Question

How should the path integral change under a dilation?

Answer 1

Let's say I have a two-point function of a scalar field in flat space:

⟨ϕ(x)ϕ(y)⟩=∫Dϕϕ(x)ϕ(y)eiS[ϕ]⟨ϕ(x)ϕ(y)⟩=∫Dϕϕ(x)ϕ(y)eiS[ϕ]

Then I dilate things:

⟨ϕ(ax)ϕ(ay)⟩=∫Dϕϕ(ax)ϕ(ay)eiS[ϕ]⟨ϕ(ax)ϕ(ay)⟩=∫Dϕϕ(ax)ϕ(ay)eiS[ϕ]

If I field redefine ψ(x)=ϕ(ax)ψ(x)=ϕ(ax) then S[ϕ]=S[ψ]S[ϕ]=S[ψ] since for the action the change to ψψamounts to just a coordinate change, under which the action is invariant if constructed with the usual d4x−g−−−√d4x−g. Meanwhile, the measure picks up a functional Jacobian:

⟨ϕ(ax)ϕ(ay)⟩=∫Dψ∣∣∣δϕδψ∣∣∣ψ(x)ψ(y)eiS[ψ]⟨ϕ(ax)ϕ(ay)⟩=∫Dψ|δϕδψ|ψ(x)ψ(y)eiS[ψ]

Given that dilations aren't a symmetry of flat space, I don't expect the two-point function to be the same after the dilation, so the Jacobian determinant shouldn't be one.

I can write

ϕ(x)=ψ(xa)⟹δϕ(x)δψ(y)=∫d4yδ4(y−xa)ψ(y)=δ4(y−xa)ϕ(x)=ψ(xa)=∫d4yδ4(y−xa)ψ(y)⟹δϕ(x)δψ(y)=δ4(y−xa)

but then I don't know how to calculate this determinant.