Habilitationsschrift: Renormalization and Knot Theory

TL;DR

The work develops an algebraic framework for renormalization in perturbative quantum field theory, showing that planar ladder and related topologies can be organized via one-loop building blocks. A central result is a strong link between the topology of Feynman graphs (via knot and link diagrams) and the appearance of transcendental numbers (Euler sums, zeta values) in counterterms, with simple topologies yielding rational Z-factors and more complex topologies generating knots and associated transcendentals. The author supports this knot-number dictionary with combinatorial proofs, Dyson–Schwinger arguments, and numerous multi-loop examples, including applications to φ^4 theory and quenched QED, and highlights significant cross-disciplinary implications for number theory and topology. The work also sketches future directions, such as connecting to chord diagrams, Drinfeld associators, and Kontsevich formalisms, suggesting a topology-driven route to predicting renormalization constants at high loop orders.

Abstract

We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory as well as number theory and report on recent results in support of this connection.

Figures (61)

• Figure 1: It is easy to see that states like $<\!k_1,k_2|k_3,k_4\!>$ can be graphically represented as above. Their normalization demands either $k_1=k_3,\;k_2=k_4$ or $k_1=k_4,\;k_2=k_3$. In general, multiparticle states $<\!k_1,\ldots,k_n|q_1,\ldots, q_m\!>$ do vanish for $n\not= m$, and for $n=m$ their normalization delivers all pairings of the $k_i$ with the $q_j$. There is no interaction between the straight lines corresponding to these pairings, which is indicated by straight lines in the figure. The straight line represents an incoming particle with momentum $k_i$ propagating undisturbed to become an outgoing particle with momentum $q_j=k_i$.
• Figure 2: The full Green function $G$ is a series starting with the bare Green function $G_0$, denoted by a straight line in the above. Interaction (for example with a classical background field $\cong$ dotted lines) takes place at various points, which we describe by vertices. We have to integrate over these points. Asymptotically, we assume that we have incoming and outgoing plane waves.
• Figure 3: A graphical illustration of Eq.(\ref{['gr']}). Upon Fourier-transformation we obtain overall momentum conservation $p_1+p_2+p_3+p_4=0$, where $\Delta_F(x_i-y)= \int d^4p_i e^{ip_i(x_i-y)}/(p_i^2-m^2+i\eta)$.
• Figure 4: Increasing the number of interaction terms amounts to a loop expansion of the theory. This time we consider an interaction of the form $g \phi^3$, so that three free propagators merge at a vertex. Each extra propagator provides a factor $g^2$, so that a $n$-loop graph provides a correction of order $g^{2n}$. These loop graphs have to be integrated over internal loop momenta. This corresponds to a sum over all possibilities for the momenta of the internal particles. Momentum conservation at each vertex, and the resulting overall momentum conservation for external particles, are not able to fix these internal loop momenta. They have to be integrated over the full four dimensional $d^4q$ space.
• Figure 5: Such a one-loop graph demands an integration of internal momenta. It turns out to be ill-defined. This is a well-known obstacle in pQFT.