Feynman’s Model of QED

The following group of discussions will try to follow some of the examples laid out in a series of lectures by Richard Feynman that we shall consider in terms of basic ‘photon interactions’ and electron interactions’ . However, in this discussion, we shall start by considering some of the most basic assumptions underpinning the  classical model in terms of the probability of light quanta, i.e. photons. For example, one of the most fundamental concepts in classical physics, which describes the ‘nature of light is the idea of reflection from a surface, i.e. the angle of incident equals the angle of reflection. However, this classical description masks an issue that be might label as ‘partial reflection’ as illustrated in the diagram below

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The diagram above show the basics of an experiment to measure the partial reflection of light, where the source of the light is assumed to be a stream of individual photons. On measurement, for every 100 photons fired from the light source, 4 photons of light are reflected from the top surface of the glass block, as detected at [A], while the remaining 96 photons are detected at [B]. On this basis, we might quantify the probability of partial reflection to be 4%, i.e. 0.04, which based on previous discussion of the wave function would suggest some sort of probability amplitude [Ψ=0.2] as defined in [1]:

[1]      1

However, given what we now know about the sub-atomic structure of the glass block, it is very difficult to explain what determines whether an individual photon of light is reflected or passes into the glass. Initially, it was assumed that partial reflection resulted from the fact that the surface of the glass was essentially made up of space between the atoms, which let most of the light through.

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Without making any assumptions at this stage, the experiment simply suggests that we cannot predict whether a given photon will arrive at [A] or [B]; although we can quantify the probability. While the idea of probability is not unknown in classical physics, e.g. statistical mechanics of thermodynamics, there is a suggestion that the issue of cause and effect is not really understood at the quantum level. So with this somewhat unsettling thought in mind, let us now consider a slight modification of the experiment above.

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The goal of this revised experiment is to measure the partial reflection from both the top and bottom surface of the glass block. Based on the first experiment, we may have good reason to assume that the bottom surface would act like the top surface and reflect 4% of the 96% of photons passing through the top surface. Unfortunately, this assumption does not appear to be upheld in the experiment, where it seems that detector [A] can received between 0-16% of the source photons depending on the thickness of the glass, as shown below

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The results above suggest that the partial reflection oscillates between 0-16% as the glass thickness is increased, from zero, to some unspecified thickness, such that we are left with questions:

How does the back surface affect the probability at A?
What happens if we introduce more surfaces ?

Clearly, this result is problematic from the perspective of a particle model, where light is described in the form of a photon. Of course, given previous discussions of the ‘wave-particle duality’ in quantum theory, we might consider whether this has something to do with a wave-like attribute of the photon. However, it seems that Feynman did not really try to explain the results of the last experiments in terms of ‘cause and effect’, only its probability, which is now wrapped up in the theory of quantum electrodynamics. In fact, it is this probabilistic approach that is now used to explain all fundamental interaction between photons and electrons. However, while most descriptions nose-dive back into the mathematics at this point, Feynman did attempt to explain the basic ideas within QED using vector ‘arrows’, the length of which is defined by the ‘probability amplitude’ given in [1], but which might be illustrated as follows :

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1a
6 In practice, an experiment may be the
result of a series of probabilistic events,
which are added together to form
the final arrow, which reflects the
probability of the measured outcome
.

However, we have started to run ahead of Feynman’s explanation, for we have to first understand how the direction of each arrow is determined. In the context of the previous experiments describing partial reflection, Feynman describes each arrow in terms of a hand on a stopwatch that turns very rapidly. The watch is started at the source and stopped when the photon reaches either [A] or [B], such that the direction on the watch face defines the direction of the arrow.

Note: It would seem that Feynman often preferred to discuss QED in terms of the semantics of particle interactions without making any obvious reference to the wave-like nature of these particles. However, the following explanation of Feynman’s stopwatch is clearly predicated on the wave property of red light. 

Feynman only indirectly describes this process in terms of a figure given as 36,000 rotations per inch for red light, which we might elaborate a little. The rate at which the ‘stopwatch’ rotates is linked to the frequency of red light and the time taken to travel 1 inch.

[2]      2

Of course, the total number of rotations, and therefore the final orientation of the arrow, is determined by the total path length from the source [S] to either [A] or [B]. However, in order to explain the observe partial reflection from the top and bottom of the glass block, another  rule is introduced that is not really explained, or for that matter, logically obvious. First, we will introduce the rule and then outline the explanation before returning to the logic supporting this rule.

In order to compute the answer correctly, the direction of the arrow associated with a photon reflected off the top surface will be the reverse of that indicated by the ‘stopwatch’; while the direction associated with the bottom surface will be maintained.

Let us consider a conceptual experiment where the thickness of the glass block approaches zero. Therefore, in the diagrams below, we see examples of a photon arriving at [A] coming from top and bottom surface, conceptually separated by a distance approaching zero.

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As such, we might conclude that the total path length between source and [A] from the top or bottom surface must be approximately the same and therefore the direction of the arrow by the stopwatch analogy would be approximately the same. However, the diagrams reflect the additional rule above, where the direction from the top surface is reversed, such that the sum of these two arrows becomes increasingly small.

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So the conclusion of the diagram above suggests, quite logically, that the probability of partial reflection from a sheet of glass approaching zero thickness, will also approach zero. However, while the conclusion appears logical, the rule to reverse the direction of the arrows from the top and bottom surfaces appears far from obvious. Classically, it can be highlighted that when light goes from one medium of index [n1] to another [n2], the reflected light at that interface undergoes a phase change as follows:

[3]      3

Therefore, the rule is consistent with the transition from air to glass in the experiment, but does this really explain why this rule is applicable in the current quantum description. However, given that this is a well verified experiment, it is assumed that [3] is equally applicable to quantum case, where this phase shift presumably takes place within the concept of a single photon!

OK, but have we explained the variation of probability with thickness?

Let us consider the case when the glass is just thick enough to allow the bottom pathway to correspond to another half turn (180o). As suggested by the diagram below, both arrows now point in the same direction, such that the probability amplitude is the sum, i.e. 0.2+0.2=0.4, which by [1] gives an overall probability of 16%.

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On this basis, we might reasonably predict, and confirm by experiment, that as we increase the thickness of the glass, the top reflection will align to an arrow changing in direction, while the bottom arrow rotates with increasing thickness. This phase shift of the top arrow to the bottom arrow will rotate through 360o, such that the sum of the arrows oscillates between a minimum-maximum of 0-16%, as the following diagram attempts to illustrate:

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However, the diagram above also tries to highlight that period of this cyclic nature is a function of the frequency of the light source, which drives the rotation rate of the stopwatch used to define the direction of the arrows.

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So, at this point, we have replicated the essential arguments of Feynman’s introduction to quantum electrodynamics, which although clearly a gross simplification of the mathematical ‘reality’ of this subject, may provide some useful visual insights before applying this model to photons and electron interaction.