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February 11, 2012
Continuing about the New Directions in the Foundations of Physics conference in Washington DC, the bulk of the quantum mechanics presentations were in the information approach. (You can check my earlier post about building “black holes” in the lab here.)
Quantum mechanics is strange. Really strange. But it turns out quantum mechanics is not as strange as it could be, and looking at things from the information point of view does help clarify some issues.
Before presenting the talks, I want to set the stage with a few key facts about this area. A basic ingredient in this area is entanglement which is viewed as a resource able to pull impossible feats in classical mechanics, like teleportation. The origin of entanglement is well understood: it is the dimensionality of the Hilbert space N2 which is larger than the dimensionality of an equivalent classical phase space N1. This in turn means that after interaction, two systems have collectively more information about them then each subsystem; in other words, the whole is bigger than the parts and the joined state is not factorizable. EPR paper was strongly against this and proposed that quantum mechanics is incomplete and hidden variables are responsible for its indeterminacy. After EPR, the foundational area remained dormant for a while, until John Bell broke the deadlock by finding inequalities which were satisfied by the classical physics (and hidden variable theories), but violated by quantum mechanics and the violations were subsequently observed by the Aspect experiment.
Fast forward again, and in 1980 Tsirelson asked the question: if Bell gave us the classical mechanics/hidden variables inequalities, what are the corresponding quantum mechanical inequalities? In the standard CHSH inequality, Bell bound is 2, while Tsirelson’s bound in 2sqrt(2). Independent of this, Aharonov proposed that the axioms of quantum mechanics should be relativistic causality and nonlocality.
Was Aharonov’s conjecture correct? No. In 1994, Popescu and Rohrlich proved that wrong by discovering a hypothetical device, now called the PR-box which reaches the maximum bound of 4 in the CHSH inequality. And so the plot thickens, because as weird as quantum mechanics is, it is not as weird as a PR-box and this begs the question why nature does not allow one?
It turns out, that for the simplest case, the state space geometry describing classical mechanics, quantum mechanics, and the PR box is an eight dimensional polytope. At the conference a remarkable talk by Valerio Scalani presented a Nature paper which identified a new physical principle able to separate classical mechanics, as well as standard non-relativistic quantum mechanics from the supposedly unphysical states corresponding to “super-quantum” correlations.
What is this principle? It was coined “Information Causality”. Here is what one can do when this principle is violated. Suppose I have a CD with 10 songs, each song is 1 mega bit long, and I have a classical communication channel which can transmit only 1 megabit. If I have a PR box, and I can super-correlate the original 10 megabits with the receiver’s state, then the receiver can pick and choose to download any of the 10 songs (up to the classical bandwidth of 1 mega bit), and not be locked into his original choice. Quoting from the paper: “Maximally strong no-signalling correlations would allow Bob access to any m bit subset of the whole data set held by Alice. If only one bit is sent by Alice (m = 1), this is tantamount to Bob being able to access the value of any single bit of Alice's data (but of course not all of them).” Just imagine the possibilities: this would be a spy’s heaven.
Information causality demands on the other hand that after receiving n bits of classical information, no amount of processing can increase this amount of information, and so music industry seems to be made safe from pesky downloads by quantum mechanics. If only Napster would have had a PR box…
(Entanglement cartoon from www.madprime.org.)
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Quantum mechanics is strange. Really strange. But it turns out quantum mechanics is not as strange as it could be, and looking at things from the information point of view does help clarify some issues.
Before presenting the talks, I want to set the stage with a few key facts about this area. A basic ingredient in this area is entanglement which is viewed as a resource able to pull impossible feats in classical mechanics, like teleportation. The origin of entanglement is well understood: it is the dimensionality of the Hilbert space N2 which is larger than the dimensionality of an equivalent classical phase space N1. This in turn means that after interaction, two systems have collectively more information about them then each subsystem; in other words, the whole is bigger than the parts and the joined state is not factorizable. EPR paper was strongly against this and proposed that quantum mechanics is incomplete and hidden variables are responsible for its indeterminacy. After EPR, the foundational area remained dormant for a while, until John Bell broke the deadlock by finding inequalities which were satisfied by the classical physics (and hidden variable theories), but violated by quantum mechanics and the violations were subsequently observed by the Aspect experiment.
Fast forward again, and in 1980 Tsirelson asked the question: if Bell gave us the classical mechanics/hidden variables inequalities, what are the corresponding quantum mechanical inequalities? In the standard CHSH inequality, Bell bound is 2, while Tsirelson’s bound in 2sqrt(2). Independent of this, Aharonov proposed that the axioms of quantum mechanics should be relativistic causality and nonlocality.
Was Aharonov’s conjecture correct? No. In 1994, Popescu and Rohrlich proved that wrong by discovering a hypothetical device, now called the PR-box which reaches the maximum bound of 4 in the CHSH inequality. And so the plot thickens, because as weird as quantum mechanics is, it is not as weird as a PR-box and this begs the question why nature does not allow one?
It turns out, that for the simplest case, the state space geometry describing classical mechanics, quantum mechanics, and the PR box is an eight dimensional polytope. At the conference a remarkable talk by Valerio Scalani presented a Nature paper which identified a new physical principle able to separate classical mechanics, as well as standard non-relativistic quantum mechanics from the supposedly unphysical states corresponding to “super-quantum” correlations.
What is this principle? It was coined “Information Causality”. Here is what one can do when this principle is violated. Suppose I have a CD with 10 songs, each song is 1 mega bit long, and I have a classical communication channel which can transmit only 1 megabit. If I have a PR box, and I can super-correlate the original 10 megabits with the receiver’s state, then the receiver can pick and choose to download any of the 10 songs (up to the classical bandwidth of 1 mega bit), and not be locked into his original choice. Quoting from the paper: “Maximally strong no-signalling correlations would allow Bob access to any m bit subset of the whole data set held by Alice. If only one bit is sent by Alice (m = 1), this is tantamount to Bob being able to access the value of any single bit of Alice's data (but of course not all of them).” Just imagine the possibilities: this would be a spy’s heaven.
Information causality demands on the other hand that after receiving n bits of classical information, no amount of processing can increase this amount of information, and so music industry seems to be made safe from pesky downloads by quantum mechanics. If only Napster would have had a PR box…
 |
(Entanglement cartoon from www.madprime.org.)
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This strikes me as potentially very fundamental. I am not sure exactly how to comment as yet. However, I will give a stab at something, even if it is wrong. This is a generalization of the Bell inequalities result. Here the equality as I read is a quantum equality, which itself can in general be violated. So this means that quantum mechanics is in some ways more "more nonlocal" than previously thought.
So there is potential application with quantum gravity. The AdS~CFT result defines the CFT on the boundary of the AdS, as a sphere. Yet if we consider correlations with black holes, a black hole in an AdS which preserves quantum information, then for dim > 3 there is a Weyl curvature, and further the temperature of the BH at extremal conditions is nonzero. So there are additional physics involved here, which I think generalizes the black hole complementarity principle. The extra degrees of freedom involved are I think given by this temperature measure. I then ponder whether there is some generalization of this sort with the quantization of black holes which determines interior and exterior states of a black hole.
Cheers LC
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So there is potential application with quantum gravity. The AdS~CFT result defines the CFT on the boundary of the AdS, as a sphere. Yet if we consider correlations with black holes, a black hole in an AdS which preserves quantum information, then for dim > 3 there is a Weyl curvature, and further the temperature of the BH at extremal conditions is nonzero. So there are additional physics involved here, which I think generalizes the black hole complementarity principle. The extra degrees of freedom involved are I think given by this temperature measure. I then ponder whether there is some generalization of this sort with the quantization of black holes which determines interior and exterior states of a black hole.
Cheers LC
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Lawrence,
This is post 2 of 4 from the conference review. I could not put my finger on it either, but also I do feel this may turn out to be a significant principle.
In the meantime I wanted to convey the core results in this very active area: Tsirelson bound, PR box and super-correlations and make them clear to a general audience. The striking things from the talks were how much those results were quoted by everyone again and again.
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This is post 2 of 4 from the conference review. I could not put my finger on it either, but also I do feel this may turn out to be a significant principle.
In the meantime I wanted to convey the core results in this very active area: Tsirelson bound, PR box and super-correlations and make them clear to a general audience. The striking things from the talks were how much those results were quoted by everyone again and again.
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Vaidman and other have worked out a quantum model of the Tsirelson bound . This all looks interesting as a way of looking at some generalization of QFT. The Aharonov postulates of relativistic causality and nonlocality appear violated by the Popescu and Rohrlich device or “box.” As yet I don’t know enough about the PR-box to comment in great depth, but I conjecture this might be related to the removal of QFT locality assumptions of field amplitudes on dim = n – 1 surfaces in n dimensional manifolds with signature [-1, 1, …,1]. This is something to explore, for if the connection exists here this generalization of nonlocality might then be a foundation for quantum cosmology.
I will be back next week.
Cheers LC
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I will be back next week.
Cheers LC
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Lawrence,
I don't quite understand the Vaidman paper. What exactly happens in a post selection? And is this not a trivial thing? Imagine this: I have an ensable of linear functions and I "post select" them using a Taylor series rule to approximate any nonlinear function. Hence I "achieve" something that is completely beyond the initial character of the linear functions.
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I don't quite understand the Vaidman paper. What exactly happens in a post selection? And is this not a trivial thing? Imagine this: I have an ensable of linear functions and I "post select" them using a Taylor series rule to approximate any nonlinear function. Hence I "achieve" something that is completely beyond the initial character of the linear functions.
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The pre and post selection states are those states which evolve under a weak measurement. This is a way to understand the Hardy paradox, where an interferometer can permit an particle-antiparticle state to exhibit no annihilation. Curiously the stretched horizon can do something similar to this.
I have only read the beginning of some of these papers, including the Vaidman paper. I spent a short weekend-holiday in the wilderness so I am just now getting back into these things. I will try to complete reading this in the next few days.
Cheers LC
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I have only read the beginning of some of these papers, including the Vaidman paper. I spent a short weekend-holiday in the wilderness so I am just now getting back into these things. I will try to complete reading this in the next few days.
Cheers LC
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SAndu Popescu has written on how Weak measurements just got stronger, which connects some with the problem of post selection of states in a weak measurement.
Cheers LC
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Cheers LC
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So the upper bound on the achievable CHSH value is 2 for classical mechanics, 2sqrt(2) for quantum mechanics, and 4 for mechanics with a PR box. How big is the bound if you have an unrestricted faster-than-light-telegraph box, (FTLT box), or instantaneous action at a distance?
Apparently still only 4, based on four terms each with maximum value 1.
Is there some other inequality which will distinguish physics with PR boxes from physics with FTLT boxes?
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Apparently still only 4, based on four terms each with maximum value 1.
Is there some other inequality which will distinguish physics with PR boxes from physics with FTLT boxes?
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Jim,
4 is the maximum value. Why you would want a FTLT box, a PR box is unphysical as is, what good a "more unphysical" box would do?
Maybe I can explain a bit more about the geometry involved here. Imagine a square with a side of 4 drawn with the base parallel with the bottom of the paper. Then draw a circle (with radius 2sqrt(2)) with the center at the center of the square. This circle touches the square at all 4 points. Last, draw a larger square rotated 45 degrees from the first square touching the smaller square at all midpoints of its sides. The top vertex of the larger square is the PR box. The smaller square is classical mechanics/hidden variables, and the circle is QM. Connect the highest point on the circle with the PR box vertex. Any point X on this line can be transformed onto any other point on a horizontal line passing through X (and inside the larger square) by data manipulation. Information Causality covers the "wings" of the drawing from the top point of the circle.
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4 is the maximum value. Why you would want a FTLT box, a PR box is unphysical as is, what good a "more unphysical" box would do?
Maybe I can explain a bit more about the geometry involved here. Imagine a square with a side of 4 drawn with the base parallel with the bottom of the paper. Then draw a circle (with radius 2sqrt(2)) with the center at the center of the square. This circle touches the square at all 4 points. Last, draw a larger square rotated 45 degrees from the first square touching the smaller square at all midpoints of its sides. The top vertex of the larger square is the PR box. The smaller square is classical mechanics/hidden variables, and the circle is QM. Connect the highest point on the circle with the PR box vertex. Any point X on this line can be transformed onto any other point on a horizontal line passing through X (and inside the larger square) by data manipulation. Information Causality covers the "wings" of the drawing from the top point of the circle.
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Hi Florian,
Thanks for your illuminating reply and confirming my calculation. I tried to look up the original paper and also found this one,
http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.3464v3.pdf
which shows the squares and circle you mention.
Why do I bring up FTLT or instantaneous action at a distance, unpopular even with Isaac Newton? I am trying to understand nonlocality. FTLT is “more nonlocal” than PR, and perhaps “backward-in-time” is even more extreme. These more-nonlocal models apparently do not “describe our world” but I am interested if they can be placed on the same scale as PR, QM and classical mechanics in a mathematically consistent and precise way.
Jim Graber
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Thanks for your illuminating reply and confirming my calculation. I tried to look up the original paper and also found this one,
http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.3464v3.pdf
which shows the squares and circle you mention.
Why do I bring up FTLT or instantaneous action at a distance, unpopular even with Isaac Newton? I am trying to understand nonlocality. FTLT is “more nonlocal” than PR, and perhaps “backward-in-time” is even more extreme. These more-nonlocal models apparently do not “describe our world” but I am interested if they can be placed on the same scale as PR, QM and classical mechanics in a mathematically consistent and precise way.
Jim Graber
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Dear Jim,
Thanks for your reference, I was not aware of it. In the simplest case, one has an 8-dimensional polytope, and apparently, in different slices, information causality does not recover the QM limit, and hence it seems that there is room for finding out a new physics principle. However, this is not what I got from the talk, and I was very surprised to see that this paper was uploaded on the archive last year. (I guess I got carried away listening to the talk and I did not caught the disclaimers)
I am also interested to understanding non-locality, and in the post 4 of 4 from the conference (to appear) I am dicussing the controversy between Tumulka's results and Conway's free will theorem. Who is right and who is wrong hinges on one's understanding on non-locality, and the issues are rather subtle.
I have some preliminary unpublished results about the impossibility of FTL and time travel and I hope to expand on those results in the next year or so. The main thrust of the proof is that there is no consistent way of incorporating that into any "physical" theory. Of course, what represents a "physical" theory is a matter of debate and I want to make the argument as airtight as possible.
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Thanks for your reference, I was not aware of it. In the simplest case, one has an 8-dimensional polytope, and apparently, in different slices, information causality does not recover the QM limit, and hence it seems that there is room for finding out a new physics principle. However, this is not what I got from the talk, and I was very surprised to see that this paper was uploaded on the archive last year. (I guess I got carried away listening to the talk and I did not caught the disclaimers)
I am also interested to understanding non-locality, and in the post 4 of 4 from the conference (to appear) I am dicussing the controversy between Tumulka's results and Conway's free will theorem. Who is right and who is wrong hinges on one's understanding on non-locality, and the issues are rather subtle.
I have some preliminary unpublished results about the impossibility of FTL and time travel and I hope to expand on those results in the next year or so. The main thrust of the proof is that there is no consistent way of incorporating that into any "physical" theory. Of course, what represents a "physical" theory is a matter of debate and I want to make the argument as airtight as possible.
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Hi dear Jim,
It's very interesting all that.
Here is my humble point of vue.
The hidden variables are merely stages of analysis, difficult to analyze and perceive due to our young age.
The system is always and everywhere in its locality in three dimension, the intricacies are an illusion of extrapolations.
It is obvious and wise to recognize that inequality is violated, because quantum mechanics and its pure determinism is and will be, but will evolve in its fundamental principles.Like a division of mass towards the planck scale and the ultim energy.
Can we prove otherwise, not well understood.
This causality and determinism are the keys to these particles and waves.
Whatever calculate its angle or its invariance?
If a force of potential quantum influences these gravitational codes(stables furthermore in evolution)it is the space and the duration and therefore the special relativity which are not respected, even gravity, even general relativity.
The locality is so poorly understood with these constants.
The particle, informations and waves are correlated but rationally, physically speaking.
So I think that this force is a mistake as non-local hidden variables.
It is important to not confuse, waves, informations, particles.
As it is essential to differentiate the school Copenaghen and extrapolations without limit of philosophical mathematics.
It's all our uniqueness and its intrinsic constants that are under maths involving variables and potential strengths.But not in the reality...can we superimpose or add like we want when we speak about the physicality, no of course.
The incompleteness seems a nice enemy of the rationalism of Copenaghen.
Regards
Steve
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It's very interesting all that.
Here is my humble point of vue.
The hidden variables are merely stages of analysis, difficult to analyze and perceive due to our young age.
The system is always and everywhere in its locality in three dimension, the intricacies are an illusion of extrapolations.
It is obvious and wise to recognize that inequality is violated, because quantum mechanics and its pure determinism is and will be, but will evolve in its fundamental principles.Like a division of mass towards the planck scale and the ultim energy.
Can we prove otherwise, not well understood.
This causality and determinism are the keys to these particles and waves.
Whatever calculate its angle or its invariance?
If a force of potential quantum influences these gravitational codes(stables furthermore in evolution)it is the space and the duration and therefore the special relativity which are not respected, even gravity, even general relativity.
The locality is so poorly understood with these constants.
The particle, informations and waves are correlated but rationally, physically speaking.
So I think that this force is a mistake as non-local hidden variables.
It is important to not confuse, waves, informations, particles.
As it is essential to differentiate the school Copenaghen and extrapolations without limit of philosophical mathematics.
It's all our uniqueness and its intrinsic constants that are under maths involving variables and potential strengths.But not in the reality...can we superimpose or add like we want when we speak about the physicality, no of course.
The incompleteness seems a nice enemy of the rationalism of Copenaghen.
Regards
Steve
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No, but what is true, already that our technology is young and childish.
If we must spend more money for research without realism, then there Where are we going ?Where are we going?
Billions for machines for traveling through time, while of course we can only travel in space.
I wonder how it is possible and plausible to identify our physicality in these repositories purely hypotheticals and imaginary.
Physics is physics, math is simply a tool and a language, but nevertheless useful but secondary to physics.Whebn maths are without limits, that becomes difficult for pragmatic observations correlated with the pure mechanic and thermodynamic.
We can invent, with our imagination, a universe with more planets and stars and moons in reality
This is not a reason to affirm the realism of that number ....there we can understand between the infinity and our young age at the universal scale, thus the real understanding of our referentials and uniqueness.
Just a thought
Steve
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If we must spend more money for research without realism, then there Where are we going ?Where are we going?
Billions for machines for traveling through time, while of course we can only travel in space.
I wonder how it is possible and plausible to identify our physicality in these repositories purely hypotheticals and imaginary.
Physics is physics, math is simply a tool and a language, but nevertheless useful but secondary to physics.Whebn maths are without limits, that becomes difficult for pragmatic observations correlated with the pure mechanic and thermodynamic.
We can invent, with our imagination, a universe with more planets and stars and moons in reality
This is not a reason to affirm the realism of that number ....there we can understand between the infinity and our young age at the universal scale, thus the real understanding of our referentials and uniqueness.
Just a thought
Steve
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Here are some papers about the super-quantum correlations, their meanings, and related communication complexities:
Quantum Mechanical Realization of a Popescu-Rohrlich Box S. Marcovitch, B. Reznik, L. Vaidman http://arxiv.org/abs/quant-ph/0601122
Abstract: We consider quantum ensembles which are determined by pre- and post- selection. Unlike the case of only pre-selected ensembles,...
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Quantum Mechanical Realization of a Popescu-Rohrlich Box S. Marcovitch, B. Reznik, L. Vaidman http://arxiv.org/abs/quant-ph/0601122
Abstract: We consider quantum ensembles which are determined by pre- and post- selection. Unlike the case of only pre-selected ensembles,...
view entire post
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Yes, it is a very active area and my feeling was that everyone is having a wonderful time just by playing around and inventing all sort of games and algorithms and trying to see what may come out of it.
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I have this idea about how this is related to holography. The physics is related to the Hardy paradox and how e-e^+ pairs in an interferometer can exist without annihilation. So the e + e^+ - -> x + y serves a model for the PR-box, or so it seems. An observer watching a particle and anti-particle pair approach a black hole will observe their transverse directions grow. This is basically the growth in the cross section for a particle at higher energy. Here the velocity of these ingoing particles is v = dr/dt = g_{tt}/g_{rr} which approaches zero as r - -> 2m, which illustrates how the pair at high energy appear frozen on the event horizon. The two particles on the horizon are in a state similar to the e-e^+ particles in the interferometer.
As yet I am still reading these papers to make sense of things to see if this idea has some element of sanity to it. It does seem as if these people have found some theoretical area that is rather fun to work on.
Cheers LC
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As yet I am still reading these papers to make sense of things to see if this idea has some element of sanity to it. It does seem as if these people have found some theoretical area that is rather fun to work on.
Cheers LC
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This is highly unlikely because Hardy's paradox is just standard QM and I bet that a careful analasys will recover only Tsirelson's bound. (If indeed you get a PR box, then this is a major find.)
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Dear James Putnam,
Reading your earlier post of May 3rd, 2010 with Ian Durham I was struck by the questions you have raised regarding temperature and thermodynamic entropy.
You write,
"Saying that thermodynamic entropy is energy in transit divided by temperature is not, I think, an answer to: What is thermodynamic entropy? What did Clausius discover? Whatever it is, it requires the passage of time. Clausius allowed for absorption of energy under conditions of equilibrium. Statistical expressions do not include this dependence upon time."
"Temperature is an indefinable property with indefinable units of measurement. My point is that temperature is not yet explained. "
I must admit I have not thought about thermodynamics all that much, but in one of my short papers ( The Temperature of Radiation ) I propose a definition of temperature which I believe answers some of your objections. It is defined as the ratio of the 'amount of accumulation of energy' over 'time '. This definition seems to be equivalent to the thermodynamic temperature T and plays the exact same role as T plays in Planck's Law. (see 'Planck's Law is an Exact Mathematical Identity' )
The quantity 'accumulation of energy' again comes up naturally as the 'primary physical' quantity. (see 'Prime physis and the Mathematical Derivation of Basic Law', 'The Meaning of psi: An Interpretation of the Schoedinger's Equation' )
Constantinos
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Reading your earlier post of May 3rd, 2010 with Ian Durham I was struck by the questions you have raised regarding temperature and thermodynamic entropy.
You write,
"Saying that thermodynamic entropy is energy in transit divided by temperature is not, I think, an answer to: What is thermodynamic entropy? What did Clausius discover? Whatever it is, it requires the passage of time. Clausius allowed for absorption of energy under conditions of equilibrium. Statistical expressions do not include this dependence upon time."
"Temperature is an indefinable property with indefinable units of measurement. My point is that temperature is not yet explained. "
I must admit I have not thought about thermodynamics all that much, but in one of my short papers ( The Temperature of Radiation ) I propose a definition of temperature which I believe answers some of your objections. It is defined as the ratio of the 'amount of accumulation of energy' over 'time '. This definition seems to be equivalent to the thermodynamic temperature T and plays the exact same role as T plays in Planck's Law. (see 'Planck's Law is an Exact Mathematical Identity' )
The quantity 'accumulation of energy' again comes up naturally as the 'primary physical' quantity. (see 'Prime physis and the Mathematical Derivation of Basic Law', 'The Meaning of psi: An Interpretation of the Schoedinger's Equation' )
Constantinos
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Florin,
As I study into this my doubts about the whole enterprise mount. Though I am not sure it is completely a waste of time. There seems to be some reason to pose the problem: If we assume a PR box exists plus spacetime physics (or gauge theory), then what are the elements which constrain the PR-box to information causality and the Tsirelson's bound. The Tsirelson's bound seems to be something similar to what we might call a "quantum horizon." So we might think of nature as a PR-box plus additional physics (spacetime causality, AdS~CFT, SUSY or ... ) which then constrains the PR-box, or observable physics, into what we identify as "ordinary quantum mechanics."
This stuff is not easy reading, so it will take me some time to get through more of it.
Cheers LC
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As I study into this my doubts about the whole enterprise mount. Though I am not sure it is completely a waste of time. There seems to be some reason to pose the problem: If we assume a PR box exists plus spacetime physics (or gauge theory), then what are the elements which constrain the PR-box to information causality and the Tsirelson's bound. The Tsirelson's bound seems to be something similar to what we might call a "quantum horizon." So we might think of nature as a PR-box plus additional physics (spacetime causality, AdS~CFT, SUSY or ... ) which then constrains the PR-box, or observable physics, into what we identify as "ordinary quantum mechanics."
This stuff is not easy reading, so it will take me some time to get through more of it.
Cheers LC
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Lawrence,
Apparently information causality recovers QM only in some polytope slices, and the question becomes: is there another principle needed in the other slices, or one can move actualy violate the Tsirelson bound in there? Beying 8 dimensional, the geometry is not easy to understand in terms of information protocols. What is missing from this approach is the C* algebra. Otherwise all sorts of non-classical "QM behaviors" can be obtained via unphysical toy models.
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Apparently information causality recovers QM only in some polytope slices, and the question becomes: is there another principle needed in the other slices, or one can move actualy violate the Tsirelson bound in there? Beying 8 dimensional, the geometry is not easy to understand in terms of information protocols. What is missing from this approach is the C* algebra. Otherwise all sorts of non-classical "QM behaviors" can be obtained via unphysical toy models.
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Florin,
Agreed, the IC does not recover the entire bound except at the extreme end for α^2 + β^2 > 1/2 near equality for α = 0. So I agree there needs to be some sort of additional physics which recovers QM, or QFT as we understand it. I am just now getting into the polytope construction at this time, and I probably will not have much worthwhile to comment on that until Sunday. I do find it interesting this is a 24 vertex polytope in four dimensions, which is the number for the F_2 Coxeter-Weyl root representation, but this curious polytope is in 8 dimensions. Curious? With the C* algebra, maybe the Serre-Swan theorem, Gelfand-C* modules and the rest have to be imposed.
More later,
Cheers LC
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Agreed, the IC does not recover the entire bound except at the extreme end for α^2 + β^2 > 1/2 near equality for α = 0. So I agree there needs to be some sort of additional physics which recovers QM, or QFT as we understand it. I am just now getting into the polytope construction at this time, and I probably will not have much worthwhile to comment on that until Sunday. I do find it interesting this is a 24 vertex polytope in four dimensions, which is the number for the F_2 Coxeter-Weyl root representation, but this curious polytope is in 8 dimensions. Curious? With the C* algebra, maybe the Serre-Swan theorem, Gelfand-C* modules and the rest have to be imposed.
More later,
Cheers LC
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Florin,
I found the following lecture “The common symmetries underlying quantum theory, probability theory, and number systems” by Philip Goyal. You might find this to be of some interest.
Cheers LC
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I found the following lecture “The common symmetries underlying quantum theory, probability theory, and number systems” by Philip Goyal. You might find this to be of some interest.
Cheers LC
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Lawrence,
Thanks for the lecture, I started to view it. Sounds interesting.
Florin
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Thanks for the lecture, I started to view it. Sounds interesting.
Florin
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Lawrence,
BINGO! I understand completely what Philip is doing, and I think his very similar way of looking at QM (compared with what I am working on) would provide an answer to how to eliminate the requirements of Jordan algebra in axiomatizing QM. THANKS FOR THE LINK!!!
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BINGO! I understand completely what Philip is doing, and I think his very similar way of looking at QM (compared with what I am working on) would provide an answer to how to eliminate the requirements of Jordan algebra in axiomatizing QM. THANKS FOR THE LINK!!!
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Hi all,
I'm a little late in reading all this great stuff. I'm trying to catch up but I recommend reading Ahronov and Rohrlich's interesting book "Quantum Paradoxes."
As for the specifics, I'll have to read through all this more carefully soon.
Nice entry Florin!
Ian
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I'm a little late in reading all this great stuff. I'm trying to catch up but I recommend reading Ahronov and Rohrlich's interesting book "Quantum Paradoxes."
As for the specifics, I'll have to read through all this more carefully soon.
Nice entry Florin!
Ian
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Hi Ian,
Thanks for the good words, I am only reporting what I saw. Two more parts are comming.
Florin
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Thanks for the good words, I am only reporting what I saw. Two more parts are comming.
Florin
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If at the light speed time disappear how than that photon moves through space-time?
Photon should move through space only!
yours amrit
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Photon should move through space only!
yours amrit
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Florin,
The point of the Jordan algebra is not to axiomatize QM, but rather as an approach to quantum gravity. Of course the Jordan matrix algebra has little to do with standard quantum mechanics. Goyal dismissed quaternions, and of course octonions. However, the assignment from logic to algebra can well enough be generalized to the Jordon product a*b = (ab + ba)/2, from which the entire Cayley hierarchy can be presented in this manner. The quaternions are important for gravity, for it most likely involves noncommutative geometry --- or quaternions. However, this is not the entire picture. Quantum gravity involves this holographic principle. The action for the octonions decomposes in part to include the quaternions plus a Chern Simons lagrangian. Remember, the associator for the octonions is cubic, and a piece of the cubic part involves this Chern-Simons Lagrangian part which defines a standard Lagrangian on a boundary. This is dual to a cubic nonassociative part which has a quaternionic boundary. This appears to be a form of the holographic principle. The fields which we measure are the associative quaternions, but their structure is determined by shadow states which are nonassociative.
Goyal’s idea is topological. By looking at sequences of events without metric this is a topological notion. From this perspective there is no structure at all --- the metric is imposed on top of this. This is in keeping with my sense that quantum states have a representation in spacetime, but are not intrinsically spacetime elements. Geometric structure in this might come from a generalization of the “logic to algebra” procedure with the Jordan product rule.
This lecture an others do give one something to ponder
Cheers LC
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The point of the Jordan algebra is not to axiomatize QM, but rather as an approach to quantum gravity. Of course the Jordan matrix algebra has little to do with standard quantum mechanics. Goyal dismissed quaternions, and of course octonions. However, the assignment from logic to algebra can well enough be generalized to the Jordon product a*b = (ab + ba)/2, from which the entire Cayley hierarchy can be presented in this manner. The quaternions are important for gravity, for it most likely involves noncommutative geometry --- or quaternions. However, this is not the entire picture. Quantum gravity involves this holographic principle. The action for the octonions decomposes in part to include the quaternions plus a Chern Simons lagrangian. Remember, the associator for the octonions is cubic, and a piece of the cubic part involves this Chern-Simons Lagrangian part which defines a standard Lagrangian on a boundary. This is dual to a cubic nonassociative part which has a quaternionic boundary. This appears to be a form of the holographic principle. The fields which we measure are the associative quaternions, but their structure is determined by shadow states which are nonassociative.
Goyal’s idea is topological. By looking at sequences of events without metric this is a topological notion. From this perspective there is no structure at all --- the metric is imposed on top of this. This is in keeping with my sense that quantum states have a representation in spacetime, but are not intrinsically spacetime elements. Geometric structure in this might come from a generalization of the “logic to algebra” procedure with the Jordan product rule.
This lecture an others do give one something to ponder
Cheers LC
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Lawrence,
What Goyal is doing is very similar with Grgin's approach: impose consistency conditions which would restrict what is allowed in nature. He recovers the eliptic, parabolic, and hyperbolic case and then he isolates QM by imposing a repetability of experiments condition under some additional conditions. I don't see his argument as topological, but as algebraic. His primitive concepts are the additivity of experiments (the ability to do them in parallel) and the composition of experiments. This is a standard categorical approach. To restrict his choices, he further considers the concept of probability. But this is too big a leap at this point. He also does two conceptual mistakes: first, his modeling choice starting from uncertainty principle to justify a pair of real numbers is not quite right, the real root cause of this is the correspondence in nature between observables and generators. Second, his final eliptic case is not a proof of the necessity of complex numbers in QM. The eliptic case can be further realized by reals, complex, quaternions, octonions, and quantions.
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What Goyal is doing is very similar with Grgin's approach: impose consistency conditions which would restrict what is allowed in nature. He recovers the eliptic, parabolic, and hyperbolic case and then he isolates QM by imposing a repetability of experiments condition under some additional conditions. I don't see his argument as topological, but as algebraic. His primitive concepts are the additivity of experiments (the ability to do them in parallel) and the composition of experiments. This is a standard categorical approach. To restrict his choices, he further considers the concept of probability. But this is too big a leap at this point. He also does two conceptual mistakes: first, his modeling choice starting from uncertainty principle to justify a pair of real numbers is not quite right, the real root cause of this is the correspondence in nature between observables and generators. Second, his final eliptic case is not a proof of the necessity of complex numbers in QM. The eliptic case can be further realized by reals, complex, quaternions, octonions, and quantions.
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There is this matter of dark energy these days. Dark energy might be best seen as a cosmological analogue of the Lamb shift in QED. There are of course two differences. The first is that the pressure term is negative, which is related to the negative heat capacity of spacetime, and secondly, which is here heat capacity plays a role, the quantum fluctuations have a measure on event horizons. The signature change means that with the analogue of the lamb shift we see a coarse grained version of it and see only noise. The splitting between the S_1/2 and P_1/2 levels appear not as some discrete spitting in atomic levels, but as some blurring out of signals our detectors receive.
Gravitation is not a unitary system. The dynamics involves hyperbolic equations and Sobelov systems. Attempts to reduce quantum gravity to unitary evolution are doomed --- utterly. I have been studying these mathematics papers by Charles Frances on the conformal completion of AdS. The Taub-NUT spacetime has a discrete structure which defines a measure zero discrete time operator. The TN spacetime has certain maps to the AdS spacetime. So the discrete system of conformal completion of AdS has a homeomorphism to the Frances approach with Klein group systems on AdS. The completion of the AdS spacetime is an “Einstein universe” which under various perturbations is a de Sitter spacetime. The energy involved with this perturbation, which is related to the discrete time operator on the TN spacetime, is this thing we called dark energy.
Dark energy is really just a rather different version of the Lamb shift.
Cheers LC
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Gravitation is not a unitary system. The dynamics involves hyperbolic equations and Sobelov systems. Attempts to reduce quantum gravity to unitary evolution are doomed --- utterly. I have been studying these mathematics papers by Charles Frances on the conformal completion of AdS. The Taub-NUT spacetime has a discrete structure which defines a measure zero discrete time operator. The TN spacetime has certain maps to the AdS spacetime. So the discrete system of conformal completion of AdS has a homeomorphism to the Frances approach with Klein group systems on AdS. The completion of the AdS spacetime is an “Einstein universe” which under various perturbations is a de Sitter spacetime. The energy involved with this perturbation, which is related to the discrete time operator on the TN spacetime, is this thing we called dark energy.
Dark energy is really just a rather different version of the Lamb shift.
Cheers LC
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Florin,
You say"The eliptic case can be further realized by reals, complex, quaternions, octonions, and quantions."
It's interesting,could you affirm that all these numbers are inside a sphere, if not proove me here please.
And also could you represent the definition of these systems(qantions, quaternions, octonions,....)
What is the other form above this sphere.and after you explain the meaning of pi,you know 3.1415.....
Thanks
Steve
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You say"The eliptic case can be further realized by reals, complex, quaternions, octonions, and quantions."
It's interesting,could you affirm that all these numbers are inside a sphere, if not proove me here please.
And also could you represent the definition of these systems(qantions, quaternions, octonions,....)
What is the other form above this sphere.and after you explain the meaning of pi,you know 3.1415.....
Thanks
Steve
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........his modeling choice starting from uncertainty principle to justify a pair of real numbers is not quite right, the real root cause of this is the correspondence in nature between observables and generators..............
http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.2507v2.pdf
....
.Quantum mechanics cannot be more non-local with measurements that respect the uncertainty principle. In fact, the link between uncertainty and non-locality holds for all physical theories.More specifically, the degree of non-locality of any theory is determined by two factors -- the strength of the uncertainty principle, and the strength of a property called ``steering'', which determines which states can be prepared at one location given a measurement at another.......
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http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.2507v2.pdf
....
.Quantum mechanics cannot be more non-local with measurements that respect the uncertainty principle. In fact, the link between uncertainty and non-locality holds for all physical theories.More specifically, the degree of non-locality of any theory is determined by two factors -- the strength of the uncertainty principle, and the strength of a property called ``steering'', which determines which states can be prepared at one location given a measurement at another.......
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Dear Anonymous,
Funny you mention http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.2507v2.pdf
I am puting the final touchs on a new blog post about this. Please stay tuned for the comming post and comment on that upcoming thread.
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Funny you mention http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.2507v2.pdf
I am puting the final touchs on a new blog post about this. Please stay tuned for the comming post and comment on that upcoming thread.
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Curiously I read this paper yesterday, though I have yet to read the supp.matl. This addresses some of the Tsirelson bound issues we discussed last May. The Bell theorem, or CHSH as presented here, is a funny thing, and in some ways both are in a way trivial. There seems to be almost nothing terribly surprising in the fact that ΔO = sqrt{ - ^2}, which is the Berry phase or a measure of the fibration of H over PH, is what determines the locality of QM.
Cheers LC
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Cheers LC
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Dear Anonymous,
Funny you mention http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.2507v2.pdf
I am puting the final touchs on a new blog post about this. Please stay tuned for the comming post and comment on that upcoming thread.
-------------
i will, mr. florin
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Funny you mention http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.2507v2.pdf
I am puting the final touchs on a new blog post about this. Please stay tuned for the comming post and comment on that upcoming thread.
-------------
i will, mr. florin
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Thank you for information http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.2507v2.pdf
It is a very interesting paper; There is a deep connection between non-locality, uncertainty principle and Non-local correlations (teleportation).
However, the paper is wrong about that "Quantum mechanics as well as classical mechanics obeys the no-signalling principle, meaning that information cannot travel faster than light". The wormhole and warp drive theories prove the opposite opinion.
Sincerely,
Constantin
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It is a very interesting paper; There is a deep connection between non-locality, uncertainty principle and Non-local correlations (teleportation).
However, the paper is wrong about that "Quantum mechanics as well as classical mechanics obeys the no-signalling principle, meaning that information cannot travel faster than light". The wormhole and warp drive theories prove the opposite opinion.
Sincerely,
Constantin
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Florin's got a new post about Oppenheim and Wehner's Science paper "The uncertainty principle determines the non-locality of quantum mechanics" here.
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Non-locality, quantum mechanical phenomena (it is all essentially quantum stuff), and Bell's proof ultimately require the following:
Space has to be both invisible and visible and it has to be flattened/contracted and expanded/stretched (on balance) in order to incorporate/unite/balance increased inertia with/as decreased gravity. This generally balances attraction and repulsion and gives us quantum gravity in conjunction with centered/averaged, fundamental, and stabilized distance in space.
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Space has to be both invisible and visible and it has to be flattened/contracted and expanded/stretched (on balance) in order to incorporate/unite/balance increased inertia with/as decreased gravity. This generally balances attraction and repulsion and gives us quantum gravity in conjunction with centered/averaged, fundamental, and stabilized distance in space.
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Space has to be both invisible and visible and it has to be flattened/contracted and expanded/stretched (on balance) in order to incorporate/unite/balance increased inertia with/as decreased gravity. This generally balances attraction and repulsion and gives us quantum gravity in conjunction with centered/averaged, fundamental, and stabilized distance in space.
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