Dear Lawrence
Your quote from comment to Platan essay:
"What do you think of algebraic curves over [0, 1, ∞] and the Langlands program?"
By coincidence i used long time ago similar trick with pseudoscalar mesons.See my essay. I signify as 1 mass of proton and then observed what position would take place other pseudoscalar mesons.That i got phenomenon 18 degrees.
Do you see some explanation?
Yuri
The number 18 is important in Jewish mysticism. I am not sure that has any bearing on what you are saying though.
Is your number something similar to a Cabibbo angle or Weinberg angle?
I'll take a look at your paper later today.
Cheers LC
It is not common with a Cabibbo angle or Weinberg angle
Just put the values of mass on y=tanx plot, then exploring angles.
Dear Laurence,
(I copy the reply to your post on my page)
Your post is very stimulating. I need time to look at this possibility of relating black-hole physics and entanglement, and non-associativity. On the other hand, I don't consider that entanglement is a primary category in non-local/contextual questions. It may be that conformal arguments adapted to Grothendieck's approach may approach the subject you are talking about. I should say that I am not familiar enough with black-hole physics to have a motivated opinion I intend to read and understand this Maldacena-Susskind paper before discussing more with you on this topic. Meanwhile, may be you can have a look at recent papers by Frédéric Holweck and co-authors (we are now working together) about entanglement and algebraic geometry.
Thanks and best wishes,
Michel
Dear Lawrence, apologies if this does not apply to you. I have read and
rated your essay and about 50 others. If you have not read, or did not
rate "link:fqxi.org/community/forum/topic/1756] my essay The
Cloud of Unknowing[/link] please consider doing so. With best wishes.
Vladimir
Sorry - here is the link: my essay The Cloud of Unknowing
The program of finding physics with [0, 1, ∞] can be found with the SL(2,C) group and the linear fractional transformation (LFT)
f(z) = (az b)/(cz d),
which has a correspondence with matrices of SL(2,C). The Mobius transformation or LFT is an automorphism group on the Argand plane, and this is equivalent to PSL(2,C). This projective linear group is then the automorphism group of C. If we let the constants a, b, c, d be points in C then the LFT
f(z) = [(z - z_1)/(z - z_2)][z_3 - z_2)/(z_3 - z_1)]
is for the identity f(z) = z a case where z_1 = 0, z_3 = 1, and z_2 = ∞. A matrix representation may be found by dividing through by z_i and taking the limit z_i --- > ∞.
From this comparatively simple example we may move up to SL(2,H) and SL(2,O). In the case of SL(2,O) ~ SO(9,1), there is an embedding of SO(9) ~ B_4. This in turn is defined with the short exact sequence
F_4: 1 --- > B_4 ---> F_{52/36} ---> OP^2 --- > 1
where the strange symbol in the middle means that the 52 dimensions of F_4 - the 36 dimensions of B_4 ~ SO(9) defines the OP^2 projective Fano plane or OP^2 ~ F_4/B_4.
The B_4 group is the SUSY group that Susskind employs with the holographic principle.
The group F_4 is a centralizer in the E_8, which means it commutes with the automorphism of E_8, which is G_2. We then have a somewhat Rococo form of the same construction. A projective form of SL(2,O), PSL(2,O), defines matrices ~ aut(O) ~ G_2 which map three points to [0, 1, ∞] with the action of the 7 elements in the Moufang plane. I think I can find this matrix in the near future.
Unfortunately I am moving shortly, so that is complicating plans to do much analysis. If I do this in the immediate future it will have to be in the next week.
Cheers LC
I read your essay sometime bzck. I have a list of these papers and which I have scored. I would probably have to reread or at least refresh myself about your paper. As I recall it is a bit of a metatheory.
Cheers LC
Lawrence
Could you please explain where is your theory connected with Golden ratio?
See part Symmetries... PSL(2,Z)etc
http://en.wikipedia.org/wiki/Golden_ratio
Yuri
Lawrence,
I loved your explanation of modal logic in causality. Very succinct! You brought back pleasant memories of doing symbolic logic in university.
The notion that "causality is necessary incomplete" can also be appreciated from a quantum information point of view using a complex valued system. When an EPR state is prepared, all entropies are conditional on the observer, who is a subsystem in an "EPR-triplet". However, in making a measurement, the observer throws away her entanglement information so that the subsystem of the EPR pair is no longer conditional. (See my essay "A Complex Conjugate Bit and It".)
A quantum correlated system thus becomes a classically correlated system. Using Venn diagrams, it becomes apparent that this process can be interpreted as a change in associativity.
Best wishes,
Richard
Yuri,
The polytope for the E8 grop, the Grosette polytope with 240 roots can be decomposed into the icosian of 120. The icosian or 120-cell has two quaternions with length (1/2)(1 +/- sqrt{5}) where the plus one has length 1.618..., which is the golden mean. In fact these quaternions define something called the golden field in a Galois ring. This is related to the Fibonacci sequence.
Cheers LC
Richard,
Thanks for the kinds words. I agree that quantum measurements and even quantum teleportation involve the destruction of entanglement. Maybe better put the entanglement is transferred to a reservoir of states in an unpredictable manner. The entropy of the system is then indeed conditional, and a measurement loses this.
I propose that the incompleteness has to do with associativity in QFT. My argument then involves the situation of fields near an event horizon. There is a profound difference between the classical case and the quantum case. I am not clear how this plays with standard quantum measurements. Zeh, as I recall, talks of a quantum horizon. Maybe there is some parallel situation there which makes associators play a role.
I will try to read your essay soon. I am rather slowly getting around to as many as I can.
Cheers LC
Dear Lawrence
Could you please find out explanation of symmetric angles picture between mass of proton and pseudoscalar mesons? This is simple parametrization proceeding.
My essay
http://fqxi.org/community/forum/topic/1818
It could be due to some aspect of the eigenvalues for gluons in a supergroup. The icosian has quaternions (roots) that have magnitude given by φ. The icosian is in a sense half of the roots space of the E8 group. The masses of hadrons is determined by the quark masses, which is induced by the Higgs field, and by the confinement properties of the QCD gauge field, called gluons. The differences in these fields in the Y-B plane is given by certain roots, and those roots in some cases have the magnitude of the φ = (1 sqrt{5})/2
That is about the best I can conjecture at this point. There might in some way be some semblance of reason for this.
LC
Hi Lawrence,
I read your essay some time ago but only now got around to rating it -- was delayed in moving from ITB/Bandung to Thailand. In any case I very much liked the use of associators/non-associativity in your essay. Two FSU colleagues of mine (Merab Gogberashvili and Vladimir Dzhunshaliev) have worked on trying to include non-associativity into QFT. Merab in particular had some mantra about the connection of different algebras to physical quantities to the effect--> real numbers are connected with mass; complex numbers with charges; non-commutative numbers with spin; non-associative numbers with the quantum wave-function (actually I do not recall exactly what the last connection was but it was something to do with the quantum nature of matter). In any case you might find Merab's work of interest. Sorry it took so long to finally read your essay.
Also I noticed your address (or one of them) is in Hungary. In this regard you might be interested in a conference Elias is arranging in Prague, Sept. 1 --5. It is a fairly large and broad conference but some of the workshops would seem related to your line of work. In any case the conference site is http://www.icmsquare.net/
I hope this is not considered "advertising". Anyway if this post is not accepted we'll know :-).
Best,
Doug
Doug,
I have read several papers by Vladimir Dzhunshaliev on octonion field theory, and Merab Gogberashvili is a familiar name as well. Trying to understand how nonassociative mathematics of operators fits into physics is really the hard part. I think that quantum mechanics is purely complex, or C. Of course classical mechanics is R. Gauge theory can be written according to quaternions H. A lot of gauge theory is done though in standard vector form without quaternions. It is interesting though that Maxwell formulated electromagnetism, the first gauge field theory, in quaternions. Field operators in a second quantization act on a Fock space basis to give quantum amplitudes. So we have a relationship that might be heuristically written as π:H --- > C. The question is then whether there is some sort of higher level structure π:O --- > H.
Spacetime I think offers a clue. A black hole horizon has some quantum uncertainty on a scale near the string or Planck length. There will then be an associative uncertainty with three quantum fields, where one of those fields is identified near the horizon. The standard approach to QFT is to assign a harmonic oscillator at every point in space, impose equal time commutators on that spatial surface with the Wightman criterion for commutation, and work from there. Yet that spatial surface on a small scale will have some noncommutative structure and this will lead to a host of uncertainties in assigning QFT operators. If there are event horizons this should lead to an associative uncertainty.
The above "maps" between C, H and O, where a similar map π:C --- > R would be the relationship between quantum mechanics and classical mechanics, are really just forms of the Hopf fibration. The relationship between quantum and classical mechanics is of course a difficult subject in its own right. With each of these "ladders" on the Hopf fibration there is some increased uncertainty. Quantum mechanics saved physics from the UV divergence that classical mechanics predicted with the hydrogen atom. Similarly this may protect physics from divergences with black holes, such as the singularity and maybe with the current big problem of firewalls.
Thanks for the good word. I had a computer crash (virus attack etc) that erased my voting code. I also had it written down on a paper that also went missing. I have not been able to vote on papers for about a week. The FQXi people have so far not serviced my request that it be retransmitted. I have also been a bit slow in reading papers this contest cycle. I see that you have a paper in the list. I seem to remember that last year your paper was riding fairly high, where mine in contrast tanked.
Cheers LC
Dear Lawrence,
The title of your essay intrigued me because I think that Wheeler wanted us to recognise the same thing.
I read your essay with much interest (I didn't need to understand the math, because the text was clear) and the conclusion that biology and EVEN consciousness will have to play a role is one that I took as essential inmy own contribution.
After reading the essay it is always informative to read your reactions on the posts, which are very informative. Especially one reaction ( of may 20 02.42 GMT was in full correspondence with my own perception :
"Any scheme for causality is going to be incomplete, it will not be able to encode ALL physical states. As a result it means that there exists a DEEPER FOUNDATION to the Universe" This incompleteness I am trying to describe in the infinite number of tones of grey between the digital entities zero and one.
I really hope that can spare some time to read/comment and rate my essay : "THE QUEST FOR THE PRIMAL SEQUENCE" , I am sure you will find some thoughts we share together for a future approach of reality.
respectfully
Wilhelmus
Wilhalmus,
I suffered recently a big computer virus crash. This machine had my voting code on it. I have not been able to get the code back in spite of my petition to FQXi.
I think that consciousness has some epiphenomenology with generating the appearance of measurement outcomes. In the MWI context the mind may be what generates the appearance of being on one of the split off worlds. In the Bohr Copenhagen interpretation I think consciousness may play a similar role in the so called collapse. In your idea of created reality (creatality) it might be that this is in some ways a mentally generated illusion.
I will score essays again once I get my voting code back.
Cheers LC
Outstanding Paper. Very Logically based. Extremely competent writing.
Similar conclusion as Torsten Asselmeyer-Maluga (incomputable).
I loved the modal logic foundation of this paper, and the expression of Godel's second theorem in Modal logic, which I had not seen before.
Your proposal to remove associativity as a physical axiom is a profoundly interesting idea. I see how this introduces a different interpretation for quantum nonlocality (similar in some sense to the paper by Ken Wharton?)
But where you come out clearly ahead in here (my favorite quote from the paper):
"This argument employs sufficient and necessary conditions in a tensed fashion, forwards and backwards in time, to give a causal chain."
Brilliant.
Your figure (on page ? 3) however, seems to assume an irreversible monotonicity in the order of t1, t2 and t3. Is this what you intended to compare to causal set theory? Was this diagram intended to imply a forwards and backwards in time causal chain?
Nice description of the history of S-Matrix, and a very thought-provoking conclusion regarding black holes.
There is also a wealth of tid-bits of mathematical hints in the paper: very insightful. A very worthy read.
Your conclusion that this (causality and undecideability) is a prospect that may play a role in the emergence of biology and even consciousness is very brave. I hope the orthodoxy does not try to dismiss you for this ;-)
I look forward to seeing more of your work in this area.
Paul,
Thanks for the very good word on my paper. I see tht you squeeked in a paper right at the deadline. I have entered several essays and I have generally found that submitting a paper around the middle of the time period for entries is about the best. I will read your paper soon. I had a couple of weeks ago a major virus attack on the machine which held my password. I can't vote for essays until FQXi honors my request to have it retransmitted to me.
My essay was inspird in large part by reading David Foster Wallace. He was a philosopher who managed to actually say something. He was also a good writer with his novels "The Broom of the System" and "Infinite Jest." I was reading his essay on the refutation of Taylor's argument for fatalism, and that sort of inspired my FQXi essay. I was originally planning to sit out this essay cycle. I also use the past tense with Wallace for he committed suicide in 2008 since he suffered major depression. I got a deeper appreciation for this problem because my brother suffered from this as well, and the past tense indicates his death by suicide a little over a year ago. He was a molecular biologist of some standing in that community.
I have been in some exchanges with Asselmeyer-Maluga on these issues. There is a theorem that the number of exotic R^4s is uncountable via the Cantor diagonalization and Godel-Turing undecidability. I worked up some calculations on Hopf links and knots with respect to the exotic smoothness of R^4 spaces. He has been on vacation of late, but should have returned this week or so. Generally I tend to leave people at peace during vacations.
As for your question about the mother-daughter relationship between events, which is in Foster's paper, the modal logic employed has no particular sense of reversibility or irreversibility. One could imagine some register or physical memory which permits one to reverse the direction of this diagram.
Cheers LC
