality and a morphism m : H -> Omega such that the following relation holds:

m(|psi_a> + |psi_b>) = (m|psi_a>) + (m|psi_b>) = A_aA_b members of Omega (10)

with A_a and A_b two points in the topological space. From this it is not hard to prove that exact local-realistic completion of any arbitrary entangled state is always possible. At this point, it is not clear if Eq. 10

can always be achieved, and this is a very interesting open problem."

I wasn't aware it was an open problem; however, it is simply proved by consequence of Joy's condition of 4pi (vice 2pi) rotation, q.e.d. Details from my results in the "Quantum music ..." blog, 16 August, text and attached figure.

His whole framework is, as he has claimed, is complete in its generalization.

Tom

report post as inappropriate

Hi Tom,

Joy could derive that in only a few cases related to the division algebras. My point was that I would like to see an actual proof that it works in general. I did not see any complete proof in any of his papers. Remember that the topology of the space Omega could be very convoluted in general. So either show that the topology is only the one related to the division algebras, or show that the property holds under any circumstance.

Florin

report post as inappropriate

Joy could derive that in only a few cases related to the division algebras. My point was that I would like to see an actual proof that it works in general. I did not see any complete proof in any of his papers. Remember that the topology of the space Omega could be very convoluted in general. So either show that the topology is only the one related to the division algebras, or show that the property holds under any circumstance.

Florin

report post as inappropriate

Hi Tom and Florin,

I think Florin has a point here. Explicit calculation for any general quantum state is a formidable task. So that is clearly not a smart way to go about this. However, I now have a much stronger position. I am now convinced that the only relevant topology is that of a parallelized 7-sphere, which of course subsumes all other parallelized spheres. But as I have already shown, for a parallelized 7-sphere Eq. (10) automatically holds, because octonions form a quasi-group. For all other topologies the correlations would be weaker-than-quantum (i.e., unphysical) as well as nonlocal (i.e., unphysical).

Joy

report post as inappropriate

I think Florin has a point here. Explicit calculation for any general quantum state is a formidable task. So that is clearly not a smart way to go about this. However, I now have a much stronger position. I am now convinced that the only relevant topology is that of a parallelized 7-sphere, which of course subsumes all other parallelized spheres. But as I have already shown, for a parallelized 7-sphere Eq. (10) automatically holds, because octonions form a quasi-group. For all other topologies the correlations would be weaker-than-quantum (i.e., unphysical) as well as nonlocal (i.e., unphysical).

Joy

report post as inappropriate

Florin and Joy,

Understood. My point is, for classical pair correlations to the cosmological limit, clearly physical spacetime ( and that's what I mean by " ... complete in its generalization") subsumes any quantum configuration space. And Omega does not have to be smooth, only simply connected.

The assumption of physical events on S^3 combined with the topology of S^7 gives the 2-point boundary of continuous observer-observed entanglement a superior role in physical reality over pair correlations by quantum entanglement, because the 3 + 1 spacetime of S^3 oriented by the 7 + 1 spacetime of S^7 orients antipodal points of S^3 at ANY arbitrary distance from the (complex) origin up to the cosmological limit.

It struck me that this is only possible to see, by assuming 4pi rotation over the S^2 complex manifold because, since every point of S^3 is homeomorphic to S^3, 2pi rotation guarantees correlation only 1/2 the time for each event (as quantum entanglement for the Bell-Aspect program predicts), while pairs of events correlated ALL the time by a 2-point continuous function must be calibrated to locate the initial condition. This pair of points is actually a single complex point, sqrt -1, or i, which returns to itself only after four full rotations on the Riemann sphere (four exponential iterations) with its one simple pole at infinity. This corresponds to Joy's "Pacman" analogy -- we can only calculate where we "really" are in spacetime by seeing the event simultaneously disappear and reappear at predicted antipodal points of the horizon, a continuous real function self similar to quantum bilocation. Arbitrary quantum configuration space is subsumed by physically real spacetime.

If you examine my results, and diagram, you should see that hyperbolic projection from the fixed point ( - 1) to all points of S^3 shows perfect correspondence between the quaternion and octonion algebras and real analytical results of Lebesgue measure. Another insight I had to absorb from Joy's research program, in order to be convinced of its validity, is that the calculational artifacts of S^7 only support the physical spacetime of S^3, and are not physically real.

This reply is pretty slapdash; however, my aims are twofold:

1. To provide a basis to determine the exact experimental setup that corresponds to Joy Christian's framework.

2. Lay groundwork for a theorem that pair correlations in physical spacetime are independnet of pair correlations in quantum configuration space.

All best,

Tom

report post as inappropriate

Understood. My point is, for classical pair correlations to the cosmological limit, clearly physical spacetime ( and that's what I mean by " ... complete in its generalization") subsumes any quantum configuration space. And Omega does not have to be smooth, only simply connected.

The assumption of physical events on S^3 combined with the topology of S^7 gives the 2-point boundary of continuous observer-observed entanglement a superior role in physical reality over pair correlations by quantum entanglement, because the 3 + 1 spacetime of S^3 oriented by the 7 + 1 spacetime of S^7 orients antipodal points of S^3 at ANY arbitrary distance from the (complex) origin up to the cosmological limit.

It struck me that this is only possible to see, by assuming 4pi rotation over the S^2 complex manifold because, since every point of S^3 is homeomorphic to S^3, 2pi rotation guarantees correlation only 1/2 the time for each event (as quantum entanglement for the Bell-Aspect program predicts), while pairs of events correlated ALL the time by a 2-point continuous function must be calibrated to locate the initial condition. This pair of points is actually a single complex point, sqrt -1, or i, which returns to itself only after four full rotations on the Riemann sphere (four exponential iterations) with its one simple pole at infinity. This corresponds to Joy's "Pacman" analogy -- we can only calculate where we "really" are in spacetime by seeing the event simultaneously disappear and reappear at predicted antipodal points of the horizon, a continuous real function self similar to quantum bilocation. Arbitrary quantum configuration space is subsumed by physically real spacetime.

If you examine my results, and diagram, you should see that hyperbolic projection from the fixed point ( - 1) to all points of S^3 shows perfect correspondence between the quaternion and octonion algebras and real analytical results of Lebesgue measure. Another insight I had to absorb from Joy's research program, in order to be convinced of its validity, is that the calculational artifacts of S^7 only support the physical spacetime of S^3, and are not physically real.

This reply is pretty slapdash; however, my aims are twofold:

1. To provide a basis to determine the exact experimental setup that corresponds to Joy Christian's framework.

2. Lay groundwork for a theorem that pair correlations in physical spacetime are independnet of pair correlations in quantum configuration space.

All best,

Tom

report post as inappropriate

Tom,

I have been following both Joy's and Florin's forums from the start. It has been fascinating and it seems to me that your analysis is "spot on". Florin has represented the opposing view admirably, and I believe Joy benefits from the communications he's received from both of you. Unfortunately, my math skills aren't sufficient to contribute or even distinguish between the "subtle" and the "malicious", but it does make me want to learn more.

Keep up the good work,

Dan

report post as inappropriate

I have been following both Joy's and Florin's forums from the start. It has been fascinating and it seems to me that your analysis is "spot on". Florin has represented the opposing view admirably, and I believe Joy benefits from the communications he's received from both of you. Unfortunately, my math skills aren't sufficient to contribute or even distinguish between the "subtle" and the "malicious", but it does make me want to learn more.

Keep up the good work,

Dan

report post as inappropriate

Dear Florin,

I miss your participation. Out of ignorance, I ask for further clarification. Joy used geometric algebra. I am still looking into that. However, I assumed that geometric algebra would have already had the connection that you mention here:

"All I would say is that at some point some consistent translation mechanism has to be found from Joy's formulation to standard statistics. I disagree with Joy not on his math, but on his interpretation. And if his math is right I am confident such a mechanism does exist and will be eventually discovered. In other words, if Joy found an equivalent way of formulating QM, Born rule's translation into the new formalism will follow. It is just not done at this point."

Does your point have to do with something other than geometric algebra or are you saying that there is no current link between geometric algebra and standard statistics? If so, how important is this missing link? Missing links seem very important to me. As I have said to Joy, if my questions or remarks are off the mark please just say so and I will study your messages again.

James

report post as inappropriate

I miss your participation. Out of ignorance, I ask for further clarification. Joy used geometric algebra. I am still looking into that. However, I assumed that geometric algebra would have already had the connection that you mention here:

"All I would say is that at some point some consistent translation mechanism has to be found from Joy's formulation to standard statistics. I disagree with Joy not on his math, but on his interpretation. And if his math is right I am confident such a mechanism does exist and will be eventually discovered. In other words, if Joy found an equivalent way of formulating QM, Born rule's translation into the new formalism will follow. It is just not done at this point."

Does your point have to do with something other than geometric algebra or are you saying that there is no current link between geometric algebra and standard statistics? If so, how important is this missing link? Missing links seem very important to me. As I have said to Joy, if my questions or remarks are off the mark please just say so and I will study your messages again.

James

report post as inappropriate

Dear James,

I am actively working on Joy's theory and please let me answer your question only after I will be done. In the meantime I can explain how geometric algebra works, it is really easy.

You start with 3 elements: e1, e2, e3 corresponding to the 3 directions: x, y z. Because rotations are non-commutative, e1,2,3 do not commute in geometric algebra. So one can have expressions like 7 e2 + 32 e1e3. There are 2 basic rules to simplify horrible things like e2e2e1e3e1e3e2e1e2:

- rule 1: e1e1 = e2e2 = e3e3 = 1

- rule 2: e1e2 = -e2e1, e2e3 = -e3e2, e1e3 = -e3e1

Applying the rules on e2e2e1e3e1e3e2e1e2 you get: e1e3e1e3e2e1e2 (by e2e2 = 1), then -e3e1e1e3e2e1e2 (by e1e3=-e3e1) then -e3e3e2e1e2 then -e2e1e2 then +e1e2e2 and finally e1

There are the following un-simplifyable elements: scalars, vectors: (e1,32,e3), bivectors (e1e2, e2e3, e3e1), and trivector (pseudo-scalar) e1e2e3

As a notation bivectors can be written as e1e2 = e1^e2 with ^ (hat) the exterior product. Also e1e2e3 = e1^e2^e3 = I

This is all there is to it from the algebra side, ther rest are just easy consequences of the 2 rules.

report post as inappropriate

I am actively working on Joy's theory and please let me answer your question only after I will be done. In the meantime I can explain how geometric algebra works, it is really easy.

You start with 3 elements: e1, e2, e3 corresponding to the 3 directions: x, y z. Because rotations are non-commutative, e1,2,3 do not commute in geometric algebra. So one can have expressions like 7 e2 + 32 e1e3. There are 2 basic rules to simplify horrible things like e2e2e1e3e1e3e2e1e2:

- rule 1: e1e1 = e2e2 = e3e3 = 1

- rule 2: e1e2 = -e2e1, e2e3 = -e3e2, e1e3 = -e3e1

Applying the rules on e2e2e1e3e1e3e2e1e2 you get: e1e3e1e3e2e1e2 (by e2e2 = 1), then -e3e1e1e3e2e1e2 (by e1e3=-e3e1) then -e3e3e2e1e2 then -e2e1e2 then +e1e2e2 and finally e1

There are the following un-simplifyable elements: scalars, vectors: (e1,32,e3), bivectors (e1e2, e2e3, e3e1), and trivector (pseudo-scalar) e1e2e3

As a notation bivectors can be written as e1e2 = e1^e2 with ^ (hat) the exterior product. Also e1e2e3 = e1^e2^e3 = I

This is all there is to it from the algebra side, ther rest are just easy consequences of the 2 rules.

report post as inappropriate

James,

What missing link? There is no "link" missing. The link between observed statistics and my model is already established in my latest paper, which Florin seems to have completely ignored. Geometric algebra is only a tool---a mere representation of the true essence of my model. If we restrict to the EPR correlations, then my model says that they are nothing but correlations between the numbers +1 and -1 occurring within a parallelized 3-sphere, which, in turn, is simply a classical topological space. It is convenient to use geometric algebra to bring these facts out, but only as a tool. Florin's attempt to find quantum mechanics within my model is deeply misguided. In particular, I find the following sentence of his quite bizarre: "...if Joy found an equivalent way of formulating QM, Born rule's translation into the new formalism will follow." What equivalent way? There is no quantum mechanics, non-locality, or contextuality of any kind within my framework, so why would one look for a "Born rule"? A rule to connect what and what? The definite measurement outcomes, +1 or -1, are already well defined by my equations (16) and (17), and the correlations between these outcomes (i.e., between the numbers +1 and -1) are then given by equation (33). Moreover, these equations are easily generalizable for the general case of 7-sphere. So I have no idea what "link" Florin is looking for.

Joy

report post as inappropriate

What missing link? There is no "link" missing. The link between observed statistics and my model is already established in my latest paper, which Florin seems to have completely ignored. Geometric algebra is only a tool---a mere representation of the true essence of my model. If we restrict to the EPR correlations, then my model says that they are nothing but correlations between the numbers +1 and -1 occurring within a parallelized 3-sphere, which, in turn, is simply a classical topological space. It is convenient to use geometric algebra to bring these facts out, but only as a tool. Florin's attempt to find quantum mechanics within my model is deeply misguided. In particular, I find the following sentence of his quite bizarre: "...if Joy found an equivalent way of formulating QM, Born rule's translation into the new formalism will follow." What equivalent way? There is no quantum mechanics, non-locality, or contextuality of any kind within my framework, so why would one look for a "Born rule"? A rule to connect what and what? The definite measurement outcomes, +1 or -1, are already well defined by my equations (16) and (17), and the correlations between these outcomes (i.e., between the numbers +1 and -1) are then given by equation (33). Moreover, these equations are easily generalizable for the general case of 7-sphere. So I have no idea what "link" Florin is looking for.

Joy

report post as inappropriate

Indeed, I was going to make the same point. Geometric algebra is not a magic wand, and neither has Joy waved it -- or his hands -- in that manner. The real significance of the result is completeness -- mathematical completeness of the same kind which characterizes relativity. The mathematical model is completely independent of experiment, and of quantum mechanics (which is why I continue to regret that it is positioned aside Bell-Aspect).

In the next day or two, I plan to post my own graphic interpretation of Joy's experimental framework, which will show why there are no gaps.

Tom

report post as inappropriate

In the next day or two, I plan to post my own graphic interpretation of Joy's experimental framework, which will show why there are no gaps.

Tom

report post as inappropriate

Tom,

I have to acknowledge your superior skill. I read your previous powerpoint presentation. It was somewhat abrupt for me. However, you presentation:

"Let's see how this makes out:

joy christian local realism experiment"

is way too thin for me. However, I am satisfied just to see what the professionals say in response.

Thank you,

James

report post as inappropriate

I have to acknowledge your superior skill. I read your previous powerpoint presentation. It was somewhat abrupt for me. However, you presentation:

"Let's see how this makes out:

joy christian local realism experiment"

is way too thin for me. However, I am satisfied just to see what the professionals say in response.

Thank you,

James

report post as inappropriate

James,

Did it look okay? -- I mean technically, as a PPT. I created it in Windows, but on my MacBook Pro it opens with errors in formatting.

Tom

report post as inappropriate

Did it look okay? -- I mean technically, as a PPT. I created it in Windows, but on my MacBook Pro it opens with errors in formatting.

Tom

report post as inappropriate

James,

It looks thin, modulo mathematical details, which would run into scores of pages.

Tom

report post as inappropriate

It looks thin, modulo mathematical details, which would run into scores of pages.

Tom

report post as inappropriate

Tom,

"It looks thin, modulo mathematical details, which would run into scores of pages."

I understand. I am hoping that more is made clear when others evaluate what you have posted. You noticed that I chose a different thread to post my response. I wanted your message to remain visable as long as possible. Thank you for posting it.

James

report post as inappropriate

"It looks thin, modulo mathematical details, which would run into scores of pages."

I understand. I am hoping that more is made clear when others evaluate what you have posted. You noticed that I chose a different thread to post my response. I wanted your message to remain visable as long as possible. Thank you for posting it.

James

report post as inappropriate

Andy M.,

Thank you, quoting Florin from 'Part 2 of To Be or Not To Be (a Local Realist)':

"For the interest of time and space let me only discuss Bell’s criticism of von Neumann and Gleason in this paper. Von Neumann had enormous influence in quantum mechanics due to his seminal work on axiomatizing it. He produced a “no-go” theorem on hidden variable theories which Bell found...

view entire post

Thank you, quoting Florin from 'Part 2 of To Be or Not To Be (a Local Realist)':

"For the interest of time and space let me only discuss Bell’s criticism of von Neumann and Gleason in this paper. Von Neumann had enormous influence in quantum mechanics due to his seminal work on axiomatizing it. He produced a “no-go” theorem on hidden variable theories which Bell found...

view entire post

report post as inappropriate

James,

This was a quick cramming done by the master bluffer in response to my scathing criticism here:

"Let me begin by pointing out a number of alarming confusions in Florin's reasoning---not only concerning my own work, but also concerning the significance of Bell's theorem itself. The worse of these confusions occurs in his understanding of the notion of contextuality. Florin has painted this notion as something to be avoided at all cost. But that sharply differs from how Bell himself viewed the notion. Bell vigorously defended contextuality in his very first paper. He famously stressed that "the result of an observation may reasonably depend not only on the state of the system (including hidden variables) but also on the disposition of the apparatus." I could not agree more! He goes on in the paper to strongly criticise the theorems by von Neumann, Jauch and Piron, and Gleason for neglecting contextuality, and argues that the demand of strict non-contextuality implicit in these theorems is "quite unreasonable" from the physical point of view. The idea of contextuality, he argues, should not be confused with the idea of realism. These are two completely separate notions."

Joy Christian

report post as inappropriate

This was a quick cramming done by the master bluffer in response to my scathing criticism here:

"Let me begin by pointing out a number of alarming confusions in Florin's reasoning---not only concerning my own work, but also concerning the significance of Bell's theorem itself. The worse of these confusions occurs in his understanding of the notion of contextuality. Florin has painted this notion as something to be avoided at all cost. But that sharply differs from how Bell himself viewed the notion. Bell vigorously defended contextuality in his very first paper. He famously stressed that "the result of an observation may reasonably depend not only on the state of the system (including hidden variables) but also on the disposition of the apparatus." I could not agree more! He goes on in the paper to strongly criticise the theorems by von Neumann, Jauch and Piron, and Gleason for neglecting contextuality, and argues that the demand of strict non-contextuality implicit in these theorems is "quite unreasonable" from the physical point of view. The idea of contextuality, he argues, should not be confused with the idea of realism. These are two completely separate notions."

Joy Christian

report post as inappropriate

Also worth noting are my comments to Lawrence on Nov. 14, 2011 @ 15:51 GMT:

"Your comments are naive, at least with regard to the sociology of physics. Any new evidence in physics---no matter how starkly presented---is almost never easily accepted. It is often misinterpreted, flatly denied, effectively neutralized, or simply ignored if the physics community does not like it (just as you don't like my ideas because they would invalidate yours).

Recall how von Neumann's theorem against general hidden variables was believed in by the physics community for 30 years (yes, 30 years) despite its clear-cut refutation by Grete Hermann in 1935 (and despite the existence of explicit counterexample to the theorem by Bohm). It was not until John Bell rediscovered Hermann's objection in 1965 that the importance of her work began to be appreciated by the community. You grossly underestimate the power of dogma within physics.

Your second error is to think that my framework, or any hidden variable framework for that matter, implies that quantum mechanics is incorrect. This is your main blind spot."

report post as inappropriate

"Your comments are naive, at least with regard to the sociology of physics. Any new evidence in physics---no matter how starkly presented---is almost never easily accepted. It is often misinterpreted, flatly denied, effectively neutralized, or simply ignored if the physics community does not like it (just as you don't like my ideas because they would invalidate yours).

Recall how von Neumann's theorem against general hidden variables was believed in by the physics community for 30 years (yes, 30 years) despite its clear-cut refutation by Grete Hermann in 1935 (and despite the existence of explicit counterexample to the theorem by Bohm). It was not until John Bell rediscovered Hermann's objection in 1965 that the importance of her work began to be appreciated by the community. You grossly underestimate the power of dogma within physics.

Your second error is to think that my framework, or any hidden variable framework for that matter, implies that quantum mechanics is incorrect. This is your main blind spot."

report post as inappropriate

The so called "scathing criticism" included the idiotic assertion that general relativity is a contextual theory. But how to say to Joy: hey, you are an imbecile? I had to do a large detour to frame the discussion correctly and clear up his propaganda smoke screen.

report post as inappropriate

report post as inappropriate

I found these remarks about general relativity as a contextual theory. I think some deeper thought needs to be given to this matter. The concept of epistemological contextualism is that the frame from which one makes an account of knowledge is a determining factor in that knowledge. In quantum mechanics this relates to naïve realism on the nature of observables. A particle, such as the...

view entire post

view entire post

report post as inappropriate

"Thus quantum mechanics is observer independent ..."

QM measurements are observer-created. Classical physics is observer-dependent, by orientation entanglement -- as Joy says, non-anthropcentric.

Lawrence, I used to think exactly as you do. Modern research by Hestenes, Lamport and Christian has changed my mind. Every challenge to classical realism can be met with their results.

Take "What can we say about general relativity? Contextuality in general relativity is not compatible with the basic format of coordinate independent spacetime physics."

If it weren't compatible, Hestenes would not be able to translate GA to spacetime algebra to Minkowski space. This is plenty good enough for me. It's stunning, in fact, and fulfills Hestenes' promise to simplify physics language.

"A spacetime with no Killing vector or isometries in a time direction does not provide a conservation law for energy."

As I said before, you are comparing limited mathematical methods with a mathematically complete theory. That is not a fair comparison -- the space of Joy's theory, based on octonionic algebra, is 8 dimensional. Even Einstein did not object to adding dimensions if there exist good physical reasons to do so.

"The coordinate independent nature of spacetime physics precludes any underlying contextuality."

Coordinate free spacetime geometry is exactly WHY Hestenes can convert GA to Minkowski space, the underlying spacetime of general relativity.

Tom

report post as inappropriate

QM measurements are observer-created. Classical physics is observer-dependent, by orientation entanglement -- as Joy says, non-anthropcentric.

Lawrence, I used to think exactly as you do. Modern research by Hestenes, Lamport and Christian has changed my mind. Every challenge to classical realism can be met with their results.

Take "What can we say about general relativity? Contextuality in general relativity is not compatible with the basic format of coordinate independent spacetime physics."

If it weren't compatible, Hestenes would not be able to translate GA to spacetime algebra to Minkowski space. This is plenty good enough for me. It's stunning, in fact, and fulfills Hestenes' promise to simplify physics language.

"A spacetime with no Killing vector or isometries in a time direction does not provide a conservation law for energy."

As I said before, you are comparing limited mathematical methods with a mathematically complete theory. That is not a fair comparison -- the space of Joy's theory, based on octonionic algebra, is 8 dimensional. Even Einstein did not object to adding dimensions if there exist good physical reasons to do so.

"The coordinate independent nature of spacetime physics precludes any underlying contextuality."

Coordinate free spacetime geometry is exactly WHY Hestenes can convert GA to Minkowski space, the underlying spacetime of general relativity.

Tom

report post as inappropriate

General relativity is a contextual theory par excellence.

Cf. Section 2 of this paper of mine.

report post as inappropriate

Cf. Section 2 of this paper of mine.

report post as inappropriate

"General relativity is a contextual theory par excellence.

Cf. Section 2 of this paper of mine."

Good, take it to the bank.

report post as inappropriate

Cf. Section 2 of this paper of mine."

Good, take it to the bank.

report post as inappropriate

TH Ray wrote: "Overcoming spacetime curvature -- now, that's an accomplishment."

Many of your masters have overcome it but did not inform you:

http://www.newscientist.com/article/mg20026831.500-what-

makes-the-universe-tick.html

"It is still not clear who is right, says John Norton, a philosopher based at the University of Pittsburgh, Pennsylvania. Norton is hesitant to express it, but his instinct - and the consensus in physics - seems to be that space and time exist on their own. The trouble with this idea, though, is that it doesn't sit well with relativity, which describes space-time as a malleable fabric whose geometry can be changed by the gravity of stars, planets and matter."

Pentcho Valev pvalev@yahoo.com

report post as inappropriate

Many of your masters have overcome it but did not inform you:

http://www.newscientist.com/article/mg20026831.500-what-

makes-the-universe-tick.html

"It is still not clear who is right, says John Norton, a philosopher based at the University of Pittsburgh, Pennsylvania. Norton is hesitant to express it, but his instinct - and the consensus in physics - seems to be that space and time exist on their own. The trouble with this idea, though, is that it doesn't sit well with relativity, which describes space-time as a malleable fabric whose geometry can be changed by the gravity of stars, planets and matter."

Pentcho Valev pvalev@yahoo.com

report post as inappropriate

Jason Wolfe wrote: "Some may argue that the relationship between gravity and light is already well known. Let me be more specific. There is more to learn, by experiment, about phase/frequency shift of light AND gravitational time dilation."

Consider this:

http://student.fizika.org/~jsisko/Knjige/Klasicna%20Meh

anika/David%20Morin/CH13.PDF

David Morin: "The equivalence principle has a striking consequence concerning the behavior of clocks in a gravitational field. It implies that higher clocks run faster than lower clocks. If you put a watch on top of a tower, and then stand on the ground, you will see the watch on the tower tick faster than an identical watch on your wrist. When you take the watch down and compare it to the one on your wrist, it will show more time elapsed."

Is it true? Clever Einsteinians know that there is no gravitational time dilation. The gravitational redshift arises from the change in the speed of light signals in the presence of gravitation:

http://www.amazon.com/Relativity-Its-Roots-Banesh-Hoffmann/d

p/0486406768

Banesh Hoffmann: "In an accelerated sky laboratory, and therefore also in the corresponding earth laboratory, the frequence of arrival of light pulses is lower than the ticking rate of the upper clocks even though all the clocks go at the same rate. (...) As a result the experimenter at the ceiling of the sky laboratory will see with his own eyes that the floor clock is going at a slower rate than the ceiling clock - even though, as I have stressed, both are going at the same rate. (...) The gravitational red shift does not arise from changes in the intrinsic rates of clocks. It arises from what befalls light signals as they traverse space and time in the presence of gravitation."

Pentcho Valev pvalev@yahoo.com

report post as inappropriate

Consider this:

http://student.fizika.org/~jsisko/Knjige/Klasicna%20Meh

anika/David%20Morin/CH13.PDF

David Morin: "The equivalence principle has a striking consequence concerning the behavior of clocks in a gravitational field. It implies that higher clocks run faster than lower clocks. If you put a watch on top of a tower, and then stand on the ground, you will see the watch on the tower tick faster than an identical watch on your wrist. When you take the watch down and compare it to the one on your wrist, it will show more time elapsed."

Is it true? Clever Einsteinians know that there is no gravitational time dilation. The gravitational redshift arises from the change in the speed of light signals in the presence of gravitation:

http://www.amazon.com/Relativity-Its-Roots-Banesh-Hoffmann/d

p/0486406768

Banesh Hoffmann: "In an accelerated sky laboratory, and therefore also in the corresponding earth laboratory, the frequence of arrival of light pulses is lower than the ticking rate of the upper clocks even though all the clocks go at the same rate. (...) As a result the experimenter at the ceiling of the sky laboratory will see with his own eyes that the floor clock is going at a slower rate than the ceiling clock - even though, as I have stressed, both are going at the same rate. (...) The gravitational red shift does not arise from changes in the intrinsic rates of clocks. It arises from what befalls light signals as they traverse space and time in the presence of gravitation."

Pentcho Valev pvalev@yahoo.com

report post as inappropriate

Pencho,

You said, "Is it true? Clever Einsteinians know that there is no gravitational time dilation. The gravitational redshift arises from the change in the speed of light signals in the presence of gravitation: "

How can we make a technological leap forward if we disagree on fundamental physics? The speed of light is invariant for the observer (and the emitter). Gravitational time dilation was proven experimentally.

http://en.wikipedia.org/wiki/Hafele%E2%80%93K

eating_experiment

Hey, if you think you know more about space-time than the experts, then why don't you build your own global positioning system.

report post as inappropriate

You said, "Is it true? Clever Einsteinians know that there is no gravitational time dilation. The gravitational redshift arises from the change in the speed of light signals in the presence of gravitation: "

How can we make a technological leap forward if we disagree on fundamental physics? The speed of light is invariant for the observer (and the emitter). Gravitational time dilation was proven experimentally.

http://en.wikipedia.org/wiki/Hafele%E2%80%93K

eating_experiment

Hey, if you think you know more about space-time than the experts, then why don't you build your own global positioning system.

report post as inappropriate

Richard Gill,

In my opinion you are an incompetent mathematician who is not able to do a very elementary and simple calculation. My one page paper is self-contained, and in fact it is only half a page long. Yet you have been claiming that

"Joy is unable to fix the gap between (6) and (7) in his one page paper."

This is a shameless, unadulterated, maliciously generated, and...

view entire post

In my opinion you are an incompetent mathematician who is not able to do a very elementary and simple calculation. My one page paper is self-contained, and in fact it is only half a page long. Yet you have been claiming that

"Joy is unable to fix the gap between (6) and (7) in his one page paper."

This is a shameless, unadulterated, maliciously generated, and...

view entire post

attachments: 48_Gill.pdf

report post as inappropriate

I am glad seeing this continuation, it is very much informative where I can be motivated.

technical content

report post as inappropriate

technical content

report post as inappropriate

Login or create account to post reply or comment.