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, ℏ, 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 ℏ 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.