Smolin: Extending dualities to trialities (deepens dynamics)

[marcus]

Looks like something special happening here:

http://arxiv.org/abs/1503.01424
Extending dualities to trialities deepens the foundations of dynamics
Lee Smolin
(Submitted on 4 Mar 2015)
Dualities are often supposed to be foundational, but they may come into conflict with background independence, because a hidden fixed structures is needed to define the duality transformation. This conflict can be eliminated by extending a duality to a triality. This renders that fixed structure dynamical, while unifying it with the dual variables.
To illustrate this, we study matrix models with a cubic action, and show how breaking its natural triality symmetry by imposing different compactifications yields particle mechanics, string theory and Chern-Simons theory. These result from compactifying, respectively, one, two and three dimensions. This may explain the origin of Born's duality between position and momenta operators in quantum theory, as well as some of the the dualities of string theory.
12 pages

 

[marcus]

Background independence emerges as a key issue.
From the conclusions paragraph:
==quote Smolin 1503.01424 ==
This [i.e. matrix Chern-Simons theory] is a theory based on a fundamental triality. This illustrates our main claim, which is that the basic dynamics of relativistic particles and strings follows from breaking that basic triality, which holds at a background independent level, to a duality.
The breaking from a triality to a duality introduces background structures which allow the dynamics of string and particles to be defined ...
==endquote==

 

[Paulibus]

...To find string theory, we compactly on a two-torus...
Now we compactly on a three-torus,...

Please, somebody, translate for me what seems to me to be an illiterate phrase in an interesting article.

 

[marcus]

the spell checker does not know the mathematical term "compactify" which means to make compact.

I just tried to write that word and the spell checker quickly changed it to "compactly".

sometimes when one is typing one does not notice that the spell checker has changed what one has typed to stuff that does not make sense

 

[marcus]

Notice the terms in the abstract related to that verb:
=======
...To illustrate this, we study matrix models with a cubic action, and show how breaking its natural triality symmetry by imposing different compactifications yields particle mechanics, string theory and Chern-Simons theory. These result from compactifying, respectively, one, two and three dimensions...
========
A simple example of a compactification in point-set topology is the "one point compactification of the real line" by adding a point at infinity. The real line ℝ is not compact (a technical term Wikipedia probably has a definition) but with one point added, which takes the role of both plus and minus infinity, it becomes topologically equivalent to the circle.

by the way, try typing "trialities" if your spell checker is like mine it will change it to "trivialities"

do you know how to turn off the spell checker?

 

[wabbit]

Oh dear. This [Smolin's article] sounds so reasonable, deep, and powerful. But I have so very little hope of actually understanding it, that in a way I'm almost hoping it isn't true : ) - or at least that someone will find a way to make it accessible to mere mortals...

 

[wabbit]

by the way, try typing "trialities" if your spell checker is like mine it will change it to "trivialities

Now waiting for the press release : "World-renowned physicist claims trivialities deepen the foundations of physics : a ray of hope for amateur theorists ?"

 

[Jimster41]

Can you please expand a bit on why this is so interesting... it feels like it's kindof a dramatic proposal w respect to fundamentals of formalism.

 

[wabbit]

Background-independent quantum mechanics.
The easiest bits (p-q-t triality) may not be deeper than the parametrized view of QM where time becomes a dynamic variable, but the approach seems so broad and general...
And also, background-independent string theory.
Probably other things, but the wabbit is impressed. Not understanding it, but impressed.
Now if someone (hint hint) would find a way to elaborate as you say, I'm all ears !

 

[Jimster41]

okay, so I think I'm connecting to this topic, when I say I groaned in the Susskind "Theoretical Minimum" book when he sort of said, "pay no attention to the assumption of the 3+1 decomposition" (pq+t). And I keep wishing he would explain how increasing Entropy is included in the Lagrangian.

 

[Demystifier]

I'm all ears !

Not unusual for a wabbit.

 

[wabbit]

￣(∵)￣

 

[Demystifier]

￣(∵)￣

\ /
(∵)

 

[wabbit]

Back on the topic, Demystifier have you had a chance to look at the article ? Any thoughts ? The crowd is gathering here and waiting to hear from someone knowledgeable : )
Thanks !

 

[Demystifier]

I looked at the paper and I wasn't impressed. It looked too vague and speculative to me. But of course, it is very likely that I haven't understood it correctly.

 

[wabbit]

Oh OK perhaps I got carried away by not understanding enough of it, but since I am eagerly awaiting the Second Coming of Quantum Mechanics and the Revelation about the Unification of QM and GR, this sort of things is bound to happen : )

 

[marcus]

I wonder if I should comment. Don't understand this particular paper of Smolin's well enough to comment responsibly. But in case it might contribute to the discussion I'll say how it strikes me and what it makes me reflect on.
For me, it harks back to the "Relative Locality" papers by Laurent Freidel et al (one of which Smolin co-authored on, but it's mainly Freidel's baby)
and something which grew out of Relative Locality which I think of as the FLM (Freidel Leigh Minic) revolution in QG and string. It's where the local momentum space is curved and eventually the phase space is not only curved but dynamic.
It's a different ontology, there isn't any global spacetime geometry that all observers could agree on---that's a figment or a useful convention.
FLM is either fundamentally crazy or fundamentally revolutionary. I don't especially like it, personally, but that means nothing.

Smolin paper is not a good entry point---it rather represents Smolin giving his blessing to the little incipient bandwagon and kind of partially getting aboard and allowing that some of his early string theory work was in a way a precursor to FLM. It's interesting, but it is NOT a good introduction.
This is just my humble opinion.

A possibly better entry point:
http://arxiv.org/pdf/1307.7080v1.pdf
==excerpt page 4==
Conclusion: We have presented a new viewpoint on string theory, with wide ramifications and applications ranging from the stringy uncertainty principle [6, 8, 9] to “non-compact” T-duality [25], including the vacuum problem in string theory. Our main point is: The fundamental symmetry of string theory contains diffeomorphisms in phase space. In this formulation both elements (η, J ) of the chiral structure are dynamical. The solutions are labelled by bi-Lagrangians and spacetime is a derived dynamical concept. The fundamental mathematical structure is encoded in the new concept of Born geometry and the choice of bi-Lagrangian structure and the induced metrics on space-time L as well as on momentum space L ̃. This manifestly implements Born reciprocity and it implies a dynamical, curved phase space, including a dynamical, curved momentum space [2], thus providing a generalization of locality. We note that this formulation can be consistently quantized [20]. ...
==endquote==
and also a presentation for NON-SPECIALISTS here
http://arxiv.org/abs/1405.3949
==excerpt from abstract==
In a natural extension of the relativity principle we argue that a quantum theory of gravity involves two fundamental scales associated with both dynamical space-time as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase space and in which space-time is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. ...
==endquote==

 

[wabbit]

Thanks for your thoughts marcus (and apologies for barging in the Dawn thread to call you here ), I shall now ponder them in silence for a while...

 

[marcus]

I'm not suggesting my reflections on this are WORTH pondering. That's the trouble with saying anything. Basically people should know about these papers, but be prepared to shrug them off as too vague or radical,... Demystifier would know how to put the shrug in a polite and cautiously dismissive way. But I feel obliged to say something. A couple of interesting points--simple observations really.
A. Born reciprocity says you can swap x↔p and Freidel takes it seriously and says that since GR implies spacetime (x) geometry is dynamical therefore momentum (p) space geometry must be dynamical---and he goes on from there and says phase space geometry must be dynamical. So the whole kit and kaboodle is kurvey!
B. The early FLM paper of July 2013 refers to two papers IN PROGRESS, by title, which have not appeared under those titles and I think may be subsumed in one that DID just appear---the "Metastring" paper http://arxiv.org/abs/1502.08005

The titles of the two papers in progress, which so far never appeared, are worth noting---as well as what it says about their results in the text---because they sketch or suggest a research PROGRAM.
[15] L. Freidel, R.G. Leigh and D. Minic, Phase Space String Theory, in preparation.

[20] L. Freidel, R.G. Leigh and D. Minic, Quantization of Phase Space String Theory, in preparation.

So it looks like in 2013 the FLM had a concept of "Phase Space String Theory" where there was a curved dynamical phase space geometry---and probably the strings didn't have the usual sort of target space to wiggle around in. Could the Metastring paper be the retitled combination of those two papers that, at least so far, have not appeared?

 

[marcus]

Let's ask what development prompted Smolin's recent triality paper, and perhaps we can understand it better. I'm guessing it is the FLM program (summed up in that essay for non-specialists I mentioned and) so far culminating in the February 2015 FLM paper
http://arxiv.org/abs/1502.08005
Metastring Theory and Modular Space-time
Laurent Freidel, Robert G. Leigh, Djordje Minic
(Submitted on 27 Feb 2015)
String theory is canonically accompanied with a space-time interpretation which determines S-matrix-like observables, and connects to the standard physics at low energies in the guise of local effective field theory. Recently, we have introduced a reformulation of string theory which does not rely on an a priori space-time interpretation or a pre-assumption of locality. This metastring theory is formulated in such a way that stringy symmetries (such as T-duality) are realized linearly. In this paper, we study metastring theory on a flat background and develop a variety of technical and interpretational ideas. These include a formulation of the moduli space of Lorentzian worldsheets, a careful study of the symplectic structure and consequently consistent closed and open boundary conditions, and the string spectrum and operator algebra. What emerges from these studies is a new quantum notion of space-time that we refer to as a quantum Lagrangian or equivalently a modular space-time. This concept embodies the standard tenets of quantum theory and implements in a precise way a notion of {relative locality}. The usual string backgrounds (non-compact space-time along with some toroidally compactified spatial directions) are obtained from modular space-time by a limiting procedure that can be thought of as a correspondence limit.
68 pages

Freidel italicized and highlighted in red those two terms in the abstract, and he put brackets around "relative locality" in third line from bottom as if he was considering an \hlt{...} or at least an \it{...}
He is a very creative intense guy. This is the first LONG paper about this---the others were short 4-9 page jobs. Freidel seems to feel that they are setting up a new field of research---a significant reformulation of string.
Making it background independent in a profound sense of no fixed prior geometry either for the spacetime or for the momentum.
What would it make sense for Smolin to be doing at this point, in connection with this? He is alert and sensitive to new developments. He think this is important. So he could be starting to find a more ALGEBRAIC way of handling what Freidel is groping for geometrically.
You always have to algebraicize geometric developments to enable at least a semblance of calculability. Or so I think. There is safety in C* algebras. Probably I'm wrong about what Smolin's paper represents, but at the moment that's how I see it. I think you don't have to pay attention to these developments but if you want to then proceed at your own risk and begin by understanding the FLM gambit, geometrically, and in mental images. Then if you like that, see how Smolin's paper fits in.
My two cents of wild guesswork

 

[wabbit]

Thanks, this does in fact help clarify things. In addition to Demystifier's and your comments about the general character of the paper, it also seems to be more deeply embedded in String Theory than I initially perceived, and that is something I follow only from a safe distance if at all, for several reasons including a fairly complete lack of understanding.

So until I hear more of a clamor about this, I'll go back to other papers such as
Wallace : Fields as Bodies: a unified presentation of spacetime and internal gauge symmetry

 

[Macuario]

Smolin's paper is a joke, pure crackpottery. It is just blablabla and it has NOTHING to do with string theory. It is incorrect at so many levels that I cannot even begin to describe. For example:

- He says that (1) is a symplectic structure. This is obviously incorrect: a symplectic structure is a non-degenerate closed two-form. I guess he has heard about the symplectic structure in classical mechanics, which is the symplectic structure always present in the phase space of a mechanical system, but it has nothing to do with (1).

- Then he says (about background independence)

"This principle asserts that the laws of physics not depend on any, fixed, non-dynamical background structures"

Well, QFT depends on a fixed minkowski metric and it works pretty well. Yang Mill theories have a fixed topology and they also work pretty well. For example, the topology of a SU(2) gauge bundle over a four-dimensional base dependes only on the second Chern class which is non-dynamical and thus fixed. These are two examples of theories which have fixed background structures and work really well.

Equation (5) is simply laughable. A well defined derivative is always a covariant derivative. The only thing that happens is that in some trivial cases it can be writeen locally just as a partial derivative.

In general the paper shows a profound lack of mathematical elementary background and it is simply rubbish. No one in ST would give a dang for that paper. This guy should be forbidden to use the word string theory in his papers, he only does it to falsely increase the relevance of his papers.

 

[marcus]

Not sure anything here requires response. I do admire the emphatic tone

Smolin's paper is a joke, pure crackpottery. It is just blablabla and it has NOTHING to do with string theory. It is incorrect at so many levels that I cannot even begin to describe. For example:

- He says that (1) is a symplectic structure. This is obviously incorrect: a symplectic structure is a non-degenerate closed two-form. I guess he has heard about the symplectic structure in classical mechanics, which is the symplectic structure always present in the phase space of a mechanical system, but it has nothing to do with (1).

- Then he says (about background independence)

"This principle asserts that the laws of physics not depend on any, fixed, non-dynamical background structures"

Well, QFT depends on a fixed minkowski metric and it works pretty well. Yang Mill theories have a fixed topology and they also work pretty well. For example, the topology of a SU(2) gauge bundle over a four-dimensional base dependes only on the second Chern class which is non-dynamical and thus fixed. These are two examples of theories which have fixed background structures and work really well.

Equation (5) is simply laughable. A well defined derivative is always a covariant derivative. The only thing that happens is that in some trivial cases it can be writeen locally just as a partial derivative.

In general the paper shows a profound lack of mathematical elementary background and it is simply rubbish. No one in ST would give a **** for that paper. This guy should be forbidden to use the word string theory in his papers, he only does it to falsely increase the relevance of his papers.

 

[Macuario]

You can start by answering this:

- He says that (1) is a symplectic structure. This is obviously incorrect: a symplectic structure is a non-degenerate closed two-form. I guess he has heard about the symplectic structure in classical mechanics, which is the symplectic structure always present in the phase space of a mechanical system, but it has nothing to do with (1).

Or maybe this:

"This principle asserts that the laws of physics not depend on any, fixed, non-dynamical background structures"

The topology of a SU(2) gauge bundle over a four-dimensional base depends only on the second Chern class of the bundle, which is non-dynamical and thus fixed. These are two examples of theories which have fixed background structures and work really well.

This two comments are two examples of the many things incorrect in that paper, which will be completely disregarded by the ST community. If he would really accomplishes what he says in the abstract, that would be a huge result in ST and many people would cite that paper. You will see that absolutely no one in ST will cite that paper.

 

[MTd2]

Some philosophic thoughts:
Triality is something that pops up everywhere in Mathematics rather than dualities.
Just think about the pattern:

Newtonian mechanics was essentially a monad. All is essentially an absolute and all movements can be mapped to one another. There is no essential auxiliary entity.

General Relativity and Quantum Mechanics are essentially dual. The former with tangent and cotangent spaces or up and down indices. Quantum mechanics is dual, wave particle duality and Fourier transform. The same can be said of string theory.

But math is full of trialities. So, it's maybe the step we need.

 

[jim hardy]

do you know how to turn off the spell checker?

no,

but is there a "Add to dictionary" in the dropdown? In mine it's right after the bold offerings.

 

[Demystifier]

Smolin's paper is a joke, pure crackpottery. It is just blablabla and it has NOTHING to do with string theory. It is incorrect at so many levels that I cannot even begin to describe. For example:

- He says that (1) is a symplectic structure. This is obviously incorrect: a symplectic structure is a non-degenerate closed two-form. I guess he has heard about the symplectic structure in classical mechanics, which is the symplectic structure always present in the phase space of a mechanical system, but it has nothing to do with (1).

- Then he says (about background independence)

"This principle asserts that the laws of physics not depend on any, fixed, non-dynamical background structures"

Well, QFT depends on a fixed minkowski metric and it works pretty well. Yang Mill theories have a fixed topology and they also work pretty well. For example, the topology of a SU(2) gauge bundle over a four-dimensional base dependes only on the second Chern class which is non-dynamical and thus fixed. These are two examples of theories which have fixed background structures and work really well.

Equation (5) is simply laughable. A well defined derivative is always a covariant derivative. The only thing that happens is that in some trivial cases it can be writeen locally just as a partial derivative.

In general the paper shows a profound lack of mathematical elementary background and it is simply rubbish. No one in ST would give a **** for that paper. This guy should be forbidden to use the word string theory in his papers, he only does it to falsely increase the relevance of his papers.

Let me guess, Lubos Motl is your idol.

 

[Demystifier]

You will see that absolutely no one in ST will cite that paper.

And if someone in LQG cites him, I guess you will say they are all crackpots, am I right?

 

[wabbit]

Hmm I was wondering if you were referring to my comment or to someone else's... But yes my safe distance comment has to do with staying far away from the cultish aspects of the ST world. I imagine some parts of ST presumably will fit somewhere in the big picture of GR+QM so actually I'm happy to hear about ST from authors who don't follow The Only True Path : )

Edit- digressing from this thread, do you think Conne's paper(s) about GR+SM arising somewhat naturally in NCG can possibly be made somewhat accessible to a layman ?

 

[sbrothy]

Let me guess, Lubos Motl is your idol.

Well, Smolin actually cites Lubos in his paper so maybe that's as it should be. :)

 

[Demystifier]

Well, Smolin actually cites Lubos in his paper so maybe that's as it should be. :)

Ah, that's why Lubos didn't criticize this paper on his blog.
And that's why Smolin cited Lubos, to avoid his criticism.

 

[marcus]

Freidel's two co-authors Leigh and Minic are string theorists. Smolin's paper seems to have a lot to do with the recent FLM, his reference [1].
It wouldn't be surprising for the next FLM to cite this one of Smolin. I guess this is fairly obvious and not so entertaining as the Lubosiness, still maybe worth pointing out.

 

[dx]

I too am unable to understand what Smolin means by triality. He writes

$$ S = \int d\tau p_\mu \frac{dx^\mu}{d\tau}$$

and calls this the 'symplectic structure.' This formula of course is true, but I don't understand why he wants to call it the symplectic structure, when the conventional meaning of that word refers to the form
$$dp \wedge dx$$.

Also, the duality (Born reciprocity) between p and x corresponds to the duality between m and τ. So

$$p \leftrightarrow x$$

$$m \leftrightarrow \tau$$

Can anyone shed light on this, particularly why Smolin wants to include d/dτ in the duality between p and x and make it a triality?

 

[dx]

If anything, It would seem more natural to consider

$$(p, x, S)$$

as the triality, since the 'hidden structure' which connects x and p is simply S:

$$P = \frac{\partial S}{\partial X}$$

S here enters into the relationship between x and p, just like † enters into the relationship between |a> and <a|, however smolin says at some point that "d/dτ is on the same footing as x and p". What does he mean by this?

 

[Prathyush]

This is a strange paper for several reasons.
In the introduction section he suggests that, Bra Ket duality, Momentum-Position duality and String dualities are related in some possible way.

However he offer no reasons for such a claim to be true.

I don't understand the claim that
$$S = \int d\tau p_\mu \frac{dx^\mu}{d\tau}$$

Has 3 elements, τ is simply a dummy variable, and it seems like one can do a simple re-parametrization.