= Solution
Split the free <interaction picture> field as $\phi=\phi^{(+)}+\phi^{(-)}$, where the first part contains <annihilation operators> and the second <creation operators>. <Time ordering> places the operator at the later time on the left, while <normal ordering> places all <creation operators> on the left. For $x^0>y^0$, commuting the annihilation part of the first field through the creation part of the second gives
$$
\phi(x)\phi(y)=:\phi(x)\phi(y):+[\phi^{(+)}(x),\phi^{(-)}(y)].
$$
The <commutator> is the c-number
$$
[\phi^{(+)}(x),\phi^{(-)}(y)]
=\int\frac{d^3p}{(2\pi)^3}\frac{e^{-ip\cdot(x-y)}}{2E_{\mathbf p}}.
$$
For $y^0>x^0$, the same calculation interchanges $x,y$. Combining the two time orders proves the <two-field Wick contraction identity>
$$
\boxed{T\phi(x)\phi(y)=:\phi(x)\phi(y):+D_F(x-y)},
$$
where
$$
D_F(x-y)=\langle0|T\phi(x)\phi(y)|0\rangle
=\int\frac{d^4p}{(2\pi)^4}\frac{i\,e^{-ip\cdot(x-y)}}{p^2-m^2+i0}.
$$
The equality with the four-dimensional integral follows by closing the energy contour at the positive-energy pole for $x^0>y^0$ and the negative-energy pole for the reverse order. This fixes the <scalar Feynman propagator pole prescription>; in this convention $D_F$ itself includes the factor $i$.
Because these interactions have no derivatives, $H_{\rm int}=-\int\mathcal L_{\rm int}d^3x$. Expand the <Dyson series>
$$
S=T\exp\left(i\int d^4z\left[\frac{\mu}{3!}\phi(z)^3-\frac{\lambda}{4!}\phi(z)^4\right]\right).
$$
Apply the <Wick theorem> to each term together with the external insertions. Every paired field contributes a propagator, and the remaining normal-ordered vacuum expectation vanishes unless no fields remain. Representing the propagators by Fourier integrals and integrating each vertex position gives the <four-momentum conservation> delta function. The $3!$ and $4!$ contractions into labelled vertex slots cancel the factorials in the interaction density. The resulting <scalar-field cubic and quartic Feynman vertices> are
$$
\boxed{\text{propagator: }\frac{i}{p^2-m^2+i0},\qquad
\text{cubic vertex: }i\mu,\qquad
\text{quartic vertex: }-i\lambda}.
$$
Each vertex also has $(2\pi)^4\delta^{(4)}(\sum p)$, each independent internal <momentum> is integrated with $d^4p/(2\pi)^4$, and repeated equivalent contractions give the usual <Feynman-diagram symmetry factor>. For the unamputated correlation functions, retain the propagators joining external insertions to vertices. The displayed unnormalized numerator includes disconnected <vacuum bubbles>; dividing by $\langle0|S|0\rangle$ cancels them. Amputated <scattering amplitudes> additionally use the <LSZ reduction formula>.
For stability, examine the classical <scalar potential>
$$
V(\phi)=\frac12m^2\phi^2-\frac{\mu}{6}\phi^3+\frac{\lambda}{24}\phi^4.
$$
The tree-level boundedness criterion is \b[$\lambda>0$], with arbitrary real cubic coupling. If $\lambda=0$, boundedness requires $\mu=0$ and $m^2\ge0$; if $\lambda<0$, or $\lambda=0$ with $\mu\ne0$, the potential is unbounded below. Thus a positive quartic coupling permits a stable ground-state vacuum, although it need not be the perturbative vacuum at $\phi=0$.
For $\lambda>0$ and $m^2\ge0$, the <stability of a cubic-quartic scalar potential> determines when zero is a global minimum:
$$
V(\phi)=\phi^2\left[\frac{\lambda}{24}\left(\phi-\frac{2\mu}{\lambda}\right)^2
+\frac{m^2}{2}-\frac{\mu^2}{6\lambda}\right].
$$
The square bracket is nonnegative for every field value exactly when
$$
\boxed{\mu^2\le3\lambda m^2}.
$$
For a nontrivial equality case there is a second degenerate minimum at $\phi=2\mu/\lambda$. If $m^2>0$ and $\mu^2>3\lambda m^2$, zero remains a local minimum but is a <false vacuum>, with a lower minimum elsewhere. If $m^2=\mu=0$ and $\lambda>0$, the origin is the quartic minimum despite its vanishing quadratic curvature. These are classical or tree-level stability statements; renormalized couplings and the <quantum effective potential> refine the quantum analysis.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-49-scalar-vacua.png]
{title=Stable, degenerate and metastable scalar vacua}
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