Interactions

Free fields describe particles that travel through space without ever affecting each other. They never scatter, never decay, never combine. A universe of free fields would be a universe of eternal, non-interacting ghosts, mathematically tractable but physically sterile. All the richness of physics comes from interactions: the coupling between fields that allows particles to scatter, transform, and create new particles.

Interaction Lagrangians

We add interaction terms to the free Lagrangian. The total Lagrangian splits as:

$$\mathcal{L} = \mathcal{L}_\text{free} + \mathcal{L}_\text{int}$$

For quantum electrodynamics (QED), the quantum theory of electrons and photons, the interaction is:

$$\mathcal{L}_\text{int} = -e\bar{\psi}\gamma^\mu\psi A_\mu$$

This couples the Dirac field $\psi$ (electrons/positrons) to the electromagnetic field $A_\mu$ (photons) with coupling constant $e$ (the electron charge). Every electromagnetic interaction (light scattering off electrons, pair production, atomic spectra) originates from this single term.

For a scalar field with self-interaction, a common choice is:

$$\mathcal{L}_\text{int} = -\frac{\lambda}{4!}\phi^4$$

This "$\phi^4$ theory" is the simplest interacting field theory and serves as a theoretical laboratory for developing techniques.

The S-Matrix

In a scattering experiment, particles come in from far away (the "in" state), interact briefly, and fly apart (the "out" state). The S-matrix (scattering matrix) connects them:

$$|\text{out}\rangle = \hat{S} |\text{in}\rangle$$

All experimentally measurable quantities (cross sections, decay rates, branching ratios) are extracted from S-matrix elements:

$$\mathcal{M}_{fi} = \langle f | \hat{S} | i \rangle$$

where $|i\rangle$ and $|f\rangle$ are specific initial and final states with definite particle content and momenta. The amplitude $\mathcal{M}$ is what we compute; the probability is $|\mathcal{M}|^2$.

Perturbation Theory

Since we cannot solve interacting theories exactly, we expand in powers of the coupling constant. The S-matrix becomes:

$$\hat{S} = T\exp\left(-i\int d^4x \, \mathcal{H}_\text{int}(x)\right) = \mathbf{1} + (-i)\int d^4x \, \mathcal{H}_\text{int} + \frac{(-i)^2}{2!}\int d^4x \, d^4y \, T\{\mathcal{H}_\text{int}(x)\mathcal{H}_\text{int}(y)\} + \cdots$$

Each term involves more powers of the coupling constant. For QED, $e^2/4\pi \approx 1/137$, so each additional power of $e$ suppresses the contribution by roughly a factor of 10. This is why perturbation theory works so spectacularly well for QED.

Using Wick's theorem, each term in the perturbative expansion can be reduced to a sum of products of propagators, the free-field two-point functions. This is where Feynman diagrams enter.

Feynman Diagrams

Feynman diagrams are not mere illustrations. They are a precise computational tool. Each diagram corresponds to a specific mathematical expression contributing to the amplitude. The rules are:

Reading a Feynman Diagram

Consider the simplest QED process: electron-electron scattering ($e^- e^- \to e^- e^-$), known as Møller scattering. At lowest order (tree level), there are two diagrams:

Each diagram has two vertices (each contributing a factor of $-ie\gamma^\mu$), one internal photon propagator, and four external fermion lines. The total amplitude is the sum of both diagrams, and because the external electrons are identical fermions, the two amplitudes come with a relative minus sign.

Cross Sections and Decay Rates

The amplitude $\mathcal{M}$ connects to experiment through cross sections (for scattering) and decay rates (for unstable particles).

Cross Sections

For a $2 \to 2$ scattering process in the center-of-mass frame:

$$\frac{d\sigma}{d\Omega} = \frac{|\mathcal{M}|^2}{64\pi^2 s}$$

where $s = (p_1 + p_2)^2$ is the center-of-mass energy squared. The cross section $\sigma$ has dimensions of area and represents the effective target size for the interaction.

Decay Rates

For a particle of mass $M$ decaying to $n$ particles:

$$d\Gamma = \frac{1}{2M} |\mathcal{M}|^2 \, d\Pi_n$$

where $d\Pi_n$ is the $n$-body Lorentz-invariant phase space. The total decay rate $\Gamma$ gives the particle's lifetime: $\tau = 1/\Gamma$.

Tree-Level vs. Loop Diagrams

The diagrams we've seen so far are tree-level: they have no closed loops of internal lines. Tree diagrams give the leading contribution to any process, at order $e^{2n}$ where $n$ is the number of vertices.

Higher-order corrections involve loop diagrams: internal lines that form closed loops, requiring integration over the unconstrained loop momentum. For example, the one-loop correction to electron-electron scattering includes:

Example: Compton Scattering

Compton scattering (a photon scattering off an electron, $\gamma e^- \to \gamma e^-$) has two tree-level diagrams. The amplitude gives the famous Klein-Nishina formula for the cross section:

$$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{2m_e^2}\left(\frac{\omega'}{\omega}\right)^2 \left(\frac{\omega'}{\omega} + \frac{\omega}{\omega'} - \sin^2\theta\right)$$

where $\omega$ and $\omega'$ are the initial and final photon energies, and $\theta$ is the scattering angle. This formula, derived from QED Feynman diagrams, matches experiment with extraordinary precision.

The Power and Limits of Perturbation Theory

Perturbative QED is the most precise theory in all of science. The anomalous magnetic moment of the electron has been computed to five-loop order (over 12,000 Feynman diagrams!) giving:

$$a_e = \frac{g-2}{2} = 0.001\,159\,652\,182\,032(720)$$

This agrees with experiment to better than one part in a trillion, the most accurate prediction in physics.

But perturbation theory has limits. When the coupling constant is not small (as in QCD at low energies), the perturbative expansion fails. And some phenomena, like confinement of quarks, are inherently non-perturbative.

Key Insights

• Interactions are described by coupling terms in the Lagrangian that connect different fields
• The S-matrix encodes all scattering and decay information; the LSZ formula relates it to field correlation functions
• Feynman diagrams are a systematic, diagrammatic representation of the perturbative expansion. Each diagram is a precise mathematical expression
• Tree-level diagrams give leading-order results; loop diagrams provide quantum corrections suppressed by powers of the coupling constant
• QED perturbation theory achieves part-per-trillion agreement with experiment, the most precise test of any physical theory
• Perturbation theory fails when the coupling constant is of order one, necessitating non-perturbative approaches

Looking Ahead

Loop diagrams bring a fundamental problem: the integrals over loop momenta often diverge. The momentum circulating in a loop can be arbitrarily large, and the integrals blow up. Far from being a defect, handling these infinities led to one of the deepest ideas in theoretical physics: renormalization. That is our next topic.