Space and Spaces

This is the title of a talk given today by Graeme Segal at Oxford’s Mathematical Institute. He gave this same talk last year as the Presidential Address for the London Mathematical Society, which means there’s a nice written version online here. I mainly mention this with the hope of one day understanding it properly, so forgive the sparsity of details.

One of the key ideas of the talk was to explain why we observe particles in nature, even though our best description of (most of) fundamental physics, quantum field theory, says that fields are the fundamental physical objects. To answer this question, one needs to think about the relevant spaces of states and observables, and one sees that noncommutative geometry enters essentially. The following nice quote touches on this:

In one way quantum field theory is enormously simpler than its classical counterpart, for there are no particles: the manifold of configurations is simply a space of smooth fields on M. In exchange for this simplification, the state-space can no longer be interpreted as an ordinary space: it is an object of non commutative geometry, and that is why when we look at the world we sometimes see particles and sometimes see fields.

To really understand what’s going on, it seems that some reasonably high-level mathematics is needed—Segal had hoped to formulate an explanation for non-mathematicians, but he couldn’t come up with one. A key result seems to be a theorem—a generalisation of the Stone-von Neumann theorem—that explains why we sometimes see particles and not waves: the theorem says that the algebra of observables on the state space of fields contains an algebra isomorphic to the observables on the tangent bundle of the configuration space of a collection of particles. I don’t understand the proof or surrounding discussion well enough to explain it further, so I won’t try, but merely plan to try sometime in the future. There’s much more in the paper about the notions of physical and mathematical spaces from the algebraic topologists perspective, but that’s even farther outside of my understanding.

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Visualising Higher Dimensional Spheres

I often try to think of how one might imagine spatial objects that have dimension greater than three. That this problem is nontrivial seems to come from the basic fact that the human mind developed to interpret and analyse the lowly three spatial dimensions that we appear to inhabit–our ancestors never had to navigate 5-dimensional mazes to escape from 4-dimensional tigers and therefore visualising in 3-dimensions was perfectly sufficient for successfully passing on our genes.

Despite this physiological shortcoming, there are various clever ways of trying to get a grip on four-dimensional shapes–see here, for example, for some nice visualisations. I recently learnt of a neat way to construct an n-dimensional sphere (henceforth, n-sphere) from an (n-1)-dimensional sphere by “glueing cones together”. (The relevant article also mentions two other visualisation methods and tries to link the method described below to Dante’s description of the universe in his Divine Comedy.) The basic construction goes like this (try it with n=1 or 2):

  • First fill in the interiors of two (n-1)-spheres to get two n-balls. These n-balls, which will be the two hemispheres of our n-sphere, should be thought of as being made of an infinite number of nested (n-1)-spheres of decreasing radii.
  • Now superpose the two n-balls, identifying their coincident boundaries–this boundary is the equator of our n-sphere.
  • Finally, mentally drag the insides of the two n-balls in opposite directions and away from the equator, such that an interior (n-1)-sphere further from the n-ball’s boundary gets dragged further away.

If you try using this to construct the 4-sphere from two 3-balls, you’ll realise that there’s no fourth dimension in which to drag the insides of the 3-balls. You didn’t expect to be able to actually visualise a 4-sphere all in one go, did you? Nevertheless, it seems like there’s something to be gained from this perspective, even if it won’t prevent you becoming a snack to that 4-dimensional tiger.