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Use the following expansion of effective action $$\Gamma[\phi_c]=\int d^4x [-V_{eff}(\phi_c)+(\partial_\mu\phi_c)^2A(\phi_c)+...]$$ where $...$ represents higher order derivatives of $\phi_c$. When $J\rightarrow 0$, $\phi_c(x)=constant$. This constant is the VEV $\langle\phi_c(x)\rangle=\phi_0$ and we obtain from above $$\Gamma[\phi_0]=-VT. ...


3

You should work out the minimum energy state of your system (classically) to find the vacuum expectation value. I assume you're working with the standard $\phi^4$-Lagrangian $$\mathcal L=\frac{1}{2}(\partial \phi)^2-\frac{1}{2}m^2\phi^2-\frac{\lambda}{4}\phi^4 $$ which corresponds to the Hamiltonian $$\mathcal ...



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