Interactions and perturbation theory

Use $\hbar=c=1$, metric $(+---)$, and a real scalar with $\mathcal L_{\mathrm{int}}=-\lambda\phi^4/4!$. The free Hamiltonian evolves interaction-picture operators, while the interaction evolves the states. For this interaction without derivative couplings, $H_I(t)=\int d^3x\,\lambda\phi_I^4/4!$. The sign follows from the Legendre transform. For derivative interactions, simply writing $H_I=-L_I$ can fail and must be checked.

The interaction-picture evolution operator obeys a first-order operator differential equation. Operators at different times generally do not commute, so the solution is not an ordinary exponential with its ordering ignored. Time ordering makes the later Hamiltonian act on the left. Expanding the ordered exponential inserts increasing numbers of interaction vertices, with integrals over every possible insertion time and position.

An amplitude is complex, and a probability is its modulus squared. If a transition has no free contribution and its first amplitude is proportional to $\lambda$, the leading probability is proportional to $\lambda^2$. If a free amplitude is already present, its interference with a first-order correction can instead change the probability at order $\lambda$. Counting vertices without identifying the observable can therefore give the wrong leading power.

For scalar quartic interactions in four spacetime dimensions, $\lambda$ has zero mass dimension. A loop often introduces a factor of order $1/(16\pi^2)$, but that is a useful estimate rather than a universal convergence theorem. Large logarithms of scale ratios, many available states, thresholds, or long evolution times can enhance nominally higher-order terms. The meaningful question is whether the neglected contribution is small relative to the retained prediction in the regime being studied.

At coincident field insertions, perturbative expressions develop ultraviolet divergences. A regulator makes intermediate expressions defined; renormalization relates parameters to measurements and organizes the allowed counterterms. A divergent bare integral is not itself an infinite prediction for a detector. Conversely, discarding a divergence without specifying a renormalization prescription does not produce a uniquely meaningful finite answer. The next module develops this distinction systematically.

Even after renormalization, many QFT series are asymptotic rather than convergent. More terms can improve an estimate up to an appropriate order without promising convergence at arbitrarily high order. Near strong coupling or nonperturbative effects, expansion around the free vacuum may miss the important physics. We will state tree-level or leading-order assumptions explicitly and avoid interpreting a truncated expansion as a complete microscopic history.