On page 352 of Peskin and Shroeder.
The athors show the proof of Goldstone's theorem.
A general continuous symmetry transformation has the form ϕa→ϕa+αΔa(ϕ),
where α is an infinitesimal parameter and Δa is some function of all the ϕ's. Specialize to constant fields; then the derivative terms in L vanish and the potential alone must be invariant under (11.12). This condition can be written V(ϕa)=V(ϕa+αΔa(ϕ))orΔa(ϕ)∂∂ϕaV(ϕ)=0.Now differentiate with respect to ϕb, and set ϕ=ϕ0: 0=(∂Δa∂ϕb)ϕ0(∂V∂ϕa)ϕ0+Δa(ϕ0)(∂2∂ϕa∂ϕbV)ϕ0.The first term vanishes since ϕ0 is a minimum of V, so the second term must also vanish. If the transformation leaves ϕ0 unchanged (i.e., if the symmetry is respected by the ground state), then Δa(ϕ0)=0 and this relation is trivial. A spontaneously broken symmetry is precisely one for which Δa(ϕ0)≠0; in this case Δa(ϕ0) is our desired vector with eigenvalue zero, so Goldstone's theorem is proved.
Why in spontaneously broken symmetry Δa(ϕ0)≠0?
And on the other hand when the symmetry is not spontaneously broken, Δa(ϕ0)=0?
Thanks.
No comments:
Post a Comment