Space and spaces

The idea of space is central in the way we think. We organize our perceptions in physical space, we think of time as a one-dimensional continuum, and we carry spatial notions over to any number of our conceptual constructs. Nevertheless, in some sense space is ‘our’ technology, wonderfully evolved for dealing with our experience of the physical world. It is probably only an approximation to reality. Long ago I happened to be looking at an account of quantum field theory written ‘for the general reader’ by Freeman Dyson, one of my favorite writers about science. I came upon a passage where he said that there are ‘well-understood mathematical reasons’ why when we quantize a wave-like physical system such as the electromagnetic field the result is best described in terms of particles. “Unfortunately,” he added, “the reason for this cannot be explained in non-mathematical terms.” I felt rather let down, for I don’t like the idea of mathematics as an arcane mystery where even the basic ideas can only be explained to initiates. I also wondered exactly which well-understood mathematical reasons Dyson had in mind. I have thought about the question a lot since then, but haven’t come up with anything that would be of much use to non-mathematicians; I leave it as a challenge. It seems to me a challenge even to give a clear mathematical account. I suppose that Dyson was thinking of the traditional statement that a free field can be regarded as a system of independent harmonic oscillators — which mystified me so much when I first encountered it in physics lectures as an undergraduate — but I believe that to understand the essential point involves thinking carefully about how we use the concept of space. That is the aim of this talk.