Paul BaldufMar 10
#paperOfTheDay : "#Renormalization of a scalar field theory in strong coupling" from 1972.
Recall that phi^6 theory in 3 dimensions is a perturbatively renormalizable scalar #QuantumFieldTheory model, and in perturbation theory (using #FeynmanIntegral s), one expects there to be counterterms for the phi^6, phi^4, and phi^2 interactions to accommodate their anomalous scale dependence. In the present paper, Wilson uses a different approach, and introduces an approximation scheme for the quantum effective action, which is not inherently related to conventional perturbation theory. In this, he finds that the so-approximated model only acquires anomalous flow for phi^2, but not for phi^4 and phi^6. The approximation is relatively coarse, so one should not take this as a "solution" of phi^6 theory, but rather as a concrete example of what could in principle happen in a strongly coupled interacting field theory.
Note that with such results, there is no contradiction with perturbation theory: Low-order perturbation theory describes a behaviour very close to a free theory, but perturbation series are divergent and asymptotic. This means that the true functional form only emerges after resummation, and is in general very different from "inserting a large number for the coupling into the low-order perturbation series".
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.6.419
Paul BaldufFeb 21
My #paperOfTheDay is the article "Asymptotically free solutions of the scalar mean field flow equations" from 2022. It concerns scalar #quantumFieldTheory . It is well known that the scalar phi^4 theory is "trivial" in 4 dimensions in the sense that if one imposes it at some high energy scale, then the interaction disappears at lower energy scales. This is different from e.g. quantum chromodynamics, which describes the strong force and is "asymptotically free": It can have non-vanishing interaction at low energy even if the high-energy theory is free.
This behaviour strongly depends on the particular interaction terms. In perturbation theory, one assumes that only the quartic phi^4vertex is present at high energy (because otherwise the perturbation series can not be renormalized). The method of #renormalization group flow equations, however, allows for an analysis of more general settings. The present article demonstrates that scalar field theories can have interesting, non-divergent, solutions even if they contain non-renormalizable interactions. #dailyPaperChallenge https://link.springer.com/article/10.1007/s00023-022-01194-w
Asymptotically Free Solutions of the Scalar Mean Field Flow Equations - Annales Henri Poincaré

The flow equations of the renormalisation group permit to analyse rigorously the perturbative n-point functions of renormalisable quantum field theories including gauge theories. In this paper, we want to do a step towards a rigorous nonperturbative analysis of the flow equations (FEs). We restrict to massive scalar (one-component) fields and analyse a mean field limit where the Schwinger functions are considered to be momentum independent and thus are replaced by their zero momentum values. We analyse smooth solutions of the system of FEs for the n-point functions for different sets of boundary conditions. We will realise that allowing for nonvanishing irrelevant terms permits to construct asymptotically free nontrivial smooth solutions of the scalar field mean field FEs.

SpringerLink
Paul BaldufFeb 21
My #paperOfTheDay for Friday was "The background field method and the non-linear sigma model" from 1988. It concerns #renormalization in theoretical #physics. In a theory with non-linear interactions, the observed quantities generally are in a non-linear relation with the "input parameters" (such as a coupling strength) of the theory. Hence, one can not immediately measure the input parameters. "Renormalization" is the procedure to disentangle these relations, so that one can use an experimentally measured value to determine parameters of the theory, and then predict all further observable outcomes (think of accelerating a ball that is immersed in water. From the required force, one can not immediately deduce the density or viscosity of water, but it is possible in principle after some calculations.). The "background field method" is one out of several approaches how to carry out renormalization in a field theory. In the present article, the authors demonstrate that this method can be used for the non-linear sigma model on an arbitrary curved surface, even if it is a bit more complicated herethan for other field theories that had been studied before. #dailyPaperChallenge https://doi.org/10.1016/0550-3213(88)90379-3
Paul BaldufJan 28
Today in the #dailyPaperChallenge I read (parts of) "critical \( \phi^4_{3,\epsilon} \)" from 2002. This paper deals with the #renormalization of a quartic interacting #fieldTheory in 3 dimensions, but with a non-standard (long-range) propagator \(1/p^{\frac 3 2} \). The methods they use are quite different from what I am accustomed to, but there are two points of contact with my work: Firstly, this theory is an example of a marginally coupled \(\phi^4 \) theory with non-integer propagator power. The #tropicalFieldTheory we are currently developing is also of that type. And secondly, the algebraic/combinatorial operations they use seem to fit nicely into a Hopf algebra description a la Connes-Kreimer (probably, someone has already worked that out in the 20 years since). Besides that, this paper also includes one section that is just a sequence of 24 lemmas, which would be more typical for Wittgenstein's tractatus than for a physics paper. What I also liked was that the paragraphs have individual titles, which makes the structure of arguments very easy to follow. https://link.springer.com/article/10.1007/s00220-003-0895-4
Critical (Φ4)3,ε - Communications in Mathematical Physics

The Euclidean (φ4)3,ε model in R 3 corresponds to a perturbation by a φ4 interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter ε in the range 0≤ε≤1. For ε=1 one recovers the covariance of a massless scalar field in R 3 . For ε=0, φ4 is a marginal interaction. For 0≤ε<1 the covariance continues to be Osterwalder-Schrader and pointwise positive. We consider the infinite volume critical theory with a fixed ultraviolet cutoff at the unit length scale and we prove that for ε>0, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. We construct the stable critical manifold near this fixed point and prove that under Renormalization Group iterations the critical theories converge to the fixed point.

SpringerLink
John TinkerOct 5

Renormalization and Power

What advice can a large language model deliver about the post-genocide situation(s) that so many are struggling to bring to an end?
https://johntinker.substack.com/p/renormalization-and-power #Gaza #genocide #ai #internationallaw #renormalization #chatbot

Renormalization and Power

I am pondering over everything that was not done to prevent the genocide in Gaza, and that so many are complicit in having made it happen, or in having allowed it to happen.

John’s Substack
Khurram Wadee ✅May 20, 2025

#xkcd: #Renormalization

https://xkcd.com/3091/

#Humour #Cartoon

Renormalization

xkcd
JMLRMay 19, 2025

'Stabilizing Sharpness-Aware Minimization Through A Simple Renormalization Strategy', by Chengli Tan, Jiangshe Zhang, Junmin Liu, Yicheng Wang, Yunda Hao.

http://jmlr.org/papers/v26/24-0065.html

#sgd #minimization #renormalization

Stabilizing Sharpness-Aware Minimization Through A Simple Renormalization Strategy

Paul BaldufMar 5, 2025
In #physics, #quantum field theory is used to describe the behavior of elementary particles. #Feynman diagrams are used to visualize, and compute, the "elementary" processes that can happen. However, the processes that really occur in nature are a sum of infinitely many Feynman diagrams. Of course, in an actual computation, one can only include finitely many processes, and all the other ones need to absorbed into some "effective" parameters, such as effective charges. This is called #renormalization, and it involves a freedom regarding how exactly one defines the effective parameters. Two renormalization schemes are common in high energy physics: In "kinematic renormalization", one defines the effective parameters as the actually measured values of a certain scattering process. In "minimal subtraction", one chooses the effective parameters such that the computation is as easy as possible, regardless of what the parameters mean concretely.
Certain infinite sums of Feynman diagrams, called "rainbows" (see picture), had been computed in minimal subtraction 30 years ago. In a recent preprint https://arxiv.org/abs/2503.02079 I computed the analogous sums in the minimal subtraction scheme. The solution is structurally similar to the known one, but they involve slightly more complicated functions.
The sum of rainbows by itself is not a physically relevant observable. But since it is one of the few infinite classes of Feynman diagrams that can be solved exactly, it is often used as a model to describe qualitative features, such as how quickly these sums grow if one includes more and more terms.
Ladders and rainbows in Minimal Subtraction

In dimensional regularization with $D=D_0-2ε$, the minimal subtraction (MS) scheme is characterized by counterterms that only consist of singular terms in $ε$. We develop a general method to compute the infinite sums of massless ladder or rainbow Feynman integrals in MS at $D_0$. Our method is based on relating the MS-solution to a kinematic solution at a coupling-dependent renormalization point. If the $ε$-dependent Mellin transform of the kernel diagram of the insertions can be computed in closed form, we typically obtain a closed form expression for the all-order solution in MS. As examples, we consider Yukawa theory and $ϕ^4$ theory in $D_0=4$, and $ϕ^3$ theory in $D_0=6$.

arXiv.org
Paul BaldufMay 20, 2024
Two years ago, I began writing my #doctoralThesis in theoretical #physics. Most effort went into giving a very detailed pedagogical account of what the #renormalization #HopfAlgebra in #QuantumFieldTheory does, and why it is natural and transparent from a physical perspective.
One year ago, my referees recommended in their reports to publish the thesis as a book, and today I received the printed copies!
It was exciting to go through all the steps of actually publishing a book, and I hope that it will be of use to convince physicists that the Hopf algebra structure in #QFT is not a weird mathematical conundrum, but it actually encodes the very way physicists have been thinking of renormalization since the 1950s: Parametrize a theory by quantities one can actually measure, instead of fictional expansion parameters.
https://link.springer.com/book/10.1007/978-3-031-54446-0
Dyson–Schwinger Equations, Renormalization Conditions, and the Hopf Algebra of Perturbative Quantum Field Theory

This book offers a systematic introduction to Hopf algebra of renormalization in quantum field theory, with a special focus on physical motivation

SpringerLink
KilleansRow 🇺🇲 🇺🇦🍀Feb 8, 2024
#Physics Errata & Poetry Dept:
Related to that special way that velocity and mass are not independent we know that time can *dilate* - temporal windows can shrink or expand. Our gauge metrics often appear to be wholly independent of the temporal but we often refuse to do the obvious thing...and then ignore what we are doing in lieu of the obvious thing...and so weakly interacting regimes heckle us when time is distance...or not. In other news and questions why does #Renormalization work ?