# Can Feynman diagrams be used to represent any perturbation theory?

In Quantum Field Theory and Particle Physics we use Feynman diagrams. But e.g. in Schwartz's textbook and here it is shown that it applies to more general cases like general perturbation theory for differential equations.

Can Feynman diagrams be used to represent any perturbation theory?

My thoughts are that we might be able to use this for perturbation theory in coupled equation systems like we get in fluid dynamics or astrophysics. But is it still possible to write down Feynman rules as a pictorial representation? And if it still does work in this case are there cases where that exhibit a different algebraic structure that Feynman rules can not represent?

Note that this is not a question about perturbation theory in Quantum Field theory (since there the use of Feynman diagrams is well known), but in a more general context. If the answer to the question is simply "no", could one at least give classes perturbation theories that it applies to?

The answer is no in general. Not all perturbation theories can be organized using diagrammatic rules (example given below). The subtlety here is that is not because you can write down diagrams to describe each terms of the perturbation theory, that you know what the diagrammatical rules are.

What I mean here is that it might happen that after calculating a perturbation theory at a given order (using standard analytical approaches), you can then rewrite the result as a sum of diagrams, instead of an equation. But it is not because you know the diagrams at a given order, that you immediately know what will be the diagrams of the next order, and in particular the numerical coefficients (what would be the symmetry factors in standard QFT perturbation theory). This changes everything, because if you do not know the diagrammatic rules, you cannot write easily all the diagrams at each order automatically, and you have to do the calculation the hard way. Of course, in standard QFTs, because the diagrams come from the average over a gaussian measure, we know the diagrammatic rules, and Feynman diagrams are very useful to organize the perturbation theory.

The counter example I have is taken from a statistical physics problem, where one wants to compute the high-temperature expansion of the Gibbs free energy (effective action) of an Ising model. There are several ways to do the calculation, and at the end of the day one can draw diagrams, but the diagrammatic rules are unknown (that is, when one goes from an order to the next, the coefficients in front of similar diagrams do not seem to follow a rule). In that case, I think that the problem comes from the fact that the unperturbed action is not quadratic (you don't have the Wick theorem for example). Whether rules indeed exist and they have just not been found, or whether they simply don't exist is, to my knowledge, an open question. The reference for this calculation is A. Georges and J. S. Yedidia, J. Phys. A 24, 2173 (1991).

• Even when the diagrammatic rules are unknown it can still be stated what they are in terms of cumulants, though the computation of the cumulants is only approximate (or perhaps even intractable). It is frequently in physics that one cannot calculate things exactly.... – Arnold Neumaier Mar 27 '16 at 18:06
• @ArnoldNeumaier Yes, of course, one can write the perturbation theory I mention in terms of cumulants. But these can then be expressed (exactly, at each order) in terms of "propagators" and "vertices", which looks very much like Feynman diagrams, though without diagrammatical rules. My point is just that it is not because one can write nice diagrams (that help a lot to condense the notations), that one knows how to organize a perturbation theory in term of those. – Adam Mar 27 '16 at 18:21

Diagram machinery works also for perturbation theory in classical statistical mechanics and classical field theories. Generally, various kinds of diagrams constitute a pictorial way of talking about tensor products and their contractions while hiding the multi-linear algebra from the layman. In the simplest case,

• vertices (or blobs) represent vectors, matrices, tensors;
• vertices have ingoing and/or outgoing ports;
• ingoing ports denote (contravariant) ro indices;
• outgoing ports denote (covariant) co indices;
• directed arcs between blobs give a pair of equal indices summed over;
• different base spaces $\Leftrightarrow$ different arc types;
• symmetric tensors $\Leftrightarrow$ undirected diagrams;
• no labels are needed for internal lines.

In many cases, the directed arcs are decorated (as thin, thick, broken,wavy, curly lines), each decoration indicating the presence of a so-called propagator, a function of its label to use as a weight in the sum, which may become an integral if the label is continuous, or a sum over an integral if the label has a discrete and a contiuous part.

In physics, diagrams typically appear in perturbative expansions of solutions to equations containing a small parameter, solvable exactly if this parameter is set to zero. The resulting expansion in powers of the small parameter contains terms made up of tensors. The use of diagrams in the perturbation theory of classical partial differential equations is discussed, e.g., in the papers

• V. Mathieu, A.H. Mueller and D.N. Triantafyllopoulos, The Boltzmann equation in classical Yang-Mills theory, Eur. Phys. J. C 74 (2014), 1-15.
• S. Jeon, The Boltzmann equation in classical and quantum field theory, Phys. Rev. C 72 (2005), 014907.
• R. Penco and D. Mauro, Perturbation theory via Feynman diagrams in classical mechanics, Eur. J. Phys. 27 (2006), 1241-1249.
• R.C. Helling, Solving classical field equations, Manuscript (undated).

Diagrams also appear in classical and quantum statistical mechanics to describe corrections to ideal gas or mean field behavior. This is documented in every statistical mechanics textbook.

In quantum field theory, the diagrams are heavily used as Feynman diagrams; here the indices $i$ typically become continuous momentum labels (with additional discrete indices in case of particles with spin), the sums become integrals (and sums over additional discrete indices accounting for spin). These Feynman diagrams in the perturbative description of S-matrix elements. In perturbation theory, the classical version of a quantum field theory only contains tree diagrams, while the quantum version also contains diagrams with one, two, or more loops. After renormalization, these give the quantum corrections to the classical theory.

The fact that Feynman tree diagrams also arise in the classical field theories obtained by taking the limit $\hbar\to0$ in the corresopnding quantum field theories shows that the interpretation of the internal lines of a Feynnman diagram as virtual particles is of a purely formal nature, without any intrinsic meaning (which would be inherited by the classical field theories).

• Excellent answer. Even the quadratic equation $y=x+ax^2$ for $x$ can be solved with Feynman diagrams. This would be the simplest case of the method explained in my article "Feynman diagrams in algebraic combinatorics" emis.de/journals/SLC/wpapers/s49abdess.html It is actually instructive to do this explicitly, see trees and Catalan numbers emerge and finally recover the usual formula with the square root. The method also works for PDEs as can be found in works of Giovanni Gallavotti, Yakov Sinai for Navier-Stokes or Michael Christ on nonlinear Schrodinger. – Abdelmalek Abdesselam Mar 28 '16 at 13:41
• @Numrok: Adam's example is only about the lack of explicit Feynman rules, not about the nonexistence of them, and he admitted it in his comment. There are examples of perturbation theory around hard sphere gases, and they are carried out in terms of diagrams, in spite of the failure of Wick's theory for them! – Arnold Neumaier Mar 29 '16 at 19:23
• @ArnoldNeumaier: "perturbation theory around hard sphere gases, and they are carried out in terms of diagrams" is an example of a case where you might have diagrammatic rules, which just tells you that in some cases, even without Wick theorem, you can find the rules. In a quantum case (strong coupling expansion in the Bose-Hubbard model), one can also do that (see PRA 79, 013614). This doesn't mean that you can always do it, which is my point. Of course, I cannot prove that the rules do not exist, but if you don't know them, diagrams don't help for the calculation of the next order. – Adam Mar 29 '16 at 19:54
• @Numrok: One knows them (and can calculate them approximately) as cumulants. Of course, their approximation might be too time-consuming to be realist, but the same can be said of lattice calculation on a ny very fine lattice. Computability and usefulness is something quite different from existence. – Arnold Neumaier Mar 29 '16 at 20:08
• @ArnoldNeumaier: I think you are missing the point. Feynman rules are useful not because they make nice drawings, but because they allow to generate all terms of a perturbation theory of a given problem. Otherwise, drawing diagrams is just a rewriting of an equation you have computed with another method. You could also write this equation in plain english, but although less compact, it would be as useful as diagrams without rules. – Adam Mar 30 '16 at 6:24

If we stretch the definition of a Feynman diagram, then yes: the technique can be applied to any problem where you use perturbation theory. But if by Feynman diagram you mean the exact same philosophy behind QFT, then in principle the answer is no: it only works in those problems where you have the same algebraic structure of QFT.

In QFT, there are two approaches to Feynman diagrams: canonical quantisation and path integrals. The latter can be summarised in the gaussian-type integrals $$g_n\equiv\int_{-\infty}^{+\infty}x^{2n}\mathrm e^{-\frac{1}{2}ax^2}=\sqrt{\frac{2\pi}{a}}a^{-n}(2n-1)!! \tag{1}$$

Feynman diagrams arises when we try to compute $g_n$ using combinatorial arguments together with the generating integral $$z_j=\int_{-\infty}^{+\infty} \mathrm e^{-\frac{1}{2}ax^2+jx} \tag{2}$$ where, for example, $g_2=[\partial^2z_j]_{j=0}$. For more details, see Mathematical Ideas and Notions of Quantum Field Theory.

This means: whenever you can invoke combinatorics to some problem at hand, then in principle you can use diagrams to represent the different combinations, which in turns means you can restate the problem through Feynman diagrams.

Canonical cuantisation, on the other hand, deals with operators with the algebraic structure $$[a_i,a_j]=0\qquad [a_i,a_j^\dagger]\sim \delta_{ij} \tag{3}$$

In this case, you can use this algebraic structure to prove Wick's theorem, which is (almost) the same as Feynman diagrams. This means: if you have any theory on some Hilbert space, where the operators satisfy $(3)$ (e.g., some Sturm-Liouville problem), then in principle you can use Feynman diagrams to solve problems.

The trivial example is, of course, the differential operator $a:L^2(\mathbb R)\to L^2(\mathbb R)$ $$a f(x)=\left(x+\frac{\mathrm d}{\mathrm dx}\right)f(x) \tag{4}$$ i.e., the ladder operator of the quantum harmonic oscillator. You can use Feynmal diagrams to calculate any "expectation value" $$\int_{-\infty}^{+\infty} p(x) f(x)\mathrm dx \tag{5}$$ where $p(x)$ is any polynomial. As the eigenfunctions of $a^\dagger a$ are (Hermite) polynomials times a gaussian exponential, we get $(1)$ back.

To sum up: if we extend the notion of Feynman diagrams, then I guess you could use them for most problems (though I'm not sure the usefulness of this). On the other hand, the standard meaning of Feynman diagrams can only be used if you are doing the same thing they were invented for, that is, path integrals, or something that shares the same algebraic structure.

A nice example of the use of Feynman diagrams to general problems is given in Solving Classical Field Equations, where the author explains how diagrams can be used to solve non-linear PDE's perturbatively. In this sense, one could say many problems in physics can be solved using Feynman diagrams, because of the ubiquity of differential equations in physics.

Another example of the use of Feynman diagrams for perturbation theory is given in this nice post by QMechanic, where you can see that Feynman diagrams can be used in (non-relativistic) quantum mechanics to simplify the evaluation of higher-order terms in perturbation theory.

Feynman diagram machinery can be utilized for ANY purturbation theory (where coupling constants are smaller than unity, e.g. QED) as well as non-perturbative theories (where coupling constants are larger than unity, e.g. QCD at low energies or large scales). This is pointed out by Bjorken and Drell,

The Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand. Although the statement of the theory in terms of graphs may imply perturbation theory, use of graphical methods in the many-body problem shows that this formalism is flexible enough to deal with phenomena of nonperturbative characters … Some modification of the Feynman rules of calculation may well outlive the elaborate mathematical structure of local canonical quantum field theory …

However, let's remind ourselves too that these diagrams (taken individually) are only a representative of the reality in the sense that they represent the trajectories of particles in intermediate stages of any scattering process. This literally means that it is their summation that would represent the reality itself.

• I think the OP question is whether Feynmann diagrams may be used to represent any perturbation theory, not just perturbation theory in Quantum Field Theory or Statistical Mechanics. – giulio bullsaver Mar 26 '16 at 14:53

Consider the differential equation as a constraint on the involved dynamical variables. You can then consider a free field Lagrangian where you implement the constraints using Lagrange multipliers. Clearly all the properties of the solution can then be extracted from such a field theory. Also, it is clear that one can treat the problem using perturbation theory by treating the constraint terms as a perturbation (multiply them by $g$ and expand in powers of $g$).