Sharp Blue: The Higgs field

Somebody on the Orion’s Arm mailing list asked about Higgs bosons (wondering whether they can be used to keep open wormholes). I have no idea about the wormhole part (although I suspect not), but here’s a (slightly edited for clarity) primer on the Higgs field that I wrote for them:

All of our theories of particle interactions are based on gauge symmetries. The idea here is that we see certain “global” symmetries in our systems of equations for non-interacting particles, and then modify the equations to make the symmetry “local” (i.e. so mathematical transformations that leave the system physically unchanged vary from place to place and from time to time). The only way to do this is to introduce further fields that compensate for the changes under which the system is supposed to be symmetric. It’s these “gauge fields” that lead to interactions, and the particles associated with them, gauge bosons, are the particles that carry forces. We also get out a conservation law for each such gauge symmetry.

This is all a bit abstract, so here’s an example. The equations of motion for a single particle are invariant under a global change of some thing called “phase”. To make this global symmetry a local one, we have to introduce a gauge field that turns out (when we calculate the equations describing its dynamics) to be the electromagnetic field. The associated gauge boson is the “photon”, and the conservation law associated with symmetry under local changes of phase is the conservation of electric charge. (The charge itself measures the strength of the coupling between the particle and the electromagnetic gauge field.)

Now, this is all well and good, but there’s a problem: it doesn’t work for particles with mass. To overcome this difficulty, we have to replace mass terms in our equations with a coupling between the matter field and a “Higgs field” - if we do this then we get out something that looks like a mass but we also preserve gauge invariance. The particular mass of a given type of particle depends on the strength of its coupling to the Higgs field (and to the strength of the Higgs field itself). (This is why people are wrong when they say that the Higgs field explains the origins of mass - instead it replaces the question “why do particles have different masses?” with the question “why do particles have different couplings to the Higgs field?”)

Of course, I’ve glossed over various parts of the discussion (and I haven’t discussed the important concept of spontaneous symmetry breaking at all), but that’s roughly how it works!