A former colleague of mine tossed me an email today in the hopes I’d know why a connection on a vector bundle over a 4-manifold whose curvature is anti-self-dual is referred to as an “instanton”.

My best guess is that “-on” comes from the interpretation of curvature components as gauge bosons in gauge field theories, and that “instant-” might have something to do with the support or polology of the propagator associated to the fields, but really I don’t know for sure. Anyone around here know?

Was the term first used to describe a connection or a particle? If it’s the latter, it might be that the particles were named so for some property they have, and the mathematical term came from an association of that particle with that object.

That’s pretty much what we don’t know in the history of the term. We both came at it from purely differential topological/geometric grounds, either in the theory of connections on principal fiber bundles or in “nonstandard” R[sup]4[/sup]s – spaces which are topologically the same as real 4-d space, but with a smooth structure different from the standard one.

Basically, they’re a consequence of there being multiple vacuum states in quantum field theory. You can thus have situations where you start with one particular vacuum and then the system quantum mechanically tunnels to another vacuum. This tunneling is rather localised in (Euclidian) time, so you can think of the situation as “nothing” happening, then suddenly something weird happens and then you’re back to “nothing” happening again. 't Hooft drew an analogy between such solutions being localised in time and everyday particle states being localised in space, hence the name.
They were very much part of the flourishing of topological ideas in QFT at the time and they’re a common notion to run across in the subject. I’ve even known someone who did their Ph.D. looking for evidence of them in collider experiments.

The classic introduction to the subject is Sidney Coleman’s “The uses of instantons” in his Aspects of Symmetry, which I highly recommend in general. It includes references to the original papers.

Is there a quick summary for a differential geometer? Go on – for once you don’t have to dumb down the math here.

Is the bit about Euclidean time meaning you’re working this out in a locally Euclidean 4-fold rather than a locally Minkowskian one (say, after Wick rotating)? In that case, why “instant”, since you can’t tell space from time anymore?

I did realise that However, since you already had a formal definition, it did seem more useful to indicate how I handwavily visualise the physics. Still, that doesn’t really motivate why instantons were defined in the first place.
Before going into that, we’ll get the minor issue out of the way.

Yep. I can see how it might seem a little odd to a mathematician, but physicists do tend to remember which axis is the time one even after Wick rotating. For a start, you may want to be rotating back at some later stage. Also the time axis will still be singled out in some way by the physics of the situation; usually there’s still a difference in the boundary condition wrt that direction.

We’re going to end up working semi-classically and pick out particular solutions of the classical field equations, so we can think classically. Consider some non-Abelian gauge theory, say QCD. As you no doubt know, solutions which are pure gauge fields at infinity can be classified into different homotopy classes characterised by a different topological winding number (a Pontryagin index).
Now consider a situation where we have vaccum states at t = +/- infinity and we want to calculate the amplitude for the transition between these two states. That’s just doing a path integral over the usual action for the gauge fields. Semi-classically the amplitude is an exponential of the integral of the action over the 4D volume involved. (There are numerous minor points to do with whether it’s a finite or infinite 3D space; I’ll ignore these.) We’re interested in field configurations that minimise this integral. One then shows that the condition this implies is that the fields are self-dual or anti-self-dual.
An instanton is thus a solution to the classical field equations that are self-dual or anti-self-dual, which are pure-gauge fields at t = +/- infinity and which have a non-zero finite action.

The vacuum states at t = +/- infinity are pure-gauge fields by definition. However, there’s nothing to say that these have the same winding number. And, in fact, an instanton solution is just a transition between two vacuum states with different winding numbers. (There’s an elegant result for the transition amplitude in terms of the difference of the winding numbers, together with the gauge coupling.)
Because there’s now a non-trivial probability for a transition between any two vacuum states, we can no longer regard the vacuum as simple: it’s some sort of superposition over all these different states, which are all related by instantons. That then has physical consequences - the theta problem - which ought to be observable, but aren’t and so are still rather mysterious.