Why Feynman Diagrams Represent

Abstract

There are two distinct interpretations of the role that Feynman diagrams play in physics: (i) they are calculational devices, a type of notation designed to keep track of complicated mathematical expressions; and (ii) they are representational devices, a type of picture. I argue that Feynman diagrams not only have a calculational function but also represent: they are in some sense pictures. I defend my view through addressing two objections and in so doing I offer an account of representation that explains why Feynman diagrams represent. The account that I advocate is a version of that defended by Kendall Walton, which provides us with a basic characterization of the way that representations in general work and is particularly useful for understanding distinctively pictorial representations – in Walton’s terms, depictions. The question of the epistemic function of Feynman diagrams as pictorial representations is left for another time.

Acknowledgements

A sizable portion of the work in this paper was achieved with the help of an SSHRC Doctoral Research Grant. Kathleen Okruhlik, Patrick Maynard, Wayne Myrvold, Sona Ghosh and other members of the Philosophy Department at the University of Western Ontario provided invaluable help on early versions of this paper. Many thanks also to David Kaiser, an anonymous reviewer and James McAllister for their extensive and extremely helpful suggestions, which have improved the paper considerably, as well as Andrew Fenton and the Philosophy Department at Dalhousie University for their continued support of my work.

Notes

[1] There are, of course, many examples of aesthetic concepts being applied in non‐representational contexts. Appeals to aesthetic values, such as beauty, are made by scientists (McAllister 1996, especially ch. 3, provides us with a list) and philosophers of science (see, for instance, Cartwright 1999, 18–19).

[2] I follow John Willats’s use of ‘mark’, which he defines as follows: ‘The actual physical entities, such as blobs of paint or pigment, used in drawings, paintings, and so on to represent picture primitives [the elementary units of shape information in a picture]’ (Willats 1997, 368). In general, Willats’s taxonomy of visual representations is excellent, offering a subtle and comprehensive lexicon for thinking and talking about visual images.

[3] This is assuming that the diagrams of the Swiss physicist, E. C. G. Stückelberg von Briedenbach, which apparently inspired Feynman (1949b, 749), are not properly speaking Feynman diagrams. Why this is a thorny, but ultimately uninteresting issue with no clear answer will become apparent, see p. 45.

[4] The photon is virtual because its ontological status is debatable.

[5] This is something of an oversimplification as Feynman diagrams are used in not only calculations concerning positions, but also calculations concerning momentums. It is certainly more intuitive to think of the diagrams as representations of positions, so this is what I address here.

[6] The annihilation of an electron positron pair is easier to understand when one knows that the positron is the antiparticle of the electron. When particle and antiparticle meet, they annihilate each other, producing energy. Feynman took the idea for pictorially representing a positron as an electron going backward in time from Stückelberg (Feynman 1949b, 749).

[7] The qualification I make here is due to the fact that the integrals may diverge, resulting in an infinite and thus meaningless result for the probability amplitude. While, on the basis of present calculations, it appears that if the expansion series were carried out indefinitely the integral would not diverge, there is as yet no conclusive proof of this.

[8] It is worth noting that while ‘representation’ can be used synonymously with these terms, they both have totally different objects. A visualization is a mental object, a type of mental representation, and an illustration is, in this context, a physical object consisting of marks on a surface that depict a state of affairs, a pictorial representation.

[9] Brown also mentions the similarity in appearances between Feynman diagrams and cloud‐chamber pictures and, arguing that Feynman diagrams do not represent, he maintains that this similarity produces the confusion of thinking that they do (Brown 1996, 265). I consider Brown’s view in more detail in note 10 and when I address Objection 2, below.

[10] Brown appears to be confused in this way, writing, ‘[Feynman] diagrams do not picture physical processes at all. Instead, they represent probabilities (actually, probability amplitudes). The argument for this is very simple. In quantum mechanics (as normally understood), the Heisenberg uncertainty relations imply that no particle could have a position and a momentum simultaneously, which means there are no such things as trajectories, paths, through space‐time. So the lines in a Feynman diagram cannot be representations of particles and their actual paths through space‐time’ (Brown 1996, 265–267). Harré appears to be confused in this way in his discussion of representation in QED. For instance, he writes: ‘… from Cartwright’s standpoint, in which models are seen as often detached from the reality they represent, Feynman diagrams are a fine example of a class of image which should not be confused with a representation’ (Harré 1988, 61). As I have suggested, an image’s being ‘detached from reality’ implies nothing about its status as a representation. Kemp also appears to confuse issues of ontological status, resemblance and representation in a brief passage on Feynman diagrams (Kemp 2006, 311).

[11] I thank an anonymous reviewer for pointing out this perspicuous way of expressing the matter.

[12] I have applied Walton’s views to analyze anatomical drawings elsewhere and there have given a somewhat similar overview of his work (Meynell 2008).

[13] ‘Fictional’ does not have the implication of falsity in Walton’s system; fictional truth is entirely independent of actual truth or falsity (Walton 1990, 79).

[14] This is, in fact, something of an oversimplification, which I adopt for the sake of clarity. Walton adds another distinction – that between the game world and what is imagined. It is easy to confuse the two, as both the game world (whether authorized or unauthorized) and what is imagined are fundamentally dependent on the viewer. The viewer assents (implicitly or explicitly) to the rules that are characterized as principles of generation and by so doing creates the game world and her mind is the locus of what is imagined. However, there is the possibility that though she recognizes and assents to the rules of the game world, she fails to imagine what, by her own lights she is supposed to imagine. Here, I conflate game world and what is imagined for the sake of simplicity and clarity and because I think the inclusion of this distinction would burden rather than enrich the discussion.

[15] Brown recognizes this possibility, briefly mentioning the existence of pseudo‐phenomena that can be drawn. ‘E. O. Wilson’s Sociobiology, for example, has almost no photographs but has several beautiful drawings of animals in various activities. One of these shows two dinosaurs fighting. Needless to say, this was seen by no paleontologist. It is not a datum, but a phenomenon. But is it a real phenomenon?’ (Brown 1996, 264–265). Certainly, the question as to whether such images picture phenomena is open, but the question of their being representations is not. If in this case the pictured pseudo‐phenomena are still representations, then there is no prima facie reason to think that it should be any different for Feynman diagrams. Perhaps they do picture pseudo‐phenomena, but this is irrelevant to their representing. Thus, by Brown’s own standards, the images may indeed be inconsistent with the Heisenberg uncertainty relations but nonetheless be representations of QED.

[16] In the note to the figure Feynman writes: ‘The Schrödinger (and Dirac) equation can be visualized as describing the fact that plane waves are scattered by a potential. Figure 1 [Figure 4, above] (a) illustrates the situation in the first order’ (Feynman 1949b, 751; emphasis mine). In the preceding text he writes: ‘We can imagine that a particle travels from point to point, but is scattered by the potential U. Thus the total amplitude for arrival at 2 from 1 can be considered as the sum of the amplitudes for various alternative routes’ (Feynman 1949b, 751; emphasis mine).

[17] Kaiser explains that some, notably Douglas Hofstadter, found the diagrammatic diversity ugly, arbitrary and vague (Kaiser 2005, 314–315). Indeed, Feynman himself warned against believing calculations in meson theory that used Feynman diagrams (Kaiser 2005, 201).