Shahn Majid

After last week’s imaginative speculation, I’d better tell you something concrete. How about the solution to quantum gravity that has been eluding us for some 90 years? Here it is … er … with one minor catch. We’ll have to suppose that spacetime is 3 dimensional, i.e. one time and only two space directions rather than three.

He's never heard of this "3rd spatial dimension"!

There is a tradition, starting I think with Edwin A. Abbott’s 1880 tale ‘Flatland’, where we suppose that we are not 3-dimensional beings but, let us say, ants, constrained to live forever on some two-dimensional surface. We tend to visualize a surface — imagine, say, the surface of a sphere or doughnut — within three dimensions, but don’t be fooled by that. That is just an aid to visualization. An ant crawling about on the surface, moving along ’shortest paths’ (the analogue of a straight line on a flat space) could fully map out the geometry of the surface without ever leaving it.

I am speaking here of the spatial geometry. We will assume that time is a further linear dimension, making spacetime 3-dimensional, mapped out as the 2-dimensional surface evolves in time.

Actually, we won’t assume any of this, since as I explained in previous blogs, there is no evidence of an actual spacetime continuum of any dimension. But we will take it as a commonly accepted starting point and then I will explain carefully where we have to make the quantum leap to throw all that away to get to actual quantum gravity. This will also give you a bit of insight into the guts of the way that scientific revolutions work in practice.

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Shahn Majid

So far in these posts we have looked at testable quantum gravity effects, but I have not said much about the ultimate theory of quantum gravity itself. There is a simple reason: I do not think we have a compelling theory yet.

Rather, I think that this deepest and most long-standing of all problems in fundamental physics still needs a revolutionary new idea or two for which we are still grasping. More revolutionary even than time-reversal. Far more revolutionary and imaginative than string theory. In this post I’ll take a personal shot at an idea — a new kind of duality principle that I think might ultimately relate gravity and information.

The idea that gravity and information should be intimately related, or if you like that information is not just a semantic construct but has a physical aspect is not new. It goes back at least some decades to works of Beckenstein and Hawking showing that black holes radiate energy due to quantum effects. As a result it was found that black holes could be viewed has having a certain entropy, proportional to the surface area of the black hole. Lets add to this a `black-holes no-hair theorem’ which says that black holes, within General Relativity, have no structure other than their total mass, spin and charge (in fact, surprisingly like an elementary particle).

What this means is that when something falls into a black hole all of its finer information content is lost forever. This is actually a bit misleading because in our perception of time far from the black hole the object never actually falls in but hovers forever at the edge (the event horizon) of the black hole. But lets gloss over that technicality. Simply put, then, black holes gobble up information and turn it into raw mass, spin and charge. This in turn suggests a kind of interplay between mass or gravity, and information. These are classical gravity or quantum particle arguments, not quantum gravity, but a true theory of quantum gravity should surely explain all this much more deeply.

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First off, newly syndicated readers who want to have access to my previous posts can find them archived here as well as listed on my own site here.

After last week’s speculations on time I would like to ask an even deeper question: why is there time?

My 4 year old daughter would be proud. What I mean is, why do things evolve in the first place? It seems to me that fundamental physics has to answer not only ‘what’ questions but also ‘why’ questions if it claims to provide understanding. I think I have an answer, or a glimpse of one.

The answer has to do with quantum anomalies; no not the large (not very quantum, then) things that seem to turn up in every other episode of Star Trek Voyager, but what physicists mean by this, which I am afraid is much more dry and dusty. In fact, I’m going to have to ask you to dust off your high school calculus books, just for a minute.

I explained in a previous post that even if nobody at the moment knows how to reconcile quantum theory and gravity, quantum spacetime should emerge as an effect coming out of any unknown theory. Typically, the coordinates x,y,z of space would also be quantum variables, so space alone should typically form some kind of symbolic algebra. Due to quantum effects, the order of the variables in this algebra will matter, xy will typically not coincide with yx. One says that the algebra is ‘noncommutative’.

Click to enlarge

Now, what about differential calculus on such a quantum space? If you remember any high school calculus it means things like dx, dy, dz as the ‘infinitesimal differences’. Newton and Leibniz both considered such things as numbers which are then made arbitrarily small. Hands up if your high school calculus class contained a picture like the one shown at left. It defines differentiation of a function f in the x direction as a limit of the slope df/dx of the triangle as dx gets small.

So to develop quantum gravity effects in physics we also need ‘quantum differentials’ dx, dy, dz. They should enjoy the properties that differentials enjoy in Newtons theory except, since xy and yx need not coincide, similarly y dx need not coincide with dx y, etc. Now, here is the remarkable thing one finds as you dig deeper into this world of quantum geometry:

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If the real world, at its base, is quantum, then should we not think with quantum logic?

Shahn Majid discusses how the notion of quantum symmetry coming out of modern ideas on space and time could provide clues to the workings of a truly quantum computer.

Have you ever sat through a really boring flow chart presentation and to pass the time found yourself wondering the following: See the way that flow chart arrow crosses that other flow chart arrow:

Does it matter whether the arrow ‘passes under’ the other arrow or ‘jumps over’ it?

If you are an engineer you could ponder the same question for a schematic for the wiring of a computer. In fact you could ponder the question when actually building a computer: does it matter if this wire connecting to that chip jumps over or under this other wire? If you thought it did matter, you would have discovered quantum computers as well as quantum symmetry!

Nice work.

Let me start with the symmetry. Truth, symmetry, beauty! The cornerstones of mathematics, some would argue of the very concept of knowledge. Surely, nothing could be deeper or more self-evident than the notion of symmetry — of finding patterns. But what if our usual conception of symmetry was not quite right? As scientists we should not be afraid to question even the most basic of assumptions. After all, Nature does not know or care what maths is in maths books, and maybe Nature is just a lot more imaginative than anything we have so far thought of.

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