A look at Richard Feynman’s QED Lectures: Part 3

This is part 3 of the four part summary of Richard Feynman’s lectures on Quantum Electrodynamics. This lecture focuses on describing the transmission and reflection of photons, as well as providing an introduction to his famous Feynman Diagrams which describe how subatomic particles (e.g. electrons, protons, neutrons) interact.

This lecture also includes a basic introduction to his famous Feynman diagrams and the underlying principles to understand them.

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Lecture 3: Electrons and their interactions

In the last lecture, Feynman introduced us to calculating the probability of compound events, that is events with multiple steps. Following on from that, Feynman presents us with the following rules for our calculations:

Rules of Composition

These rules are used for calculating probabilities of events with multiple steps (under ordinary circumstances).

1. If something can happen in alternative ways, we add the probabilities for each of the different ways.
2. If the event occurs as a succession of steps, then the probabilities are multiplied for each of the steps.

That ends the probability of compound events (for now). Next, Feynman moves onto an explanation of something much more complicated- the strange behaviour of light.

The behaviour of light

Light is strange, Feynman tells us, and to demonstrate this, he uses the following set-up is used:

A source, S, emits weak light of one colour (monochromatic light)- one photon at a time (very weak). Directly opposite is a detector, D. Between S and D is a screen with two holes several millimetres away from each other: hole A and hole B. Hole A is in line with S and D, while B is slightly upwards. So how many photons travels through hole A and how many photons travel through hole B?

Close off hole B, count the number of clicks detector D makes to give the number of photons coming through hole A, and vice versa. For simplicity, Feynman says for every 100 photons that are emitted from S, 1 goes through A and 1 goes through B.

But something odd happens if both holes are open. Interference happens. This means that the detector clicks from 0 to 4% depending on the separation between hole A and hole B. At standard separation the percentage of photons going through A and B is 1% + 1% = 2%. However, when the separation between A and B is changed, the percentage can go between 0 and 4% (maximum).

So, in a strange way, opening a second hole doesn’t necessarily increase the amount of light reaching D. But wait, there’s more.

Say we wanted to leave both A and B open and detect whether a photon went through either A or B to get to D. In his example, Feynman places special detectors (that detects whether a photon goes through it without stopping the photons movement like a regular detector would) in hole A and B.

Theoretically, this should be able to tell us which hole the photon goes through. But as soon as the special detectors are used in the experiment, the percentage becomes constant at 2%. There is no more interference so the percentage doesn’t fluctuate.

Graph (a) shows the percentage of light reaching D when there there are no special detectors, and graph (b) shows the percentage of light reaching D when the special detectors are in place.

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So, to calculate probabilities we split events up into several steps. But, Feynman asks, how far can we split these steps up? What are the basic steps that can be split no more? To better understand basic steps, you can think about driving a car. Steps to get to the shops might be to drive down the street, turn left, drive into the car park and stop the car. These steps can be broken down further: turning left could be split into slowing down, waiting for the lights to change, and turning the wheel. And these steps could be broken down into simpler steps, and so on until we reach steps that cannot be broken down.

So what are the basic actions needed to produce all of the phenomena associated with light and electrons? According to Feynman, there are three:

• Action 1: A photon goes from place to place.
• Action 2: An electron goes from place to place.
• Action 3: An electron emits or absorbs a photon.

There are only two “actors” as Feynman puts it: photons and electrons. These are two of the biggest elements of Feynman diagrams, the very basics of which are shown below. So now let’s put those “actors” on a stage.

The use of space-time diagrams to represent movement of particles and photons

The stage on which photons and electrons perform is space and time, and can be represented by a set of axis, with time on the y-axis and space on the x-axis. To begin with a simple example, imagine a static baseball. On the space-time diagram the baseball is represented with a column because as time progresses its position in space (X0) does not change.

The same baseball, now moving to hit a stationary wall, can be represented with a bent line. So to summarise, in space-time diagrams static objects are vertical lines and moving objects are everything else (broadly speaking).

Instead of a baseball, lets now look at a photon, represented using a squiggly line (technical term) from point A to point B. This is the first basic action: a photon goes from point to point.

Action 2, an electron going from point to point, and Action 3, an electron emitting or absorbing a photon (in this case emitting), are shown below. Electrons are represented using a straight line and (from above) photons are represented using a squiggly line. Another point to note is what Feynman calls a “coupling”- an electron either emitting or absorbing a photon (this comes from the fact that in physics particles that interact with each other are said to be coupled).

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The probability of electrons moving through space-time

Lets calculate the probability electrons get from points 1 and 2 to 3 and 4. There are many different ways to go about it, some more complex than the rest.

Above, are the two simplest routes because they are direct and because no photons have been emitted or absorbed.

Moving onto two more possible routes, we seen there has been a photon emitted from point 5 and absorbed at 6 in each case. Since the electrons came from points 1 and 2 and ended up at points 3 and 4, these routes are indistinguishable from the simpler routes above.

So far everything has been standard. Confusing, but logical. But what if a photon were to travel backwards in time? This sounds like science-fiction, but this is what happens in example (c) below. Point 5 (where the photon in emitted) is higher up than point 6 (where the photon is absorbed). Therefore, the photon was emitted later in time than it was absorbed.

Example (b) shows a photon that is emitted at at the same time as it is absorbed (also odd to think about), while example (a) is the standard case we looked at above with emission and absorption happening “in order”.

While the idea of a photon travelling backwards in time seems odd, to Nature all these routes are the same, so we say that a photon is simple “exchanged” without worrying too deeply into it (at least not in this lecture).

And it’s not just photons that can move backwards in time. We also get backwards-moving electrons. These backwards-moving electrons appear the same as normal electrons, with the exception that they are attracted to normal electrons due to its positive charge- for this reason it is called a positron and is commonly known as the antiparticle (a particle with the same mass but opposite charge) of the electron.

In fact, every particle in nature can move backwards, therefore every particle has a corresponding antiparticle. Even photons sort of do- they are their own antiparticle.

Electrons in atoms

In this lecture, Feynman tells us to approximate the behaviour of a nucleus as a particle that can move from one point in space to another. Since the nucleus is so heavy when compared with an electron (thousands of times heavier), due to its slow movement in the examples coming up it is shown as staying in the same place through time.

Here is the representation of a hydrogen atom. A hydrogen atom is made up one proton and one electron, the proton being the vertical line to the left and the electron being the wavy line to the right.

Photon exchanges (the squiggly lines) keep the electron within a certain distance of the nucleus. This is an atom with just one proton- the more protons in the nucleus of an atom the more complicated these diagrams would quickly become!

Revisting the Partial Reflection of light

When talking about the partial reflection of light in lecture 2, Feynman mentioned the reflection off a front surface and off a back surface when calculating the probabilities. In this lecture, he not only includes a front surface and a back surface in the calculations, Feynman splits the glass up into 6 separate layers. Which, as you can imagine, complicates things a great deal.

So to start off, lets look at our light source. The source of light used in this experiment is a monochromatic (one colour) source, which greatly simplifies things as a monochromatic source emits photons at regular, calculable intervals. This means we are able to calculate the direction the probability amplitude arrow is facing at a certain time relatively easily.

If we look at this on a space-time graph we see while the photons are emitted at the same angle, the directions of the arrows representing each of the photons are angled differently as time progresses. The turning of the arrows with time matches fairly well with the stopwatch analogy Feynman provided us with in the first lecture. Something to note is the rate of turning- it changes depending on the colour of light. The direction of the probability amplitude arrows for blue light turns nearly twice as quickly as that of red light.

Once a photon has been emitted the arrow no longer turns and the direction the arrow points becomes fixed.

Lets now look at the glass. Feynman split it into 6 very thin sections. Below I’ve added his diagram in an attempt to aid understand of a very complicated idea.

In diagram (b) a stationary of light source, S, emits 6 photons as time progresses. The first emitted photon (labelled as T60 hits the back layer of glass, labelled X6. The second photon hits X5 and so on until the last emitted photon (labelled T1) hits the front layer, X1.

Now, to calculate the probability we need to calculate the probability for each photon then, once we have out probability arrows, we can combine them to find our resultant arrow.

We can split the path each photon takes into 4 steps:

1. The photon leaves the source, S.
2. The photon travels to one of the points on the glass (which point depends on which photon we are looking at).
3. The photon is scattered by an electron at that point (shown by the wide thick line on the diagram. The scattering of light is tricky to understand so for this lets just say that the old photon goes in and a new photon comes out).
4. The new photon travels to the detector, A.

Going back to the end of lecture 2, Feynman introduced some rules for the reflection and transmission of light by glass:

1. Reflection from air back to air (reflection off a surface) involves a shrink of 0.2 and half turn.
2. Reflection from glass back to glass (off a back surface- reflection inside the glass material) involves a shrink of 0.2 but no turning.
3. Transmission from air to glass or from air to glass involves a shrink to 0.98 and no turning.

So we must now go through the whole process of shrinking and turning the arrows for each step. For example, while the arrows for step 2 and 4 stay the same, the arrow for step 3 is shrunk and turned. So after all this shrinking and turning has taken place, we can now multiply the 4 step arrows to get the probability amplitude arrow for just one of the photons. (I haven’t included any values here as I’d rather focus more on the theory.) We must then repeat this process for all 6 of the photons.

Now we have our six probability amplitude arrows, we can then combine them.

Our resultant arrow has sides of length 0.2 and 0.2 so the length of the resultant arrow is around 0.28. The probability of an event is the length of the resultant arrow squared so 0.28 * 0.28 = 0.08 or 8% exactly. If you think back to lecture 1, the percentage of light reaching the detector varied from 0% – 16% depending on the thickness of glass, with 8% as our average value. So the maths works out.

Needless to say, it’s very complicated.

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Thank you for reading! The next (and final) lecture is summarises some of the issues with QED and how it ties into other areas of physics such as quantum chromodynamics (QCD), the weak interaction and gravity.

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Credit to QED: The Strange Theory of Light and Matter for use of its diagrams.