So far, we've studied free field theory where particles don't interact. Real physics involves interactions: photons scattering off electrons, quarks binding into protons, Higgs bosons decaying into lighter particles.

💡Free vs Interacting Theory

Free theory: Particles created, propagate independently, never scatter. We solved it exactly! But it's boring - no physics happens.

Interacting theory: Particles scatter, decay, create new particles. This is where real physics happens! But we can't solve it exactly - need perturbation theory.

Key idea: Start with free theory (we understand it), add small interactions, compute corrections order by order. This is the essence of perturbative QFT!

1.2 Interacting Lagrangian

We split the Lagrangian into free and interacting parts:

$$\boxed{\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_{\text{int}}}$$

Examples of interaction terms:

φ⁴ theory: $\mathcal{L}_{\text{int}} = -\frac{\lambda}{4!}\phi^4$
Scalar self-interaction, prototypical toy model
Yukawa theory: $\mathcal{L}_{\text{int}} = -g\bar{\psi}\psi\phi$
Fermion-scalar coupling, used in nuclear physics
QED: $\mathcal{L}_{\text{int}} = -e\bar{\psi}\gamma^\mu\psi A_\mu$
Electron-photon interaction, most precisely tested theory!

The coupling constants λ, g, e are small - this is why perturbation theory works!

1.3 Interaction Picture

In QM, we have three pictures:

Schrödinger Picture

States evolve, operators fixed

$\frac{d}{dt}|\psi\rangle = -iH|\psi\rangle$

Heisenberg Picture

Operators evolve, states fixed

$\frac{d}{dt}\hat{O} = i[H,\hat{O}]$

Interaction Picture

Both evolve (hybrid)

$H = H_0 + H_{\text{int}}$

In the interaction picture, we use H₀ for "easy" evolution:

\begin{align*} |\psi_I(t)\rangle &= e^{iH_0 t}|\psi_S(t)\rangle \\ \hat{O}_I(t) &= e^{iH_0 t}\hat{O}_S e^{-iH_0 t} \end{align*}

States evolve according to:

$$\boxed{i\frac{d}{dt}|\psi_I(t)\rangle = H_{\text{int},I}(t)|\psi_I(t)\rangle}$$

The interaction picture operators evolve like free fields! This makes calculations much easier.

1.4 Time Evolution Operator

The solution to the interaction picture equation is:

$$|\psi_I(t)\rangle = U(t,t_0)|\psi_I(t_0)\rangle$$

where U(t,t₀) is the time evolution operator:

$$\boxed{U(t,t_0) = T\exp\left(-i\int_{t_0}^t dt' H_{\text{int},I}(t')\right)}$$

T is the time-ordering operator:

$$T[A(t_1)B(t_2)] = \begin{cases} A(t_1)B(t_2) & t_1 > t_2 \\ B(t_2)A(t_1) & t_2 > t_1 \end{cases}$$

This ensures operators are ordered with later times to the left.

1.5 The S-Matrix (Scattering Matrix)

In scattering experiments, we prepare particles in the distant past (t → -∞), let them interact, and measure them in the distant future (t → +∞).

💡Physical Setup

Imagine a particle accelerator experiment:

Prepare two beams of particles (initial state |i⟩)
Collide them in the interaction region
Detect final products far from collision point (final state |f⟩)

The S-matrix gives the probability amplitude for |i⟩ → |f⟩!

The S-matrix is the time evolution from t = -∞ to t = +∞:

$$\boxed{S = U(+\infty, -\infty) = T\exp\left(-i\int_{-\infty}^{+\infty} dt H_{\text{int},I}(t)\right)}$$

Scattering amplitude from initial state |i⟩ to final state |f⟩:

$$\boxed{S_{fi} = \langle f|S|i\rangle}$$

Properties of S-Matrix

Unitary: S†S = 𝟙 (probability conservation)
Lorentz invariant: Same result in all reference frames
Analytic: Analytic properties encode causality
Crossing symmetry: Particles ↔ antiparticles

1.6 Dyson Series (Perturbative Expansion)

We can't compute S exactly, but we can expand in powers of the coupling:

$$S = 1 + \sum_{n=1}^\infty \frac{(-i)^n}{n!}\int dt_1 \cdots dt_n \, T[H_{\text{int}}(t_1)\cdots H_{\text{int}}(t_n)]$$

This is the Dyson series. Each term has a Feynman diagram interpretation:

n = 0: No interaction (free propagation)
n = 1: Single interaction vertex
n = 2: Two interactions (virtual particle exchange)
n = 3: Three interactions, etc.

💡 Key Insight

Each term in the Dyson series corresponds to a specific "history" of particle interactions. Feynman diagrams provide a visual way to organize these terms!

1.7 LSZ Reduction Formula

How do we connect S-matrix elements to Green functions (correlation functions)? The Lehmann-Symanzik-Zimmermann (LSZ) formula:

$$\langle p_1,\ldots,p_n|S|k_1,\ldots,k_m\rangle = \prod_{i=1}^n \sqrt{Z}\int d^4x_i e^{ip_i \cdot x_i}(i\square_i + m^2) \times$$
$$\times \prod_{j=1}^m \sqrt{Z}\int d^4y_j e^{-ik_j \cdot y_j}(i\square_j + m^2) \langle 0|T\{\phi(x_1)\cdots\phi(x_n)\phi(y_1)\cdots\phi(y_m)\}|0\rangle$$

This incredibly important formula says:

S-matrix elements = amputated, on-shell Green functions
"Amputated" = remove external propagators (apply Klein-Gordon operator)
"On-shell" = external momenta satisfy p² = m²
Z is the field strength renormalization

This is how we actually compute scattering amplitudes: calculate Green functions using Feynman diagrams, then apply LSZ!

🎯 Key Takeaways

Interacting Lagrangian: ℒ = ℒ₀ + ℒ_int (free + interaction)
Interaction picture: States evolve by H_int, operators by H₀
S-matrix: S = U(+∞, -∞) connects asymptotic states
Dyson series: Perturbative expansion in powers of coupling
Each term = sum of Feynman diagrams at that order
LSZ formula: Connects S-matrix to time-ordered correlation functions
Next: Path integrals provide systematic way to compute correlation functions!

Quantum Field Theory Course

Interacting Theories & S-Matrix

🔗Course Connections

📚Prerequisites

Video Lecture

1.1 Why Do We Need Interactions?