Smolin: Extending dualities to trialities (deepens dynamics)

[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.

 

[Berlin]

I am curious if we have already seen influences of this triality for the normal point particle view. Allthough I cannot find it explicitely in the paper, I suppose that Heisenberg's p-x uncertainty relation as well as the Born probability both need modification. If this would require some function of d/dtau input instead of identity matrices this could result in a non-commutive spacetime view. I will scan the Connes papers.. "From triple spectrum to triality". Nice titel for a paper!

berlin

 

[Ralph Dratman]

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?

You don't want to turn it off, just turn off the automatic correction. After typing, look for red underlines and click to get suggestions.

 

[MTd2]

Smolin starts with something that is not classical, not quantum mechanical and certainly that does not belong to string theory, a priori. He aims to get all those things at some limit.
Besides, from the beginning when wants to go beyond the probability axiom by using the hermitian product as something not fixed. So, there could be a complex time, complex position, in principle. So, it is not true that an action can be obtained by a reparametrization, since volume element is violated, so other constrains must be imposed. Because of this, it is sensible, I think to seek a topological invariant as way to constrain.

 

[Quantomg]

Looking at this as a distant observer, I can clearly see the semblance of something like the last salvo in the war between Loop Quantum G T and String T in an attempt by Loop Quantum T to widen enough its framework in order to absorb String T. Given that String T has repeatedly claimed that they can drop the whole of LQG T into their own framework.

So the trialities are going after the dualities to gulp them up, leaving alone the Unities, which are too “symplectic” to be “dynamically compactified” (aka to deserve any attention)! In this seductive world of abstractions and catchy terminology that physics has become, one is left to wonder where is the meat? Where does all this hit verifiably the physics of nature? This would be otherwise so interesting if not so tragic, so costly and so unacceptably mystifying to the “Wabbits” of the world, I argue.

There is a simple test that this barrage of abstraction on both sides can meet for their own luster and justification: give me a physical constant, just one; for so much elaboration about spacetime and its fabric, the vacuum and the vacua, give me G constant. For so much quantization of the vacua and entropy, give me h (Planck constant). Please give me the mass of the simplest of all fermions, just that of the electron. These and the string of other constants of nature are the key validators of all intellectual framework attempting a granular description of physical nature. Seems to me that the time has now run out for this dual 40-year old corpus of abstractions and hypotheses about universal architecture and its building blocks, now that there is this new Quanto-Geometry vision out there, able enough to convincingly derive all those constants and others.

I say to the “Wabbits” of the world that Q-G is simple, effective, speaks with common and familiar words indexed in our dictionaries, and delivers falsifiable physics. Very much worth exploring. Yet I am not dismissing or antagonizing ST or LQG T, I only say to them that the argument that there is nothing else or nothing better out there than their hypotheses holds no more.

 

[wabbit]

I say to the “Wabbits” of the world that Q-G is simple, effective, speaks with common and familiar words indexed in our dictionaries, and delivers falsifiable physics. Very much worth exploring. Yet I am not dismissing or antagonizing ST or LQG T, I only say to them that the argument that there is nothing else or nothing better out there than their hypotheses holds no more.

This wabbit tends to agree, though he wouldn't be able to make such a cogent argument as yours in the above post, and also he finds LQG closer to being predictive than ST

 

[Jimster41]

Pardon my presumptuousness... here.

His statements about (1) seemed straightforward to me. Configuration and momentum integrated over times slices represent a measurement of the classical phase space (I thought that was just what he was talking about). It seems intuitive to me he's saying a way of defining a shape one can hold up to that "space" - and see how big something is, or what shape. Time is the fixed spine of the shape - an assumption.

As I wiki through the other examples I get more or less inklings of what he's pointing at.

The problem with fixed backgrounds - like time in the case of the classical phase space, seems plain to me, how do you know they are fixed? Don't we hate hidden assumptions that can't be justified? And pardon my naivete but I thought that one problem with QFT (at least) is that GR hasn't been integrated. So it works well, other than that seems kind-of a legitimate complaint.

I thought in eq (5) he was proposing that a step towards not cheating and assuming some fixed spine for the referencing shape (the Symplectic), in the case of the back ground time index in classical p,q would be to say it could at least be articulated in some dynamic way through the connection. Free to move but subject to control...

I am reading Smolin and Unger's book on the "Singular Universe". I'm still only on Unger but I'm at the part where he is proposing radically inclusive "prefferred cosmic time", and discussing the perceived conflict between such an Idea and GR (relativity of simultaneity). I'm pretty curious as to how this Triality thing might fit when I get to Smolin.

The part I'm not getting at all is how having three things moving rather than just two makes things easier... but then who could have predicted that adding -t to the theretofore stable and happy euclidean concept - would make things easier.

 

[wabbit]

but then who could have predicted that defining c as a constant in all inertial frames and adding -t to the theretofore stable and happy euclidean concept - would make things easier.

This is quite the opposite of an a priori postulation if I understand it correctly. The constancy of c was first observed (and also predicted from Maxwell's equations), not hypothesized - then, taking into account that observation and reconciling classical mechanics with electromagnetism led to SR and the necessity of merging space and time. Nothing arbitrary there.

 

[Jimster41]

I was just reading that history this morning in https://www.amazon.com/dp/B008JRJ1VK/?tag=pfamazon01-20 . The book also says that, "what troubled Einstein most... was the idea of Galilean Relativity, which states there is no preferred frame of reference... Einstein trusted this principle to such an extent... and the notion of the stationary ether (Maxwell's medium) was incompatible with Galilean Relativity... Either Maxwell's theory or the Galilean transformations had to be wrong..." Granted both of those were observable, but then so are QFT and GR - which seems like at least part of what's bothering Smolin.

All I was trying to say with that lame comparison anyway, is that it seems to me what Smolin is proposing, to the extent I can follow, is sort of a fundamental de-constraining of the algebra, which although possibly synonymous with "complexifying" it, may dislodge some obstacles to discovery, maybe even uncover obscured simplicity, or at least elegance, (like SR and GR were before c)... This made some sense to me.

Trying to read further into the paper, it seems kindof cool how he uses the harmonic oscillator as the connected background index for the clock in classical mechanics then he goes on to contextualize that in the relativistic case using using the string, which he then contextualizes with a Membrane

 

[john baez]

The surprising thing to me about Smolin's paper on trialities is that he doesn't mention the most famous concept of triality, even though he is discussing matrix models, and the most famous form of triality gives rise to the exceptional Jordan algebra, which Smolin used a while ago to formulate a matrix model.

 

[Demystifier]

The surprising thing to me about Smolin's paper on trialities is that he doesn't mention the most famous concept of triality, even though he is discussing matrix models, and the most famous form of triality gives rise to the exceptional Jordan algebra, which Smolin used a while ago to formulate a matrix model.

Perhaps he thought that this purely technical kind of triality contains nothing conceptually deep?

 

[yoron]

It is interesting. But after reading you all I'm still stumped to what he means here with a triality? Any one that can break it down for me?

 

[Jimster41]

There is a pretty dense Wiki page for "Triality" (lost me quickly). I think it provides the canonical context for the word, mathematically. My ability to follow what Smolin is after is too hand waving to help much.

After trying the wiki again, and looking at "duality" - both seem pretty deep in the arcana of functional analysis, topological spaces etc. Just so abstract.

Just trying to clarify my cartoon vis-a-vis the difference between a triality and "3 dimensions': A Triality is three dimensional but it is "three dimensions of relation" or an Algebraic world using of 3 different maps all functioning together.

 

[Jimster41]

From the wiki:
http://en.wikipedia.org/wiki/Triality

A duality between two vector spaces over a field F is a non-degenerate bilinear form

i.e., for each non-zero vector v in one of the two vector spaces, the pairing with v is a non-zero linear functional on the other.

Similarly, a triality between three vector spaces over a field F is a non-degenerate trilinear form

i.e., each non-zero vector in one of the three vector spaces induces a duality between the other two.

 

[yoron]

A 'relational space' sounds nice to me :) and thanks. We have this "dynamical phase space and in which space-time is a derived concept." by which i assume it to mean that SpaceTime becomes a expression of something more fundamental, related to the concept of a dynamical phase space in some mean 'constructing' both a background as well as the dynamical spaceTime we observe. It's a mathematical concept, so maybe it's not translatable? But what I'm really wondering about is if it is a intrinsic approach to describing how a SpaceTime becomes? As if you can think of it as 'becoming' solely locally defined? Because one of the premises seems to be that there are no 'fifth dimension' joining observer dependencies, if I now got this right? instead we then find this dynamic phasespace from which our time and room express itself to us, and locally so, right?

 

[Jimster41]

Below are some links to the wiki pages on the specific definitions of some terms that seem (to me) to be relevant to what I think you are wondering about (which seems to me to be a lot). I have a whole folder full of these links. I am constantly referring to them, over and over sometimes, often getting further down each page than the last time.

http://en.wikipedia.org/wiki/Phase_space
Phase space
From Wikipedia, the free encyclopedia
In mathematics and physics, a phase space of a dynamical system is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. The concept of phase space was developed in the late 19th century byLudwig Boltzmann, Henri Poincaré, and Willard Gibbs.[1]

http://en.wikipedia.org/wiki/Spacetime
Spacetime
From Wikipedia, the free encyclopedia
In physics, spacetime (also space–time, space time or space–time continuum) is any mathematical model that combines space and time into a single interwoven continuum. The spacetime of our universe is usually interpreted from a Euclidean space perspective, which regards space as consisting of three dimensions, and time as consisting of one dimension, the "fourth dimension". By combining space and time into a singlemanifold called Minkowski space, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic andsubatomic levels.

http://en.wikipedia.org/wiki/Quantum_nonlocality
Quantum nonlocality
From Wikipedia, the free encyclopedia
In theoretical physics, quantum nonlocality is the phenomenon by which the measurements made at a microscopic level necessarily refute one or more notions (often referred to as local realism) that are regarded as intuitively true in classical mechanics. Rigorously, quantum nonlocality refers to quantum mechanical predictions of many-system measurement correlations that cannot be simulated by any local hidden variable theory. Manyentangled quantum states produce such correlations when measured, as demonstrated by Bell's theorem.

If you want a really sweet headache-of-bizarre check out the link to "Bell's Theorem". Better yet go to Amazon and search for the same. There are lots of good non-mathematical descriptions of it by practitioners of theoretical physics and good teachers.

 

[yoron]

I know, pretty hard to understand me when I'm not sure myself how to express it. A dynamical SpaceTime is something updating at 'c' to me. Dimensions is what we define it from (3+1). Now, if a phase space is a expression of a 'system', what creates the 'system'? Our definitions of a SpaceTime? Using SR local definitions rule this SpaceTime, from repeatable experiments to any measurement of 'c'. Although assuming any part of a volume of it to represent the same laws it shouldn't matter how you restrict your system, as all volumes/portions of a SpaceTime must be equivalent relative the laws, rules, constants, etc, existing. To me there are two ways of thinking of it, from what I call a 'container definition', which is the one in where we are 'joined' into one common SpaceTime. The other being a strict local definition, which to me also points to a discreteness to exist, as in 'quanta'. That's part of why I'm wondering about Smolins et al 'trinity' and the phase space it builds on.
=

Both assumptions work, but with the one using 'quanta', scaling it up into our SpaceTime, you will need something that connects them to each other. and the way it connects is what interests me. Without a background 'locality' (as in scaling) becomes a natural choice to me, assuming a background I get a headache. I don't really use the classical definition of a local cause and effect, instead exchanging it for scales where 'locality' becomes whatever defines a quanta or locality, as well as the way Einsteins used your 'frame of reference) . The difference is one of something 'measurably defined to some position scaling' relative ones macroscopic definition of a clock and ruler. Two ways of looking at it there too.

 

[Jimster41]

I really like the "updating at c" phrase. I understand you better now I think. Your container sounds like "Bulk/Brane" or like the "holographic" models I have heard descriptions of. Frankly, I wonder if essentially the canonical physics more or less describes this "container" but just has so much hate for calling it that - because that would be to resign at some level to it's mystery. I can understand that resistance. But I heard some things on anti-particles the other day, saying how normal they ware and something about that sounded... Like a medieval treatise on the Bodily Humors, ridiculous. Described as "going backward in time", among other things, they seem to me more like the hypothetical "reactivity" of the extra-dimensional bulk. But I don't understand them at all, and I want to.

I haven't read a single additional page of Smolin and Unger's "Sigular Universe" since this thread started. Primarily because I'm tryng to make my way through some self teaching textbooks on SR, GR and QM. I get the sense you are somewhat like me. You see a lot of the puzzle but to a significant degree it's through a set of your own metaphors and images. I find it frustrating when I want to talk with people here. Seems a lot of them have training and know the standard ways of looking at things, which of course are rich and rewarding, powerful, partly because they are shared and "canonized" and lots of smart people work on them and think in those terms. Hence the self study. But then as I'm reading these textbooks sometimes I get stuck because I'm like why in the world would you look at it that way... When the reason is probably just, History. That was the way that was invented when there wasn't another way, and now it's a strange looking stone in the foundation. Pretty tough to remove, or do without.

Cheers man.

 

[yoron]

You're quite right in that it always will be giving to go through the history of how something get defined. One reason will be that what was more or less inexplainable at that time, except mathematically, with age will have been treated by so many minds searching for ways to translate it that it actually becomes meaningful, although not necessarily simpler. And yes, that's how I expect the container to exist naively. The premise is that it should build locally, 'c' being what we find connecting it, but to me 'c' isn't just a speed, as I define it through Planck scale as also being your local clock. Doing so, accepting the premise, 'c' split to Planck scale becomes a 'quanta of time' :) and a (local) constant. So you have both the clock, the ruler, and that 'speed', all of them becomes constants, although a 'speed', treating it this way is a macroscopic definition. And this 'container concept' we intuitively presume has puzzled me for some time, once I realized how much we take it for granted, as when we argue about what infinity should mean for example. But building a universe intrinsically, as if it constantly gets 'seeded locally', then infinity becomes a very weird idea. No way to to measure it as with this kind of universe you can't 'leave it', and to get a definition of size that is what you need to do. If it builds intrinsically, without a outside, how could it be any other way than 'infinite' from such a definition? The tree dimensions we call the room goes two ways in any defined system, but the arrow has only one way. We're in the never measurable 'now', constantly so, in much the same way as to how we define our macroscopic clock and ruler, all of them theoretical and philosophical concepts, and it builds on SR, but thinking the way I do GR too must accept 'c' as a constant, to keep it the way I prefer, as simple as possible :) that is.

And there's one more thing to it. It makes scaling important to me, because thinking this way it is scales that builds this universe, not probable 'sizes' etc . Also it allows me to think of what you meet at Planck scale, or further 'down' depending on where you set the limits, (But I use Plank scale for those three constants) as 'coexisting', meaning that where there is no arrow defined everything 'coexist'. Looked at that way scaling makes a lot of sense to me, defining a size to this universe very little.

( btw :) If you're like me I'm sure you try to look up the original definitions as good as you can, because sometimes those is the clearest. Which then makes my first proposition of it becoming clearer with time questionable, or if you do as me, they too 'coexist' :)

forgot to add that looking at relativity from a local perspective there are no time dilations, neither any length contractions. Those are results of me comparing my frame of reference to some other, and that is not what locality means (to me). If you use action and reaction you also introduce a container concept.

So there are a lot of common points to me, where Smolin and his friends thoughts becomes interesting, even though they look at it from another perspective.