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For $H=H_0+H_{\mathrm{int}}$, the interaction picture moves the free evolution into operators and leaves states to evolve with $H_I(t)=e^{i{H}_{0}t}{H}_{\mathrm{int}}{e}^{-i{H}_{0}t}$. Its evolution operator is the time-ordered Dyson series.

The interaction-picture Hamiltonian density is ${\mathcal{H}}_I=\lambda \varphi {\Phi }^{†}\Phi$. The first-order Dyson series term cannot connect the four external particles, so the leading contribution isBy the Wick theorem, the nonzero terms annihilate the incoming complex particle using the annihilation part of $\Phi$, create the outgoing complex particle using the creation part of ${\Phi }^{†}$, annihilate the incoming real particle using the annihilation part of $\varphi$, and create the outgoing real particle using its creation part. The remaining $\Phi$ and ${\Phi }^{†}$ form an internal Feynman propagator. Interchanging which vertex absorbs the incoming real scalar gives the two contractions; this factor of two cancels the Dyson factor $1/2$.

There are therefore an $s$-channel internal momentum $p+k$ and a crossed channel internal momentum $p-k^{\prime }$. With covariantly normalized external states,whereEquivalently,with $s=(p+k{)}^2$ and $u=(p-k^{\prime }{)}^2$.

For a Schrödinger-picture state satisfyingdefine the interaction picture byDifferentiation givesThe formal solution from $t_0$ is the Dyson seriesThrough quadratic order,Differentiating giveswhich checks the equation to the requested order.

At the endpoints, $|i{⟩}_I=e^{-i{H}_{0}T/2}|i{⟩}_S$ and $|f{⟩}_I=e^{i{H}_{0}T/2}|f{⟩}_S$. ThereforewhereFor ${\varphi }_1(p_1)\to {\varphi }_2(p_1^{\prime }){\varphi }_2(p_2^{\prime }){\varphi }_2(p_3^{\prime })$, the leading term isThere are $3!$ contractions of the identical outgoing fields. With covariantly normalized states,Equivalently, the interaction vertex from $-\lambda {\varphi }_1{\varphi }_2^3$ is $-i3!\lambda$.

With a nonconstant potential energy, momentum no longer diagonalizes the Hamiltonian operator. The operator derivation must use the energy eigenfunctions of the full Hamiltonian, a Dyson series, or a time-sliced Trotter product formula. In the classical derivation, the straight paths are replaced by every solution of the nonlinear Euler-Lagrange equation with the specified endpoints. The semiclassical propagator becomes a sumwhere the prefactor is the Van Vleck determinant and ${\mu }_{\mathrm{cl}}$ is a Maslov index. Unlike a quadratic theory, the classical-path sum is generally only an asymptotic approximation: the exact path integral includes fluctuations of every order. On the circle, the sum must still include all winding number sectors.

The interaction $-g\varphi {\Psi }_i^{\ast }{\Psi }_i$ gives one cubic vertex $-ig$ joining a real-scalar line to a particle-antiparticle pair of either complex scalar species. The second-order term of the Dyson series isWick theorem contracts the incoming ${\Psi }_1{\overline{\Psi }}_1$ pair at one vertex, the outgoing ${\Psi }_2{\overline{\Psi }}_2$ pair at the other, and the two $\varphi$ fields with each other. The two assignments of $x,y$ cancel the factor $2!$. Thus there is one connected tree-level Feynman diagram, the $s$-channel exchangeWith standard relativistic external-state normalization,up to the physically irrelevant common sign convention for $\mathcal{M}$.

In the center-of-momentum frame, $q=(E_1,-\mathbf{p})$ and $p=(E_1,\mathbf{p})$, soPair creation is kinematically possible exactly when $s\ge 4M_2^2$. The threshold incident momentum is therefore

Wick theorem expresses a time-ordered product of free fields as its normal-ordered product plus the sum over every possible contraction. It turns the Dyson series into Feynman diagrams.