5.3 Interaction and intersection
5.3.6 Feynman diagrams
It is hard to imagine today’s practice in physics without Feynman diagrams. They are used everywhere as a tool to handle complicated algebraic calculations. What is more, their ubiquitous presence as well as their pictorial character has lead some to believe that they are more than mere tools and indeed show how nuclear processes look like. This section will discuss Feynman diagrams between these two poles: tool and pictorial representation. I will first present how the diagrams enter physics and then go through the arguments that have been put forward for one or the other of the two possibilities. The conclusion will be that Feynman diagrams cannot be regarded as giving insights into nature in their own right, beyond what is already known from the formalism of QFT without diagrams.
The place where Feynman diagrams appear in the canonical quantisation formalism is the perturbation expansion of the S-matrix, 5.3.33, which consists of time-ordered products of field operators.24 According to Wick’s theorem (cf. Greiner & Reinhardt
1996, ch. 8.5) such products can be reformulated and computed through sums of contractions, defined as
23A topic that I don’t want to go in further here is the very popular image of the vacuum filled with
virtual particles constantly coming into existence and vanishing again. Furthermore, it is often supposed that they have a measurable effect, namely the Casimir effect, by putting pressure on two plates close to each other. I have only brief comments on that. First, contrary what seems to be the popular understanding, the vacuum state is not literally nothing, but merely the ground state of a field. Second, fluctuations of the vacuum state are due to the Heisenberg uncertainty principle and thus a property of every state in quantum physics. Virtual particles coming into existence and vanishing after some time again would have to be described by an appropriate dynamics, that is, a state and an evolution operator for that state. The latter, however, are not part of contemporary QFT. Thus, vacuum fluctuations are fundamentally different from, e.g., particle creation and annihilation in scattering experiments. Third, it is not at all clear whether virtual particles, or even vacuum fluctuations are part of the explanation of the Casimir effect (cf. Jaffe 2005).
24Here I only intend to describe Feynman diagrams as they were used originally. Today they are
applied in many different places, which I will not present, since my argument does not depend on it. Kaiser (2005) gives a detailed discussion of the migration of Feynman diagrams from their origins into other areas, and captures this under the heading of the ‘plasticity’ of the diagrams.
ˆ 1(x1) ˆ2(x2) :=h0|T n ˆ 1(x1) . . . ˆ2(x2) o |0i i.e., vacuum expectation values.
Taking again QED, with the Lagrangian 5.3.28, as an example, the S-matrix can be expanded as (cf. Greiner & Reinhardt 1996, ch. 8.6)
S =I + 1 X n=1 S(n) = 1 X n=1 1 n!( ie) nˆ d4x 1. . . d4xnT h : ˆ (x¯ 1) 1µ (xˆ 1) ˆAµ1· · · ˆ¯ (xn) µ n (xˆ n) ˆAµn : i with : . . . : denoting the normal product in which negative frequency parts of the operators are always written on the left of positive frequency parts. Now one finds that the first order term does not contribute to the S-matrix. The second order term, however, does contribute and is explicitly given as
S =1 2!( ie) 2ˆ d4x 1d4x2 : ˆ (x¯ 1) µ (xˆ 1) ˆ (x¯ 2) ⌫ (xˆ 2) ˆAµ(x1) ˆA⌫(x2) : (a) +1 2!( ie) 2ˆ d4x 1d4x2 : ˆ (x¯ 1) µ (xˆ 1) ˆ (x¯ 2) ⌫ (xˆ 2) ˆAµ(x1) ˆA⌫(x2) : (b) +1 2!( ie) 2ˆ d4x 1d4x2 : ˆ (x¯ 1) µ (xˆ 1) ˆ (x¯ 2) ⌫ (xˆ 2) ˆAµ(x1) ˆA⌫(x2) : (c) +1 2!( ie) 2ˆ d4x 1d4x2 : ˆ (x¯ 1) µ (xˆ 1) ˆ (x¯ 2) ⌫ (xˆ 2) ˆAµ(x1) ˆA⌫(x2) : (d) +1 2!( ie) 2ˆ d4x 1d4x2 : ˆ (x¯ 1) µ (xˆ 1) ˆ (x¯ 2) ⌫ (xˆ 2) ˆAµ(x1) ˆA⌫(x2) : (e) +1 2!( ie) 2ˆ d4x 1d4x2 : ˆ (x¯ 1) µ (xˆ 1) ˆ (x¯ 2) ⌫ (xˆ 2) ˆAµ(x1) ˆA⌫(x2) : (f) +1 2!( ie) 2ˆ d4x 1d4x2 : ˆ (x¯ 1) µ (xˆ 1) ˆ (x¯ 2) ⌫ (xˆ 2) ˆAµ(x1) ˆA⌫(x2) : (g) +1 2!( ie) 2ˆ d4x 1d4x2 : ˆ (x¯ 1) µ (xˆ 1) ˆ (x¯ 2) ⌫ (xˆ 2) ˆAµ(x1) ˆA⌫(x2) : (h)
To obtain a result from this long expression, it is now convenient to first draw Feynman diagrams according to the following Feynman rules:
1. Every point x corresponds to a vertex, that is, a point where other lines meet. 2. Each fermion field operator corresponds to an external straight line, directed
either into a vertex or coming from one.
3. Each boson field operator corresponds to a external wavy line, without direction, but with a vertex on one end.
4. Each contraction between two fermion field operators, ˆ (x1) ˆ (x¯ 2), corresponds
to an internal line with direction from x1 to x2.
5. Each contraction between two boson field operators, ˆAµ(x1) ˆA⌫(x2), corresponds
to an undirected25 internal wavy line between x
1 and x2.
Accordingly, term (a) above can be translated into two diagrams, each with two external fermion and one external boson line. For all eight terms of the expansion above, this leads to the following schematic diagrams, where time is the vertical axis:
Figure 5.2: Schematic Feynman diagrams, corresponding to the second order per- turbation expansion of QED. In: Greiner & Reinhardt (1996, p. 239) The specific form of the contributing diagrams depends then on the interaction one is describing. If, for example, the process one wishes to compute is the scattering between two electrons or positrons or one electron and one positron, then only diagram (d) will contribute to the S-matrix, since only this diagram has the right number of particles in the initial and final state. More precisely, depending on whether one considers either (i) electron-electron scattering, or (ii) positron-positron scattering, or (iii) electron-positron scattering, the schematic diagram (d) above will take on one one of the following forms:
Figure 5.3: Feynman diagrams corresponding to term (d) of the second order per- turbation expansion of QED. In: Greiner & Reinhardt (1996, p. 240) Finally, each element of a diagram corresponds to an algebraic expression, e.g., internal wavy lines to the Feynman propagator for photons, which can then be set together to give the entire S-matrix.
Since the appearance of Feynman diagrams in the path integral formalism is very similar to that in the canonical quantisation formalism, I will not go into details here. Suffice it to note that from the perturbative expansion of the Green’s function
25This is due to the symmetry of the photon propagator, that is, D
5.3.42 Feynman diagrams can be created according to the following rules: (i) The number of vertices is given by the order of the perturbation series. (ii) Propagators correspond to internal lines. (iii) Source term correspond to external lines. Of course, since we are now in the path integral formalism, propagators are now Feynman path integrals of classical field configurations, and source terms also consist of classical fields only.
I will now turn to the question how Feynman diagrams have to be interpreted, that is, in an instrumentalist or realist fashion. The first argument I want to consider is directed against the realistic interpretation of Feynman diagrams in general. Similar to virtual particles, Feynman diagrams only show up in the perturbative approach to QFT. Since perturbative methods are not compulsory and can often be avoided, one might want to conclude that the diagrams cannot contribute to the interpretation of QFT. This conclusion, however, would be overly hasty. The perturbation expansion is not completely detached from the rest of the formalism, that is, some parts of the structure of QFT are also visible in the expansion. One therefore cannot exclude the possibility to learn something about QFT in general from the perturbative method. Likewise, in the preceding section I have only argued against a literal understanding of virtual particles as particles, but not against their existence as forces.
Consequently, I will follow Wüthrich (2012) and adopt a more cautious approach in answering the question on Feynman diagrams. As he notes, there is not only the choice between either dismissing or affirming Feynman diagrams as a whole, but there is also the possibility that they partially represent the world.26 More precisely,
diagrams have at least three properties, each of which may independently of the others either represent or not. The properties are: 1) The diagrams have parts, that is, they consist of external lines, internal lines and vertices. 2) The parts are connected at the vertices. 3) The parts are localised in space. In what follows I will introduce the arguments for different interpretations of the diagrams based on this distinction.
The first trait of Feynman diagrams I want to discuss is their localisation in spacetime, i.e., the property of being constructed out of lines and points. Harré (1988) highlights that Feynman diagrams resemble particle tracks as they show up in measurement devices like cloud chambers and this, as Harré argues, might be the most important reason why one might mistakenly believe that the diagrams give an exact picture of physical processes. That this belief indeed would be a mistake has several reasons. First, as I have argued in chapter 5.2.3, tracks in cloud chambers are not particle tracks at all, but rather a series of interactions. What is more, the particles whose tracks supposedly compose the diagrams have a well defined momentum and thus, according to the Heisenberg uncertainty principle cannot have a well defined position at the same time (cf. Meynell 2008). Thus, it follows at least that particles cannot be as exactly localised as the lines in the diagrams.
26At this point, I am not interested in a detailed account of how partial representation could work.
Suffice it to say that there are worked out theories of partial representation, for example as given by da Costa & French (2003).
Next is the property of the diagrams having parts, which is independent of the fact that these parts cannot be as localised as well as in the diagrams. And indeed, this is a property in which Feynman diagrams represent processes, since it rests on the structure of the Lagrangian. The straight and wavy lines in the diagrams correspond to the fermion and boson fields, appearing in the Lagrangian. The distinction between external and internal parts, on the other hand, is determined by which process is calculated and consequently which particles appear in the initial and final states and which one only in the intermediate state.
Also the property that the parts are connected at vertices can at least partially be seen to represent. The first Feynman rule, stated above, tells that the number of vertices equals the number of space-time coordinates in the perturbation term. This can be regarded as reflecting the locality of the Lagrangian, that is, interactions between fields are always evaluated point by point at one space-time coordinate only and there is no action at a distance. However, since the space-time coordinates are integration variables and usually the integration runs over all of space-time, the vertices cannot represent insofar as they are points. Rather, they only represent locality and nothing more.
Wüthrich (2012, p. 178), however, goes a step further and claims that Feynman diagrams “allow us to explain certain aspects of the behaviour of the system which they represent”. According to him, diagrams like that on the right-hand side of picture (iii) in figure 5.3 for electron-muon scattering “allow us to explain that, most of the time, the electron and the muon deflect each other only a little.” (Wüthrich 2012, p. 178) This explanation supposedly rests on the facts that long range forces lead to a high reaction rate and that a force can only have a long range if the electron and muon deflect each other only little. Nevertheless, I have to disagree with Wüthrich when he claims that it is the Feynman diagram that explains the specific deflection angle. The problem is that the premises and the conclusion of Wüthrich’s argument are not part of the Feynman rules and are thus not used in their construction of diagrams nor can they be read of from them. The range of the forces, the angles between meeting lines and so on have no importance for the construction of Feynman diagrams and thus the diagrams cannot represent those features of the world. Even though Wüthrich’s explanation is correct, the explanation can be given perfectly without any reference to Feynman diagrams. Therefore, it is not the Feynman diagram doing the explanatory work here and they do not represent that what is explained.
In conclusion, Feynman diagrams can represent certain features of physical pro- cesses, however, they are of little help for the interpretation of QFT. Since the properties of the diagrams that do represent are all derived from the underlying formalism, diagrams simply tell us nothing new about the physical process. Instead of taking the detour of Feynman diagrams, any interpretation of QFT should rely directly on the mathematical formalism. I will therefore not consider them further in this work.