Given a Quantum field theory, for a scalar field ϕ with generic action S[ϕ], we have the generating functional Z[J]=eiW[J]=∫Dϕei(S[ϕ]+∫d4xJ(x)ϕ(x))∫DϕeiS[ϕ].
The one-point function in the presence of a source J is.
ϕcl(x)=⟨Ω|ϕ(x)|Ω⟩J=δδJW[J]=∫Dϕ ϕ(x)ei(S[ϕ]+∫d4xJ(x)ϕ(x))∫Dϕ ei(S[ϕ]+∫d4xJ(x)ϕ(x)).
The effective Action is defined as the Legendre transform of W
Γ[ϕcl]=W[J]−∫d4yJ(y)ϕcl(y), where J is understood as a function of ϕcl.
That means we have to invert the relation ϕcl(x)=δδJW[J] to J=J(ϕcl).
How do we know that the inverse J=J(ϕcl) exists? And does the inverse exist for every ϕcl? Why?
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