In the paper "A Prehistory of $n$-Categorical Physics" J. Baez and A. Lauda give an account of the use of category theory throughout physics. In section “Penrose (1971)” starting from page 25 they explain how one can use the language of monoidal categories to interpret Feynman diagrams from a categorical point of view (using the fact that all representations of a group and their intertwiners form a monoidal category). Then on page 29-30 it is said that the divergence of loop diagrams is related to the fact that the relevant unitary representations of the Poincaré group are infinite-dimensional (and loops in the diagrams give the dimension of the representation).

I was working on a project for university based on this paper and when talking about the part mentioned above, my professor said that this was not completely correct. He mentioned that when including supersymmetry, hence enlarging the relevant symmetry group, certain loop diagrams become finite and this seems to be in contradiction with the dimension argument from Baez and Lauda.

Now I was wondering what the formal explanation behind this phenomenon is? Is it because the super-Poincaré group does have finite-dimensional unitary representations or because the transition from dimensions to super-dimensions introduces a cancellation which keeps the loops finite?


The divergences of Feynman diagrams have nothing to do with the infinite dimensionality of the unitary representations of the Poincaré Group (PG). I agree with the argument given by your professor. And you don't even need SUSY to argue that the claim in the paper is misleading/wrong. There are non-supersymetric models in lower dimensions which are perfectly finite (e.g., Glimm & Jaffe's $\phi^4_2$), yet they have reps of PG which are infinite-dimensional (as long as $d>0$, all unitary reps are infinite-dimensional).

And, more importantly, Feynman diagrams know nothing about the unitary representations of the Poincaré Group.

The PG appears in two different ways in QFT (cf. e.g. this PSE post):

  • Particles, described by unitary (and hence infinite-dimensional) reps of PG, and

  • Fields, described by finite-dimensional (and hence non-unitary) reps of PG.

Feynman diagrams encode the properties of fields, not particles, and therefore they carry the information of finite-dimensional (non-unitary) representations. The unitary representations appear when using the LSZ formula which, in short, amputates external legs and attaches a polarisation vector carrying the one-particle state information.

This remains true when considering the super-Poincaré Group (SPG):

  • Particles are organised into multiplets, which are unitary reps of SPG, and which can be thought of as collections of unitary reps of the standard PG. They are still infinite dimensional (recall that the infinite dimensionality is required because the group is non-compact; the culprit is the subgroup of translations, which is also present in the super case, and whose eigenvalues are momenta; supermultiplets also carry momentum quantum numbers, and this is where the infinite dimensionality comes from).

  • Fields are organised into superfields, which are finite-dimensional reps of SPG, and which can be thought of as collections of finite-dimensional reps of the standard PG.

The improved UV behaviour of super-theories has nothing to do with the dimension of the representations; indeed, the particles are still infinite-dimensional, and the fields are still finite-dimensional. It has to do with cancellations, or with more subtle properties of supersymmetry (e.g., the so-called non-renormalisation theorems; in short, divergences must be supersymmetric, but sometimes one can prove that there is no counterterm with the required symmetry/divergence structure, and so the divergence is not there to begin with, cf. e.g. this PSE post).

So what do Baez & Lauda mean? hard to tell, but my guess is the following: loops indeed are associated to traces over a representation of PG (times a representation of an internal group, like colour), and so they are in a sense proportional to the dimensionality of the rep. But the rep is that of the field associated to the line, not a particle, and so it is finite-dimensional. For example, gluon loops typically grow like $N^2$, and quark loops like $N$; this is because gluons live in the adjoint, and quarks in the fundamental. These are finite-dimensional representations. So the authors are either confused or I didn't understand their point.

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    $\begingroup$ ... so, where do the divergences come from? you may want to check What is the issue with interactions in QFT? for a discussion. $\endgroup$ Aug 18 '19 at 15:18
  • $\begingroup$ Oh ok i see. What I thought was that they meant that for internal loops one just traces over all possible particle types and that this induced the divergences. $\endgroup$
    – NDewolf
    Aug 18 '19 at 15:39
  • $\begingroup$ Which would be correspond to the momentum integrals which are used in the computation of loop diagram $\endgroup$
    – NDewolf
    Aug 18 '19 at 16:45

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