Electron Interactions

In the previous introduction to Feynman’s QED model covering the basics of photon interaction, it was suggested that this model could not predict whether a given photon would be reflected or transmitted by a glass block, only that we might calculate the probability. In this context, the following probabilistic rules were defined:

  1. If an interaction can happen in alternative ways, we add the probabilities associated with each possible way.
  2. If an interaction can be described as a succession of steps, then we multiply the probability of each step.

Again, we are not being too rigorous in any of these definitions as the goal is simply to provide some visual counterpoint to the mathematical complexity that is really required to describe any of these processes. Therefore, this discussion of basic electron-photon interactions will also continue with the premise of Feynman’s model based on rotating arrows, which allows the overall probability to be defined as the square of the final arrow length, i.e. probability amplitude.

[1]      1

From our previous discussion of the ‘particle modelwe might described the ‘particle’ components in terms of fermions, rather than bosons-like photons that act as the mediator of an interaction. Under the heading of fermions, we might focus on the fact that most stable atoms consist of just 3 particles, i.e. protons, neutrons and electrons, while noting that protons and neutrons are not fundamental particles in the sense that they are ultimately described as an arrangement of up/down quarks. However, it has already been highlighted that the interaction of quarks is described by another aspect of quantum field theory called quantum chromodynamics, not quantum electrodynamics, which is the primary focus of Feynman’s model. While, we shall only use Feynman’s model to provide a general description of the interaction of electrons mediated by photons, the full description of QED covers all the leptons under the heading of fermions. As such, the interaction of this model reduces to just 3 actions:

  1. Photons can go from one point in spacetime to another.
  2. Electrons can go from one point in spacetime to another.
  3. Electrons can emit or absorb a photon at a given point in spacetime.

 
Action-1: Photon can go from one point in spacetime to another
While previous discussions have outlined some of the basic concepts, we now need to consider the wider aspects of the 3 actions starting with the idea of photons moving in spacetime. From the perspective of a space-time diagram, we would expect the photon to be shown propagating with a velocity [c], such that it might be represented as a wavy line at an angle of 45o, as shown below:

1

We might also described the dynamics of this photon in terms of its probability amplitude to move from [x1,t1] to [x2,t2], which we might denote P(AB); although in-line with previous descriptions of optical effects, we would have to consider the sum of all possible paths between [A] and [B]. However, we might also note that the QED model was not based on an assumption that light travels in a straight line, only that that the final arrow, i.e. probability amplitude, supports this observation as a net result.

So might QED consider pathways implying slower and faster values of [c]?

Theoretically, it would appear so, although like non-straight paths, these time variant paths would imply a non-standard value of [c] that have low probability amplitudes in comparison to the central pathway conforming to [c]. Equally, the slower and faster pathways effectively cancel each other out over any measurable macroscopic distance.


Action-2: Electrons can go from one point in spacetime to another
The QED model basically describes a particle, such as the electron, moving between two points in spacetime as a line between [A] to [B], although the velocity of the particle can now range between v(0..c), such that the angle of the line might now range from the vertical to 45o.

2

It seems likely that the movement of a particle, like an electron, might also be described in terms of some probability amplitude, which we might label E(AB). However, while we might reasonably assume a straight-line path in the absence of a force, as illustrated above, we might also suspect that other less obvious pathways might also have to be taken into consideration. In the diagram below, we see two permutations of a 2-stage path [ACB, AC’B] as well as a 3-stage path [ADEB]. Conceptually, within this model, the electron can change its path between [AB] in an infinite number of ways through an infinite number of stages.

3

The implications of having an infinite number of staged pathways is only indirectly addressed within this QED model, but is linked to a mathematical process called perturbation theory:

Perturbation theory can be described as an approximation method of a more complicated quantum system. The idea is to start with a simple system, for which a mathematical solution is known, to which is added an additional ‘perturbing’ Hamiltonian that represents a weak disturbance to the system. If this disturbance is not too large, the physical quantities associated with the perturbed system can be extrapolated based on the idea of continuity, i.e. as small 'corrections' to the simpler system. These corrections, being 'small' compared to the size of the quantities themselves can be calculated using approximate methods, such as an asymptotic series. As such, more complicated system can therefore be studied based on knowledge of the simpler one.

The formula for calculating E(AB) is actually very complicated, so the following ‘explanation’ is little more than a characterization of the process. As implied, the value of E(AB) is the sum of every conceivable way the electron could go from [A] to [B], which parallels the process previously outlined for photons. However, in the case of the electron description involving multiple steps, there is an additional probability factor applied for each step, i.e. [n2].

[2]     2 

Initially, we shall simply state that the factor [n] can be less than one, such that the exponentials, i.e. n2, n4,  might quickly converge to zero for pathways involving multiple steps. While an explanation of [n] is of key importance, we shall defer any further discussion until after outlining the basic  idea of an interaction between an electron and a photon.


Action-3: Electrons can emit or absorb a photon at a given point in spacetime
In the context of an atomic model, e.g. the Bohr model, an electron sits in a define energy orbital. However, in quantum mechanics, an electron can transition into another energy orbital through the emission or absorption of a photon. In QED, and especially Feynman diagrams, this interaction is described as a ‘junction’ or ‘coupling’ point, which the following diagram might suggest is taking place between an electron and a photon. However, this diagram is somewhat misleading, when applied to a free electron outside an atomic structure, because in normal circumstances a free-electron is not allowed to emit or absorb a photon based on the arguments of the conservation of energy – see discussion on ‘virtual particles. However, the diagram below does have an important role to play in Feynman diagrams, which will be outlined in more detail in subsequent discussions.

4

So, for now, we will simply accept the implication of the diagram above at face value, while noting that some important caveats need to be applied at a later stage, which we might also link to another type of ‘coupling constant. So let us continue the review of some of the examples raised within Feynman’s lectures, starting with the idea of 2 electrons at initial points (1) and (2), which we might perceive to end up at points (3) and (4). Of course, just how many permutations have to be considered is a key issue:

5

In the diagram on the left, we see a perfectly logical starting assumption, although we might realize that if all electrons are essentially identical, then the outcome on the right may be equally valid. As such, both diagrams are valid paths, which we need to resolve in terms of some probability amplitude. Therefore, to calculate the probability of electrons at points (1) and (2) ending up at points (3) and (4), we first determine the ‘arrow’ for (1) going to (3) and (2) going to (4) and, as suggested by the diagram, the ‘arrow’ for (1) going to (4) and (2) going to (3). Finally, we add the two arrows as an initial approximation of the final probability amplitude arrow.

But what other paths might have to be considered?

Based on earlier descriptions, it seems that almost every conceivable pathway has to be considered, which in the case of electron-photon interaction includes the possibilities as outlined below:

6

In the diagram above, we see the suggestion that as the electron proceeds from point (1)  to (3), a (virtual) photon is emitted at point (5) and then absorbed at point (6). From the perspective of the initial and final states, the diagram above might be said to be identical to the previous diagram discussed and therefore is representative of an alternative path with its own probability amplitude. While this may already appear to be getting impossibly complex, we might hope that the convergence suggested by perturbation theory might come to the rescue. However, before discussing how this might help, we need to consider yet another complication in terms of the next diagram:

7

It has been suggested that a photon can have probability amplitudes associated with a velocity being both slower and faster than [c]. Another non-intuitive possibility is suggested in option (c) above, reflecting the conceptual idea that the photon emitted at [5] may occur after it is absorbed at [6], i.e. photons can travel backwards in time. While we will not pursue this idea in any detail, at this stage, it is highlighted that there is an interpretation, known as the ‘Transaction Interpretation, which explores the implication of this idea. However, for now, we will simply assume that we are only dealing with a mathematical method for calculating probability. One other illustrative example from Feynman’s lecture involving an electron and photon interaction, which might also be described in terms of the scattering of light analogous to ‘Compton scattering’.

8

If described in terms of a particle collision, involving an electron and photon, we might have an expectation of the electron and photon coming together and ricocheting of each other in different directions. However, while inset (a) might appear to be generally representative of this idea, what QED appears to be describing in inset (a) is the inference that the electron ‘absorbs’ the photon, continues for a little bit, then splits back into a electron and photon. In contrast, inset (b) suggests an alternative in which the electron first emits a photon before then absorbing another photon. Finally, inset (c)  suggests an even stranger sequence in which the electron first emits a photon, as per (b), but then travels backwards in time before absorbing another photon. However, an electron which appears to move backwards in time can be described in a different way, as an anti-particle of an electron, which has positive charge, i.e. a positron. Given that we are simply introducing some of the general ideas of QED, we will just say that every fermion particle has an anti-particle, which might be said to have a probability amplitude that travels backwards in time and when colliding with its counterpart, annihilates releasing energy in the form of a photon-pair. Clearly, there can be many more permutations, which now involve photons as the electron goes between two points that introduce another type of couplings, as originally suggested above, i.e.

 910

While returning to one part of the original example above, i.e. inset left, we might also speculate on even more complex pathways. If we return to the form of equation [1], we see that there was a coupling index [n] assigned to each stage of the pathways between source and destination. In essence, we see a similar situation in the diagrams above, although we might realise that any coupling involving an electron and photon might be defined in terms of a different set of physical attributes, such that we might label it [j] instead of [n]. Let us consider  the example on the left first:

[3]       3

In [3], we see the probability amplitude of the electron probability E(15) and E(53) coupled via [j] to the emission of a photon probability P(56), which is absorbed and affects the probability amplitude of the path E(26) and E(64). Of course, it is clear that the inset diagram on the right has more combinations, which we might quantify in the following way:

  • Every electron path, i.e. straight line, has a probability amplitude E(mn).
  • Every photon path, i.e. wavy line, has a probability amplitude P(mn).
  • Every coupling point has a [j], i.e. [5,6,7,8]

So in the second diagram, we have four (4) electron paths, two (2) photon paths and four (4) coupling points, which are all combined as per [2]. However, if we review all the previous diagrams showing the possible paths between [1,2] and [3,4], we started with a diagram with no photons, i.e. no [j] coupling points, then one with two [j], as shown in equation [2] and finally the diagram on the right above would have four [j]. Without really explaining the nature of [j], it has a value [j<0.1], such that [j2<0.01] and [j4<0.0001], such that we might recognise that while adding increasingly improbable paths is possible, they do not necessarily add much to the final probability of an observed outcome.

But what physical interpretation is given to the couplings [n] and [j]?

The coupling indices [n] and [j] will be touched upon in slightly more detail in a subsequent discussion, while the goal at this point is simply to suggest a physical basis of the interaction processes previously described.

  • Coupling Parameter [n]:
    In a way, the parameter [n] might be described as the rest mass of an ideal electron, which goes from [A] to [B] via the most direct path. However, the previous examples have indicated that this single path assumption is false, such that ‘real’ rest mass [me] of an electron, as measured experimentally always includes the corrections of the lower probability paths.

  • Coupling Parameter [j]:
    The second coupling parameter used by Feynman is denoted as [j] and is connected with the charge [e] of the electron. However, today, it is more commonly discussed in terms of the fine structure constant [α], where [j2=α] and [α] is proportional to [e2].

So, as implied, if electrons only went from point to point by the most direct route in spacetime, [n] and [me] would be the same for an electron. However, this appears not to be the case, as ‘real’ electrons are said to emit and absorb a photon, which suggests that its probability amplitude also depends on [j]. As such, the experimentally measured values of electron mass [me] and charge [e] differ from the coupling parameters [n] and [j] used in the calculations of the probability amplitude. Equally, when making these calculations, all possible coupling points must be taken into consideration, i.e. to a conceptual resolution where these points are effectively separated by zero distance. Of course, once the idea of zero is introduced into a mathematical equation, the risk is that it can yield an infinite value, which is actually what happens in QED. So there is an aspect associated with both [n] and [j] that leads to infinities in the calculations of the probability amplitude, which needs to be discussed in the wider context of ‘QED renormalisation.