#tropicalfieldtheory — Public Fediverse posts

In December, there is the 15. Latin American Symposum on nuclear #physics and applications in São Paulo, Brazil. #LASNPA 2026.
The goal is to strengthen collaboration among Latin American and international institutes in nuclear research and applications, topics are ranging from hadronic physics and fundamental symmetries to instrumentation and medicine.
Registration and submission of abstracts is open until 10 July.
https://indico.global/event/15832/
#scientificConference
(coincidentally, the "tropical" in #tropicalFieldTheory or tropical semirings is also named after São Paulo, but that's unrelated to nuclear physics)

Monday's #paperOfTheDay is "Renormalization Group flows between Gaussian Fixed Points" from 2022. This preprint concerns scalar #quantumFieldTheory with different choices of the propagator. Conventionally, one has (in a massless theory) a propagator of the form 1/p^2, corresponding to a kinetic term of second derivatives. However, there could be (i.e. it is generated by quantum fluctuations) also 2-point interactions proportional to more derivatives, in particular a fourth derivative. This raises the question whether one can equivalently use that term as the propagator, i.e. assign the value 1/p^4 to edges in #FeynmanIntegral s, and use the other term as an interaction vertex. In principle that works, but it leads to a number of technical issues such as having states with negative norm (ghosts).
The present preprint takes a different perspective: At low energies (consider e.g. plane waves with long wavelength), a fourth derivative will be numerically small, while it dominates at high energy. One can therefore view the transition from one choice of propagator to the other as a #renormalization group flow that starts in the UV with a fourth derivative, and arrives at a second derivative in the IR. An analogous argument has long been known for a mass term (i.e. 2-point term with zero derivatives): In the UV, the kinetic term p^2 determines the behaviour of the field (e.g. UV convergence of Feynman integrals), whereas at low energy, every propagator is essentially constant 1/m^2. Notice that all these transitions are taken at fixed spacetime dimension, whereas #tropicalFieldTheory is an analogous limit to zero derivatives in zero dimensions, which gives a different result.
https://arxiv.org/abs/2207.10596v1

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

Thursday's #paperOfTheDay is "Tropical Mathematics" from 2009.
I'm currently developing a version of #QuantumFieldTheory called #tropicalFieldTheory . The present article is background on what "tropical" means in #mathematics : This term has first appeared in the context of #computerScience in the 1970s, and it was coined in honor of the early work being done in São Paulo, Brasil. The basic idea is to consider a special type of (mathematical) ring: A typical example of a ring would be the real numbers, together with addition and multiplication. Now, the "tropical semiring" is the real numbers and infinity, but "addition" is replaced by "taking minimum", while "multiplication" is replaced by "addition". This strange object behaves well in many ways. For example, in the usual ring of real numbers one would have
7 + 2*3 = 7+6 = 13
in the tropical semiring, the same equation becomes
min{ 7, 2+3 } = min { 7, 5 }= 5.
The tropical semiring is only SEMI because taking minimum does not always have an inverse: There is no x such that min {x,5}=8 .
In the following decades, tropical arithmetics has been developed into a full mathematical theory. In particular, ome has tropical polynomials, where the conventional addition of monomials is replaced by taking minimums. This is exactly what we do in tropical field theory: The #FeynmanIntegral s are integrals over rational functions, and we replace their denominators and numerators by tropical polynomials.
Today's article was written before tropical field theory, but it discusses a nice application from #biology : One can compute phylogenetic trees with the help of tropical algebraic geometry.
https://arxiv.org/abs/math/0408099

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

#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

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

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

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

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

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

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