Feynman Diagrams Do Not Represent Fundamental Interactions

Nearly anyone who has read a good book on theoretical physics written for a general audience came across Feynman diagrams. In this blog post I will attempt to argue for a particular interpretation of these objects, starting with a brief introduction for those who aren’t familiar with them. I will try to explain everything in simple terms, although some knowledge of mathematics and physics will be, of course, beneficial.

If you are familiar with Feynman diagrams, you can skip to „The main thesis”.

Feynman diagrams – introduction

We will be concerned with particle interactions. Particle interactions are simple: we have the „initial” state, which is a certain configuration of elementary particles, such as electrons, photons (quanta of light), Higgs bosons, etc. These particles then „interact”: they may change their configuration, some of them may be destroyed and new ones created. In the „final” state we could observe an entirely different, almost unrecognizable, set of particles.

According to quantum mechanics, we usually cannot predict what are we going to get at the end nor can we know what happenned „in between”: we can only perform calculations describing the probabilities of certain end-states, and we can measure experimentally which end-state was, in fact, reached. This is where Feynman diagrams come into play: they show what kind of interactions are possible – see Fig. 1. below.

 

Fdiagram

Fig. 1. Feynman diagram for an interaction between an electron and anti-electron, with an exchange of a photon; source

In order to perform the calculations of probabilities, we employ the principles of quantum field theory, which often constitutes an integral part of a theoretical physics course. Shortly speaking, we must start from a field in spacetime, which is defined by its mathematical properties. These mathematical properties enable us (at least in theory) to describe which end-states will be achieved from which initial states with what probabilities. Feynman diagrams constitute a way to represent terms in mathematical expressions that arise in the course of these calculations.

More precisely, Feynman diagrams are a way of representing certain elements of an approximation to the path integral – one of the most important mathematical objects a particle physicist works with. For the sake of completeness, I will write the path integral here, although it is not necessary to understand what the expression means:

1

It has been claimed that Feynman diagrams are something more than what I said above. It has been claimed that they represent the way interactions really take place.

The main thesis

The tone of the previous paragraph suggests that I do not agree with these claims. Indeed: in the remaining part of this entry, I will give my reasons for believing the following proposition:[1]

Feynman diagrams do not represent the way particle interactions „really” take place. Instead, each diagram is merely a pictorial representation of a term in a certain mathematical expression. Thus, internal lines of a Feynman diagram cannot be interpreted as particles.

I believe I have several good arguments for the above claim. Let me present them one by one.

1. The proper origin and motivation of a Feynman diagram is a graphical representation of mathematical expressions.

In modern particle physics, particles are not seen as fundamental objects that are put into the theory from the start. Theoretical models are instead defined in the language of fields, while the relationship between particles and fields is like the one between wrinkles and a carpet. More precisely, individual particles may be seen as the simplest excitations of a more fundamental object – the field.

The view according to which Feynman diagrams display the reality, on the other hand, would imply that particles are fundamental. This contradicts the spirit of modern physics, as explained above. In modern quantum field theory, Feynman diagrams arise from the language of fields, and they are used primarily to simplify certain calculations. Any interpretation in terms of particles is, at best, secondary.

2. According to standard interpretations of quantum mechanics, the interaction did not have to follow any particular Feynman diagram.

The laws of quantum mechanics imply that if a process cannot be observed, there is no „actual path” the system took, during this process, from its initial to the final state. For example, the scattering of an electron and an anti-electron could be described by a sum of two Feynman diagrams, one of which suggests a photon was exchanged, and the other – that the particles annihilated to create a photon which then decayed into electron-antielectron pair. But it would be a mistake to assume that the electrons must have chosen only one of these options; instead, the actual path was something like a superposition of both diagrams. Since the interaction does not take place according to any particular Feynman diagram, it is hard to understand how these Feynman diagrams could be taken as representations of the way interactions actually take place.

3. The infinite series of Feynman diagrams usually does not converge (ie. the diagrams cannot be summed up).

If each Feynman diagram represented physical reality, then (assuming the difficulty from point 2 is brushed away) in order to find the total probability of two particles interacting in a certain way, one could consider the contributions from all the possible physical trajectories (each trajectory represented by a Feynman diagram) and simply add them up. But more often than not, it simply cannot be done. The reason for it is the subtle difference between a Taylor series and an asymptotic series.

A Taylor series is an infinite series converging on the value of a function everywhere in a certain region (let’s say: near zero). Intuitively: we can approximate a function at a given point by a finite sum of simple terms (x^n), and the more terms we take, the better the approximation is going to be.

An asymptotic series is a symbolic expression which, in a certain sense, also approximates a function. Let’s say we approximate f(x) near x=0, by an expression which is increasingly more accurate as x → 0. Suppose the error in this approximation is of order x. Then we can produce a more accurate approximation by introducing a term identical to this linear component, thus cancelling it out. The new approximation’s error as x → 0 might be now of the order x² (which is smaller than x for sufficiently small x). I think you can see where this goes: we eventually get ourselves talking about an infinite series of approximations, each one working better than the previous one. Nonetheless, at a fixed x > 0, such approximations do not necessarily converge to f(x)!

Feynman diagrams are, formally, terms of some asymptotic series, not Taylor series. Thus we often cannot find the value of physical observable by summing up all the (infinitely many) Feynman diagrams, because the asymptotic series these diagrams represent does not converge to a limit. But if they did represent what really takes place, then surely the contributions from all the different, manifestly real processes would give a meaningful answer. Thus we have achieved a contradiction, implying that our assumption – that each of the processes described by Feynman diagrams is „real” – was incorrect.

4. Feynman diagrams also occur in the context of solid state physics, where they obviously cannot literally represent particle interactions.

In particle physics, we work in a spacetime which usually has 4 dimensions (3 for space and 1 for time). Imagine for a moment that we eliminate one spatial dimension, so we now have 2+1 dimensions. Now, some of you may be familiar with the idea that time is akin to imaginary space, as in the Hartle-Hawking model. If we are dealing with a quantum theory, we can take the expression for the path integral, formally change time into space, and obtain a theory living in 3 space dimensions, with no time! It is described by what is called a partition function:

2

This theory could model a 3-dimensional solid body (for example a magnetized slab of iron) and is mathematically very similar to the previous one (although physically and metaphysically different, since time is not space). Now comes the crucial point: since the partition function has a form resembling the path integral, this means that mathematical formalism is essentially the same, and we can still work with Feynman diagrams! Yet it should be obvious that in the context of solid state physics there is no such thing as „particles that mediate interactions”, and hence Feynman diagrams cannot be given the realist interpretation. Since Feynman diagrams do not have the realist interpretation in this context, why would they have the realist meaning in the context of particle physics? I think the simplest hypothesis is that Feynman diagrams have the same status – the antirealist one – in both theories.

5. The existence of a particle depends on the reference frame.

If an inertial observer claims to detect a particle, then all inertial observers will agree. Otherwise it would be possible to detect inertial motion in space, contradicting Special Relativity.

But it is well known in theoretical physics that an accelerating observer will see a picture that is altogether different, detecting particles that a stationary observer would say do not exist at all. Unruh effect describes this kind of situation: an observer who accelerates uniformly is going to see a thermal bath of particles with certain temperature.

This means that the existence of particles is not absolute: what one observer would see as a particle, another would see as an absence of it. While all inertial observers will agree that they observe the vacuum state, in General Relativity we learn that defining the vacuum state (of a field) is an ambigous procedure when accelerations and curvature play any role. Since we cannot uniquely define what we mean by a state with zero particles, the surprising conclusion is that we cannot give any absolute definition of a particle! (That is, the definition of what state of a field constitutes a particle and what state doesn’t cannot be absolute. As a side note, my understanding is that this is the deep reason behind the Hawking radiation; a stationary observer detects particles emanating from the black hole even though an inertial, infalling observer sees the vacuum.)

Since we cannot give any unambiguous definition of a particle, I think interpreting Feynman diagrams in realist terms would be a stretch; but I will leave the details of this argument to the reader.

In conclusion, I think Feynman diagrams do not represent fundamental interactions. It is just a mathematical tool. Perhaps there are arguments against this thesis, and there is a possibility that I will discuss these on another occasion – for now, a declaration that I haven’t found any convincing one must suffice. In a future post I might also discuss the implications of the antirealist interpretation on other areas of the philosophy of physics, for example on the claim (put forward by Lawrence Krauss and others) that the quantum vacuum is the true nothingness.

 

[1] This might be called the antirealist interpretation of Feynman diagrams.

Further reading:

See arXiv/1711.03790.

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