Thursday, 23 April 2020

lagrangian formalism - Defining quantum effective/proper action (Legendre transformation), existence of inverse (field-source)?


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)|ΩJ=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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