# nLab quantum electrodynamics

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## Surveys, textbooks and lecture notes

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• Axiomatizations

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• -theorem

• Tools

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• Structural phenomena

• Types of quantum field thories

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• examples

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# Contents

## Idea

Quantum electrodynamics (“QED”) is the perturbative quantum field theory of the electromagnetic field coupled to a Dirac field via the electron-photon interaction: it describes the quantum theory of photons and electrons.

## Definition

The Lagrangian density defining QED is the sum of

1. the free field contribution of the electromagnetic field;

2. the free field contribution of a Dirac field;

3. the electron-photon interaction term with coupling constant given by the fine structure constant.

The underlying free field theory admits a quantization via the corresponding Wick algebra. QED is, by definition, the perturbative quantum field theory obtained from this by finding a perturbative S-matrix for the electron-photon interaction.

As a result, scattering amplitudes for electron-photon interactions in QED are then expressed in terms of Feynman amplitudes:

###### Example

( in – example of )

In , are labeled that encode , called (this prop.) which in turn contribute to for physical processes – :

The are the summands in the -expansion of the

$\mathcal{S} \left( S_{int} \right) = \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} \frac{1}{(i \hbar)^k} T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots , S_{int}}} )$

of a given $L_{int}$.

The Feynman amplitudes are the summands in an expansion of the $T(\cdots)$ of the with itself, which, away from coincident vertices, is given by the of the $\Delta_F$ (this prop.), via the contraction

$T(S_{int}, S_{int}) \;=\; prod \circ \exp \left( \hbar \int \Delta_{F}^{a b}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}(y)} \right) ( S_{int} \otimes S_{int} ) \,.$

Each in a corresponds to a factor of a in $T( \underset{k \, \text{factors}}{\underbrace{S_{int} \cdots S_{int}}} )$, being a ; and each corresponds to a factor of the at $x_i$.

For example in involves (via this example):

1. the modelling the , with called the (this def.), here to be denoted

$\Delta \phantom{AAAA} \text{electron propagator}$
2. the modelling the , with called the (this prop.), here to be denoted

$G \phantom{AAAA} \text{photon propagator}$
3. the

$L_{int} \;=\; \underset{ \text{interaction} }{ \underbrace{ i g (\gamma^\mu)^\alpha{}_\beta } } \, \underset{ { \text{incoming} \atop \text{electron} } \atop \text{field} }{\underbrace{\overline{\psi_\alpha}}} \; \underset{ { \, \atop \text{photon} } \atop \text{field} }{\underbrace{a_\mu}} \; \underset{ {\text{outgoing} \atop \text{electron} } \atop \text{field} }{\underbrace{\psi^\beta}}$

The for the alone is

where the solid lines correspond to the , and the wiggly line to the . The corresponding is (written in -notation)

$\underset{ \text{loop order} }{ \underbrace{ \hbar^{3/2-1} } } \underset{ \text{electron-photon} \atop \text{interaction} }{ \underbrace{ i g (\gamma^\mu)^\alpha{}_\beta } } \,. \, \underset{ {\text{incoming} \atop \text{electron}} \atop \text{propagator} }{ \underbrace{ \overline{\Delta(-,x)}_{-, \alpha} } } \underset{ { \, \atop \text{photon} } \atop \text{propagator} }{ \underbrace{ G(x,-)_{\mu,-} } } \underset{ { \text{outgoing} \atop \text{electron} } \atop \text{propagator} }{ \underbrace{ \Delta(x,-)^{\beta, -} } }$

Hence a typical Feynman diagram in the induced by this looks as follows:

where on the bottom the corresponding is shown; now notationally suppressing the contraction of the internal indices and all prefactors.

For instance the two solid between the $x_2$ and $x_3$ correspond to the two factors of $\Delta(x_2,x_2)$:

This way each sub-graph encodes its corresponding subset of factors in the :

graphics grabbed from Brouder 10

A priori this is defined away from coincident vertices: $x_i \neq x_j$. The definition at coincident vertices $x_i = x_j$ requires a choice of to the locus. This choice is the of the .

## References

### Original articles

The term “quantum electrodynamics” is due to

• Paul Dirac, The quantum theory of emission and absorption of radiation, Proc. Roy. Soc. London A 114, 243, 1927 (spire, pdf)

This left some issues with the quantization of the radiation field. These were addressed, and the causal propagator for the electromagnetic field was first found in

A comprehensive, albeit informal, theory of QED and of perturbative quantum field theory in general was eventually developed by

For more on the history see

and (Scharf 95, section 0.0).

Discussion of vacuum stability in QED includes

The rigorous construction of perturbative QED in causal perturbation theory is worked out in

and refined to perturbative AQFT in

and generalized to possibly non-abelian Yang-Mills theory in

The weak adiabatic limit of QED was established in

• P. Blanchard and R. Seneor, Green’s functions for theories with massless particles (in perturbation theory), Ann. Inst. H. Poincaré Sec. A 23 (2), 147–209 (1975) (Numdam)

• Paweł Duch, Massless fields and adiabatic limit in quantum field theory (arXiv:1709.09907)

The local net of algebras of observables and hence the algebraic adiabatic limit was worked out in

Further discussion of the adiabatic limit and infrared divergences includes

### Review

Mathematically rigorous discussion in causal perturbation theory/perturbative AQFT is in

### Phenomenology

Comparison to experiment is reviewed in

• Tests of QED (pdf)

Specifically via the Lamb shift:

• Savely G. Karshenboim, D. I. Mendeleev, The Lamb Shift of Hydrogen and Low-Energy Tests of QED (arXiv:hep-ph/9411356)

Last revised on January 24, 2018 at 14:17:27. See the history of this page for a list of all contributions to it.