# Quantum Electro-Dynamics

As has been outlined, quantum field theory can be described as a
composite of theories that attempts to explain the particle model in
terms of quantum dynamics. In this respect quantum electrodynamics is
a description of the most fundamental elements of the particle interaction
model, i.e. electrons and photons, and is therefore an appropriate point
to start. As a generalization, QED is said to describe the dynamics
of electrically charged particles, e.g. electrons, interacting by
means of an exchange of photons. However, mathematically, QED is described
in terms of the complexity of a perturbation theory within an electromagnetic
quantum vacuum. In an attempt to address this inherent complexity,
Richard Feynman, one of the key contributors to the development
of QED, gave *a series of lectures* on this subject, which he hoped could
be understood by a wider audience. These lectures were transcribed and
published in 1985 and entitled* ‘The strange theory of light
and matter’. *In part, this opening section of discussions
will attempt to review some of the examples outlined in this series
of lecture, which start with the reassuring premise that the complexity
of QED can be described in terms of just three simple actions.

- Photon go from one point in spacetime to another.
- Electrons go from one point in spacetime to another.
- Electrons emit or absorb a photon at a given point in spacetime.

These actions are then represented in a visual form using a
notation that will eventually be described in terms of
*Feynman diagrams* governed
naturally enough by *Feynman rules*,
but which
we shall simply introduce at this point as follows:

Without going into the detail, it is possibly worth highlighting from the outset that these diagrams should not necessarily be interpreted as existing in physical spacetime. Equally, while the photon is represented, by convention, as a wavy line, it is not necessarily trying to promote a wave-like interpretation, in fact, Feynman might be said to be biased towards a particle-like description. Finally, in many ways, QED does not necessarily explain how these event-interactions happen, but rather constrains its goal to calculating the probability of an given event-interaction. So, by using the basic elements defined in the diagram above, we might represent an electron in a given point in spacetime, e.g. [A], and a photon in another point in spacetime, e.g [B], and consider how this system might evolve:

One possibility is that the electron will move from [A] to [C], while
the photon moves from [B] to [D]. If we can calculate the probabilities
of each of these subprocesses, i.e. E(AC) and P(BD), then theory suggests
the probability of E(AC) plus P(BD) happening can be calculated as a
product of the individual results. However, the diagram above also
suggests that there are other possible permutations, as such, the probability
of more complex processes has to be calculated as a *‘superposition’*
of all possible outcomes. In practice, it might appear that there are
an infinite number of possibilities, which might suggest that this
process could turn into an impossible task. However, perturbation theory linked
to the idea of a *‘coupling constant’* allows this complexity
to
be reduced in scope depending on the accuracy required.

*But how is this description of probability reconciled with earlier
descriptions? *

As has already been outlined, quantum mechanics embodies a number
of ideas based on probability, which do not align with *‘real
numbers’* but rather ‘*complex numbers’*, which are then referred
to as ‘*probability amplitudes’ *. However, in the following review
of Feynman’s lectures on QED, the mathematics of complex numbers is
side-stepped using the concept of an ‘*arrow*’ that has magnitude
and direction, i.e. it is essentially a vector, where the square of
the arrow length corresponds to the probability amplitude, which
in-turn reflects the
probability of an event-interaction. As indicated, QED does not really
offer up any explanation of these arrows as a physical description of
reality, only that they appear to give the correct result. So, for a
given process, if two probability amplitudes, e.g. [v] and [w], are involved,
the probability of the process is defined by either:

[1] .

We might initially try to quantify the rules regarding adding or multiplying the arrows [v,w] in the same way we might add or multiply probability amplitudes, which are complex numbers in a complex vector space, e.g.

Addition and multiplication are known operations in the theory of
complex numbers. While the sum can be described in terms of the parallelogram
method illustrated, the same result can be obtained by simply adding
the start of one ‘*arrow*’ to the tail of the previous ‘*arrow*’.
The product of two arrows is an arrow whose length is the product of
the two lengths, while the direction of the product is found by adding
the angles associated with each. The relevance of this description will
be expanded in the context of Feynman’s examples. It should also be noted
that the fact that both photons and electrons can be polarized is not
really taken into account in these examples. In practice, the probability
P(AB) will actually consists of 16 complex numbers or probability amplitude
arrows, which also require some further consideration of a quantity
that Feynman refers to as [j], which is linked to the concept of a coupling
constant. However, this introduction is hopefully enough for us to proceed
to ‘*Feynman’s model of QED’*