Feynman Diagrams

Prerequisites: Canonical Quantum Field Theory, Perturbation Theory. Full path (21 pages)

• Intuitive
• Concrete
• Abstract
• History

Intuitive

Feynman diagrams are a pictorial way to keep track of terms in quantum field theory calculations. Each element of a Feynman diagram represents a different term in our calculation.

The thing is that in most interacting quantum field theories we can't calculate things exactly, and the standard method is a perturbation approach (Taylor series in the coupling). For a small coupling, the first term in this approximation to the correct result yields the biggest contribution and thus is the most important part.

An example can be seen on the right-hand side. In this example two electrons scatter. They do this by exchanging a photon $\gamma$, which is denoted by a wiggly line. After the exchange of the photon the two electrons move away from each other with different momenta.

In the first order approximation (i.e. when we only consider the first term in the approximation series mentioned above) the two electrons scatter by only exchanging a photon.

However, at the next order we also take into account that the photon can become a virtual electron-positron pair during this exchange. This will not happen with a high probability but is possible. Even more unlikely, but also possible, is that the virtual electron interacts with one of our in-going electrons via another photon. The total probability amplitude for the scattering to happen is the sum over all such possibilities.

Concrete

• Feynman Diagrams for Beginners by Krešimir Kumericki
• Feynman Diagrams For Pedestrians by Thorsten Ohl
• David Griffith: Elementary Particles contains an absolutely painless introduction to Feynman diagrams.
• For another nice introduction see chapter 6 in An Invitation to Quantum Field Theory by Luis Álvarez-Gaumé
• R.D. Mattuck: A Guide to Feynman Diagrams in the Many-body Problem
• For the derivation of the Feynman rules for a given theory see section 2 of Diagrammar by Veltman and 'tHooft

Take note that the usage of Feynman diagrams is not limited to Quantum Mechanics. See:

• A. Thorndike, “Using Feynman diagrams to solve the classical harmonic oscillator,” Am. J. Phys. 68 (2), 155-159 (2000)
• R. Penco and D. Mauro, “Perturbation theory via Feynman diagrams in classical mechanics,” hep-th/0605061.

Feynman Rules Resources:

• "When performing a full calculation within the Standard Model or its extensions, it is crucial that one utilizes a consistent set of signs for the gauge couplings and gauge fields. Unfortunately, the literature is plagued with differing signs and notations. We present all Standard Model Feynman rules, including ghosts, in a convention-independent notation, and we table the conventions in close to 40 books and reviews." from A resource for signs and Feynman diagrams of the Standard Model by Jorge C. Romao
• No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, chapter "Amplitude Technology"

Abstract

The combinatorial content of Feynman diagrams is already visible in finite-dimensional Gaussian integrals. Let $A$ be a positive definite symmetric $n\times n$ matrix and $$S(x)=\tfrac12\langle x,Ax\rangle-\sum_{k\ge3}\frac{g_k}{k!}B_k(x,\dots,x),$$ with symmetric $k$-linear forms $B_k$. Consider $$Z(\hbar)=\int_{\mathbb{R}^n}d^nx\;e^{-S(x)/\hbar},\qquad Z_0=\int_{\mathbb{R}^n}d^nx\;e^{-\langle x,Ax\rangle/2\hbar}.$$ Expanding the interaction exponential and evaluating each Gaussian moment with Wick's theorem (the moment $\langle x_{i_1}\cdots x_{i_{2m}}\rangle$ is the sum over all pairings of the product of $\hbar A^{-1}$ entries) gives the asymptotic expansion $$\frac{Z(\hbar)}{Z_0}\sim\sum_\Gamma\frac{\hbar^{E(\Gamma)-V(\Gamma)}}{|\operatorname{Aut}\Gamma|}\,F_\Gamma ,$$ where the sum runs over isomorphism classes of graphs with vertices of valence $\ge3$ (including the empty graph), $E$ and $V$ count edges and vertices, and $F_\Gamma$ is obtained by placing $g_kB_k$ at each $k$-valent vertex and $A^{-1}$ on each edge and contracting all indices. $\operatorname{Aut}\Gamma$ is the group of permutations of vertices and half-edges that preserve the graph. This is the precise content of the Feynman rules. Three standard consequences:

• $\log(Z/Z_0)$ is the same sum restricted to connected graphs.
• For a connected graph the exponent of $\hbar$ is $E-V=L-1$, where $L=E-V+1$ is the number of independent loops (the first Betti number). The expansion in $\hbar$ is the loop expansion.
• After adding a source term $\langle J,x\rangle$ and passing to the Legendre transform of $W(J)=\hbar\log Z(J)$, the effective action $\Gamma(\phi)$ is $S(\phi)$ plus corrections of order $\hbar$ whose Taylor coefficients at $\phi=0$ are sums over connected one-particle-irreducible graphs (graphs that stay connected when any single edge is cut) with amputated external legs.

The series is asymptotic and in general divergent: the coefficient of $g^V$ grows factorially with $V$. For example, $\int dx\,e^{-x^2/2-gx^4/4!}$ is not analytic at $g=0$ since the integral diverges for $g<0$.

In quantum field theory $x$ becomes a field, $A$ a differential operator such as $-\partial^2+m^2$ (in Euclidean signature), $A^{-1}$ its Green's function, and the contraction of indices becomes integration over vertex positions, or over loop momenta after a Fourier transform. The finite-dimensional statements then hold as formal power series identities, but the individual $F_\Gamma$ are typically divergent integrals because they involve products of distributions at coinciding points. Renormalization assigns finite values to them in a way consistent with the graph structure: the BPHZ forest formula subtracts divergences of all subgraphs. Connes and Kreimer showed that this subtraction can be organized by a Hopf algebra of Feynman graphs, whose coproduct sums over divergent subgraphs, with renormalization appearing as a Birkhoff factorization of the regularized Feynman rules viewed as a character of this Hopf algebra. - Feynman Diagrams for Pedestrians and Mathematicians by M. Polyak

History

It struck Feynman that everyone had a favorite principle or theorem and he was violating them all… Feynman knew he had failed. At the time, he was in anguish. Later he said simply: “I had too much stuff. My machines came from too far away.”[…] Schwinger’s students at Harvard were put at a competitive disadvantage, or so it seemed to their fellows elsewhere, who suspected them of surreptitiously using the diagrams anyway. This was sometimes true… Murray Gell-Mann later spent a semester staying in Schwinger’s house and loved to say afterward that he had searched everywhere for the Feynman diagrams. He had not found any, but one room had been locked…from "Genius" by Gleick

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Newtonian Mechanics
10. Variational Calculus
11. Configuration Space
12. Euler-Lagrange Equations
13. Special Relativity
14. Lagrangian Formalism
15. Phase Space
16. Classical Field Theory
17. Hamiltonian Formalism
18. Schrödinger Equation
19. Canonical Quantum Mechanics
20. Canonical Quantum Field Theory
21. Perturbation Theory

Leads to: Mass Insertion Approximation, Phenomenology, Renormalization, Spinor-Helicity Formalism, Virtual Particles.