Classical electroweak field theory

[bcrowell]

A question I'm sure I've seen asked here and/or elsewhere is why there doesn't seem to be any classical force corresponding to the weak interaction. I came up with the following, and am wondering whether this seems correct and satisfying to others.

Basically, being able to write down a Lagrangian density isn't the same thing as being able to describe the classical theory that is the counterpart of a quantized system. In particular, it seems like this can't possibly work for unstable particles. For example, the Lagrangian density for muon decay has a constant in it, G_F, the Fermi coupling constant. The half-life of the muon goes like $h/G_F^2$. In the classical limit, the half-life goes to zero, so the classical theory of muons is a theory with no muons in it.

 

[fzero]

Classical EM would serve quite well to describe the flight of atmospheric muons to a detector at sea level; the finite lifetime is not a significant problem. The reason there is no classical weak force (analogous to Coulomb's law) is because the effective range of the weak force is $r_W \sim 1/M_W$, which is orders of magnitude smaller than the Compton wavelength of any low-energy probe. This is of course ignoring that EM and strong interactions between the probes we do have available would make it hard to isolate the weak interaction in the first place.

 

[bcrowell]

Classical EM would serve quite well to describe the flight of atmospheric muons to a detector at sea level; the finite lifetime is not a significant problem.

I'm talking about the fundamental underlying field theory.

The reason there is no classical weak force (analogous to Coulomb's law) is because the effective range of the weak force is $r_W \sim 1/M_W$, which is orders of magnitude smaller than the Compton wavelength of any low-energy probe.

I'm interested in the reasons why there isn't even in principle any classical electroweak field theory.

(Why would the relevant thing be the Compton wavelength, h/mc, rather than the de Broglie wavelength, h/p, that was relevant?)

 

[Bill_K]

I'm interested in the reasons why there isn't even in principle any classical electroweak field theory.

To have a classical field you need to accumulate a large number of bosons, and the W/Z bosons are unstable.