Mastodawn

Thursday's #paperOfTheDay: "Almost zero-dimensional quantum field theories" from 1992. That paper considers the behaviour of #quantumMechanics and #quantumFieldTheory close to zero spacetime dimensions. The actual limit D=0, the zero-dimensional field theory, is well understood. The authors now study a (radially symmetric) Schrödinger equation, and then a free field theory, close to D=0. They find that this limit exists (i.e. there is a continuous family of theories for real parameters D which interpolates between the physical and the 0-dimensional theory), and the linear approximation in D already gives numerically meaningful estimates of the physical theory.
This setup is in the same spirit as our #tropicalFieldTheory , but the difference is that the older paper varies D alone, whereas the tropical limit arises when one reduces D and the power of the kinetic term (i.e. spacial decay rate of propagators) simultaneously. Secondly, we now have a much better understanding of analytical properties of #FeynmanIntegrals than 30 years ago, so that we can perform this limit in a mathematical clean way for all graphs of an interacting field theory. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.46.5557

Paul Balduf Mar 1

This Sunday's #paperOfTheDay is "Solutions to Nonlinear Fractional Duffing Oscillator using MsDTM" from 2026.
One possibility to derive #tropicalFieldTheory is a scaling limit of long range interacting field theory, see https://paulbalduf.com/research/tropical-field-theory/tropical-motivation/ . The equation of motion for such theories includes, in place of an ordinary Laplacian (i.e. second derivative), a non-integer derivative operator. There are numerous different ways to define such operators, and they have been studied in various areas of #mathematics and #physics in recent years. Roughly speaking, having a non-integer derivative in a differential equation is equivalent to an integral operator, so that the solutions show much stronger memory and non-locality effects than usual differential equations.
The present paper is concerned with a novel method to solve non-integer differential equations numerically. I really like the authors' exposition of the background and broader context. They then display several numerical solution curves they computed, which are nice illustrations of the qualitative effects of the non-integer differential equation under consideration. This paper is a follow-up on another paper that actually introduced the method. The authors make numerous comments how the new method is superior to existing ones, and I agree with this intuitively, but unfortunately they miss the opportunity to show concrete plots or numbers for comparison. Nonetheless interesting work, you learn something new every day (in particular when you do a #dailyPaperChallenge ) https://link.springer.com/article/10.1007/s10773-025-06210-3

Motivation for Tropical Field Theory – Paul-Hermann Balduf

Paul Balduf Feb 23

#paperOfTheDay in my #dailyPaperChallenge is "Modular resurgent structures" from 2024. There are many relevant functions in #physics and #mathematics that are not expressible as convergent Taylor series (for a simple example, think of the square root function around the origin). If one attempts to compute these functions in perturbation theory, the resulting series are divergent, and it makes no sense to "insert a value" into them. However, it turns out that they do in fact contain a lot, and sometimes even all, information about the true function they should represent. "Resurgence" is the method to recover this information. The present paper analyzes a somewhat controlled restricted case, namely, when the Borel transform of the function in question has only one (infinite) sequence of simple pole or logarithmic singularities. Then, one can rearrange the various sums to expose a number-theoretic function, the L-function, of the residues of the poles ("Stokes constants"). This situation does in fact occur in certain physical models.
Unfortunately for me, the structure of #tropicalFieldTheory is more complicated (namely, there is one infinite sequence of singularities, but they are much more complicated than being simple poles), so I can not use this method directly for my own research. Nevertheless, I find it a very interesting and novel approach to consider generating functions of Stokes constants. https://arxiv.org/abs/2404.11550

Modular resurgent structures

The theory of resurgence uniquely associates a factorially divergent formal power series with a collection of exponentially small non-perturbative corrections paired with a set of complex numbers known as Stokes constants. When the Borel plane displays a single infinite tower of singularities, the secondary resurgent series are trivial, and the Stokes constants are coefficients of an $L$-function, a rich analytic number-theoretic fabric underlies the resurgent structure of the asymptotic series. We propose a new paradigm of modular resurgence that focuses on the role of the Stokes constants and the interplay of the $q$-series acting as their generating functions with the corresponding $L$-functions. Guided by two pivotal examples arising from topological string theory and the theory of Maass cusp forms, we introduce the notion of modular resurgent series, which we conjecture to have specific summability properties as well as to be related to quantum modular forms.

arXiv.org

Paul Balduf Feb 4

My #paperOfTheDay is "A Nonlocal Schwinger Model" from 2024. The ordinary Schwinger model is a 2-dimensional #quantumFieldTheory consisting of electrons and photons, it is a toy model for confinement (which, in 4 dimensions, is the mechanism that prevents loose quarks from coming out of protons). In this nonlocal version of the Schwinger model, one instead allows the photon to have an arbitrary dimension 2<d<4. This leads to a lot of surprising effects. While in the 2-dimensional theory, the photons "condensate" and become massive, this no longer happens at d>2, and the would-be value of the mass is instead a complex number that moves around as d increases. Some of these effects are reminiscent of what we observe in #tropicalFieldTheory when the kinetic term of a scalar field theory is given a non-integer power. #dailyPaperChallenge https://arxiv.org/abs/2412.02514

A Nonlocal Schwinger Model

We solve a system of massless fermions constrained to two space-time dimensions interacting via a $d$ space-time dimensional Maxwell field. Through dimensional reduction to the defect and bosonization, the system maps to a massless scalar interacting with a nonlocal Maxwell field through a $F ϕ$-coupling. The $d=2$ dimensional case is the usual Schwinger model where the photon gets a mass. More generally, in $2<d<4$ dimensions, the degrees of freedom map to a scalar which undergoes a renormalization group flow; in the ultraviolet, the scalar is free, while in the infrared it has scaling dimension $(4-d)/2$. The infrared is similar to the Wilson-Fisher fixed point, and the physically relevant case $d=4$ becomes infrared trivial in the limit of infinite ultraviolet cut-off, consistent with earlier work on the triviality of conformal surface defects in Maxwell theory.

arXiv.org

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