Mastodawn

Friday's #paperOfTheDay is "The functional f(R) approximation" from 2022. This is a review article about a certain approach to #quantum #gravity in #physics . Namely, the "asymptotic safety" scenario, which asserts that although the Einstein-Hilbert action is perturbatively not renormalizable, it will at high energies give rise to an interacting fixed point, so that the observables in fact stay finite.
In principle, the behaviour of high-energy quantum gravity can be computed with the methods of functional #renormalization group equations. In practice, a number of assumptions and approximations are required, for example choosing a suitable splitting of the full metric field into a background and quantum-fluctuations, and choosing an IR cutoff functional that leads to analytic simplifications. Of particular importance is the choice of truncation for the effective action: The effective action is the generating function of correlation functions, it contains all physical information. In gravity, these correlation functions can potentially depend on all possible tensor structures, and have an arbitrary dependence on momenta. The earliest truncation, used in the 1990s, was to assume that there are only two terms: One proportional to the cosmological constant, and one proportional to the curvature R. By now, many further terms have been included. The present review analyzes the case where an arbitrary function f(R) of the curvature is allowed. This includes arbitrary powers R^n, but also trans-monomials like exp(1/R).
https://arxiv.org/abs/2210.11356

The functional $f(R)$ approximation

This article is a review of functional $f(R)$ approximations in the asymptotic safety approach to quantum gravity. It mostly focusses on a formulation that uses a non-adaptive cutoff, resulting in a second order differential equation. This formulation is used as an example to give a detailed explanation for how asymptotic analysis and Sturm-Liouville analysis can be used to uncover some of its most important properties. In particular, if defined appropriately for all values $-\infty<R<\infty$, one can use these methods to establish that there are at most a discrete number of fixed points, that these support a finite number of relevant operators, and that the scaling dimension of high dimension operators is universal up to parametric dependence inherited from the single-metric approximation. Formulations using adaptive cutoffs, are also reviewed, and the main differences are highlighted.

arXiv.org

Paul Balduf 5d ago

#paperOfTheDay for Wednesday is "Form factors in quantum gravity: Contrasting non-local, ghost-free gravity and Asymptotic Safety" from 2022.
Unlike all other elementary forces, #gravity does not straightforwardly make sense as a perturbative #quantumFieldTheory . This has given rise to a number of alternative approaches over the decades, two of which are being compared in today's paper.
The first one is "asymptotic safety", which, roughly, asserts that the conventional Einstein-Hilbert action is indeed the correct low energy description, but at higher energies, it does not simply blow up as could be expected from naive power counting. Instead, the strong gravity interaction at high energy (or equivalently at short scale) produce a state that is essentially scale invariant: An interacting fixed point. To study this behaviour, one usually resorts to numerical integrations of flow equations of the functional renormalization group.
The second approach is non-local ghost free gravity, where one assumes that, in perturbation theory, the propagator secretly has an exponentially decaying factor that only becomes relevant at high energies. This renders the theory renormalizable because it eliminates UV divergences.
The two approaches can also be interpreted in terms of two different, momentum-dependent, wave-function #renormalization factors. They correspond to rather different high-energy behaviour, which, however, is far beyond current range of experimental data.
https://www.sif.it/riviste/sif/ncc/econtents/2022/045/02/article/3

Paul Balduf Mar 31

Since yesterday I've been on a conference about functional #renormalization group in #physics , taking place in #Trento in Italy. Consequently I've been learning about many old and new papers on that topic. We start with something old. Monday's #paperOfTheDay is "Renormalization Group Equation for Critical Phenomena" from 1973. This is (one of) the very first papers to introduce what is now know as functional renormalization group methods, namely, the idea to solve the path integral by integrating out only one momentum shell at a time, while keeping track of an effective action that "flows" from the classical action to the full quantum effective action. This particular paper works with spins and also examines the limit N->infinity of the O(N) symmetric model.
What I found particularly interesting was the general argument why the right-hand side of such a flow equation can contain at most second derivatives of the effective action: This has to do with expectation values of products of many spins decaying quickly in the continuum limit. Also the tropical loop equation in #tropicalFieldTheory has second derivatives (as has the analogous equation for 0-dimensional QFT). There, this property was obvious from Feynman diagrams: Cutting a loop in a Feynman diagram amounts to cutting exactly one propagator, and every propagator has exactly two ends, therefore this necessarily produces a diagram with two more legs, hence a second derivative in the generating function.
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.8.401

Paul Balduf Mar 26

#paperOfTheDay for Wednesday is "Dimensional renormalization: The number of dimensions as a regularizing parameter" from 1972. As the title suggests, this is one of the articles that first introduced dimensional regularization.
In perturbative #QuantumFieldTheory (or statistical physics), one encounters #FeynmanIntegral s which are divergent. These divergences are eventually removed through #renormalization , but in order to even get to that point, one first needs to assign some value to these integrals. This is called regularization. Various methods of regularization are known, but the typical problem is that they destroy symmetries of the theory. Dimensional regularization was a breakthrough for practical computation of Feynman integrals because it respects many symmetries.
The basic idea is to define an integral for non-integer dimension of spacetime. This is done, essentially, by analytic continuation: We know what it means to take a first, second, third etc derivative of a function, and to integrate it once, twice, thrice etc. If the function is spherically symmetric (i.e. depends only on the radius of spherical coordinates), then the "count" of the integrals or derivatives appears as an explicit number in intermediate steps. For example, the volume element in 3 dimensional spherical coordinates is r^2*dr*(angular part), where the exponent "2" represents dimension D=2+1=3. Basically, you could insert any number in place of the "2", and declare this to be the D-dimensional integral. Of course, in reality this is more sophisticated, but the basic idea is very much in this spirit.
https://link.springer.com/article/10.1007/BF02895558

Dimensional renorinalization : The number of dimensions as a regularizing parameter - Il Nuovo Cimento B (1971-1996)

We perform an analytic extension of quantum electrodynamics matrix elements as (analytic) functions of the number of dimensions of space(ν). The usual divergences appear as poles forν integer. The renormalization of those matrix elements (forν arbitrary) leads to expressions which are free of ultraviolet divergences forν equal to 4. This shows thatν can be used as an analytic regularizing parameter with, advantages over the usual analytic regularization method. In particular, gauge invariance is mantained for anyν.

SpringerLink

Paul Balduf Mar 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 Balduf Feb 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 Balduf Feb 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 Balduf Jan 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 Tinker Oct 5, 2025

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