## Wednesday, May 25, 2011

### The cube of physical theories

Stefan has a lot of books. And most of them are about physics. The other day I picked a random book and opened a random page and saw myself faced with the "cube de théories physiques" (the cube of physical theories) in the book "De l'importance d'être une constante" by Jean-Philippe Uzan and Bénédicte Leclercq. You find a nice, though very French, illustration of that cube here. There is also an English translation of that book ("The natural laws of the universe: understanding fundamental constants") which has an English, though somewhat unsightly version of the cube on page 57. For your convenience, I've redrawn the illustration:

It shows a coordinate system with three axis that depict the values of three fundamental constants, G, , and 1/c – the coupling constant of gravity, Planck's constant and the inverse of the speed of light. To our best present knowledge these constants are indeed constant, but you can imagine varying them and ask what happens to the theory then. In many cases this corresponds to some physical limit. For example, if your theory contains terms in v/c, where v is velocity, then the limit of velocities small compared to c (i.e. non-relativistic) formally corresponds to taking c to infinity, ie 1/c to zero.

The cube seems to go back to Gamow, Ivanenko and Landau about a century ago, who allegedly cooked it up for a paper to impress a girl. It was rediscovered about 50 years later by some Russian guy named Okun, and then again by the Frenchmen who wrote above mentioned book. You can find here (PDF) a translation of the original paper by Gamow, Ivanenko and Landau, together with a comment by Okun.

You'd surprise me if you've seen that cube before. It is certainly not a particularly deep illustration, but it's inspirational and it gives you food for thought.

My trouble with popular physics books, though, is that in some cases you better not think about what you read because you might get terribly confused. That's what happened to me in this case.

To begin with it isn't clear to me what, physically, corresponds to varying G which is essentially the strength by which matter deforms the background geometry. It would seem more illustrative to e.g. take a constant with dimension of an energy density, so one could think of varying it as considering higher and lower energy densities. And taking the limit G to zero does decouple all the matter but doesn't actually give Special Relativity. Special Relativity is the limit of the vanishing of the curvature tensor or, equivalently, that of flat Minkowski space. Yet General Relativity has nontrivial vacuum solutions even in the absence of matter. These are Ricci-flat, but the curvature tensor doesn't necessarily also vanish. So there's something fishy about the front, lower left corner.

Also, it is to some degree of course a question of terminology, but what is usually referred to as the Newtonian limit of General Relativity is not the limit of velocities small compared to the speed of light. Instead, it is the limit of small distortions to the background, so there's something fishy about the back, upper left corner too.

What the nonzero value of all of these constants in the top, upper left corner has to do with a unification of all particle interactions is entirely unclear to me, and what the non-relativistic limit of that theory is good for I don't really know, though it arguably exists.

There is also the question whether taking these limits does actually commute, or if not approaching a corner does depend on the direction one is coming from. You can for example reach the corner with G and equal zero while keeping the ratio (the Planck mass) fixed. Or you can let them go to zero at a different pace so that the ratio goes to zero or infinity. It seems to me the difference should play a role, yet the diagram makes no distinction.

All together, the "cube of theories" is a very appealing representation. But do not wonder if it confuses you – it has to be taken with a large grain of salt.

1. I've seen that cube before. Joy Christian was into it when he was a visitor at PI a few years back.

What bugs me about the cube, aside from the issue of whether varying hbar is the best way to think about the classical limit of quantum theory, is that there are a vast number of missing dimensions. What about varying the coupling constants of the non-graviational interactions for example? You could object that this would give lots of non-physical limits, but there are already non-physical limits on the cube, e.g. quantum Newtonian gravity. Also, I'd like to see a dimension represeting "level of classical ignorance" so that we would have a direction taking Newtonian mechanics to Lioville mechanics/statistical mechanics and quantum theory to quantum statistical mechanics.

2. I was first introduced to it by Joy Christian as well. I'd guess that Joy learned about it from John Stachel who discussed the cube in his 2003 "A brief history of space-time".

Stachel calls it a Bronstein Cube because it was Matvai Bronstein who pointed out these relationships in the 1930s.

I wasn't aware of the other references you cite, but I look forward to following up on them. I'm actually making use of the Bronstein cube in a couple of my forthcoming philosophy papers. (I argue that our knowledge of the domains of applicability of physical theories -- illustrated by the Bronstein Cube -- gives us reason to reject dualist accounts of the mind.)

-Peter Bokulich

3. Ha, I didn't know Joy talked about the cube. What was his interest in it?

And yes, there's of course lots that's not on the cube. That guy Okun for example wrote a paper about a hypercube, including k.

4. Joy's interest was the upper-right-hand corner of your cube: Non-Relativistic Quantum Gravity.

He published on this back in 1997: "Exactly soluble sector of quantum gravity."

5. "You'd surprise me if you've seen that cube before. It is certainly not a particularly deep illustration, but it's inspirational and it gives you food for thought."

I've seen it before (can't really remember where). And it doesn't look very inspirational, but that may be because I'm accustomed to think about multi-dimensional data as [hyper]cubes.

6. Discrete Scale Relativity Vindicated; Supersymmetry Strikes Out Again.

Supersymmetry has predicted that the electron is "egg-shaped".

Brand new experiments say SUSY is wrong on this prediction, as discussed in the latest issue of Nature.

Discrete Scale Relativity predicts that electrons are among the most perfectly spherical objects in the Universe.

The new electron shape research vindicates the definitive prediction of Discrete Scale Relativity, and contradicts the SUSY prediction.

So let's see, that's 400 billion of the predicted unbound planetary-mass objects and evidence for a spherical electron. It's been quite a week!

Who needs a cube, when you can observe nature and draw self-evident conclusions?

7. Robert, to how many blogs and sites do you have to crosspost your nonsense to before you realize that *NOBODY CARES* ?

9. Hi Bee,

Hmmm....just wonder what these young minds think?

Engaging perspective of the cube is with it's faces, or, from inside?

Calorimeter or Colorimetric, and what is inside has a defined coordinate? A cubed parameter detailing a "configuration point" in space?

Salvador Dalí (Spanish, 1904-1989). Crucifixion (Corpus Hypercubicus), 1953–54. Oil on canvas. 77 x 49 in. (195.6 x 124.5 cm). Gift of the Chester Dale Collection, 1955 (55.5). The Metropolitan Museum of Art, New York.

Moving from the fifth postulate, to a "non euclidean perspective" > or < then 180, is a way in which artistically Dali might have extended the cube? He might of call it heaven "geometrically" having seen Jesus die for our sins? Of course you don't have to believe in the religion, just that it was a way for Dali to move from the cube to the hypercube in a artistic way. Look at Jesus in this sense?

Best,

10. In geometry, the tesseract, or hypercube, is a regular convex polychoron with eight cubical cells. It can be thought of as a 4-dimensional analogue of the cube. Roughly speaking, the tesseract is to the cube as the cube is to the square.

Generalizations of the cube to dimensions greater than three are called hypercubes or measure polytopes. This article focuses on the 4D hypercube, the tesseract.

Polytopes are interesting too:)

11. Hi Bokulich,

Thanks, that's interesting. Best,

B.

12. Hi Nemo,

Haha, you are right of course. I should read my own blog ;-) Best,

B.

13. Is Woofy getting nervous?

Clearly he has broken off is leash.

There is a spanking coming for him at sci.astro.research

14. Robert: Please stop it. You probably think you're witty, but actually you're just annoying everybody with your off-topic comments that nobody wants to hear. Thanks,

B.

15. I've also seen this cube before on some QG paper. Maybe John Stachel's. Didn't give much thought abt it at that time. I am intrigued of the idea that taking these limits with diff order might give diff results.
Regarding Joy, all I know is that he wrote a series of papers on Bell's theorem that I've refuted on arxiv.

16. Here is a nice interactive version:
http://cjwainwright.co.uk/maths/physicscube/

I've definitely seen it, but not in that book. If I remember where, I'll tell.

@tytung: Stachel calls it Bronstein's cube:
http://www.phy.syr.edu/research/hetheory/minnowbrook/stachel.html
and others attributed it to Penrose:
http://users.physik.fu-berlin.de/~lenzk/PICS/bronsteincube.gif

17. This comment has been removed by the author.

18. I'm surprised somebody wrote a paper to impress a girl. Huh, what? Are there actually girls out there impressed by papers?

What was the girl's name, if so? Emmy Noether?

Flowers and a smile always worked best for me, but wow, yet another "pick-up" trick. Now I have to go write a paper to impress my wife, so cya.

19. Hi Bee,

So this came from a random perusing of one of Stefan’s many books, which I find in itself interesting. That is as I have quite a collection myself I’d be curious as to which ones he has and how many of the titles we share. This one for sure I know I don’t have and impressed that it be in French as of course all of mine being in English; just another non obvious advantage of being versant in several languages.

That is for instance I’ve often wondered what it would be like to read Einstein and Dirac in German or deBroglie in French. Then again those like Descartes, Galileo and Newton all wrote in Latin, as it being the language of scholars at the time and although I studied some Latin in school not enough to read these originals. I guess its just that I’ve long wondered what is lost with the translations and what perhaps just can’t be translated properly.

This has me reminded of a public lecture I attended given by Anton Zeilinger , where he said the German word for “quantum entanglement” doesn’t project the same meaning as it does in English, as in German its more as to mean a “handshake” as opposed to a shared state being “holistic”. He went on to complain as it was an initial German observation that the English speakers have things improperly conceived. I’ve often since wondered about that and thought perhaps this could be considered the other way around, as Zeilinger seemed to have a problem himself with the concept of holism when it comes to quantum phenomena.

This leads us back to the cube you’ve highlighted here and although I would agree it being a lame attempt to connect concepts, it is an attempt never the less. The thing I like about it, is that it gives dimension or actual space to ideas and thus have them, that is at least for me, more easily conceived. It also renders clues about how the author thinks and I must admit those that express themselves this way I have always had a greater bond with. So even though the book might be fraught with dangerous misinterpretation I think I might pick up the English version, as at least it’s of a form I’m more equipped to evaluate.

”But if thought corrupts language, language can also corrupt thought.”
-George Orwell

Best,

Phil

20. Hi Phil,

I think in a field where the essential statements are mathematical, the subtleties of different languages don't matter that much. Most of the important works from a century ago have been translated into English I believe, and though one or the other expression might have gotten lost in translation I don't think you've missed much.

Yes, the English word 'entanglement' doesn't mean exactly the same as the German word 'Verschränkung.' The former seems to indicate more of a general mess, whereas the latter is more orderly. Crossing your arms for example is a 'Verschränkung,' yet not actually an entanglement. Best,

B.

21. G which is essentially the strength by which matter deforms the background geometry.... These are Ricci-flat, but the curvature tensor doesn't necessarily also vanish

Somebody is not intoxicated into clamorous stupor by quantum gravitation! A GR curvature background is just as easily a teleparallel torsion background acting like Lorentz force - that is chiral, and testably so. It makes a difference how you hook the jumper cables' polarity to your rail gun. Somebody should look.

To unite classical and statistical thermodynamics you need Boltzmann's constant, as stated - a tesseract. Thermodynamics plus the Beckenstein bound is GR. Is one of the four a dependent variable? (x,y,z,t) naïvely should be fit with three.

http://en.wikipedia.org/wiki/Bekenstein_bound
http://en.wikipedia.org/wiki/Theodore_Jacobson

We live in a world of infinite possibilities. Theory enumerates what is interesting, experiment discloses what is useful. Science is observed fact being observably factual.

22. Some notes about variations of fundamental constants:

In discussion between L. B. Okun, G. Veneziano and M. J. Duff, concerning the number of fundamental dimensional constants in physics (physics/0110060). They advocated correspondingly 3, 2 and 0 fundamental constants. Why they not considering case,where only 1 constant Planck-Dirac's constant; h/2pi=1,054x10^-27ergxsec?

This will be convincingly, because c not contain mass dimension for triumvir and G not contain t for triumvir

My be h only dimensional constant of Nature? Some hint give Planck mass Mp=(hc/G)^1/2 .We simultaneously can decrease or increase c and G, but Mp remains unchanged.

As a consequence only Mp/Me=1836 true dimensionless constant?

I am sure Planck mass(energy) is eternal relevant.

I am not sure about Planck length and Planck time.

I will try why:

In the formula Planck length G/c^3 no linear link.

In the formula Planck time G/c^5 no linear link.

23. Ok, here is something more on-topic.

What if nature's actual "cube" is a tesseract, i.e.,

http://en.wikipedia.org/wiki/Tesseract

which is a cube within a cube. Now you have two sets of vertices, with discretely different values for the measures at the corresponding vertices.

The final step is to consider an unbounded tesseract, which is a cube within a cube within a cube ..., well, you get the idea.

Could nature be so perverse?

I would regard conformal symmetry as sophisticated, or perhaps elegant, but that is a matter of taste and science.

24. I think 'Verschränkung may be the orderly and more original description but entanglement gets at the dependent nature of the relationships between two or more particles. Just like trying to untie a knot there can be unforeseen consequences unless one does it very carefully.

As far as constants and which are "real" - if only people would take a clue from the fine structure constant. G is not in it while c and hbar are. I think this is important and not merely coincidence.

25. I should add, I guess, that the fine structure constant has the electric charge, e, as the third constant within it.
Perhaps e should replace G as the third dimension of the cube of reality. This doesn't seem very intuitive but there must be at least one piece of the puzzle that isn't intuitive or the quantum gravity puzzle would have been solved long ago.

We know the energy density of the universe has to change with its expansion. Perhaps the only constant that must be scaled by hand is G. Hbar, c, and e would then be the three physical constants. As Al would say, someone should look.

26. Hi Bee,

“I think in a field where the essential statements are mathematical, the subtleties of different languages don't matter that much.”

You are of course right if all that’s considered as relevant is action and outcome. It’s also true that when physics was primarily concerned with what we could see, or what we could have to be seen, the maths would take care of the rest. However today when the very existence of the vast majority of what is considered substance and force are taken as givens; not as a result of knowing what each are as to being anticipated, yet rather that our theories would crumble if asked to explain the observed action and outcome without their existence.

That is I would ask if it’s unreasonable to wonder, if our theories are explaining what we find or is what we find being decided by our theories and more importantly how we have them as interpreted; or even at times as being reluctant to having them interpreted at all, to having the maths define the limit of such.

I guess this than is what has me curious about the subtleties; that is like those found between such as Zeilinger and Bohm. That’s to wonder what if anything important is missed with thinking clasped hands of crossed arms as describing the situation better then a complex and yet indivisible whole.

"[Thought] seems to have some inertia, a tendency to continue. It seems to have a necessity that we keep on doing it. However ... we often find that we cannot easily give up the tendency to hold rigidly to patterns of thought built up over a long time. We are then caught up in what may be called absolute necessity. This kind of thought leaves no room at all intellectually for any other possibility, while emotionally and physically, it means we take a stance in our feelings, in our bodies, and indeed, in our whole culture, of holding back or resisting. This stance implies that under no circumstances whatsoever can we allow ourselves to give up certain things or change them."

-David Bohm & Mark Edwards, "Changing Consciousness"_, p. 15

"A key difference between a dialogue and an ordinary discussion is that, within the latter people usually hold relatively fixed positions and argue in favor of their views as they try to convince others to change. At best this may produce agreement or compromise, but it does not give rise to anything creative."

-David Bohm & David Peat, "Science Order, and Creativity"_, p. 241

Best,

Phil

27. Leonard Susskind:
"One of the deepest lessons we have learned over the past decade is that there is no fundamental difference between elementary particles and black holes."

Discrete Scale Relativity was there long before "the past decade". That idea and the reason for the equivalence was published in 1985. Theoretical physicists did everything in their power to prevent the development of this idea.

So eventually theoretical physicists may acknowledge the obvious, and then claim it was their unique discovery.

Same as it ever was.

28. Robert,

Leonard Susskind:
"One of the deepest lessons we have learned over the past decade is that there is no fundamental difference between elementary particles and black holes."

If you repeat it often enough maybe some of us will actually believe it. Apparently for you repetition is the the substance of a good argument.

29. A woman is riding in a car with a man.

He makes four wrong turns, but is unwilling to seek the knowkedge of others who have the insight to reach the goal.

The woman knows the correct way to the goal.

Would you recommend that the woman remain silent? Out of respect? Out of fear? Out of a desire not to rock the boat?

I think she should speak up loud and clear, and persist until her superior knowledge is appreciated.

And next time she should drive the car.

30. Robert,
My suggestion would be that the woman get her own blog, er car, and there would be no further disputes with the man over who is to drive.

31. Hi Robert,

You've taken a couple of wrong turns there and I suggest you shut up now or I'll bury your comments in digital Nirvana. Salut,

B.

32. The issue with this cube, i.m.o. is that people from high school onward are taught misleading things about units, dimensions etc. in physics. It's just a handy tool to do computations, but physics is in principle dimensionless. Many people believe that the way we have assigned (supposedly) "incompatible dimensions" is somehow fundamental, while in reality, it is just a convention.

Then this is how I see this cube. If you use natural units, you can re-introduce hbar, c and G, but now interpreted as dimensionless rescaling constants.

E.g., the correct derivation of the classical limit from special relativity should i.m.o. be a scaling argument like the one given here The usual textbook argument amounts to cheating; the non-trivial part of the derivation is skipped when using SI units.

33. I have stumbled across this particular blog article of yours - very interesting to me. Congratulations, you have a new regular follower. To try and provide in return an interesting tidbit for you to consider return, starting from free accociating about mathematically-oriented women and cubes, have you ever heard of Alicia Boole Stott? Here is a general and also a somewhat more mathematical description of her work.

Best wishes!

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