# The path integral and Feynman diagrams

This question is somewhat of a historical one, but it also contains some physics. I am curious to find how exactly the concept of Feynman diagrams arose (I assume from Feynman's path integral)?

The leap from path integrals to diagrammatic computations isn't obvious (to me, at least); I'd like to understand better how Feynman's thinking approximately developed. For instance, how did he come up with interpreting the propagator as the propagation of particles? Was there a particular analogy that can be made? Is there any understanding to be gained by learning how the technique was originally developed?

I realize that my phrasing might be a quite vague. If the question is too broad as of right now, please let me know how I can improve on it!

• I had always assumed that the Feymann path integral approach to quantum mechanics had evolved somewhat independent of Feynman's diagrammatic approach to perturbation in quantum field theory. Since they seem to fit together so well, I am interested in hearing what others have to say the history of these two ideas. – DrEntropy May 26 '14 at 6:22
• For what I know, it arose by the computation of path integrals with the Wick rule: the contractions of the Wick rules are the lines of a diagram, the fields contracted are the vertexes, the inverse of the matrix of the Gaussian (or Grassmannian for fermions) measure is the propagator. The Feynmann diagrams are just a smart representation of the Wick rule, in certain books are proposed in a completely independent way from any field-theoretic concept (see for example Non perturbative renormalization, autor Vieri Mastropietro). – giulio bullsaver May 28 '14 at 14:28
• I think there is a history of science SE site now, where this seems to fit. Then again, I don't know who frequents this boards. To get to the question, since you ask how Feynman came to his conclusions, the answer surely lies in his knowledge and for this it's crucial to have a look what he worked on before: wikipedia.org/wiki/Wheeler-Feynman absorber theory. – Nikolaj-K May 28 '14 at 14:45
• @SanathDevalapurkar Another reminder for you! Maybe the bounty will provide some more motivation? ;) – Danu May 28 '14 at 14:55
• I don't know about the history, but I wouldn't consider leap from path integral perturbation expansion to feynman diagrams very difficult: if you write down a perturbation expansion for a path integral, and explicitly compute a few terms, you will find that they are all of a certain form: namely, each term is a product of a bunch of propagators (plus some combination sources, sinks, counterterms and such). – zzz May 31 '14 at 19:33

For instance, how did he come up with interpreting the propagator as the propagation of particles?

The path integral is usually introduced as a matrix element of the time evolution operator $$\langle x_f\lvert\mathrm e^{-\frac{\mathrm i}{\hbar}\hat{H}(t_f-t_i)}\lvert x_i\rangle,$$ which is a measure of the probability of finding a system in final state and time $x_f,t_f$ when it had been in state $x_i$ at time $t_i$ initially. It is quite plausible to name it propagator as it gives immediate access to the probability that a system, maybe only a single particle, propagates from state $x_i$ to $x_f$ in time $t_f-t_i$. Probably it is more difficult to understand that this notion is still maintained when the path integral is used to calculate the grand partition sum in quantum statistics.

Is there any understanding to be gained by learning how the technique was originally developed?

The idea of symbolizing formulae by nodes and connections between them is used in many other fields and was probably not new at the time. The idea is basically that of an isomorphism between a class of graphs and, given an unambiguous translation rule, the formulae at hand. This gives intuitive connection to graph theory and eases its application, for instance when a diagram is called 'connected' or 'disconnected', meaning that the respective formula can be factorized or not. Another example of this kind that is not related to Feynman is the diagrammatic treatment of the classical Ising model.

• This is an excellent, clear answer, but it doesn't say how Feynman came to the conclusions he did, or any other historical figures for that matter. – DanielSank Oct 1 '14 at 0:43
• @DanielSank For that I recommend his original paper, which can be found here: authors.library.caltech.edu/47756/1/FEYrmp48.pdf Very readable. Enjoy! – Ellie Oct 16 '14 at 19:59

I was taught that the Feynman diagrams arose as a smart way to write down the intricate computations appearing in the perturbative approach to the path integral.

The keystone is the well known Wick rule, that allows one to compute standard and Grassmannian integrals of correlations with Gaussian measure, e.g. an expression like $$\int dx_1\cdots dx_n \ x_{j1}\cdots x_{jn} \ \exp\{- (\hat{x},A\hat{x})\}$$ is rewritten as a sum of several terms, one for each way of "contracting" all the $x_{j1}\cdots x_{jn}$ into couples. In particular each couple of $x$'s contracted, will give also a contribute proportional to an entry of the inverse of $A$.

In the path integral formulation applied to QFT we will need to compute similar integrals where the $x$ are replaced with fields, and the $A$ of the quadratic term is a less trivial object, but is assumed the Wick rule is still true. (At least, so I was taught.) The inverse of $A$ need a suitable generalization, and it is taken to be its Green function, so you see the Wick rule will make propagators appear.

In order to describe interacting theories you need additional terms in the exponential argument, like $(\hat{J},\hat{x})$ or $\hat{x}{}^4$. This ruins the game since now the Wick rule does not apply anymore. Here enters the idea of expanding the exponential of the new terms $(\exp\{f(x)\} = 1 + f(x) + f(x)^2/2 + \cdots)$, so that you find yourself with a series of integrals computable via the Wick rule. Depending on the field type, bosonic (standard integral) or fermionic (Grassmann integral), and on the terms you have put in the exponential, you can represent the contractions of the Wick rule in a pictorial way respecting some set of rules (the Feynman rules), of course the obtained drawings are the Feynman diagrams.

In general you will have vertices for each field appearing in the integral (outside the exponential of the Gaussian measure) and the contractions among couples will be represented by lines.

A reference I find very interesting is "Non-perturbative renormalization" by Vieri Mastropietro, in the section "Grassmannian measures" the Feynman diagrams are presented as a very natural way to represent the Wick rule for the Grassmannian integral, without mentioning anything about QFT.

• This is fine but the question was about historical development. – DanielSank Jun 3 '14 at 15:05

I read in either one of Feynman's books or in the biography Genius: The Life and Science of Richard Feynman by James Gleick, that Feynman was at a conference in a hotel room trying to work out some path integral, in his pajamas, and at some point found himself surrounded by a bunch of pieces of paper, each containing a term in a perturbation expansion. These basically were so-called Feynman diagrams. If I remember correctly (I don't have the book in front of me) he met with someone else at the conference who had used a similar idea and they realized that it was a good idea and shared it with others.

P.S. That book by Gleick is really good.

EDIT: According to the comments below, the story I'm remembering comes from The Pleasure of Finding Things Out. Furthermore, the actual explanation of where the diagrams come from appears in the book a few pages before the story I described.

• Pajamas are in The Pleasure of Finding Things out, p. 198 of the Penguin edition. But this was after the diagrams were already thought up, and before they were called Feynman diagrams. The pajamas are when Feynman wonders if Physical Review would print the diagrams if they were really useful. The actual answer appears one or two pages before the pajamas. – Řídící Jun 3 '14 at 15:26
• I'm not sure about it, because this particular story doesn't mention a conference or a hotel room (in relation to the invention of the diagrams). Maybe you got it somewhere else. – Řídící Jun 3 '14 at 18:26