I’ve been having some discussion with a collaborator in Sydney regarding a numerical model that we are developing. It concerns the response of the brain to pulses of magnetic field, but for the purposes of this blog entry, that is immaterial. One thing that we’ve been grappling with is ‘dealing with infinity’. Basically, in physical processes, what is happening now depends on a sum of everything that’s happened previously. Processes are causal. To understand the present, we look at the past (not the future). How far back in the past do we have to look? That depends on the problem, but, technically speaking, we have to look infinitely far back in the past. There is not cut-off time. Sure, there comes a point where what happened X seconds (or hours, years, millennia ago) isn’t going to make much of a difference, but where we take this cut-off is usually arbitrary. So often we physicists simply say ‘start from a time of minus infinity’, which means consider all things in the past, no matter how far back they happened.
What we get are integrals that run from minus infinity to now in time. And that, in the case of the model that I’ve been looking at, causes some conceptual problems. We end up with an infinite result. But that doesn’t actually turn out to be a physical problem. That’s because what we actually want to know involves integrating this result over another infinite time range, which generates an infinite sum of infinities. But here’s the neat bit. Some of these are positive infinities and some are negative infinities, and they cancel and divide out to something sensible and finite. It sounds odd, but mathematically it is all quite reasonable.
There are a few examples of this kind of thing in physics. Perhaps the most significant is renormalization in particle physics. Here, when one calculates using quantum electrodynamics how electrons and photons interact, the calculations are full of infinities. But the infinities, if considered carefully can be shown to cancel out. The positive infinities cancel with negative infinities. It was a major conceptual leap forward to accept that this sort of thing was reasonable.
I first came across the idea with my PhD – looking at how electrons interact with each other in solids. Here, there is a seminal paper by Gell-Mann and Bruekner (1957) which shows that while perturbation theory gives an infinite sum of infinite integrals, the infinities can be dealt with and a finite result remains. After a bit of on-line hunting I tracked it down here. It’s rather neat (if you’re a physicist).
The moral of all this for the non-mathematician is to be careful with the idea of infinity. Don’t treat infinity as ‘the biggest possible number’. There is no such thing. If you think you’ve found it, let me know and I’ll prove you wrong by adding one to it. It’s simply a short-hand for ‘and keep going…’