Atiyah's Quest for a New Basis of QM

[marcus]

http://online.kitp.ucsb.edu/online/strings05/atiyah/

My personal hunch is that Atiyah could be nudging towards an
extremely interesting reformulation of quantum mechanics.
a fertile new "paradigm" or way to look at it.
Liked this talk VERY much. only wish he hadnt been interrupted
by the KITP people so much, looked like he had to skip over
half his prepared transparency notes because so much of the hour
was taken up by a competing "lecture" by someone (DG) in the
audience.

Any reactions from others?

Does anyone know any other online source for these Atiyah ideas?

 

[marcus]

thanks to Not Even Wrong for pointing this Atiyah talk out
http://www.math.columbia.edu/~woit/wordpress/?p=285

Peter's blog has a lively discussion, in which I see a couple of PF people taking part.

 

[Kea]

...looked like he had to skip over half his prepared transparency notes because so much of the hour was taken up by a competing "lecture" by someone (DG) in the audience...

I believe this was largely David Gross. A pity we couldn't hear these remarks.

 

[Kea]

Marcus

I didn't think there was much substance to Atiyah's physical ideas. Even if he is literally correct in that there is a dependency on the past, taking that idea alone and working with a very classical picture in a background spacetime seems rather ugly to me. Just a thought.

Really enjoyed the general introduction! And the discussion at the end, where one can actually hear what audience members are saying. A must see.
Cheers, Kea

 

[marcus]

...
Really enjoyed the general introduction! And the discussion at the end, where one can actually hear what audience members are saying. A must see.
Cheers, Kea

Hi Kea, this is another case where you did me a favor by being on the lookout and commenting. It was your comment at Peter's blog that convinced me to take the time to watch the talk----and it was worth all 60 minutes!

Here was what you said that I found so persuasive:

Atiyah did not deny this dichotomy between continuous and discrete. On the contrary, he endorses it (a) by indicating a preference for Connes’ picture and (b) by saying explicitly that this Hale formalism for past dependent equations incorporates aspects of both.

I enjoyed the talk! Forget the physics - it’s full of subtle jibes and the sort of jokes that only very respectable old men can make.

 

[mccrone]

A fun talk but - from a systems science point of view - it makes the familiar mistake of focusing on a "single-scale" discrete~continuous dichotomy and not the more fundamental "scalefree" dichotomy of local~global.

Atiyah suggests incorporating the uncertainty of QM by making just a single Planck-sized step into the past. A scale-free approach to memory would want to model a powerlaw distribution of "pasts". In this view the Planck scale defines not the fundamentally small (and discrete) but instead the fundamentally cogent (a web of interactions/gone to equilibrium).

Start by asking what is memory in a system. It is the general context within which a succession of particular states form. It is an ambience - a prevailing equilibrium in a system of interactions. So it is the "frozen" global view. And in hierarchical fashion, it exerts a downward constraint on any localised particulars. But of course, global constraint cannot "see" everything and so the local events (particle interactions, etc) are left with degrees of freedom. What we call their inertias.

So the memory of a system is its global coherent state which reflects some average of its past and - because the coherence is self-stable - also serves to predict the future for some reasonable distance. A universe that has developed certain constants, like the speed of light, is likely to continue to roll with those constants (barring a slow underlying evolution that eventually allows some further phase change to a new global vacua).

The point is that this globalised system memory is spread over all spatiotemporal scales. It is "scalefree" because the issue of scale has been thermalised.

So rather like a hologram, a very small "bite" into the past - like Atiyah's suggested Planck-scale step - would give some kind of snapshot of the whole, but it would be very fuzzy. By definition, it would seem to give the most uncertain view!

By being the closest to the localised freedoms of the system, it would see the least of the global constraints, the global memory. It would be like trying to observe the convergence nature of the system, its tendency towards a global remembered balance, from exactly the most open and divergent viewing position. To offer another analogy, it would be like trying to gauge the temperature of a gas by measuring some particular passing molecule.

Prigogine has some arguments along these lines in his modelling of QM in "The End of Certainty".

 

[Mike2]

So the memory of a system is its global coherent state which reflects some average of its past and - because the coherence is self-stable - also serves to predict the future for some reasonable distance.

Sounds like conservation laws to me. The individual particle trajectories might change through local interactions. But the overall (global) energy is conserved. The system "remembers" what total energy it has to maintain.

 

[Juan R.]

http://online.kitp.ucsb.edu/online/strings05/atiyah/

My personal hunch is that Atiyah could be nudging towards an
extremely interesting reformulation of quantum mechanics.
a fertile new "paradigm" or way to look at it.
Liked this talk VERY much. only wish he hadnt been interrupted
by the KITP people so much, looked like he had to skip over
half his prepared transparency notes because so much of the hour
was taken up by a competing "lecture" by someone (DG) in the
audience.

Any reactions from others?

Does anyone know any other online source for these Atiyah ideas?

As it is typical in string theorists "a lot of speculation summed with lot of stuff that appears to be revolutionary for string theorists but well known for other researchers". I understand that string theoirst worked with elementary stuff but they would read that is published in others fields before doing claims and gave talks...

Atyah question "is there memory in the universe?" was already solved decades ago! in the irreversible quantum mechanics. There is still thecnical difficulties on chossing the correct formulation but the topic about the existence of memory effects is already a bit 'outdated'.

An interesting formulation that generalizes QM at a very sophisticated level is Brushels School generalized QM. There the memory effects are caused by a cosmological dissipative structures at the Big Bang.

This revolutionary approach -Prigogine theory- can be derived from more general canonical approaches.

The rest of Atiyah discussion is :zzz:

 

[marcus]

...

Atiyah question "is there memory in the universe?" was already solved decades ago! in the irreversible quantum mechanics. There is still technical difficulties on choosing the correct formulation but the topic about the existence of memory effects is already a bit 'outdated'.
...

You understand, Juan, that the vague general question "Does the Universe have a memory?" is not the issue. That is simply the provocative title of Atiyah's talk. Mathematically there are SEVERAL ways one could respond to this question. In this thread we are not interested in OTHER ways people have addressed this general question. We are interested in the SPECIFIC way that Atiyah has suggested in this talk. That is already hard enough to grasp, I find.

You seem to be implying that there is PRIOR WORK along the lines of Atiyah's talk. that would be interesting, if it were a fact.

As always in these cases, Juan, if you have any specific published references or links to primary sources, please share them with us! I am open to examining on-line material which pursues the mathematical approach Atiyah indicated. I take this approach to involve dispensing with velocity and momentum in the configuration space, so that one quantizes only those measurments WHICH DO NOT PRESUPPOSE TAKING A (time) LIMIT.

Atiyah's proposal, I gather from his talk, is to POSTPONE TAKING (temporal,
"delta tee") LIMITS until one has already quantized the system.

I believe, contrary to what I hear from you, that this is probably a risky departure----untried, untested so far by anyone, apt to fail. Perhaps I misundertood what Atiyah was driving at.

I would guess that what is proposed is remotely related to the constructive math tradition of Brower, which I know very little about---I believe it involves a kind of finitistic bias about what should be considered fundamentally meaningful---I may have the names wrong here but there is such a mathematical tradition.

But I do not want to get involved with larger issues or with general questions. I want some examples of prior work that specifically takes Atiyah's approach

If you cannot provide them then I shall continue to assume that what he proposes has not been tried yet.

I will try searching for "irreversible quantum mechanics", which are keywords I get from your post.

 

[Juan R.]

You understand, Juan, that the vague general question "Does the Universe have a memory?" is not the issue. That is simply the provocative title of Atiyah's talk. Mathematically there are SEVERAL ways one could respond to this question. In this thread we are not interested in OTHER ways people have addressed this general question. We are interested in the SPECIFIC way that Atiyah has suggested in this talk. That is already hard enough to grasp, I find.

So far i know usual standard physics including the standard model is memoryless. String theory AND M-theory are also memoryless. In fact, all that string theorists are doing in this difficult question of irreversibility is -as it is typical in string theorists- copy the work done by others...

Moreover, they often copy incorrectly. The recent attempt to solve the problem of arrow of time using a radical generalization of standard string theory has been a joke, because string theorists are specialists on nothing (except gargabe and propaganda in media). They believe that string theory IS difficult AND fundamental and the rest of disciplines a kind of second-class activity. Then they do some work on any speciality where they have no idea and the rest of people do jokes with the nonsense published...

In fact, string theorists used wrong tools developed in other fields by Prigogine and others 50 years ago, tools that the own Prigogine abandoned time ago by new more sophisticated tools that CANNOT be used in that trivial theory called string theory which relies in outdated aspects as Hilbert space mathematics.

I remark again that string theory is both outdated and wrong.

You claim that mathematically there are SEVERAL ways one could respond to this question. Of course! but we need the correct physical way and in more than 100 years nobody found one still!

You seem to be implying that there is PRIOR WORK along the lines of Atiyah's talk. that would be interesting, if it were a fact.

There is a 100-years research in the problem of time even if you newer heard this. There has had several attempts to find an arrow of time in quantum gravity and also in string theory. All material published is from the perspective of a string theorists 'fascinating'; from the perspective of people has done research in this difficult question it is :zzz:

String theorists would read has been published in a specific topic before doing irrelevant comments and proposals. This is also advice for you, Marcus before doing grandilocuents claims about the last fad on string theory -and rest of insane disciplines- you would first read has been published in that topic by other people. It is a good thecnique.

Perhaps this was the reason of the 40 years failure of string theory community for obtain anything: the implicit -asummed- fact that string theory goes beyond any other theory...

********************************************

Now i explain my reading of Atiyah talk.

Firsts items are pure string theory marketing. Some of claims of Atiyah are very discussible, for exmaple the claim that string theory has provided Fields Medals is not correct.

In item 5 does a wrong claim about causality. It is talking about causality of a reversible unitary deterministic universe. Our universe is not that. That in QM the state IS an element of Hilbert space is false. This is valid only as approximation, for example for stable particles. For instable particles one needs to work with generalized states. There is a very large literature in the topic of generalized states beginning with the classical -not very adequate- Gamow states for instable particles and radioactive nuclei.

Item 6. Ignorance of our past does not explain Heisemberg uncertainty. This is an complete exageration. Heisemberg uncertainty relationshiphs are time-simmetric.

Item 7. Retatrded differential equations are well known in physics. i) They do not explain the arrow of time ii) They cannot be solved. In fact, people is doing is just the contrary that Atiyah claims: the ABANDON of retarded differential equations. This is done, for example using powerful memory-kernel thecniques that eliminate the retardation in favor of the present state knowledge. Why are retarded differential equations ABANDONED? Because are computationally intractable. For example Zwanzig kinetic equation has been UNSOLVED until today except in the short memory regime when equation IS time local.

Item 10. Finite difference equations are also very, very, very, very well-known on physics of irreversibility. They do NOT solve arrow of time and do not work. The quantum of time is called chronon. Has has some attempt to derive decoherence and measurement quantum process from discrete approaches. Did not work.

The rest may be a joke. Unless that one remembers that 'quantization' for a mathematician can mean classical for a physicist. For example, in NC geometry mathematicians call 'quantum' to everything with a discrete structure and some models of NC geometry are really classical for physicists. Item 11 reinforces this opinion.

Item 11. Atiyah expresses his misunderstanding of what is time, what is evolution, and what is spacetime. To claim two copies of spacetime with different structure for pass and future and that is may explain the arrow of time is a pure nonsense.

Item 12. Perhaps is Atiyah asking by a Zubarev-like equation?

partial |phi(t)> / partial t = -iH |phi(t)> + e{|phi(t)> - |phi(t_0)>}

The infinitesimal source corresponds to the standard Abel kernel, it introduces an instantaneous memory kernel selecting retarded solutions (t > t_0) of the Schrödinger-like equations. This is known for decades. The memory kernel can be generalized for a finite decay of quantum correlations. It does not work! Note In rigor the above state vector CANNOT be a Hilbert vector. In fact it is not. Atiyah again proves his ignorance of those topics.

Item 13.
Yes, superposition is approximated (a simple look to above Zubarev-like equation proves that). This is ALREADY known for decades. Zubarev equation was published in the 50s (??)

Nothing of this is new! BUT retarded equations like Zubarev one do NOT solve the measurement process even if are nonlinear equations breaking the supersposition principle of the linear Schrödinger equation of QM. It is all a 'bit' more complex that Atiyah claims...

Item 14.
Is there such one thing as 'curved spacetime'? Who has proven that? In sci.physics.research i asked by the Newtonian limit of GR and nobody replied still what is the curvature of spacetime on the Newtonian limit. Carlip has done a wrong attempt to derive the Newtonian limit using a wrong metric and a wrong understanding of basis of Newtonian physics. Carlip does not know the difference between a 'potential' and a 'field' (still do no replied to my question in sci.physics.relativity of why the island universe asumption is wrong in fields but experimentally proven for potentials. Carlip simply does not understand Penrose criticism), has asumed that time in Newtonian gravity is a dimension! work with covariant derivatives which is wrong (in Newtonian physics derivatives are FLAT), etc. Moreover, even in Carlip metric the curvature scalar R is zero because it contains an extra c term into g_00. Etc, etc.

Precisely the approaches without curved spacetime: FTG (worked by Feynman), spacetime calculus (discussed here in PF some time ago), teleparallel gravity, etc. are more powerful. There is no such one kind as 'curved spacetime' on those approaches. This solves many of classical GR problems: singularities, energy, systems of references, etc.

Atiyah's statement that retarded actions give "natural answer to wave equations" is WRONG. In fact, the usual Lienard Wiechert potentials taking only the retarded part of the wave equations are wrong (i already cited many literature on the topic many times in PF). It is also well-known that retarded solution is incomplete: self-action, impulse renormalization of mass, violation of conservation of momenta, etc.

There is NO Hamiltonian associated to retarded equations. This is the reason that Wheeler-Feynmann theory NEWER was posted in Hamiltonian form and famous promised talk newer arised...

In the point 2) It appears that Atiyah still thinks that the Dirac equation is a relativistic generalization of the Schrödinger equation, which is a pure nonsense.

See Weinberg volume 1 for a discussion of why the Dirac equation is NOT a wave equation (in fact it is not in QFT, it is an identity for field operators NEWER for wavefunctions). In sci.physics.research (thread about quantum fields) you can find some related information about why the Dirac equation is not a wavefunction equation by Eugene and myself. The proofs involves sophisticated math that Atiyah may unknow (one can prove that generator of time translations in Dirac equation does not have compact support).

Items 15, 16, and 17. Bla, bla, bla.

Item 18. Point 1. Yes, time reversiblity is not correct, but way proposed by Atiyah is simply wrong. The point now is the searching of some additional effect, for example Poincaré resonances in LPS in the Brushel School approach (which are NOT retarded differential equations). The Brushel School abandoned the search of the basis of the arrow of time in retarded equation in the 60s or 70s if i remember correctly. One would read material published for learning WHY.

Point 2. Speculative. In fact some recent work in the topic suggests that momentum-positin duality holds, but a new kind of complementarity arises. Complementarity time-entropy. Precisely NC geometry of the time coordinate introduces nonunitarity.

About the ultimate description of Nature on geometrical terms, I doubt!

I believe -like Newton or Feynman- that the geometric approach is only approximated.

 

[Physics Monkey]

Juan R,

You are awfully critical of Atiyah and string theory and just about everything that isn't exactly what you believe. Do you really think that Atiyah and everyone else is so woefully unaware of your new miracle science? I'm sorry if I sound critical, but where are the results of this longstanding and profound understanding of reversability, quantum theory, etc? I think the answer is not so simple as you make it out to be, and I would like to echo marcus' reasonable request: could you please provide some specific material that discredits Atiyah's concepts. Thanks.

 

[Juan R.]

Juan R,
You are awfully critical of Atiyah and string theory and just about everything that isn't exactly what you believe. Do you really think that Atiyah and everyone else is so woefully unaware of your new miracle science? I'm sorry if I sound critical, but where are the results of this longstanding and profound understanding of reversability, quantum theory, etc? I think the answer is not so simple as you make it out to be, and I would like to echo marcus' reasonable request: could you please provide some specific material that discredits Atiyah's concepts. Thanks.

Monkey, you can by your-self (i you are able to use Google or similar) choose some of more than 1000 references available only from members of the Brushel School since 1950 or so!

Last book by Prigogine the End of certainty offers a non-technical view of the recent approaches on search of irreversibility.

I cannot be more clear, Atiyah's claim the use of retarded equation and a Hilbert space is a bit outdated, just as all the rest of string theory...
Please Monkey, if you read carefully my post, you can see that i did not call of own results on "new miracle science" which corrects Prigogine theory in some aspects.

All i said is well-known in specific advanced literature -I remember recent Solvay conference in the topic- except by string theorists, of course, who are ignorant in almost everything.

P.S: The number of string theorists in that Solvay conference on time irreversibility was naturally zero.

P.S: Perhaps a Google search is difficult for you, here some help

(Books)
http://en.wikipedia.org/wiki/Ilya_Prigogine

Articles and monographs
http://order.ph.utexas.edu/people/Prigogine.htm

In the left part, you can find a short biography of Prigogine. He is not a irrelevant guy.

Of course, Atiyah is not aware of my own research. I newer contacted with him! Why would i contact with a guy with no idea of physics? However, i collaborated with Prigogine and members of his research group in several issues.

 

[marcus]

Hi Physics Monkey,
glad you showed up in this Atiyah thread! I was reacting to only one thing Juan said. This was back in post #8, his first post on this topic:

...Atyah question "is there memory in the universe?" was already solved decades ago! in the irreversible quantum mechanics. There is still technical difficulties on choosing the correct formulation but the topic about the existence of memory effects is already a bit 'outdated'.
...

I just wanted to know if he had any evidence that Atiyah's idea was not NEW. I asked if he could back up his pronouncements by showing PRIOR WORK.

My feeling is that at this point it is irrelevant to attack or defend. when a creative mathematician starts to tell you a new notion, which is obsessing him, and which is not rigorously worked out yet, the thing to do is to open all the ears of your mind and listen very carefully.

A proposal can be obviously flawed, patently absurd, and still have a shadow or clue or blueprint somewhere in it----a sketch of something you haven't thought of yet.

Sir Michael has been a very creative mathematician (not merely awesome, but something more important: fertile, inventive, perhaps a little clairvoyant). He said the talk was intended to be provocative, and he modestly subtitled it "crazy thoughts of an old man".

As we are humans, this was not the time to consider Atiyah as belonging to the "String Camp". To launch a polemic against him because of his past associations with stringy-business would be to miss the point.

In this talk, he is not doing anything for String---he is totally "outside the box". Maybe his talk was even felt by string thinkers to be threatening----which could be why David Gross interrupted so incessantly, seeming almost desperate to stop Atiyah from saying in public whatever he was going to say.

 

[Physics Monkey]

Juan R, you seem to know of some specific material that refers to Atiyah's most recent intellectual excursion. I was hoping you could provide some specific references, but instead you simply tell me to "google it"
and refer me to the entire body of work of a famous chemist. I am able to use google, but I had hoped you might provide something a little more direct. I only asked because I was interesting in learning more about such ideas, but I see that you are not interested in assisting me. I'm sorry you feel that way.

As for the rest of your posts, here and elsewhere, you seem pretty convinced that almost the entire rest of the world is completely confused. I'm sure many string theorists, and Atiyah, and plenty of other people would have some powerful criticisms of your unsupported claims. Send me a paper over PM if your theory can predict something interesting. But I digress, all this is all off topic, so I won't discuss it anymore here. I don't want to derail the nice thread any further.

***

marcus, thanks for the welcome, I'm glad to be here. I will return later with some thoughts on Atiyah's presentation. I did like it, and it's always fun to listen to interesting ideas, especially when they are the "crazy thoughts of an old man" like Atiyah.

 

[marcus]

I think the answer is not so simple as you make it out to be, and I would like to echo marcus' reasonable request: could you please provide some specific material that discredits Atiyah's concepts. Thanks.

Yes, he has clearly way oversimplified

I will tell you why I think Atiyah's talk was disturbing, or as he hoped provocative.

First, remember that he said repeatedly during the talk "these ideas may be WORTHLESS, I have been obsessed with them, you will have two weeks to shoot them down".

1. His talk bears witness to the sense that theoretical physics has gotten off track or has gotten stalled and NEEDS A RADICALLY NEW IDEA.
Gerard 't Hooft has also said this. And remember he was the one whose reaction was "that sounds like physics" when Atiyah told him his new ideas.
There are not a few important people who are now paying LIP SERVICE to the idea that radically new concepts from a completely unexpected direction will probably be needed before real progress occurs. But Atiyah and 't Hooft are taking ACTIONS that bear witness to that need for radical new concepts. 't Hooft is supporting and even himself trying out new approaches.
David Gross may feel offended by Atiyah because his willingness to overtly grope in new directions seems like a sign that the mainstream program is bankrupt. Whether it is or not, Atiyah's behavior could be read as a SIGN that it is---and help trigger a run on the bank.

2. People have been tinkering with time in quantum mechanics for as long as they have had quantum mechanics (we don't need Juan to remind us of that! ) the point is HOW they tinker? WHAT SPECIFICALLY DO THEY PROPOSE? Lots of different formalisms have been proposed. What is interesting in the context of this thread is not the GENERAL question time and irreversibility. What is interesting is the specific proposal of Atiyah. (this could turn out to have other ramifications besides time---we should look at it without preconceptions)

when I look at what he is trying to do, it does not seem to me to be very compatible with string theory AFAIK or to be so much in the SPIRIT of perturbative quantum field theory and allied theories.

It seems to throw out the ideas of velocity and momentum, or to postpone their definition and treatment.

What this says to me is that what he is offering as his crazy ideas could actually be more compatible with the Loll triangle-gravity path integral.
That is a model of the universe where changes are NOT DIFFERENTIABLE.
You only get to talk about smooth rates of change after quantizing and not before.

this is so unclassical that (unless I have badly misunderstood Atiyah) it is extremely risky and very likely to be wrong. So? When a creative mathematician proposes something that he says has been obsessing him then I do not automatically dismiss or attack that because it is LIKELY to be wrong, or because it is boggles my mind how he could be wanting to explore into it. Not at all. then is just when i want to pay the closest attention.

 

[mccrone]

"It seems to throw out the ideas of velocity and momentum, or to postpone their definition and treatment."

The Universe is made up of both location and motion. The problem in modelling is how to account for these dichotomous extremes.

The standard approach is monadic - to believe that two opposites must be described at the same fundamental level. And normally this is the fundamentally small.

But Prigogine and others who are familiar with scale and systems would reduce opposing extremes to opposing modelling limits. So location would be reduced to the small scale model and motion to the large scale model.

From this it would be no surprise to find a Planck scale description that is discrete, reversible, substance-oriented - located atoms. But it would have to be matched by a global description that captures the continuity, the irreversibility, the forms - the expanding void.

The memory of the universe would lie in the large scale view. The fundamentally small becomes reversible because all constraining context has been stripped away - the located can no longer know which way they are pointing.

 

[Kea]

Yes, I agree that the position/momentum dichotomy was important to Atiyah. He mentioned Non-linear Fourier Transforms a few times I believe. I can't help substituting the words Pontryagin Duality which is an instance of Stone Duality http://en.wikipedia.org/wiki/Stone_duality.

 

[Kea]

If I might be even more presumptuous...here is an idea of Galileo as discussed in 1997 by F. William Lawvere in a talk he gave in Montreal, in which he said the usual idea of dynamical systems hadn't really caught up with Galileo :

States are states of Becoming. This again admits many variants: the States may involve velocities, or memories, or destinies, but in any case they themselves should be more structured than just points which abstract static Being particularised as configurations.

...quoted apparently from Birkhoff's 1919 paper, which Lawvere says led to the theory of fibre bundles.

 

[mccrone]

Yes, I agree that the position/momentum dichotomy was important to Atiyah. He mentioned Non-linear Fourier Transforms a few times I believe. I can't help substituting the words Pontryagin Duality which is an instance of Stone Duality http://en.wikipedia.org/wiki/Stone_duality.

An important question here - is duality in this sense about asymmetric extremes (defined by their being mutually exclusive) or some simpler symmetry relation (postive/negative) or an even simpler "twoness" - just a pair of things?

Asymmetry leads you naturally into a semiotic view - where a system of self-organising constraint acts to separate a vague space of possibility. So, for example, position~momentum would begin as a confused potential and by mutual constraint become divided crisply into position (without momentum) and momentum (without position).

Anyway Atiyah made some of these points a few years back - "Finally, I should like to mention that in physics what emerges very prominently are “dualities". These dualities, broadly speaking, arise when a quantum theory has two realizations as a classical theory. A simple example is the duality between position and momentum in classical mechanics. This replaces a space by its dual space, and in linear theories that duality is just the Fourier transform. But in non-linear theories, how you replace a Fourier transform is one of the big challenges. Large parts of mathematics are concerned with how to generalize dualities in non-linear situations. Physicists seem to be able to do so in a remarkable way in their string theories and in M-theory. They produce example after example of marvellous dualities that in some broad sense are non-linear versions of Fourier transforms and they seem to work. But understanding those non-linear dualities does seem to be one of the big challenges of the next century as well."
http://duch.mimuw.edu.pl/~sjack/atiyah.ps

 

[Kea]

An important question here - is duality in this sense about asymmetric extremes (defined by their being mutually exclusive) or some simpler symmetry relation (postive/negative) or an even simpler "twoness" - just a pair of things?

Actually, none of the above.

The term mutually exclusive doesn't quite apply, because Pontrjagin duality (and Stone dualities in general) rely on their being some Object that can be interpreted as either Type. Neither are these dualities exactly simple, like plus and minus.

 

[Juan R.]

Juan R, you seem to know of some specific material that refers to Atiyah's most recent intellectual excursion. I was hoping you could provide some specific references, but instead you simply tell me to "google it"
and refer me to the entire body of work of a famous chemist. I am able to use google, but I had hoped you might provide something a little more direct. I only asked because I was interesting in learning more about such ideas, but I see that you are not interested in assisting me. I'm sorry you feel that way.
As for the rest of your posts, here and elsewhere, you seem pretty convinced that almost the entire rest of the world is completely confused. I'm sure many string theorists, and Atiyah, and plenty of other people would have some powerful criticisms of your unsupported claims. Send me a paper over PM if your theory can predict something interesting. But I digress, all this is all off topic, so I won't discuss it anymore here. I don't want to derail the nice thread any further.
***
marcus, thanks for the welcome, I'm glad to be here. I will return later with some thoughts on Atiyah's presentation. I did like it, and it's always fun to listen to interesting ideas, especially when they are the "crazy thoughts of an old man" like Atiyah.

As already explained above all of Atiyah talks is outdated, trivial, or wrong. Literature on irreversiblity is very huge, therefore you would not have problem on finding material.

I do not refer you to the entire body of work of a famous chemist. In fact page contain only a small percentage of Prigogine work (order of 1/50).

My initial belief about google search proved now.

About the use of retarded equations you can see last book cited in THAT page

Nonequilibrium Statistical Mechanics Wiley-Interscience, New York 1962

Recent work by Prigogine and others proves why a retarded equation cannot solve the arrow of time, in fact its group is now working Markovian equations.

About why states in QM are not represented by Dirac states in a Hilbert space -Atiyah is very, very outdated- you can read references

1998 "Relativistic Gamov Vectors"
J. Math. Phys. 39

or

2000 "Thermodynamic limit, Hilbert space and breaking of time symmetry"
(with T. Petrosky)
Chaos, Solitons and Fractals 11, 373-382

from THAT page.

Etc, etc.

As said Atiyah's talks is outdated, trivial or wrong. The same about string theorists which have provided none serious thought about anything -in reality specialists have durely critized string theorists ignorance-

 

[Juan R.]

My personal hunch is that Atiyah could be nudging towards an extremely interesting reformulation of quantum mechanics.

Since your interest is in discrete mechanics. You would read material publshed in the topic.

For example in J. Phy. A: Math. Gen. 1997 3115-3144

you can see one of MANY formulations on discrete mechanics where derivatives are substituted by differences

dx/dt ----> { x_{n+1} - x_n } / T.

In my opinion, this kind of approach is wrong since time cannot be quantized. None "extremely interesting reformulation of QM" arises from 'discretization' procedures.