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.
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.
Now, you may ask, in this day and age, where string theorists are happy to work in 10500 dimensions: what is so special about three? In any dimension the modern way of thinking about gravity is in terms of a ‘metric’. This is a gadget which at each point of spacetime allows one to compute the distance to all nearby points. It goes back to the 19th century mathematician Riemann and was used by Einstein. The mathematicians Cartan and Weyl found a different way of thinking about this in terms of a ‘frame’ and a ‘connection’ at any point of the spacetime. Their theory works in any dimension but I am going to cut straight to a very special answer only in three spacetime dimensions.
First, think about what you can do to a rigid object in three flat dimensions (in our case one of these is time but you will be able to visualise better if you dont worry about that). Well, you can move the object around and you can rotate it. Together these form a classical symmetry group E_3 of ‘translations and rotations’ in three dimensions. The same in a three dimensional flat spacetime.
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