@Quantensalat yes, I will look into these PT symmetry papers in the next weeks, too. Somehow, this #dailyPaperChallenge has so far increased the backlog of papers I want to read, rather than reducing it...

#paperOfTheDay : "Spectrum continuity and level repulsion: the Ising CFT from infinitesimal to finite epsilon" from 2023. This article deals with the quartic interacting scalar #quantumFieldTheor across dimensions: At 4 dimensions, this theory is essentially a free theory, but one can change the dimension (mathematically) by a small number epsilon, and consider what happens. This produces power series in epsilon, which then, in principle, can be resummed to obtain predictions at 3 dimensions which are important for #physics .
This is the epsilon in the title, but the paper actually uses a different method, namely numerical conformal bootstrap. It is non-perturbative in the sense that it does not involve the expansion in some small quantities, but still it is another type of approximation (or rather truncation) scheme, not an exact solution of the full theory. This method is relatively powerful at computing predictions far away from 4 dimensions, they go as far down as 2.6. The critical exponents of various operators change with dimensions, but, similar to eigenvalues of random matrices, they never cross each other. This is hard to see from low-order perturbation theory, but it is explicitly observed with these numerical bootstrap methods. Besides the results, the paper is also nice as a high-level introduction for dimensional dependence of operators in field theory. #dailyPaperChallenge
https://link.springer.com/article/10.1007/JHEP02(2023)218

The #paperOfTheDay is the most bizarre one I have read in the whole #dailyPaperChallenge so far: It's "Geometrisches zur Abzählung der reellen Wurzeln algebraischer Gleichungen" from 1892. It is well known that a polynomial equation of order n has exactly n complex solutions. The present article deals with the question how many of these are real: Consider for example a quadratic equation, its curve is a parabola, and a parabola can have zero, one, or two intersections with the y=0 axis. These cases can be distinguished by a calculation, without drawing the curve. More generally, in equations of higher order, these things become more involved, but Klein argues that they still correspond to concrete questions in geometry.
What is so special about this article is where it appeared: In the book "Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente", published by the Deutsche Mathematiker-Vereinigung. That is, in 1892 the German mathematical association published a printed book about mechanical calculation devices and physical equipment, which also contained a few pure #mathematics research articles. The common theme of all this is that it is in some way related to visualization or geometric realization of mathematics.

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

For the #dailyPaperChallenge, I have read the #paperOfTheDay "Four-fermion interaction near four dimensions" from 1991. There are a few model systems that have been studied extensively in #quantumFieldTheory and wider theoretical #physics . Among them are the phi^4 theory (scalar field with quartic interaction) which can be treated perturbatively in 4-dimensional spacetime. The Gross-Neveu model (Fermions with quartic interaction) and nonlinear sigma model (scalars with non-monomial interaction) are renormalizable only in 2-dimensional spacetime. For each of them, a large-N expansion can be used as an alternative to a small-coupling expansion. The present paper, firstly, is a nice overview of the features of these various models. Secondly, it studies the possibility to have effective 4-fermion interactions in 4 dimensions. The Gross-Neveu model as such is not renormalizable there, but one can, essentially, couple it to a scalar field theory, which then effectively acts as a mediator to enable 4-fermion interactions. A similar mechanism is at play in the real world, in the standard model of particle physics. The difference is that the true mediators (the standard model gauge bosons) carry spin and charges and complicated interactions, whereas the present model uses a much simplified version of an intermediate particle. https://www.sciencedirect.com/science/article/pii/055032139190043W

#paperOfTheDay is "The axial vector current in beta decay" from 1960. This paper was written at a time where the #quantumFieldTheory of the weak force was not known, and the interactions of elementary particles were quite mysterious. There were all sorts of experimental observations, and theorists had to come up with models to describe these results. In practice, this means you propose that certain fields exist, and how they interact, and then you do a calculation to show that this gives the observed result.
The present paper introduced what is now known the "nonlinear sigma model". The "linear sigma model" had been known before, it gave reasonable predictions, but postulated the existence of a particle that had not been observed. The key innovation in the present paper is to require another constraint, so that the un-observed field is not actually a particle, but it gets removed from the theory through the new constraint. This constraint amounts to demanding that the field sigma only changes its direction, but has constant magnitude.
Today we know that the nonlinear sigma model does not describe neither strong nor weak force in nature. However, it has become one of the most widely studied "toy model" systems in theoretical #physics because it is relatively easy, and at the same time produces interesting non-trivial effects. #dailyPaperChallenge https://link.springer.com/article/10.1007/BF02859738

#paperOfTheDay is "Disquisitiones Generales circa seriem infinitam" by Carl Friedrich Gauss 1812. Probably one of the most influential #mathematics publications of all times. Gauss introduces a certain class of infinite series -- today known as Gaussian hypergeometric series -- which describe a new class of transcendental functions, namely the Gaussian hypergeometric function. Essentially all other transcendental functions known at that time are special cases of it. Gauss proves numerous properties and identities, including the relation to the Euler gamma function, and new identities for it. Also, he introduces the concept of contiguous relations (which today are a workhorse of perturbative #quantumFieldTheory under the name "IBP-relations"). He also comments on the monodromy arising when the function is being continued around one of the singular points of the differential equation.
I have read the German translation from 1888, which I find surprisingly clear and easy to follow. There is essentially no technical language (because Gauss was decades ahead of the community), but instead simple explicit calculations that any student can follow, and which lead to deep conclusions when interpreted cleverly. #dailyPaperChallenge https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0002_2NS

#paperOfTheDay : "The Brownian loop soup" from 2003. This is a #mathematics paper about random walks in the plane, but there is a famous concrete example from #physics : the 2-dimensional Brownian motion. Generically, the trajectory of such a random walk in a plane can intersect itself. A special case are the non-intersecting random walks. They can be obtained by taking a self-intersecting one, and whenever it intersects itself, one discards the "loop" part (shown in blue in my drawing). What remains is a random walk without self intersection (shown in black).
The purpose of the present paper is to prove that this situation has an equivalent second interpretation: One can view it as a non-intersecting random walk that proceeds through a "Brownian loop soup", namely the collection of random self-intersecting detached loops. Whenever the walk intersects a loop for the first time (green dot), one glues in this loop, and thereby obtains a self-intersecting walk. The non-trivial proof is that this is not just "similar", but in fact mathematically equivalent: The properties of distributions are identical regardless if one starts from the loop soup, or from a self-intersecting walk. #dailyPaperChallenge https://arxiv.org/abs/math/0304419

#paperOfTheDay is "1/n Expansion: Calculation of the exponent nu in the order 1/n^3 by the Conformal Bootstrap method" from 1982. This paper is the first computation of the third-order correction in 1/N of the critical exponent of phi^4 #QuantumFieldTheory They use the mapping of the theory to a sigma model, which has the advantage of making explicit the N-dependence of diagrams. Then, they use "conformal bootstrap" to determine the sought-after values. This method is quite clever: At the critical point, the theory is conformal. Therefore, one can essentially do a perturbation calculation around the conformal theory. This has the advantage that the functional dependence of propagators and vertices is under control (the full momentum dependence of these functions is infinitely complicated). My impression is that the Broadhurst-Kreimer "Hopf algebra" approach to solving Dyson-Schwinger equations is essentially a mathematical formulation of the same idea. To the best of my knowledge, this has never been discussed in the literature, probably because the BK formalism puts much emphasis on mathematical precision rather than intuition. #dailyPaperChallenge https://link.springer.com/article/10.1007/BF01015292

#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

The #paperOfTheDay is "Time fractional Schrödinger equation" from 2004. Physicists working in #QuantumFieldTheory are familiar with the concept of non-integer integrals from dimensional regularisation: Basically one computes the integral in spherical coordinates, then the radial part of the integration contains the dimension D as a parameter, and one declares the result to be the value of the integral even if the dimension is not integer. By taking derivatives of this non-integer integral, one constructs a non-integer derivative (e.g. take 2 derivatives of an integral at D=1.3, obtain a 0.7-fold derivative).
The present paper discusses what happens if one uses such a derivative in place of the 1-fold time derivative in the Schrödinger equation. Apparently, one can to some extent work with this, but the outcome is a wave function where probability is not conserved. I don't know how to make sense of this physically, but the converse is a nice conclusion: In order to have a unitary time evolution, and energy levels that stay constant in a static system, the time derivative must be of integer order. #dailyPaperChallenge https://doi.org/10.1063/1.1769611

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

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

Thursday's #paperOfTheDay (which I only finished reading in the train on Friday) was "Current algebra and Wess-Zumino model in two dimensions" from 1984. This paper is an analysis of the various constraints that exist for correlation functions in the Wess-Zumino-Witten model. The authors show that these functions must satisfy a partial differential equation, nowadays know as the Knizhnik-Zamolodchikov equation, which they solve for one particular situation.
This paper is one of the classical works of what is now known as "mathematical #physics ": The application of systematic modern mathematical methods (here predominantly algebra) to a model system that resembles some characteristics of real-world physics. Also the opposite direction exists, that is, the use of physics-inspired models in order to prove mathematical results. #dailyPaperChallenge https://doi.org/10.1016/0550-3213(84)90374-2

Today's #paperOfTheDay is "Why there is Nothing rather than something: A theory of the cosmological constant" from 1988. Like yesterday's paper, it deals with the intersection between quantum field theory and #generalRelativity, but the 30 years between them clearly show. Coleman's 1988 paper is an argument in the style of that time (which structurally is quite similar to much of the older #renormalon literature): Heuristic manipulations of formal objects such as the wave function of the universe, or divergent sums over all spacetime geometries. The outcome of this argument is that if #wormholes exist (caused by quantum effects at a scale that is much smaller than observations, but larger than the Planck scale), they can drive the cosmological constant to zero in an Euclidean path integral formulation of general relativity. As always with Coleman, the language is quite funny and frank about the paper's limitations: He writes "Although I find this theory in many ways very attractive, I must honestly stress its speculative character. It rests on wormhole dynamics and the Euclidean formulation of quantum gravity. This is doubly a house built on sand. [...] the Euclideon formulation of gravity is not a subject with firm foundations and clear rules of procedure; indeed, it is more like a trackless swamp". Observations like these have by now, 30 years later, led to a style of theoretical physics that is much more systematic and mathematical than in the 1980s, but also sometimes less intuitive. #dailyPaperChallenge https://doi.org/10.1016%2F0550-3213(88)90097-1

On Tuesday, I have read "Bell's Theorem for Temporal Order" from 2017 in my #dailyPaperChallenge. This paper is concerned with one of the big open questions in #physics : What happens when quantum mechanics and general relativity are applied simultaneously? Most theoretical work in that direction aims towards a #quantumFieldTheory of #gravity, that is, it seeks to understand the behaviour of "elementary particles of gravity" along the same lines as any other elementary particle. The present paper asks another question: In quantum theories, the causal (or temporal) order of events is well defined: Something happens first, and something else afterwards. In general relativity, whether or not something was "before" something else depends on time dilatation, which can be caused by massive objects. Such objects should in principle be subject to quantum mechanics. Hence, the proposal is: Create a specific type of an entangled state, where the entanglement refers to the causal order of operations. It is very hard to realize this with current laboratory equipment, but in principle it should be possible, and one can then decide with a concrete measurement whether quantum mechanical laws apply to causality or not, similar to the #BellInequalities. Obviously, this is all quite philosophical for now, but interesting nonetheless. #paperOfTheDay https://arxiv.org/abs/1708.00248

The #paperOfTheDay is "Renormalons and fixed points". This article from 1996 investigates the relation between #renormalons and infrared behaviour of #QCD. A renormalon is an effect that can lead to the divergence of a perturbation series, and such effects have been observed in various contexts. What has never become quite clear (at least to me) is the precise logical relation between its different incarnations: Divergence of the series, Landau poles, the peculiarities of QCD (renormalons exist in scalar theories as well!), non-trivial fixed points, and questions of uniqueness and resummability. The present paper points out some difficulties -- namely that some of the quantities involved are only defined perturbatively, or are sensitive to choices of analytic continuation. These considerations are interesting and not trivial, but I find it sometimes hard to follow the article since it has no explicit structure such as subsections or theorems, it is simply one continuous discussion. Or maybe I've just become too much of a mathematician by now.
#dailyPaperChallenge https://www.sciencedirect.com/science/article/pii/0370269396000615

Sunday's #paperOfTheDay is "Multi-loop spectra in general scalar EFTs and CFTs" from July 2025. Interacting statistical systems have the feature that close to their critical point, certain quantities behave according to power laws. For example, when a magnet is cooled towards the Curie temperature, the correlation function between the individual spins decays with the distance, and the decay constant (i.e. correlation length) increases as a power law of the temperature difference to the critical point. The corresponding exponents are called critical exponents, and they can be computed with #quantumFieldtheory. This had been done for decades, and results are scattered in the literature, and often restricted to only specific theories or quantities that were of interest to the authors at that time. The present paper makes a systematic effort to use state-of-the-art computation of Feynman integrals in order to arrive at critical exponents for various operators. The results are largely consistent with other methods, and they provide a nice overview of how various correlation functions behave at critical theories. #dailyPaperChallenge https://arxiv.org/abs/2507.12518

#paperOfTheDay is the third of the three classical amplitudes: Shapiro's generalization of the Virasoro amplitude from 1970. What I like particularly about this is that it is motivated from classical electrodynamics. This relates to my findings about Feynman integrals at large loop order, which can be reasonably approximated by electrical networks. #dailyPaperChallenge https://www.sciencedirect.com/science/article/abs/pii/0370269370902558

A few days ago in the #dailyPaperChallenge I read Veneziano's proposal for a 4-point amplitude. This Friday, my #paperOfTheDay was "Alternative Construction of Crossing-Symmetric Amplitudes with Regge Behaviour" from 1969, were another, more general, expression is proposed by Virasoro. Overall, the spirit is very similar to Veneziaon's article: Propose a formula and discuss its properties. In particular, the Virasoro amplitude reduces to the Veneziano one if an extra condition is imposed, and at the same time it is argued that this condition is not satisfied for some realistic scattering processes, and therefore Virasoro's amplitude should be expected to better reflect reality than Veneziano's. Again, such heuristic arguments have become somewhat obsolete by now since we now know #QCD as a fundamental theory, and don't have to guess amplitudes any more. Still, the Virasoro amplitude stays relevant for certain theoretical considerations. https://journals.aps.org/pr/abstract/10.1103/PhysRev.177.2309