#paperoftheday — Public Fediverse posts

The #paperOfTheDay is a book by Gauss: "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (theory of the motion of the heavenly bodies about the sun in conic sections) from 1809. Gauss wrote this in German, but had to translate it to Latin to get it published, and a few decades later various translations (including back to German) of this Latin text appeared.
The bulk of the 300 page book is a detailed practical guide how to compute orbits of celestial bodies from observations, how the various parameters of the orbits are related, etc. Many of these calculations are straightforward for us thanks to electronic computers, but back then, a major consideration was how to solve equations with the least amount of work, given only tables of logarithms and trigonometric functions.
Besides practical algorithms, the book introduces major innovations in pure #mathematics . In particular, in the process of approximately solving a certain transcendental equation, Gauss arrives for the first time at the power series of what is now known as Gaussian hypergeometric functions (whose mathematical theory he developed in a separate paper that I mentioned a few weeks ago). Also, he argues that independent experimental observations should give rise to values distributed according to what we now know as Gaussian normal distribution. Using this distribution, he develops the method of least squares to solve an over-determined system of equations in the best possible way (this method had been known to other authors previously, but apparently not widely used and systematically understood). #astronomy #physics
https://openlibrary.org/books/OL19572834M/Theoria_motus_corporum_coelestium

The #paperOfTheDay is a book by Gauss: "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (theory of the motion of the heavenly bodies about the sun in conic sections) from 1809. Gauss wrote this in German, but had to translate it to Latin to get it published, and a few decades later various translations (including back to German) of this Latin text appeared.
The bulk of the 300 page book is a detailed practical guide how to compute orbits of celestial bodies from observations, how the various parameters of the orbits are related, etc. Many of these calculations are straightforward for us thanks to electronic computers, but back then, a major consideration was how to solve equations with the least amount of work, given only tables of logarithms and trigonometric functions.
Besides practical algorithms, the book introduces major innovations in pure #mathematics . In particular, in the process of approximately solving a certain transcendental equation, Gauss arrives for the first time at the power series of what is now known as Gaussian hypergeometric functions (whose mathematical theory he developed in a separate paper that I mentioned a few weeks ago). Also, he argues that independent experimental observations should give rise to values distributed according to what we now know as Gaussian normal distribution. Using this distribution, he develops the method of least squares to solve an over-determined system of equations in the best possible way (this method had been known to other authors previously, but apparently not widely used and systematically understood). #astronomy #physics
https://openlibrary.org/books/OL19572834M/Theoria_motus_corporum_coelestium

#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/

#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/

#paperOfTheDay is Euler's "De summis serierum reciprocarum" from 1740. This article concerns the sum of inverse squares, like
1+ 1/4 + 1/9 + 1/16 + ...
Euler starts with the remark that such series had been studied intensively by various scholars, including himself, where nobody had been able to find an exact value for their sum. Now, however, he found a new approach. The key idea is to examine the power series representations of sine and cosine, such as
sin(x) = x - x^3/(1*2*3) + x^5/(1*2*3*4*5) - ...
One then studies the inverse relation, i.e. for which x is sin(x)=0? Clearly x=0, but what are the others? Geometrically, we know that these are the multiples of pi, but it must also follow from the series, by studying the solutions of
0 = 1-x^2/(1*2*3) + x^4/(1*2*3*4*5) - ....
If the right and side vanishes at some x=A, this means that it can be written with this linear factor, i.e. (x-A)*(something else). Using a sequence of clever transformations, one eventually isolates the sought-after sum of inverse squares, and finds its value to be pi^2/6. Likewise, the sums of other inverse even powers can be found, and evaluated to powers of pi, but not the sums of inverse odd powers.
From today's perspective, this is reflected in the fact that the Riemann zeta function, for even arguments, evaluates to a power of pi, whereas it gives complicated numbers for odd arguments.
The only non-trivial input needed for Euler's paper is the series representation of sine and cosine, and the geometric knowledge how these functions are periodic, and related to pi. Everything else is elementary algebra, used in an ingenious way.
#mathematics
https://scholarlycommons.pacific.edu/euler-works/41/

#paperOfTheDay is Euler's "De summis serierum reciprocarum" from 1740. This article concerns the sum of inverse squares, like
1+ 1/4 + 1/9 + 1/16 + ...
Euler starts with the remark that such series had been studied intensively by various scholars, including himself, where nobody had been able to find an exact value for their sum. Now, however, he found a new approach. The key idea is to examine the power series representations of sine and cosine, such as
sin(x) = x - x^3/(1*2*3) + x^5/(1*2*3*4*5) - ...
One then studies the inverse relation, i.e. for which x is sin(x)=0? Clearly x=0, but what are the others? Geometrically, we know that these are the multiples of pi, but it must also follow from the series, by studying the solutions of
0 = 1-x^2/(1*2*3) + x^4/(1*2*3*4*5) - ....
If the right and side vanishes at some x=A, this means that it can be written with this linear factor, i.e. (x-A)*(something else). Using a sequence of clever transformations, one eventually isolates the sought-after sum of inverse squares, and finds its value to be pi^2/6. Likewise, the sums of other inverse even powers can be found, and evaluated to powers of pi, but not the sums of inverse odd powers.
From today's perspective, this is reflected in the fact that the Riemann zeta function, for even arguments, evaluates to a power of pi, whereas it gives complicated numbers for odd arguments.
The only non-trivial input needed for Euler's paper is the series representation of sine and cosine, and the geometric knowledge how these functions are periodic, and related to pi. Everything else is elementary algebra, used in an ingenious way.
#mathematics
https://scholarlycommons.pacific.edu/euler-works/41/

The #paperOfTheDay is "The Ehrhart polynomial of a matroid specializes to the beta invariant" from 2025.
A #matroid is an abstract generalization of a set of vectors in a vector space: Consider some set of k vectors in R^n (allowing k>n). Some of these k vectors could be linearly independent, but assume that not all of them are. If some subset is linearly independent, then so is every subset of that subset. This and a few other properties jointly give the definition of a matroid: Basically, consider the same logical relations arising from linear independence, but without demanding that the objects under consideration are actually vectors in R^n.
If one declares the elements of a matroid to be unit vectors in some abstract vector space (these basis vectors are all linearly independent in that abstract space, whereass not all elements are independent in the matroid), then the maximum set of independent vectors in the matroid amounts to some subset of these abstract vectors, whose convex hull is a polytope. This polytope contains a finite number of points of Z^n. If one rescales the polytope by an integer t, the number of lattice points changes. It turns out that the number of lattice points included is a polynomial of t (not a more complicated function), this defines the Ehrhart polynomial.
Every graph gives rise to a matroid (but not every matroid can be represented as a graph). For graphs, the Tutte polynomial is a classical quantity. The linear term of the Tutte polynomial is the Crapo-beta invariant (and can also be defined for non-graphical matroids). The present paper proves that a linear term of the Ehrhart polynomial coincides with beta. https://arxiv.org/abs/2504.15518 #mathematics #graphTheory

The #paperOfTheDay is "The Ehrhart polynomial of a matroid specializes to the beta invariant" from 2025.
A #matroid is an abstract generalization of a set of vectors in a vector space: Consider some set of k vectors in R^n (allowing k>n). Some of these k vectors could be linearly independent, but assume that not all of them are. If some subset is linearly independent, then so is every subset of that subset. This and a few other properties jointly give the definition of a matroid: Basically, consider the same logical relations arising from linear independence, but without demanding that the objects under consideration are actually vectors in R^n.
If one declares the elements of a matroid to be unit vectors in some abstract vector space (these basis vectors are all linearly independent in that abstract space, whereass not all elements are independent in the matroid), then the maximum set of independent vectors in the matroid amounts to some subset of these abstract vectors, whose convex hull is a polytope. This polytope contains a finite number of points of Z^n. If one rescales the polytope by an integer t, the number of lattice points changes. It turns out that the number of lattice points included is a polynomial of t (not a more complicated function), this defines the Ehrhart polynomial.
Every graph gives rise to a matroid (but not every matroid can be represented as a graph). For graphs, the Tutte polynomial is a classical quantity. The linear term of the Tutte polynomial is the Crapo-beta invariant (and can also be defined for non-graphical matroids). The present paper proves that a linear term of the Ehrhart polynomial coincides with beta. https://arxiv.org/abs/2504.15518 #mathematics #graphTheory

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

The #paperOfTheDay is today a little book on pure #mathematics : "On Applications and Theory of Functional Equations" from 1969. Today, one would perhaps call it a proceeding, it consists of the notes of two different talks given by the author at various meetings in the 1960s. Basically, they are a discussion of classical examples of functional equations. These are, roughly speaking, equations for one or more unknown functions, expressed as relations between these functions for different argument values. Maybe the most famous one is the functional equation
f(x+y) = f(x) + f(y).
Such equations are treated by clever manipulations of the arguments. One can for example set y=0 to find f(x) = f(x) + f(0) and therefore f(0)=0. For rational values of x, the unique solution is f(x) = c*x, with some constant c. If one allows arbitrary real x, an additional condition is required, e.g. that f(x) should be bounded in bounded intervals, or differentiable.
Of course, this book is outdated, and by now the state of affairs might have changed. But at that time, apparently there were only relatively few general theorems about whether a functional equation has solutions at all, or whether it is unique, or how to construct or approximate it. Apparently, sometimes a single functional equation that involves multiple functions can determine all of these functions at once. Compare this to ordinary algebraic equations, or even differential equations, where one often knows at least whether a solution exists and how many free parameters it has, and in many cases there are even efficient numerical procedures for finding approximate solutions.
https://openlibrary.org/books/OL5512535M/On_applications_and_theory_of_functional_equations.

The #paperOfTheDay is today a little book on pure #mathematics : "On Applications and Theory of Functional Equations" from 1969. Today, one would perhaps call it a proceeding, it consists of the notes of two different talks given by the author at various meetings in the 1960s. Basically, they are a discussion of classical examples of functional equations. These are, roughly speaking, equations for one or more unknown functions, expressed as relations between these functions for different argument values. Maybe the most famous one is the functional equation
f(x+y) = f(x) + f(y).
Such equations are treated by clever manipulations of the arguments. One can for example set y=0 to find f(x) = f(x) + f(0) and therefore f(0)=0. For rational values of x, the unique solution is f(x) = c*x, with some constant c. If one allows arbitrary real x, an additional condition is required, e.g. that f(x) should be bounded in bounded intervals, or differentiable.
Of course, this book is outdated, and by now the state of affairs might have changed. But at that time, apparently there were only relatively few general theorems about whether a functional equation has solutions at all, or whether it is unique, or how to construct or approximate it. Apparently, sometimes a single functional equation that involves multiple functions can determine all of these functions at once. Compare this to ordinary algebraic equations, or even differential equations, where one often knows at least whether a solution exists and how many free parameters it has, and in many cases there are even efficient numerical procedures for finding approximate solutions.
https://openlibrary.org/books/OL5512535M/On_applications_and_theory_of_functional_equations.

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "The Diagrammar of Quantum Magnusian" from 2026.
The Magnusian is, morally, the logarithm of the S-matrix in #quantumFieldTheory . It has a perturbative expansion similarly to #FeynmanDiagrams , but these are not exactly the same diagrams, and they come with different combinatorial factors. The Magnus expansion has long been well understood for the case of differential equations, but the application to full quantum field theory is relatively new and so far had been rather ad-hoc, by expanding the corresponding series and explicitly matching terms. The present paper provides a systematic description of which diagrams, and with which combinatorial factors, appear in the Magnus expansion in QFT. Even for a scalar QFT, this is more complicated than ordinary Feynman diagrams, e.g. because the Magnus diagrams have directed edges (representing retarded propagators). https://arxiv.org/abs/2605.25473

#paperOfTheDay is "The Diagrammar of Quantum Magnusian" from 2026.
The Magnusian is, morally, the logarithm of the S-matrix in #quantumFieldTheory . It has a perturbative expansion similarly to #FeynmanDiagrams , but these are not exactly the same diagrams, and they come with different combinatorial factors. The Magnus expansion has long been well understood for the case of differential equations, but the application to full quantum field theory is relatively new and so far had been rather ad-hoc, by expanding the corresponding series and explicitly matching terms. The present paper provides a systematic description of which diagrams, and with which combinatorial factors, appear in the Magnus expansion in QFT. Even for a scalar QFT, this is more complicated than ordinary Feynman diagrams, e.g. because the Magnus diagrams have directed edges (representing retarded propagators). https://arxiv.org/abs/2605.25473

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "On the Local Borel transform of perturbation theory" from 2010. This article is about perturbative #QuantumFieldTheory , where one computes the terms of the perturbation series in terms of #FeynmanIntegral s. It is well known that the number of Feynman diagrams grows factorially with the loop order, and also there are individual classes of diagrams whose integral grows factorially. Since this is a growth rate faster than exponentially, it implies that the perturbation series does not converge.
For scalar quantum field theories in 2 or even 3 dimensions, this situation is relatively well under control because the second of the two effects is not present. In 4 dimensions, it is more tricky. It was established in the 1970s that at least the two effects do not conspire to give factorial-squared growth, but still, bounds were relatively crude. The present paper aims to give numerically more tight bounds of the magnitude of correlation functions depending on loop order, number of legs, or external momenta.
Notice that this statement is different from "Borel resummability": The latter asks whether the Borel transform is finite and not too quickly growing on the entire positive line, whereas the present article asks whether it has non-zero radius of convergence in the complex plane (e.g. bounded by singularities on the negative line).

Curiously, none of the existing articles, not even this present one, establish a factorial *lower* bound, i.e. from what is proven, it would be conceivable that the perturbation series is actually convergent. However, all numerical data we have is suggestive of factorial growth.
#physics
https://link.springer.com/article/10.1007/s00220-009-0979-x

#paperOfTheDay is "On the Local Borel transform of perturbation theory" from 2010. This article is about perturbative #QuantumFieldTheory , where one computes the terms of the perturbation series in terms of #FeynmanIntegral s. It is well known that the number of Feynman diagrams grows factorially with the loop order, and also there are individual classes of diagrams whose integral grows factorially. Since this is a growth rate faster than exponentially, it implies that the perturbation series does not converge.
For scalar quantum field theories in 2 or even 3 dimensions, this situation is relatively well under control because the second of the two effects is not present. In 4 dimensions, it is more tricky. It was established in the 1970s that at least the two effects do not conspire to give factorial-squared growth, but still, bounds were relatively crude. The present paper aims to give numerically more tight bounds of the magnitude of correlation functions depending on loop order, number of legs, or external momenta.
Notice that this statement is different from "Borel resummability": The latter asks whether the Borel transform is finite and not too quickly growing on the entire positive line, whereas the present article asks whether it has non-zero radius of convergence in the complex plane (e.g. bounded by singularities on the negative line).

Curiously, none of the existing articles, not even this present one, establish a factorial *lower* bound, i.e. from what is proven, it would be conceivable that the perturbation series is actually convergent. However, all numerical data we have is suggestive of factorial growth.
#physics
https://link.springer.com/article/10.1007/s00220-009-0979-x

Today's #paperoftheDay is all 3s: "3-loop 3PI effective action for 3D SU(3) QCD". The authors study quantum chromodynamics (#QCD, but without Fermions) in 3 Euclidean dimensions, but with an unconventional method: the 3PI effective action. These higher nPI effective actions in #quantumFieldTheory formally arise from higher-order Legendre transforms of W, the generating functional for connected correlation functions. The 1PI effective action, usually denoted gamma, is what is simply called "effective action" in most of the #physics literature, it is a functional of the classical field (i.e. the 1-point function of the quantum field theory). Analogously, a 3PI effective action is a functional of the 1-point, 2-point, and 3-point functions of the theory. This means that the true physical state of the theory coincides with the global minimum of the 3PI effective action with respect to these 3 functions (in the same way that a global minimum of the usual effective action is the true vacuum expectation value of the quantum field). Contrary to what one might think, the 3PI effective action is not the generating function for 3PI, i.e. 3-particle-irreducible, diagrams in a conventional sense, this choice of name is thus a bit unfortunate from a #graph theory perspective.
By nature, this method requires to work with momentum-dependent functions as "variables", not just a number like the (translation-invariant) field value for the 1PI action. This is handled in an interesting way: The authors use Pade approximants to parametrize these functions as a rational function, and the actual data they work with is the set of coefficients of these rational approximants.
https://arxiv.org/abs/1202.4756

Today's #paperoftheDay is all 3s: "3-loop 3PI effective action for 3D SU(3) QCD". The authors study quantum chromodynamics (#QCD, but without Fermions) in 3 Euclidean dimensions, but with an unconventional method: the 3PI effective action. These higher nPI effective actions in #quantumFieldTheory formally arise from higher-order Legendre transforms of W, the generating functional for connected correlation functions. The 1PI effective action, usually denoted gamma, is what is simply called "effective action" in most of the #physics literature, it is a functional of the classical field (i.e. the 1-point function of the quantum field theory). Analogously, a 3PI effective action is a functional of the 1-point, 2-point, and 3-point functions of the theory. This means that the true physical state of the theory coincides with the global minimum of the 3PI effective action with respect to these 3 functions (in the same way that a global minimum of the usual effective action is the true vacuum expectation value of the quantum field). Contrary to what one might think, the 3PI effective action is not the generating function for 3PI, i.e. 3-particle-irreducible, diagrams in a conventional sense, this choice of name is thus a bit unfortunate from a #graph theory perspective.
By nature, this method requires to work with momentum-dependent functions as "variables", not just a number like the (translation-invariant) field value for the 1PI action. This is handled in an interesting way: The authors use Pade approximants to parametrize these functions as a rational function, and the actual data they work with is the set of coefficients of these rational approximants.
https://arxiv.org/abs/1202.4756

#paperOfTheDay is "Complex poles and spectral functions of Landau gauge #QCD and QCD-like theories" from 2020.
In #QuantumFieldTheory , one generally wants to compute n-point correlation functions of fields. Of particular interest is the 2-point function, which can be interpreted as describing how a "particle" of that theory moves. If one uses perturbation theory, the leading order of the 2-point function is the "propagator", the quantity that represents the edges in a #FeynmanIntegral . A typical expectation is that a general, non-perturbative 2-point function should still be similar in nature, namely admit a Källen-Lehmann representation, which is qualitatively "an integral over propagators with different mass, weighted by some spectral density function". The density is supposed to describe density of states of the theory. Conversely, the 2-point function as a function of complex energy should have a branch cut on the positive real line.
The present article considers the 2-point functions in theories roughly resembling QCD, and studies their form for complex energy. They find that in addition to the branch cut, these propagators can have pairs of poles in the complex plane off the real axis. In addition, the spectral "density" function can be negative for low energies (and hence can not actually be interpreted as a density).
These findings are based on various approximations and probably not precisely correct, but similar effects have been seen in other work, too. The interpretation, basically, is "confinement": In low-energy QCD, individual quarks or gluons are not meaningful "particles", so their 2-point function is different from that of a typical particle. #physics https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.074044

#paperOfTheDay is "Complex poles and spectral functions of Landau gauge #QCD and QCD-like theories" from 2020.
In #QuantumFieldTheory , one generally wants to compute n-point correlation functions of fields. Of particular interest is the 2-point function, which can be interpreted as describing how a "particle" of that theory moves. If one uses perturbation theory, the leading order of the 2-point function is the "propagator", the quantity that represents the edges in a #FeynmanIntegral . A typical expectation is that a general, non-perturbative 2-point function should still be similar in nature, namely admit a Källen-Lehmann representation, which is qualitatively "an integral over propagators with different mass, weighted by some spectral density function". The density is supposed to describe density of states of the theory. Conversely, the 2-point function as a function of complex energy should have a branch cut on the positive real line.
The present article considers the 2-point functions in theories roughly resembling QCD, and studies their form for complex energy. They find that in addition to the branch cut, these propagators can have pairs of poles in the complex plane off the real axis. In addition, the spectral "density" function can be negative for low energies (and hence can not actually be interpreted as a density).
These findings are based on various approximations and probably not precisely correct, but similar effects have been seen in other work, too. The interpretation, basically, is "confinement": In low-energy QCD, individual quarks or gluons are not meaningful "particles", so their 2-point function is different from that of a typical particle. #physics https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.074044

#paperOfTheDay is "Use of analyticity in the calculation of nonrelativistic scattering amplitudes" from 1968. In theoretical #physics , scattering amplitudes arise in various contexts, and their calculation is usually quite hard because they involve all sorts of "oscillations": For example, a real-time path integral in #quantumFieldTheory has an oscillatory Boltzmann factor exp(i S), which is numerically unstable. Or, more intuitively: When waves "collide", a tiny inaccuracy in the relative phase has big influence on the outcome of the process. Conversely, these calculations become quite easy in imaginary time ("Wick rotation"): QFT turns into statistical physics, the path integral gets the weighting factor exp(-S), which strongly suppresses fluctuations, and the wave equation turns into the heat equation, which is numerically well behaved and stable.
This intuition is the basis of the present paper: Formally, the difference between the two (physically very different) situations is whether certain quantities are real or imaginary. One of the settings is easily computable. But the answer is a function of this complex input variable, so one can get to the "hard" situation by analytic continuation. Concretely, the paper studies non-relativistic scattering as a function of complex energy, thereby interpolating between bound states and scattering.
On the technical side, the article is noteworthy because it introduces numerical algorithms to compute rational function approximations for a function where only a finite set of evaluations f(x_i) is given (if, conversely, a finite number of Taylor coefficients of f are given, one would use a Padé approximant). https://journals.aps.org/pr/abstract/10.1103/PhysRev.167.1411

#paperOfTheDay is "Use of analyticity in the calculation of nonrelativistic scattering amplitudes" from 1968. In theoretical #physics , scattering amplitudes arise in various contexts, and their calculation is usually quite hard because they involve all sorts of "oscillations": For example, a real-time path integral in #quantumFieldTheory has an oscillatory Boltzmann factor exp(i S), which is numerically unstable. Or, more intuitively: When waves "collide", a tiny inaccuracy in the relative phase has big influence on the outcome of the process. Conversely, these calculations become quite easy in imaginary time ("Wick rotation"): QFT turns into statistical physics, the path integral gets the weighting factor exp(-S), which strongly suppresses fluctuations, and the wave equation turns into the heat equation, which is numerically well behaved and stable.
This intuition is the basis of the present paper: Formally, the difference between the two (physically very different) situations is whether certain quantities are real or imaginary. One of the settings is easily computable. But the answer is a function of this complex input variable, so one can get to the "hard" situation by analytic continuation. Concretely, the paper studies non-relativistic scattering as a function of complex energy, thereby interpolating between bound states and scattering.
On the technical side, the article is noteworthy because it introduces numerical algorithms to compute rational function approximations for a function where only a finite set of evaluations f(x_i) is given (if, conversely, a finite number of Taylor coefficients of f are given, one would use a Padé approximant). https://journals.aps.org/pr/abstract/10.1103/PhysRev.167.1411

#paperOfTheDay is another classic from the early days of #quantumFieldTheory : "A relativistic equation for bound-state problems" by Bethe and Salpeter in 1951. In this article, they derive the equation that now bears their name, an integral equation describing bound states.
The starting point is to use perturbative relativistic quantum field theory in the form of #FeynmanIntegral s (which was still a novelty in #physics at that time). But a Feynman integral describes essentially an "instantaneous" interaction, whereas a bound state between two particles means that these interact for a long time (e.g. to make an atom, nucleus and electrons need to attract each other permanently, as opposed to a scattering process, where they merely interact for a short moment). In principle, the sum over all (infinitely many) Feynman diagrams should give the full solution of the theory, including bound states, but in practice this is impossible to compute.
The Bethe-Salpeter equation is essentially a rearrangement of Feynman diagrams: One introduces an "interaction kernel" (which can be determined e.g. from solving Feynman integrals), and this kernel is then used "infinitely often". Of course, the exact kernel can not be computed with reasonable effort, but the approximation is still much better than using only a few Feynman diagrams. One can also view the Bethe-Salpeter equation as a rearranged version of a Dyson-Schwinger equation: An integral equation whose self-consistent solution represents an infinite sum of Feynman diagrams.
https://journals.aps.org/pr/abstract/10.1103/PhysRev.84.1232

#paperOfTheDay is another classic from the early days of #quantumFieldTheory : "A relativistic equation for bound-state problems" by Bethe and Salpeter in 1951. In this article, they derive the equation that now bears their name, an integral equation describing bound states.
The starting point is to use perturbative relativistic quantum field theory in the form of #FeynmanIntegral s (which was still a novelty in #physics at that time). But a Feynman integral describes essentially an "instantaneous" interaction, whereas a bound state between two particles means that these interact for a long time (e.g. to make an atom, nucleus and electrons need to attract each other permanently, as opposed to a scattering process, where they merely interact for a short moment). In principle, the sum over all (infinitely many) Feynman diagrams should give the full solution of the theory, including bound states, but in practice this is impossible to compute.
The Bethe-Salpeter equation is essentially a rearrangement of Feynman diagrams: One introduces an "interaction kernel" (which can be determined e.g. from solving Feynman integrals), and this kernel is then used "infinitely often". Of course, the exact kernel can not be computed with reasonable effort, but the approximation is still much better than using only a few Feynman diagrams. One can also view the Bethe-Salpeter equation as a rearranged version of a Dyson-Schwinger equation: An integral equation whose self-consistent solution represents an infinite sum of Feynman diagrams.
https://journals.aps.org/pr/abstract/10.1103/PhysRev.84.1232

#paperOfTheDay is "Analytic structure of three-point functions from countour deformations" from 2022. This article concerns #FeynmanIntegral s, which are the coefficients of the perturbation series for e.g. scattering amplitudes in #quantumFieldTheory . These integrals are functions of momenta and masses, and for specific values of these arguments, they show non-trivial analytic properties. The basic example is the 1-loop massive propagator correction, which starts having an imaginary part when p^2 >= (2*m)^2. Physically, this threshold is the point where the two "virtual" particles in the loop can actually be physical, i.e. there is enough energy to make their rest masses.
Today, there is an active research community in mathematical physics dealing with such questions from the perspective of algebraic geometry: One uses a representation of Feynman integrals in terms of Schwinger parameters, where the integrand is a rational function, so the singularities are a zero locus of the denominator polynomial (and these types of objects -- algebraic varieties -- are well studied in #mathematics ).
The present article, instead, uses the momentum representation. They discuss the 1-loop 2-point and 3-point scalar diagrams in great detail, and comment on generalizations. The momentum representation is more complicated than the parametric one because the integrand is less structured (the integration variables are vector-valued, and can be negative). On the other hand, it has the advantage that it makes very clear the physical reason for analytic properties, so that this article is also a nice pedagogic introduction to the basic mechanism.
https://arxiv.org/abs/2212.02515

#paperOfTheDay is "Analytic structure of three-point functions from countour deformations" from 2022. This article concerns #FeynmanIntegral s, which are the coefficients of the perturbation series for e.g. scattering amplitudes in #quantumFieldTheory . These integrals are functions of momenta and masses, and for specific values of these arguments, they show non-trivial analytic properties. The basic example is the 1-loop massive propagator correction, which starts having an imaginary part when p^2 >= (2*m)^2. Physically, this threshold is the point where the two "virtual" particles in the loop can actually be physical, i.e. there is enough energy to make their rest masses.
Today, there is an active research community in mathematical physics dealing with such questions from the perspective of algebraic geometry: One uses a representation of Feynman integrals in terms of Schwinger parameters, where the integrand is a rational function, so the singularities are a zero locus of the denominator polynomial (and these types of objects -- algebraic varieties -- are well studied in #mathematics ).
The present article, instead, uses the momentum representation. They discuss the 1-loop 2-point and 3-point scalar diagrams in great detail, and comment on generalizations. The momentum representation is more complicated than the parametric one because the integrand is less structured (the integration variables are vector-valued, and can be negative). On the other hand, it has the advantage that it makes very clear the physical reason for analytic properties, so that this article is also a nice pedagogic introduction to the basic mechanism.
https://arxiv.org/abs/2212.02515

#paperOfTheDay is "Exact evolution equation for the effective potential" from 1993 by Christof Wetterich. This is the paper where the Wetterich equation is first introduced in its modern form.
In #quantumFieldTheory , there are "quantum fluctuations" which lead to the full theory being different from the Lagrangian or action one starts with. A famous example is light by light scattering in QED: The Lagrangian in quantum electrodynamics contains basically one type of allowed interaction, namely matter (such as electrons) emitting or absorbing one photon. This can take many different forms in practice, for example Bremsstrahlung (electron being accelerated and producing photon), or electrostatic repulsion (photons being exchanged between two electrons, accelerating them away from each other). QED does not allow for an elementary interaction between photons. But such interaction does in fact take place due to #quantum fluctuations with "virtual" intermediate electrons. A central task for theoretical #physics is to compute such effects.
The effective action contains all such quantum effects, that is, if one uses the effective action as a "classical" one (without adding further quantum corrections), one obtains the full quantum answer. Many different methods are known to compute the effective action, several of them based on the renormalization group. Compared to the previous work by Wilson, Wegner, and Polchinski, the Wetterich equation is somewhat more explicit in terms of interpretation, and it has the advantage of directly giving the quantum effective action and not some proxy quantity. It has by now become the cornerstone of functional renormalization group ( #FRG ) methods. https://www.sciencedirect.com/science/article/pii/037026939390726X?via%3Dihub

#paperOfTheDay is "Exact evolution equation for the effective potential" from 1993 by Christof Wetterich. This is the paper where the Wetterich equation is first introduced in its modern form.
In #quantumFieldTheory , there are "quantum fluctuations" which lead to the full theory being different from the Lagrangian or action one starts with. A famous example is light by light scattering in QED: The Lagrangian in quantum electrodynamics contains basically one type of allowed interaction, namely matter (such as electrons) emitting or absorbing one photon. This can take many different forms in practice, for example Bremsstrahlung (electron being accelerated and producing photon), or electrostatic repulsion (photons being exchanged between two electrons, accelerating them away from each other). QED does not allow for an elementary interaction between photons. But such interaction does in fact take place due to #quantum fluctuations with "virtual" intermediate electrons. A central task for theoretical #physics is to compute such effects.
The effective action contains all such quantum effects, that is, if one uses the effective action as a "classical" one (without adding further quantum corrections), one obtains the full quantum answer. Many different methods are known to compute the effective action, several of them based on the renormalization group. Compared to the previous work by Wilson, Wegner, and Polchinski, the Wetterich equation is somewhat more explicit in terms of interpretation, and it has the advantage of directly giving the quantum effective action and not some proxy quantity. It has by now become the cornerstone of functional renormalization group ( #FRG ) methods. https://www.sciencedirect.com/science/article/pii/037026939390726X?via%3Dihub

#paperOfTheDay is "Bounding scalar operator dimensions in 4D CFT" from 2008.
A conformal field theory (#CFT ) is a #quantumFieldTheory that, in addition to the usual Lorentz symmetry, also has conformal #symmetry (a stronger version of scale invariance). This implies that the 2-point correlation function is a power law, i.e. instead of an arbitrary complicated function of distance, it is fully specified by knowing a single number, the operator's scaling dimension. Moreover, even the 3-point function is fixed once one knows the scaling dimensions of all three operators (where an operator is a polynomial in field variables and derivatives, evaluated at one point) involved in it. This illustrates the enormous importance that scaling dimensions of operators have in a CFT.
Secondly, a CFT allows for an operator product expansion (OPE), where one can rewrite a product of two operators at distinct spacetime points as an infinite sum of operators at only one of the points (morally analogous to a Taylor series for an ordinary function).
The most straightforward operator is the field variable itself. The present paper asks the question: When we know that the field has scaling dimension d, then what can we say about the scaling dimension of the leading operator that appears in the OPE of field x field? By combining the various structural features of the CFT, it turns out that one can find a strict upper bound to the possible scaling dimension of this operator.
The paper is relatively long, but very readable, since it includes detailed reviews and examples of how the various CFT constructions work. Beyond the actual result, it serves as a good introduction to CFT. #physics
https://iopscience.iop.org/article/10.1088/1126-6708/2008/12/031

#paperOfTheDay is "Bounding scalar operator dimensions in 4D CFT" from 2008.
A conformal field theory (#CFT ) is a #quantumFieldTheory that, in addition to the usual Lorentz symmetry, also has conformal #symmetry (a stronger version of scale invariance). This implies that the 2-point correlation function is a power law, i.e. instead of an arbitrary complicated function of distance, it is fully specified by knowing a single number, the operator's scaling dimension. Moreover, even the 3-point function is fixed once one knows the scaling dimensions of all three operators (where an operator is a polynomial in field variables and derivatives, evaluated at one point) involved in it. This illustrates the enormous importance that scaling dimensions of operators have in a CFT.
Secondly, a CFT allows for an operator product expansion (OPE), where one can rewrite a product of two operators at distinct spacetime points as an infinite sum of operators at only one of the points (morally analogous to a Taylor series for an ordinary function).
The most straightforward operator is the field variable itself. The present paper asks the question: When we know that the field has scaling dimension d, then what can we say about the scaling dimension of the leading operator that appears in the OPE of field x field? By combining the various structural features of the CFT, it turns out that one can find a strict upper bound to the possible scaling dimension of this operator.
The paper is relatively long, but very readable, since it includes detailed reviews and examples of how the various CFT constructions work. Beyond the actual result, it serves as a good introduction to CFT. #physics
https://iopscience.iop.org/article/10.1088/1126-6708/2008/12/031

#paperOfTheDay is "Accurate critical exponents for Ising like systems in non-integer dimensions" from 1987. This is one out of a sequence of articles with very similar titles and similar sets of authors, appearing over several years with updated numerical values.
The basic scheme is to use perturbative #quatumFieldTheory in 4-epsilon dimensions to compute critical exponents of the Ising model. This produces a power series in epsilon, which is then, in principle, evaluated at epsilon=1 in order to compute values for the physically interesting dimension 3 (computing directly in D=3 dimensions is less accurate because there, the quartic coupling is relevant, and therefore the theory is strongly interacting, whereas the expansion around 4 dimensions describes a weakly interacting system).
Of course, the power series in epsilon is divergent, therefore it would be inaccurate to simply substitute epsilon=1 (in the truncated series). Instead, one computes a Borel transform of the series data, conformally maps to the complex unit disk, and computes a Laplace integral from 0 to 1 (this is a widely known standard method for resummation of this type of divergent series). Here, the authors additionally introduce three free parameters, which they tune to improve numerical accuracy. The output is a table of resummed critical exponents between 1.5 and 4 dimensions in small steps.
Today, we know 8-loop data instead of 4, which gives rise to much more accurate values, but the basic picture has not changed much.
#physics
https://jphys.journaldephysique.org/articles/jphys/abs/1987/01/jphys_1987__48_1_19_0/jphys_1987__48_1_19_0.html

#paperOfTheDay is "Accurate critical exponents for Ising like systems in non-integer dimensions" from 1987. This is one out of a sequence of articles with very similar titles and similar sets of authors, appearing over several years with updated numerical values.
The basic scheme is to use perturbative #quatumFieldTheory in 4-epsilon dimensions to compute critical exponents of the Ising model. This produces a power series in epsilon, which is then, in principle, evaluated at epsilon=1 in order to compute values for the physically interesting dimension 3 (computing directly in D=3 dimensions is less accurate because there, the quartic coupling is relevant, and therefore the theory is strongly interacting, whereas the expansion around 4 dimensions describes a weakly interacting system).
Of course, the power series in epsilon is divergent, therefore it would be inaccurate to simply substitute epsilon=1 (in the truncated series). Instead, one computes a Borel transform of the series data, conformally maps to the complex unit disk, and computes a Laplace integral from 0 to 1 (this is a widely known standard method for resummation of this type of divergent series). Here, the authors additionally introduce three free parameters, which they tune to improve numerical accuracy. The output is a table of resummed critical exponents between 1.5 and 4 dimensions in small steps.
Today, we know 8-loop data instead of 4, which gives rise to much more accurate values, but the basic picture has not changed much.
#physics
https://jphys.journaldephysique.org/articles/jphys/abs/1987/01/jphys_1987__48_1_19_0/jphys_1987__48_1_19_0.html

#paperOfTheDay is "On the Numerical calculation of a class of definite integrals and infinite series" by G. Stokes from 1850.
This article is a follow up on the work of Airy I mentioned a few days ago, where the Airy function was introduced through its convergent integral representation. This function gives the intensity in interference patterns of light (e.g. the location of "higher order" rainbows). In #physics experiments, what one actually measures is the location of maxima or minima of intensity, which would correspond to the zeros of the Airy function. With the technology of 1850 it was hard to numerically compute integrals. Airy derived a convergent Taylor series expansion about zero, but from that, it is still hard to find higher order zeros.
The present paper introduces a new approach to such calculations. The idea is to first derive a differential equation for the Airy function, and then use it to obtain a series expansion around infinity. This expansion is divergent (which is clear since the Airy function qualitatively changes character around zero), but it gives very high numerical accuracy for not-too-small arguments. In particular, one can obtain the locations of all zeros.
The plot shows the true Airy function in black, and the first 9 terms of its convergent power series expansion around zero in red. We see that they reproduce the first zero, but completely miss all other zeros. The green curve is the first (!) term of the divergent expansion around -infinity. It is visually indistinguishable from the true function for all zeros. This high numerical accuracy of divergent series expansions is a typical feature seen in #resurgence computations.
#mathematics

https://www.cambridge.org/core/books/abs/mathematical-and-physical-papers/on-the-numerical-calculation-of-a-class-of-definite-integrals-and-infinite-series/1DA9818868338CC0E684A6CB63B2503D

#paperOfTheDay is "On the Numerical calculation of a class of definite integrals and infinite series" by G. Stokes from 1850.
This article is a follow up on the work of Airy I mentioned a few days ago, where the Airy function was introduced through its convergent integral representation. This function gives the intensity in interference patterns of light (e.g. the location of "higher order" rainbows). In #physics experiments, what one actually measures is the location of maxima or minima of intensity, which would correspond to the zeros of the Airy function. With the technology of 1850 it was hard to numerically compute integrals. Airy derived a convergent Taylor series expansion about zero, but from that, it is still hard to find higher order zeros.
The present paper introduces a new approach to such calculations. The idea is to first derive a differential equation for the Airy function, and then use it to obtain a series expansion around infinity. This expansion is divergent (which is clear since the Airy function qualitatively changes character around zero), but it gives very high numerical accuracy for not-too-small arguments. In particular, one can obtain the locations of all zeros.
The plot shows the true Airy function in black, and the first 9 terms of its convergent power series expansion around zero in red. We see that they reproduce the first zero, but completely miss all other zeros. The green curve is the first (!) term of the divergent expansion around -infinity. It is visually indistinguishable from the true function for all zeros. This high numerical accuracy of divergent series expansions is a typical feature seen in #resurgence computations.
#mathematics

https://www.cambridge.org/core/books/abs/mathematical-and-physical-papers/on-the-numerical-calculation-of-a-class-of-definite-integrals-and-infinite-series/1DA9818868338CC0E684A6CB63B2503D

#paperOfTheDay is "Dirac traces and the Tutte polynomial" from 2025. A Fermion is a particle that is subject to the Pauli exclusion principle, namely at most one Fermion can occupy any one position at a given time. All constituents of matter, such as electrons or quarks, are Fermions.
Mathematically, Fermions and their interactions are being described by certain matrices, the Dirac matrices (recall that for two matrices, A*B can be distinct from B*A, therefore matrices allow for effects such as flipping sign when being encountered in reverse direction).
In calculations, one then encounters products of such Dirac matrices. In particular, one often wants to compute the trace of them. In principle, this could be done by multiplying the matrices and computing the trace of the result matrix, however, there are 2 disadvantages: Firstly, matrix multiplication is slow, and secondly, in #quantumFieldTheory one wants to compute in a setting where the spacetime dimension is a symbolic parameter, and it is impossible to compute a matrix product when the number of rows in the matrix is an undetermined symbolic value. Hence, instead, one uses the defining properties of Dirac matrices: How they flip signs when the order of their product is changed. In this way, one can "order" an arbitrary product, so that adjacent factors are the same matrix, in which case they become a unit matrix.
The present paper shows that this combinatorial operation is encoded in a specific type of graph, and that the value of the trace is given by the Tutte polynomial of that graph.
https://link.springer.com/article/10.1007/JHEP05(2025)235 #physics

#paperOfTheDay is "Dirac traces and the Tutte polynomial" from 2025. A Fermion is a particle that is subject to the Pauli exclusion principle, namely at most one Fermion can occupy any one position at a given time. All constituents of matter, such as electrons or quarks, are Fermions.
Mathematically, Fermions and their interactions are being described by certain matrices, the Dirac matrices (recall that for two matrices, A*B can be distinct from B*A, therefore matrices allow for effects such as flipping sign when being encountered in reverse direction).
In calculations, one then encounters products of such Dirac matrices. In particular, one often wants to compute the trace of them. In principle, this could be done by multiplying the matrices and computing the trace of the result matrix, however, there are 2 disadvantages: Firstly, matrix multiplication is slow, and secondly, in #quantumFieldTheory one wants to compute in a setting where the spacetime dimension is a symbolic parameter, and it is impossible to compute a matrix product when the number of rows in the matrix is an undetermined symbolic value. Hence, instead, one uses the defining properties of Dirac matrices: How they flip signs when the order of their product is changed. In this way, one can "order" an arbitrary product, so that adjacent factors are the same matrix, in which case they become a unit matrix.
The present paper shows that this combinatorial operation is encoded in a specific type of graph, and that the value of the trace is given by the Tutte polynomial of that graph.
https://link.springer.com/article/10.1007/JHEP05(2025)235 #physics

The #paperOfTheDay is "On the intensity of light in the neighbourhood of a caustic" by George Biddell Airy, 1838.
This is a paper from the time long before anything quantum or relativity, where elementary questions of #optics were at the forefront of theoretical #physics . Since light has the character of both a wave and a ray, it is important not only where a ray can point, but also "how long" it is; one will find destructive or constructive interference depending on the difference of path length (in relation to the wave length). In the framework of geometric (=light ray) optics, this amounts to the fact that when light is reflected, a small change of the point of reflection does not change the total length of the path travelled to first order (i.e. light is reflected at such angle that the path length is stationary).
For a point of focus, this relation is true not only infinitesimally, but for the full reflecting surface: All incoming rays sent to a focal point have the same length. Airy studies the intermediate case: Not ALL rays have the same length, but still, the difference vanishes to higher than first order locally. Geometrically, this gives rise to a "caustic", basically a focal point deformed to a focal line. The existence of such line amounts to the reflecting surface satisfying a certain differential equation. Most notably, for the total intensity (in the easiest case), Airy finds the expression
integral from z=0 to z=infinity of cos(z^3 + t*z) dz , where t is a parameter. This function Ai(t) is nowadays known as the Airy function, and it constitutes the basic example of many effects in the theory of #resurgence and special functions in #mathematics .
https://www.semanticscholar.org/paper/ON-the-Intensity-of-Light-in-the-neighbourhood-of-a-Airy/8d6349998be20279220e8a3e033de1679a58a916

The #paperOfTheDay is "On the intensity of light in the neighbourhood of a caustic" by George Biddell Airy, 1838.
This is a paper from the time long before anything quantum or relativity, where elementary questions of #optics were at the forefront of theoretical #physics . Since light has the character of both a wave and a ray, it is important not only where a ray can point, but also "how long" it is; one will find destructive or constructive interference depending on the difference of path length (in relation to the wave length). In the framework of geometric (=light ray) optics, this amounts to the fact that when light is reflected, a small change of the point of reflection does not change the total length of the path travelled to first order (i.e. light is reflected at such angle that the path length is stationary).
For a point of focus, this relation is true not only infinitesimally, but for the full reflecting surface: All incoming rays sent to a focal point have the same length. Airy studies the intermediate case: Not ALL rays have the same length, but still, the difference vanishes to higher than first order locally. Geometrically, this gives rise to a "caustic", basically a focal point deformed to a focal line. The existence of such line amounts to the reflecting surface satisfying a certain differential equation. Most notably, for the total intensity (in the easiest case), Airy finds the expression
integral from z=0 to z=infinity of cos(z^3 + t*z) dz , where t is a parameter. This function Ai(t) is nowadays known as the Airy function, and it constitutes the basic example of many effects in the theory of #resurgence and special functions in #mathematics .
https://www.semanticscholar.org/paper/ON-the-Intensity-of-Light-in-the-neighbourhood-of-a-Airy/8d6349998be20279220e8a3e033de1679a58a916

#paperOfTheDay is "Minimal Model Explanations" from 2014. This article is about #philosophy of #science , which is not my area of expertise, so my reading might well be inaccurate.
From a #physics point of view, the authors discuss the philosophical underpinning of the phenomenon of universality that is well known in statistical physics: Many different physical systems behave in the same way near a phase transition when the variables and functions are appropriately identified. Famously, the Ising model (which describes binary spin variables on a cubic lattice) becomes equivalent to the quartic-interacting scalar continuum field theory, in the sense that their large-scale behaviour near the critical point is the same. Other examples include fluid dynamics: Different fluids have different shapes of molecules which can behave in complicated different ways, yet most of them obey the same form of continuum equations (Navier-Stokes equation) with merely different numerical coefficients.
The present paper asks the question what makes a model "good". The classical philosophical perspective is that the model faithfully represents key behaviour and causal relations in the system in question, and becomes better when this correspondence is more accurate. But this framework fails for "minimal models" such as the Ising model: They are NOT actually a good description of quartic field theory. The #renormalization group theory provides an answer: A minimal model is good if it lies in the same universality class, AND it is possible to identify precisely which properties (such as symmetries etc.) are crucial for class, without putting in an a priori assumption about importance of features.
https://www.cambridge.org/core/journals/philosophy-of-science/article/minimal-model-explanations/640580F4EA5571EA1AA971C5DC063FA6

#paperOfTheDay is "Minimal Model Explanations" from 2014. This article is about #philosophy of #science , which is not my area of expertise, so my reading might well be inaccurate.
From a #physics point of view, the authors discuss the philosophical underpinning of the phenomenon of universality that is well known in statistical physics: Many different physical systems behave in the same way near a phase transition when the variables and functions are appropriately identified. Famously, the Ising model (which describes binary spin variables on a cubic lattice) becomes equivalent to the quartic-interacting scalar continuum field theory, in the sense that their large-scale behaviour near the critical point is the same. Other examples include fluid dynamics: Different fluids have different shapes of molecules which can behave in complicated different ways, yet most of them obey the same form of continuum equations (Navier-Stokes equation) with merely different numerical coefficients.
The present paper asks the question what makes a model "good". The classical philosophical perspective is that the model faithfully represents key behaviour and causal relations in the system in question, and becomes better when this correspondence is more accurate. But this framework fails for "minimal models" such as the Ising model: They are NOT actually a good description of quartic field theory. The #renormalization group theory provides an answer: A minimal model is good if it lies in the same universality class, AND it is possible to identify precisely which properties (such as symmetries etc.) are crucial for class, without putting in an a priori assumption about importance of features.
https://www.cambridge.org/core/journals/philosophy-of-science/article/minimal-model-explanations/640580F4EA5571EA1AA971C5DC063FA6

#paperOfTheDay "Convergent Strong-Coupling Expansions from divergent weak-coupling perturbation theory" from 1995 studies the anharmonic oscillator, one of the most popular models for perturbation theory in theoretical #physics . Recall that with a potential V(x)= w/2*x^2, an object performs harmonic oscillations with frequency w, i.e. the trajectory (of a classical system) is a sine, or the #quantum oscillations are a complex exponential function.
One now makes this potential "anharmonic" by adding an additional term +g/4*x^4. The potential is still bounded from below, and the quantum system does some sort of oscillations, but no longer sine/cosine ones. One is interested in the energies, in particular the ground state energy, as a function of the parameter g. An obvious idea is to use perturbation theory, i.e. construct a power series expansion around the point g=0. It came as a big surprise that this power series is divergent for all non-zero g. This can be understood by considering small negative g: The potential would then be unbounded, and the quantum system instable, hence the ground state energy can not be finite (or real) in that case, hence the power series can not diverge in any radius around g.
The present paper constructs a different type of perturbation series, essentially using the frequency w instead of the coupling g as an expansion parameter. Eventually, this turns out to be equivalent to a perturbation series in the parameter 1/g, hence it is an expansion in strong coupling. This new expansion is convergent, and it can be generated from the divergent small-coupling expansion simply by reordering terms.
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.75.2787

#paperOfTheDay "Convergent Strong-Coupling Expansions from divergent weak-coupling perturbation theory" from 1995 studies the anharmonic oscillator, one of the most popular models for perturbation theory in theoretical #physics . Recall that with a potential V(x)= w/2*x^2, an object performs harmonic oscillations with frequency w, i.e. the trajectory (of a classical system) is a sine, or the #quantum oscillations are a complex exponential function.
One now makes this potential "anharmonic" by adding an additional term +g/4*x^4. The potential is still bounded from below, and the quantum system does some sort of oscillations, but no longer sine/cosine ones. One is interested in the energies, in particular the ground state energy, as a function of the parameter g. An obvious idea is to use perturbation theory, i.e. construct a power series expansion around the point g=0. It came as a big surprise that this power series is divergent for all non-zero g. This can be understood by considering small negative g: The potential would then be unbounded, and the quantum system instable, hence the ground state energy can not be finite (or real) in that case, hence the power series can not diverge in any radius around g.
The present paper constructs a different type of perturbation series, essentially using the frequency w instead of the coupling g as an expansion parameter. Eventually, this turns out to be equivalent to a perturbation series in the parameter 1/g, hence it is an expansion in strong coupling. This new expansion is convergent, and it can be generated from the divergent small-coupling expansion simply by reordering terms.
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.75.2787

#paperOfTheDay is "Criterion for Dominance of Directional over Size Fluctuations in Destroying Order" from 1999. This article is about statistical #physics , which can as an effective model also be described by #quantumFieldTheory . Many practically relevant statistical models live in D=3 or D=2 spacial dimensions, and they describe a quantity (called "order parameter") which is itself a vector with N components. It can be that N=D, but other situations are conceivable.
At low temperature, such systems are usually in an "ordered" phase, where the O(N) rotation symmetry is broken. For example, in a Magnet, the small elementary magnets align and produce some non-zero macroscopic magnetic field. Even if the fundamental theory was agnostic with respect to vector orientations, the actual ground state has a preferred direction, the symmetry is "spontaneously broken".
As the temperature increases, fluctuations increase, and eventually the symmetry is restored because all microscopic vectors are shuffled. One wants to know at which temperature this first happens, this is the critical temperature.
An old estimate for the critical temperature was the "Ginzburg criterion", which is derived from the energy contribution of typical fluctuations in the magnitude of the field. The present article demonstrates that often, the "directional" fluctuations of the field are stronger than the magnitude fluctuations. This gives rise to a new, lower estimate of the critical temperature, the "Kleinert criterion". This nicely fits with the Goldstone theorem: The "angular" modes in the broken symmetry phase are massless, and thus more important for fluctuations than the massive "radial" mode.
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.84.286

#paperOfTheDay is "Criterion for Dominance of Directional over Size Fluctuations in Destroying Order" from 1999. This article is about statistical #physics , which can as an effective model also be described by #quantumFieldTheory . Many practically relevant statistical models live in D=3 or D=2 spacial dimensions, and they describe a quantity (called "order parameter") which is itself a vector with N components. It can be that N=D, but other situations are conceivable.
At low temperature, such systems are usually in an "ordered" phase, where the O(N) rotation symmetry is broken. For example, in a Magnet, the small elementary magnets align and produce some non-zero macroscopic magnetic field. Even if the fundamental theory was agnostic with respect to vector orientations, the actual ground state has a preferred direction, the symmetry is "spontaneously broken".
As the temperature increases, fluctuations increase, and eventually the symmetry is restored because all microscopic vectors are shuffled. One wants to know at which temperature this first happens, this is the critical temperature.
An old estimate for the critical temperature was the "Ginzburg criterion", which is derived from the energy contribution of typical fluctuations in the magnitude of the field. The present article demonstrates that often, the "directional" fluctuations of the field are stronger than the magnitude fluctuations. This gives rise to a new, lower estimate of the critical temperature, the "Kleinert criterion". This nicely fits with the Goldstone theorem: The "angular" modes in the broken symmetry phase are massless, and thus more important for fluctuations than the massive "radial" mode.
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.84.286