nLab
electron-photon interaction

Contents

Context

Fields and quanta

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

In quantum electrodynamics the interaction between the Dirac field Ψ\Psi, whose quanta are electrons, and the electromagnetic field AA, whose quanta are photons, is encoded by the interaction Lagrangian density

L int=i(Γ μ) α βψ¯ αψ βa μdvol Σ \mathbf{L}_{int} \;=\; i (\Gamma^\mu)^\alpha{}_\beta \overline{\psi}_\alpha \psi^\beta a^\mu \, dvol_\Sigma

(with notation as as used at A first idea of quantum field theory, see this example).

For g swC cp (Σ)g_{sw} \in C^\infty_{cp}(\Sigma) a bump function on spacetime thought of as an adiabatically switched coupling constant, the corresponding interaction action functional is the local observable

S int iΣg sw(x)(Γ μ) α βΨ¯ α(x)Ψ β(x)A μ(x)dvol Σ(x) =iΣg sw(x)(Γ μ) α β:Ψ¯ α(x)Ψ β(x)A μ(x):dvol Σ(x), \begin{aligned} S_{int} & \coloneqq i \underset{\Sigma}{\int} g_{sw}(x) \, (\Gamma^\mu)^\alpha{}_\beta \, \overline{\mathbf{\Psi}}_\alpha(x) \cdot \mathbf{\Psi}^\beta(x) \cdot \mathbf{A}_\mu(x) \, dvol_\Sigma(x) \\ & = i \underset{\Sigma}{\int} g_{sw}(x) \, (\Gamma^\mu)^\alpha{}_\beta \, : \overline{\mathbf{\Psi}}_\alpha(x) \mathbf{\Psi}^\beta(x) \mathbf{A}_\mu(x) : \, dvol_\Sigma(x) \end{aligned} \,,

where in the first line we have the integral over a pointwise product (this def.) of three field observables (this def.), which in the second line we write equivalently as a normal ordered product, by the discusssion at Wick algebra (this def.).

(e.g. Scharf 95, (3.3.1))

The corresponding Feynman diagram is

The square of the coupling constant

α14πg 2 \alpha \coloneqq \tfrac{1}{4 \pi} g^2

is called the fine structure constant.

References

Discussion in the context of causal perturbation theory is in

Last revised on February 12, 2018 at 08:19:28. See the history of this page for a list of all contributions to it.