Wednesday, 27 September 2017

quantum field theory - The proof of Goldstone's theorem


On page 352 of Peskin and Shroeder.

The athors show the proof of Goldstone's theorem.



A general continuous symmetry transformation has the form ϕaϕa+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=(Δaϕb)ϕ0(Vϕa)ϕ0+Δa(ϕ0)(2ϕaϕbV)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.




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