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 entitledThe strange theory of light and matter’. In part, this opening section of discussion 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 a Feynman diagrams governed naturally enough by Feynman rules, but which we shall simply introduce at this point as follows:

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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 baised towards a particle-like description. Finally, in many ways, QED does not necessarily explain how these event-interactions happen, but rather  constrains its goals 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:

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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 individiual 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 probabilityamplitudes, e.g. [v] and [w], are involved, the probability of the process is defined by either:

[1]      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.

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Addition and multiplication are known operations in the theory of complex numbers. While the sum can be described in term 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