4 Quantum electrodynamics.
4.1 Electromagnetic interactions
The electromagnetic interactions follow the rules of Quantum ElectroDynamics or QED. In the
model of interactions we have proposed, a charge, for example, interacts by emitting and
absorbing virtual photons. We now examine the possibility of observing consequences of this
which are not predicted by standard quantum mechanics. Our model of an electron
interacting with an external electromagnetic field involves it in absorbing a virtual photon,
and thus changing its momentum. However, other internal interactions can occur. An
electron, of momentum p, may emit a virtual photon of momentum k, and hence continue
with a reduced momentum 𝑝 − 𝑘 until it reabsorbs the virtual photon. Similarly a photon of
momentum k may convert into a virtual electron positron pair, with the electron and positron
sharing the original momentum, until the virtual pair recombine and produce the photon
again. Indeed, more complicated cases may occur, involving combinations of emission of
virtual photons with virtual pair productions. Each coupling of a photon to a fermion line,
known as a vertex, involves a factor √𝛼 in the amplitude, where 𝛼 is the fine structure
constant defined as
𝑒2 1
𝛼= ≈
4 𝜋𝜖0 ℏ𝑐 137
To show the range of electromagnetic interactions we refer to Fig. 2.1 of [Perkins], which is
reproduced here for your convenience. The diagram in (a) represent the basic QED
interaction vertex and has a 𝑀𝑓𝑖 which is proportional to √𝛼. This interaction cannot
proceed on its own as it is not allowed by energy conservation (in the electron rest frame a
photon would be emitted and the electron retained!).
A combination of more than one of these vertexes is needed to describe a physical process.
Diagram (b) shows the QED interaction between two electrons via the exchange of a
“virtual” photon. The matrix element for each of the vertexes is proportional to √𝛼, hence
the matrix element is proportional to 𝛼. The virtual photon introduces a propagator factor
(discussed in the previous section) of 1⁄𝑞2. So the matrix element is proportional to
𝛼
𝑀𝑓𝑖 ∝ 2
𝑞
And the cross section is then proportional to
2 𝛼2
|𝑀𝑓𝑖 | = 4
𝑞
Which leads to the formula for the Rutherford scattering with a charge to the power of 4 at
the numerator and the fourth power of the momentum transfer at the denominator.
In the figure, the factor 𝛼 associated to each vertex, refer to the square of the matrix
element term, e.g. the factor that we get in the cross section formula.
Diagram (c ) shows the diagrams for the interaction of an electron and a positron. Two
different diagram, with a different time orientation, are needed to describe this process
with (slightly) different calculations needed.
Diagram (d) describes the phenomenon of bremsstrahlung, which is the emission of a
photon from an electron in interaction with a nuclear field. While diagram (e) shows how
the process 𝛾 → 𝑒 + 𝑒 −can take place in presence of a nuclear field. (Note that this process
will not be allowed in vacuum)
32
, Diagrams under (f) represent self-interaction terms. These are characteristic of a quantum
field theory and require a bit more discussion, which we will cover in the next section.
Fig. 4.1 Example Feynman diagrams in QED
33
4.1 Electromagnetic interactions
The electromagnetic interactions follow the rules of Quantum ElectroDynamics or QED. In the
model of interactions we have proposed, a charge, for example, interacts by emitting and
absorbing virtual photons. We now examine the possibility of observing consequences of this
which are not predicted by standard quantum mechanics. Our model of an electron
interacting with an external electromagnetic field involves it in absorbing a virtual photon,
and thus changing its momentum. However, other internal interactions can occur. An
electron, of momentum p, may emit a virtual photon of momentum k, and hence continue
with a reduced momentum 𝑝 − 𝑘 until it reabsorbs the virtual photon. Similarly a photon of
momentum k may convert into a virtual electron positron pair, with the electron and positron
sharing the original momentum, until the virtual pair recombine and produce the photon
again. Indeed, more complicated cases may occur, involving combinations of emission of
virtual photons with virtual pair productions. Each coupling of a photon to a fermion line,
known as a vertex, involves a factor √𝛼 in the amplitude, where 𝛼 is the fine structure
constant defined as
𝑒2 1
𝛼= ≈
4 𝜋𝜖0 ℏ𝑐 137
To show the range of electromagnetic interactions we refer to Fig. 2.1 of [Perkins], which is
reproduced here for your convenience. The diagram in (a) represent the basic QED
interaction vertex and has a 𝑀𝑓𝑖 which is proportional to √𝛼. This interaction cannot
proceed on its own as it is not allowed by energy conservation (in the electron rest frame a
photon would be emitted and the electron retained!).
A combination of more than one of these vertexes is needed to describe a physical process.
Diagram (b) shows the QED interaction between two electrons via the exchange of a
“virtual” photon. The matrix element for each of the vertexes is proportional to √𝛼, hence
the matrix element is proportional to 𝛼. The virtual photon introduces a propagator factor
(discussed in the previous section) of 1⁄𝑞2. So the matrix element is proportional to
𝛼
𝑀𝑓𝑖 ∝ 2
𝑞
And the cross section is then proportional to
2 𝛼2
|𝑀𝑓𝑖 | = 4
𝑞
Which leads to the formula for the Rutherford scattering with a charge to the power of 4 at
the numerator and the fourth power of the momentum transfer at the denominator.
In the figure, the factor 𝛼 associated to each vertex, refer to the square of the matrix
element term, e.g. the factor that we get in the cross section formula.
Diagram (c ) shows the diagrams for the interaction of an electron and a positron. Two
different diagram, with a different time orientation, are needed to describe this process
with (slightly) different calculations needed.
Diagram (d) describes the phenomenon of bremsstrahlung, which is the emission of a
photon from an electron in interaction with a nuclear field. While diagram (e) shows how
the process 𝛾 → 𝑒 + 𝑒 −can take place in presence of a nuclear field. (Note that this process
will not be allowed in vacuum)
32
, Diagrams under (f) represent self-interaction terms. These are characteristic of a quantum
field theory and require a bit more discussion, which we will cover in the next section.
Fig. 4.1 Example Feynman diagrams in QED
33
