Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part IV, Chapter 1

Interaction Picture & S-Matrix Theory

Systematic framework for computing scattering amplitudes

πŸ”—Course Connections

πŸ“šPrerequisites

QFT Part III
S-Matrix Introduction
Basic S-matrix formalism
β†’
QFT Part III
Feynman Diagrams
Diagrammatic techniques
β†’

1.1 Review: Three Pictures of Quantum Mechanics

In quantum mechanics, we have three equivalent formulations:

SchrΓΆdinger

States evolve, operators are time-independent

$i\frac{d|\psi\rangle}{dt} = H|\psi\rangle$

Heisenberg

States are fixed, operators evolve

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

Interaction

Both states and operators evolve

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

For interacting QFT, the interaction picture is most useful because it isolates the effect of interactions!

πŸ’‘Why the Interaction Picture?

We know how to solve free field theory exactly (Part II). The interaction picture lets us:

  1. Use free field operators (we understand these!)
  2. Put all the complexity into the state evolution
  3. Treat interactions perturbatively

It's like using a familiar coordinate system (free fields) and adding corrections for the complicated stuff (interactions).

1.2 Interaction Picture Definition

Split the Hamiltonian:

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

where Hβ‚€ is the free Hamiltonian (we can solve exactly) and Hint contains interactions.

States in the interaction picture:

$$\boxed{|\psi_I(t)\rangle = e^{iH_0 t}|\psi_S(t)\rangle}$$

Operators in the interaction picture:

$$\boxed{O_I(t) = e^{iH_0 t}O_S e^{-iH_0 t}}$$

The operators OI(t) evolve like free field operators! This is the key advantage.

Equation of Motion

States evolve according to:

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

where Hint,I(t) = eiHβ‚€tHinte-iHβ‚€t.

1.3 Time Evolution Operator

The formal solution is:

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

where U(t,tβ‚€) satisfies:

$$i\frac{d}{dt}U(t,t_0) = H_{\text{int},I}(t)U(t,t_0), \quad U(t_0,t_0) = \mathbb{1}$$

Integrating this differential equation gives the Dyson series:

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

where T is the time-ordering operator. Expanding the exponential:

$$U(t,t_0) = 1 + \sum_{n=1}^\infty \frac{(-i)^n}{n!}\int dt_1 \cdots dt_n \, T[H_{\text{int}}(t_1)\cdots H_{\text{int}}(t_n)]$$

Each term is an nth-order correction in the interaction!

1.4 The S-Matrix in Detail

The scattering matrix (S-matrix) evolves from t = -∞ to t = +∞:

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

Matrix elements give scattering amplitudes:

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

This is the probability amplitude for initial state |i⟩ β†’ final state |f⟩.

Probability and Cross Sections

The transition probability is:

$$P_{i\to f} = |S_{fi}|^2$$

For scattering, we extract the scattering amplitude Mfi:

$$S_{fi} = \delta_{fi} + i(2\pi)^4\delta^4(p_f - p_i)\mathcal{M}_{fi}$$

The Ξ΄fi term represents "no scattering", while Mfi is the actual scattering amplitude.

🎯 Key Takeaways

  • Interaction picture: Operators evolve by Hβ‚€, states by Hint
  • Time evolution: U(t,tβ‚€) given by Dyson series
  • S-matrix: S = U(+∞,-∞) connects asymptotic states
  • Perturbative expansion: Each order gives Feynman diagrams
  • Physical observables: Cross sections from |Mfi|Β²
  • Next: Computing cross sections from S-matrix elements!