## Feynman Diagrams are Maths not Physics

A Feynman Diagram such as the one shown above is a succinct way of summarising a mathematical calculation. However, even though it looks like a ‘cartoon’ representation of the physics, it does not describe the physical process.

After giving a talk to A level physics teachers the other weekend, I stayed around to listen to a presentation about radioactivity: it really is a pleasure listening to other people teaching!

From my disinterested perch at the back of the class I was surprised to find that the process of beta decay was described by means of a Feynman Diagram.

I was then even more surprised when teachers asked detailed questions about which type of ‘vector boson’ was involved. I began to wonder how this could make any sense to ‘A’ level students. Do they really know what a ‘boson’ is? Or a ‘vector boson’?

And an e-mail today from an AS level student asking me about these diagrams crystallised my misgivings: I realised that teaching and examining this kind of thing as physics is potentially quite misleading.

At the heart of the matter is the fact that Feynman diagrams represent an ingenious way of describing a calculation: they do not describe the physics underlying the process. You can read an excellent article about their history here.

Let me explain:

• The diagram at the head of the page describes the way that the electrical interaction of two electrons – their mutual repulsion – is calculated in an advanced theory called quantum electrodynamics (QED). The lines and vertices each have a precise mathematical interpretation.
• QED describes the repulsion between the electrons in terms of the exchange of an infinite number of ‘virtual photons’. The diagram above summarises the way the exchange of a single ‘virtual photon’ – the wiggly line in the middle – is calculated.

Now QED is an astounding theory. It has been checked thoroughly and there is an astonishing correspondence between the results of its calculations and physical reality. In other words it is in some sense ‘correct’.

But nonetheless there are two problems when using these diagrams in schools.

• Firstly I underlined the word ‘infinite’ in the bullet point above because when you see that word you can be sure you are in the realm of maths, not physics. This is because there are no infinite quantities in physics.
• The second problem is that it involves the concept of a ‘virtual photon’. Despite 35 years of exposure to this concept – I haven’t a clue what it means physically. I suspect strongly that its role is calculational rather than physical.[I searched for a comprehensible ‘link’ but there are none! Try this as a typical example.]

Some people might argue that because ‘virtual photons’ are part of the way QED works, then the accuracy of QED is in itself evidence that virtual photons ‘exist’. To these people I have a one word rebuttal: ‘Epicycles‘: just because a calculational technique improves predictions does not mean that there is a physical counterpart to the ‘calculational entities’.

Now why does any of this matter?

It matters because this stuff is being taught for all the wrong reasons. It is being taught , I guess, because it looks like  a cool cartoon, and also requires no numerical skills. We are asking students to simply imitate the marks made on blackboards by other physicists.This is bad.

Further, its inclusion has caused the exclusion of a really interesting feature of beta decay that students could appreciate directly.

Instead of electrons being emitted in the same way as alpha particles are – with a single characteristic energy and momentum, electrons emitted in beta decays have a wide range of energies, from a maximum characteristic value, all the way down to zero.

In the early days of nuclear physics this spectrum was puzzling because it seemed as though beta decay did not conserve energy or momentum. ‘A’ level students can readily appreciate both these conservation laws, and the potential significance of them being broken.

And the resolution of the apparent breakdown of the conservation laws was that there was a third particle involved – a particle with almost no mass called a neutrino. And the existence of this particle – not to be directly detected for 25 years after its existence was hypothesised – was based on the law of conservation of momentum.

So these diagrams look like physics, but they are not. And IMHO they don’t belong in an ‘A’ level physics syllabus.

### 7 Responses to “Feynman Diagrams are Maths not Physics”

1. T Roberts Says:

I could not agree more. I really struggled when I covered this at A-level, until I realised it was another part of the syllabus which required rote learning – not understanding.

I think the teaching of particle physics remains a problem at undergraduate level, where ‘toy theories’ are used in place of a full quantum field theory treatment. However, they do give more of a flavour of the real physics/maths than the part of A-level syllabus does.

I suspect the inclusion of this sort of particle physics, at A-level, is an attempt to give an idea of developments at CERN – which is certainly fashionable at the moment. I don’t think there is an easy way of doing this.

2. Jon Butterworth Says:

I share some reservations about teaching Feynman diagrams to people who haven’t learned any quantum mechanics, and I know some professional particle physicists who get misled by them sometimes. However, I disagree with your main point that they are “not physics” in some sense. Physical concepts they could be used to teach or illustrate include:

– conservation laws (charge, lepton number, baryon number etc as well as energy and momentum – so in fact they could easily be used to bring out the neutrino/missing momentum in beta decay)

– different force strength (gamma, W, Z, gluon decay) and how they correlate to conservation laws.

– resonant phenomena. This is where the virtual/off mass shell particles come in, in that when a virtual particle can have the right mass (while energy and momentum are still conserved) the process is much more likely to occur.

Of course whether they are really effect or useful depends on how the teaching is done and what they displace. Rote learning diagrams isn’t going to help anyone understand physics much, I guess. But learning conservation laws….?

Incidentally I tried to explain something about virtual particles here

http://www.theguardian.com/science/life-and-physics/2014/apr/06/off-mass-shell-pythagoras-to-the-lhc-via-einstein-and-feynman

(and I agree they are a calculational tool – but why does that make them not physics? Any theory is a calculational tool in the end, surely)

Cheers,
Jon

3. James Miall Says:

My take on this is that I’m not particularly happy with the shut-up-and-calculate approach to theories – it would be nicer if a theory offered an explanation that seemed to get us closer to what was ‘really’ happening, even though it is difficult to argue with a theory that gives the right answers.
But there’s a lot to be said for easily understandable diagrams because people are so good at dealing with visual and spatial information. In fact I suspect that if all physics could be encoded in neat diagrams rather than text or equations it would be a lot easier to learn and manipulate and more progress would be made. Such as it is easier to quickly understand a circuit diagram than a list of all the components and nodes they are connected to, and often easier to understand a complicated graph than the complicated equation it is based on.

• protonsforbreakfast Says:

I think that’s a good point. And I like the analogy with the power of circuit diagram. In fact I think that analogy goes a long way, especially with regard to the different ‘styles’ used to draw diagrams.

And I have no problem with Feynman diagrams – they’re an interesting aid to discussion.

But imagine if you had to learn circuit diagrams ‘verbatim’ and be required exact copies of the diagrams when you didn’t know what the electronic components were. How pointless would that be?

Well that’s we have in A level physics: marks given for ‘verbatim’ reproduction of the diagrams. And I think that is bad – that’s my point.

• James Miall Says:

isn’t the verbatim reproduction of low order feynman diagrams though rather like being asked to draw a simple approximation to the circuit in a black box when you know what the inputs and outputs are and that obeys the same rules as electrical circuits outside the black box? even if you don’t know the components exactly you are still reproducing something that behaves similarly to the black box and has predictive power – not a terrible thing in my view

4. David Cotton Says:

I teach A level physics and really look forward to the Feynman diagrams. Please excuse my ignorance and I welcome any corrections to my knowledge. My students learn from my mistakes! I come at what a ‘boson’ is as the explanation to avoid action at a distance. I agree that the story of Pauli and the need for a neutrino to preserve energy conservation in beta decay should be in the A level specification and it is not. There is not enough historical development of ideas in physics at A level.

Yes some of the aspects of the questions on Feynman diagrams can be learnt by rote. The students do need to understand conservation of charge, lepton number etc, and they need to follow the decay for example when a proton turns to a neutron and a positron and electron neutrino are formed, the students need to ascertain that it is a W+ they need to draw on the diagram.
I must admit that although I leap to the defence of their inclusion in the syllabus, I am keen to understand more about the fact that they are only a way of describing a calculation.

I remember reading QED and Feynman says when talking about partial reflection of light;

“I am going to show you ‘how we count the beans’ – what the physicists do to get the right answer. I am not going to explain how the photons actually ‘decide’ whether to bounce back or go through; that is not known. (probably this question has no meaning.) I will only show you how to calculate the correct probability that light will be reflected from glass of a given thickness.”

After attempting to read past page six in a few books on quantum theory I had the realisation that quantum mechanics does not have a mechanism at its heart, it is just a calculating tool. However in relation to the good old Feynman diagram I thought particles like the W and Z did exist, and even if they are virtual, does not the Casimir–Polder force show the existence of such entities?

I think I understand that you are saying this one diagram we draw is not the process it is one of an infinite number of processes that need to be summed to give the probability of the events occurrence?

The thing that confuses me is that in our booklet we have a neutron collide with a neutrino and a W+ leaves the neutrino and I would say gives the neutron its ‘positiveness’ so it can become a proton. However can we draw the Feynman diagram with the arrow the other way so a W- leaves the neutron and is absorbed by the neutrino so it becomes a negative electron?

I hope this garbled rant makes sense! I can’t wait to tell the students next week as we revise for particle physics that the lovely graphic on the board is part of an infinite series of integrals and is just maths.
Thanks
Dave

• protonsforbreakfast Says:

Dave,

Thanks for those comments which don’t sound very ‘ranty’ to me.

I think that the fact that you ‘look forward’ to teaching Feynman diagrams is probably why they are in the syllabus. They provide an element of modernity in what can sometimes seem like a syllabus rooted in 100-year old physics.

Regarding Quantum mechanics, I am not an advocate of Feynman’s ‘shut up and calculate’ physics and I do not think that physics is just applied mathematics. When thinking about the nature of tiny entities such as atoms, then I consider that thinking ‘physically’ is essential, otherwise we become epicyclists.

For example, most students don’t understand that a ‘quantum transition’ is another way of describing a resonant vibration. But it is exactly that.
Or the emission of a photon when an atom is excited by a collision is often described as ‘excitation to a higher energy level’. But it is possible to describe this physically.

1. A perturbation reaches an atom.
2. By Perturbation I mean anything with a significant electric field comparable to the immense internal fields that an electron experiences (10^10) volts per metre.
3. The Hamiltonian is now different. The perturbation ‘mixes’ the stationary eigen-states of the atom which are now non-stationary. i.e. the charge oscillates.
4. By oscillates I mean that the dipole-moment operator has a significant oscillatory term.
5. The frequency of the radiation depends on the difference in energy of the stationary atomic states.
Of course you could just say that an atom is hit and vibrates at its natural resonant frequency just like a bell does, but somehow we don’t teach our students to use the physics they understand and allow them to see that it still works in a modified way on this small scale.

And my motivation in writing the article was that the the use of Feynman diagrams seemed to be another way of taking physical thinking out of the course and putting in the words and signs that sound modern.

Regarding the Casimer-Polder force I have read about that on Wikipedia but I am frankly unconvinced. I think explanations in terms of regular electromagnetic forces may be possible and seem to me intrinsically more likely.

Anyway, good luck with your revision for the A levels: I hope the classes go well.

Happy Easter Sunday

Michael