Interaction Hamiltonian ($H_I$)

The interaction Hamiltonian generates interaction-picture time evolution. To leading order for many cubic field interactions it is the negative of the cubic interaction Lagrangian, although derivative interactions require the canonical Legendre transform to justify this relation.

For $\mathcal{L}={\dot{\varphi }}^2/2-{\mathcal{V}}_2+{\mathcal{L}}_3(\varphi ,\dot{\varphi })$, where ${\mathcal{L}}_3$ is cubic in perturbation amplitude, the canonical momentum has $\pi =\dot{\varphi }+O({\varphi }^2)$. Put $\dot{\varphi }=\pi +\Delta$ with $\Delta$ second order. Then $\pi \dot{\varphi }-{\dot{\varphi }}^2/2={\pi }^2/2-{\Delta }^2/2$, so through cubic order the Legendre transform in mechanics gives ${\mathcal{H}}_3=-{\mathcal{L}}_3$ evaluated with the free momentum. At fourth order there are extra terms, so derivative interactions do not obey this simple rule at all orders.