#quantumfieldtheory — Public Fediverse posts

The #paperOfTheDay is "Landau's Leviathans" from 2026. This paper concerns scattering #amplitudes in #QuantumFieldTheory . Theser are computed through #FeynmanIntegral s, and they express the probability of a scattering event as a function of masses and momenta of all particles involved in the process. This functional dependence is very complicated, as can be seen already in much simpler theories: If one e.g. considers an idealized pendulum under the effect of an external oscillation, the behaviour strongly depends on the frequency of the external oscillation. If it coincides with the pendulum's natural ("eigen") frequency, the pendulum will swing strongly, otherwise rather not. A "response function", as a function of external frequency, therefore has a peak at this particular frequency.
In the same way, the scattering amplitude of quantum fields has all sorts of peaks and singularities when certain combinations of the energies of incoming particles match masses of particles. For practical calculations, it is useful to know when that happens. The present draft introduces a new method, based on algebraic geometry, to determine these points. In principle, this "only" amounts to simplifying or solving systems of polynomial equations with rational coefficients, but in practice this is hardly doable (a polynomial in multiple variables can have extremely many independent coefficients). So they do the actual computations over finite fields: Instead of dealing with huge rational numbers and functions, they compute everything modulo a prime so that all numbers fit into standard integer types, repeat this many times, and finally reconstruct the true rational function. #physics https://arxiv.org/abs/2606.29612

The #paperOfTheDay is "Landau's Leviathans" from 2026. This paper concerns scattering #amplitudes in #QuantumFieldTheory . Theser are computed through #FeynmanIntegral s, and they express the probability of a scattering event as a function of masses and momenta of all particles involved in the process. This functional dependence is very complicated, as can be seen already in much simpler theories: If one e.g. considers an idealized pendulum under the effect of an external oscillation, the behaviour strongly depends on the frequency of the external oscillation. If it coincides with the pendulum's natural ("eigen") frequency, the pendulum will swing strongly, otherwise rather not. A "response function", as a function of external frequency, therefore has a peak at this particular frequency.
In the same way, the scattering amplitude of quantum fields has all sorts of peaks and singularities when certain combinations of the energies of incoming particles match masses of particles. For practical calculations, it is useful to know when that happens. The present draft introduces a new method, based on algebraic geometry, to determine these points. In principle, this "only" amounts to simplifying or solving systems of polynomial equations with rational coefficients, but in practice this is hardly doable (a polynomial in multiple variables can have extremely many independent coefficients). So they do the actual computations over finite fields: Instead of dealing with huge rational numbers and functions, they compute everything modulo a prime so that all numbers fit into standard integer types, repeat this many times, and finally reconstruct the true rational function. #physics https://arxiv.org/abs/2606.29612

#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

Registration is open for the #academicConference "High Precision for Hard Processes 2026) in #Karlsruhe Germany on 5-9 October 2026. This series of workshops is devoted to high-precision studies of hard scattering processes at hadron colliders and beyond. The main themes are recent developments and new results for theoretical computations in #quantumFieldTheory and their applications to collider phenomenology.
Abstracts need to be submitted before 15 July.
https://indico.kit.edu/event/5167/overview
#physics

Registration is open for the #academicConference "High Precision for Hard Processes 2026) in #Karlsruhe Germany on 5-9 October 2026. This series of workshops is devoted to high-precision studies of hard scattering processes at hadron colliders and beyond. The main themes are recent developments and new results for theoretical computations in #quantumFieldTheory and their applications to collider phenomenology.
Abstracts need to be submitted before 15 July.
https://indico.kit.edu/event/5167/overview
#physics

#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

Registration is open for the #KMPB summer schoool in theoretical #physics and #mathematics in #Berlin on 31 August to 4 September: Symbols, Periods, and #Resurgence . These are important themes in modern #quantumFieldTheory and mathematical physics. The event itself is free of charge, there is even limited financial support available for accommodation or travel cost.
https://indico.global/event/17166/

#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 "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

#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

#paperOfTheDay "Integrating out Gluons in Flow equations" from 1996 is another early article about the functional renormalization group #frg , but this time applied to #QCD. The article is relatively long and contains many technicalities, but the main idea is the following: Like every #quantumFieldTheory , QCD contains "quantum fluctuations" on every energy scale, which can be integrated out from high to low energy with the help of a renormalization group flow equation. Unlike the scalar field theories that are often studied as toy models, QCD contains two fundamentally different types of fields: The fermions (quarks), which represent matter, and the bosons (gluons), which are particles of the strong force. Now it turns out that one can arrange the flow equations in such a way that only one type of field is (at first) integrated out, and serves as an "input" for the flow of the other. In principle, this would be exact and yield a full solution of QCD (which still today would be a breakthrough in #physics ), but in practice of course one has to use truncations and approximations. In fact, the computations presented in the paper are rather "coarse" and don't really produce new results; the point is rather to establish the method.
What is interesting is that here, the gluons are integrated out, and one obtains an effective theory for the interaction of matter. This sounds reasonable, but it is the opposite of how lattice simulations (another well-developed approach at non-perturbative QCD) work: There, the gluon field is being simulated, and the fermions are merely a correction term.
https://arxiv.org/abs/hep-ph/9604227

#paperOfTheDay "Integrating out Gluons in Flow equations" from 1996 is another early article about the functional renormalization group #frg , but this time applied to #QCD. The article is relatively long and contains many technicalities, but the main idea is the following: Like every #quantumFieldTheory , QCD contains "quantum fluctuations" on every energy scale, which can be integrated out from high to low energy with the help of a renormalization group flow equation. Unlike the scalar field theories that are often studied as toy models, QCD contains two fundamentally different types of fields: The fermions (quarks), which represent matter, and the bosons (gluons), which are particles of the strong force. Now it turns out that one can arrange the flow equations in such a way that only one type of field is (at first) integrated out, and serves as an "input" for the flow of the other. In principle, this would be exact and yield a full solution of QCD (which still today would be a breakthrough in #physics ), but in practice of course one has to use truncations and approximations. In fact, the computations presented in the paper are rather "coarse" and don't really produce new results; the point is rather to establish the method.
What is interesting is that here, the gluons are integrated out, and one obtains an effective theory for the interaction of matter. This sounds reasonable, but it is the opposite of how lattice simulations (another well-developed approach at non-perturbative QCD) work: There, the gluon field is being simulated, and the fermions are merely a correction term.
https://arxiv.org/abs/hep-ph/9604227

#paperOfTheDay "Critical Exponents from the Effective Average Action" from 1993 is one of the early works of what is now known as the functional renormalization group #frg , called at that time "exact non-perturbative evolution equation". In #quantumFieldTheory and statistical #physics , the behaviour of a system is different for different energy scales. This change is captured by the renormalization group: Changing the energy scale gives back a similar system, but with different numerical values of couplings or masses.
The functional renormalization group equation is a "flow equation" for the quantum effective action. Basically, it expresses the change of all correlation functions under change of energy scale. One can also view it as a successive solution of the path integral, where one starts with the classical (tree-level) action, and successively integrates out high-energy modes, so that , when one reaches zero energy, the full path integral has been performed and one has found the full quantum effective action.
Of course, the functional renormalization group equation can not be solved in closed form for any meaningful theory, so one is forced to introduce approximations. One can recover the usual coupling/loop expansion ( #FeynmanIntegral s), but also other types of approximation schemes are possible, for example including only 1PI correlation functions up to a certain number of legs.
The present paper is concerned with O(N) symmetric scalar fields in D=3 space dimensions. They demonstrate that with a suitable low-order approximation of the flow equations, one can indeed compute the critical exponents of this theory to a few percent accuracy.
https://arxiv.org/abs/hep-ph/9308214

#paperOfTheDay "Critical Exponents from the Effective Average Action" from 1993 is one of the early works of what is now known as the functional renormalization group #frg , called at that time "exact non-perturbative evolution equation". In #quantumFieldTheory and statistical #physics , the behaviour of a system is different for different energy scales. This change is captured by the renormalization group: Changing the energy scale gives back a similar system, but with different numerical values of couplings or masses.
The functional renormalization group equation is a "flow equation" for the quantum effective action. Basically, it expresses the change of all correlation functions under change of energy scale. One can also view it as a successive solution of the path integral, where one starts with the classical (tree-level) action, and successively integrates out high-energy modes, so that , when one reaches zero energy, the full path integral has been performed and one has found the full quantum effective action.
Of course, the functional renormalization group equation can not be solved in closed form for any meaningful theory, so one is forced to introduce approximations. One can recover the usual coupling/loop expansion ( #FeynmanIntegral s), but also other types of approximation schemes are possible, for example including only 1PI correlation functions up to a certain number of legs.
The present paper is concerned with O(N) symmetric scalar fields in D=3 space dimensions. They demonstrate that with a suitable low-order approximation of the flow equations, one can indeed compute the critical exponents of this theory to a few percent accuracy.
https://arxiv.org/abs/hep-ph/9308214

Registration is open until 1st of June for the #amplitudes summer school in #Southampton UK. This school is targeted at doctoral candidates in theoretical #physics and it covers modern topics in #quantumFieldTheory and amplitudes at a level at detail exceeding the unsual university lectures. #academicConference
https://amplitudes.soton.ac.uk/school26/

Registration is open until 1st of June for the #amplitudes summer school in #Southampton UK. This school is targeted at doctoral candidates in theoretical #physics and it covers modern topics in #quantumFieldTheory and amplitudes at a level at detail exceeding the unsual university lectures. #academicConference
https://amplitudes.soton.ac.uk/school26/

#paperOfTheDay "Über die Eigenkräfte der Elementarteilchen I" from 1933.
This is another paper from the very early days of #quantumFieldTheory , concerned with the question of the seemingly infinite self-energy of the electron in its own electromagnetic field, namely: If the electron is point-like, then its classical electromagnetic field should be infinite at its location, which is clearly nonsense.
The present paper presents a more refined relativistic analysis, starting from the assumption that the locations where the electron "generates" the field and where it "feels" it are distinct by a small vector r. If r is space like (i.e. the two locations differ by a distance that is farther than the distance that light could travel in the same time interval), one recovers the familiar divergence. On the other hand, if r is inside the light cone (i.e. the electron "feels" its own field in its causal future or past), the divergence is absent even in the limit r->0. However, this computation only works for a classical electron in a classical electromagnetic field. Using the Dirac equation for the electron, new obstacles appear.
The present article is typical for the time when #quantum theory was being developed, but it was not at all clear how to interpret it, or whether it was even correct. Schrödinger coined the term "Zitterbewegung" for the intuition of the electron making infinitely fine random jumps at light speed; the present paper mentions this Zitterbewegung as an obvious reason for difficulties in the self-energy. Today, I would say that Zitterbewegung can be an intuitive picture, but the laws of classical #physics are simply not valid at so small scales.
https://link.springer.com/article/10.1007/BF01341363

#paperOfTheDay "Über die Eigenkräfte der Elementarteilchen I" from 1933.
This is another paper from the very early days of #quantumFieldTheory , concerned with the question of the seemingly infinite self-energy of the electron in its own electromagnetic field, namely: If the electron is point-like, then its classical electromagnetic field should be infinite at its location, which is clearly nonsense.
The present paper presents a more refined relativistic analysis, starting from the assumption that the locations where the electron "generates" the field and where it "feels" it are distinct by a small vector r. If r is space like (i.e. the two locations differ by a distance that is farther than the distance that light could travel in the same time interval), one recovers the familiar divergence. On the other hand, if r is inside the light cone (i.e. the electron "feels" its own field in its causal future or past), the divergence is absent even in the limit r->0. However, this computation only works for a classical electron in a classical electromagnetic field. Using the Dirac equation for the electron, new obstacles appear.
The present article is typical for the time when #quantum theory was being developed, but it was not at all clear how to interpret it, or whether it was even correct. Schrödinger coined the term "Zitterbewegung" for the intuition of the electron making infinitely fine random jumps at light speed; the present paper mentions this Zitterbewegung as an obvious reason for difficulties in the self-energy. Today, I would say that Zitterbewegung can be an intuitive picture, but the laws of classical #physics are simply not valid at so small scales.
https://link.springer.com/article/10.1007/BF01341363

#paperOfTheDay : "How soon after a zero-temperature quench is the fate of the Ising model sealed?" from 2013.
As is well known, several methods of #quantumFieldTheory and statistical #physics can be used to study the behaviour of systems in equilibrium, and in particular at the critical point. For example, the #Ising model describes a lattice of spin variables, and one can compute critical exponents for the correlation length, assuming that the model has reached a steady state for a fixed temperature.
The present article studies the Ising model, but in a different situation: A (somewhat low) temperature is given, but the model is initialized in a fully random state (which would be the equilibrium state at very high temperature). As the simulation starts, the model moves towards its steady state: Neighbouring spins start to align, and clusters of a certain size are formed. Qualitatively, the size of the clusters in equilibrium is known, but their precise shape and orientation depends on the particular (random) simulation. This information must therefore emerge at some point after the initialization of the simulation. The present paper asks: When? The outcome is that this happens very early, in particular, long before the equilibrium is reached. But also, it's not the first cluster that survives. Instead, several clusters emerge after very few time steps, some disappear, some rearrange, but then one configureation "wins", and for a long time all that happens is that this clustering grows into its final equilibrium shape.
This paper is a nice (full of pictures!) example for properties of the Ising model beyond the usual equilibrium critical exponents story.
https://arxiv.org/abs/1312.1712

#paperOfTheDay : "How soon after a zero-temperature quench is the fate of the Ising model sealed?" from 2013.
As is well known, several methods of #quantumFieldTheory and statistical #physics can be used to study the behaviour of systems in equilibrium, and in particular at the critical point. For example, the #Ising model describes a lattice of spin variables, and one can compute critical exponents for the correlation length, assuming that the model has reached a steady state for a fixed temperature.
The present article studies the Ising model, but in a different situation: A (somewhat low) temperature is given, but the model is initialized in a fully random state (which would be the equilibrium state at very high temperature). As the simulation starts, the model moves towards its steady state: Neighbouring spins start to align, and clusters of a certain size are formed. Qualitatively, the size of the clusters in equilibrium is known, but their precise shape and orientation depends on the particular (random) simulation. This information must therefore emerge at some point after the initialization of the simulation. The present paper asks: When? The outcome is that this happens very early, in particular, long before the equilibrium is reached. But also, it's not the first cluster that survives. Instead, several clusters emerge after very few time steps, some disappear, some rearrange, but then one configureation "wins", and for a long time all that happens is that this clustering grows into its final equilibrium shape.
This paper is a nice (full of pictures!) example for properties of the Ising model beyond the usual equilibrium critical exponents story.
https://arxiv.org/abs/1312.1712

The #paperOfTheDay is "Foundations of the new field theory" from 1934. This new field theory today goes by the name of "Born-Infeld theory", it is an alternative version of classical electrodynamics.
Recall that in 1934, #quantum mechanics had recently been developed, but there was not yet any consistent #quantumFieldTheory , let alone a fundamental theory of elementary particles. In particular, classical (Maxwell) electrodynamics predicts an infinite self-energy if one assumes the electron to be point-like, and people discussed different ways to unify the picture of microscopic #physics .
Born-Infeld theory represents one possible scenario, modeled after Einstein's general theory of #relativity . Namely, a theory of electromagnetism based on general coordinate invariance, and the assumption that there is an universal maximum electrical field strength that no system can exceed. This gives rise to a Lagrangian that is structurally similar to the Einstein-Hilbert one. The field equations are then non-linear, but reduce to the Maxwell theory for weak enough fields in flat space.
Close to the center of an electron, the field strength is large, and the new theory is substantially different from classical electrodynamics: The potential is not singular at the origin, but always stays finite.

Later, however, many of the old mysteries got resolved with the quantization of Maxwell electrodynamics. On the other hand, Born-Infeld theory (much like general relativity) is strongly non-linear and hard to quantize with existing methods.
https://royalsocietypublishing.org/rspa/article/144/852/425/3579/Foundations-of-the-new-field-theory

The #paperOfTheDay is "Foundations of the new field theory" from 1934. This new field theory today goes by the name of "Born-Infeld theory", it is an alternative version of classical electrodynamics.
Recall that in 1934, #quantum mechanics had recently been developed, but there was not yet any consistent #quantumFieldTheory , let alone a fundamental theory of elementary particles. In particular, classical (Maxwell) electrodynamics predicts an infinite self-energy if one assumes the electron to be point-like, and people discussed different ways to unify the picture of microscopic #physics .
Born-Infeld theory represents one possible scenario, modeled after Einstein's general theory of #relativity . Namely, a theory of electromagnetism based on general coordinate invariance, and the assumption that there is an universal maximum electrical field strength that no system can exceed. This gives rise to a Lagrangian that is structurally similar to the Einstein-Hilbert one. The field equations are then non-linear, but reduce to the Maxwell theory for weak enough fields in flat space.
Close to the center of an electron, the field strength is large, and the new theory is substantially different from classical electrodynamics: The potential is not singular at the origin, but always stays finite.

Later, however, many of the old mysteries got resolved with the quantization of Maxwell electrodynamics. On the other hand, Born-Infeld theory (much like general relativity) is strongly non-linear and hard to quantize with existing methods.
https://royalsocietypublishing.org/rspa/article/144/852/425/3579/Foundations-of-the-new-field-theory

#paperOfTheDay is "Correlation functions and zeros of a Gaussian power series and Pfaffians" from 2013.
This paper is a generalisation of the study of random polynomials: They consider random power series, i.e. polynomials with infinitely many terms. These have (almost always) a radius of convergence of unity, so that it only makes sense to study them in the domain (-1,1). There is an accumulation of zeros (=roots) close to the boundaries of this interval.
Given that the coefficients of the power series are random, so are the locations of zeros. The positive locations form an infinite sequence of random numbers, a point process. As such, one can ask about the mean, variance, and all other correlation functions. The main result of the article is that these quantities are given by a Pfaffian (which is an algebraic object similar to a determinant) of some explicitly known matrices.
I got interested in this observation because Pfaffians also show up in #quantumFieldTheory . For example, Isserlis theorem (sometimes called Wicks theorem by physicists) says that the expectation of a product of Gaussian variables is the Pfaffian of their covariances. Or, Pfaffians show up as the integrands in #FeynmanIntegral s in topological field theories.
#mathematics #probabilityTheory
https://projecteuclid.org/journals/electronic-journal-of-probability/volume-18/issue-none/Correlation-functions-for-zeros-of-a-Gaussian-power-series-and/10.1214/EJP.v18-2545.full

#paperOfTheDay is "Correlation functions and zeros of a Gaussian power series and Pfaffians" from 2013.
This paper is a generalisation of the study of random polynomials: They consider random power series, i.e. polynomials with infinitely many terms. These have (almost always) a radius of convergence of unity, so that it only makes sense to study them in the domain (-1,1). There is an accumulation of zeros (=roots) close to the boundaries of this interval.
Given that the coefficients of the power series are random, so are the locations of zeros. The positive locations form an infinite sequence of random numbers, a point process. As such, one can ask about the mean, variance, and all other correlation functions. The main result of the article is that these quantities are given by a Pfaffian (which is an algebraic object similar to a determinant) of some explicitly known matrices.
I got interested in this observation because Pfaffians also show up in #quantumFieldTheory . For example, Isserlis theorem (sometimes called Wicks theorem by physicists) says that the expectation of a product of Gaussian variables is the Pfaffian of their covariances. Or, Pfaffians show up as the integrands in #FeynmanIntegral s in topological field theories.
#mathematics #probabilityTheory
https://projecteuclid.org/journals/electronic-journal-of-probability/volume-18/issue-none/Correlation-functions-for-zeros-of-a-Gaussian-power-series-and/10.1214/EJP.v18-2545.full

Registration is open for the #academicConference "Effective Theories for Nonperturbative #physics " 24 August-4 September in Durham, UK. This conference continues a programme to connect and build a community of researchers in #effectiveFieldTheory and #quantumFieldTheory with emphasis on #nonperturbative aspects, that has been begun with a 2024 conference at the #MITP and a 2025 theory workshop at #CERN
Deadline is 24 June.
https://conference.ippp.dur.ac.uk/event/1546/

Registration is open for the #academicConference "Effective Theories for Nonperturbative #physics " 24 August-4 September in Durham, UK. This conference continues a programme to connect and build a community of researchers in #effectiveFieldTheory and #quantumFieldTheory with emphasis on #nonperturbative aspects, that has been begun with a 2024 conference at the #MITP and a 2025 theory workshop at #CERN
Deadline is 24 June.
https://conference.ippp.dur.ac.uk/event/1546/

The #paperOfTheDay is "The Universe Fan" from 2026. It concerns a very special type of #quantumFieldTheory , describing the cosmic microwave background, that goes by the name "wave function of the universe". The precise #physics application is maybe not so much the point, but rather, that this theory has a curious mathematical feature. Namely, the tree level amplitudes can be described in a nice way.
Recall that tree level amplitudes (i.e. correlation functions of classical field theory) have their name because their Feynman diagrams are trees without closed loops. In position-space, these trees represent integrals over internal vertices, in momentum space they are just products of propagators without any integral. The difficulty is therefore not the individual "Feynman integral", but rather the fact that with many external legs, there can be many trees with different topologies and orientations, which all depend on certain linear combinations of momenta. It is a recurrent theme in quantum field theory that the sum of all Feynman integrals is usually much simpler than one would guess, but still, one usually needs to enumerate all diagrams and compute all their integrals in order to find this sum.
In the present case, there is a shortcut: The authors consider Laplace transforms of the tree amplitudes, so that they are actually integrals. It then turns out that the individual trees correspond to different domains of integration, a so-called fan (i.e. a sum of wedges in coordinate space). Crucially, this fan can be described with a rather simple formula, thus defining the full (tree-level) amplitude without ever generating all tree graphs.
#mathematics
https://arxiv.org/abs/2602.21194

The #paperOfTheDay is "The Universe Fan" from 2026. It concerns a very special type of #quantumFieldTheory , describing the cosmic microwave background, that goes by the name "wave function of the universe". The precise #physics application is maybe not so much the point, but rather, that this theory has a curious mathematical feature. Namely, the tree level amplitudes can be described in a nice way.
Recall that tree level amplitudes (i.e. correlation functions of classical field theory) have their name because their Feynman diagrams are trees without closed loops. In position-space, these trees represent integrals over internal vertices, in momentum space they are just products of propagators without any integral. The difficulty is therefore not the individual "Feynman integral", but rather the fact that with many external legs, there can be many trees with different topologies and orientations, which all depend on certain linear combinations of momenta. It is a recurrent theme in quantum field theory that the sum of all Feynman integrals is usually much simpler than one would guess, but still, one usually needs to enumerate all diagrams and compute all their integrals in order to find this sum.
In the present case, there is a shortcut: The authors consider Laplace transforms of the tree amplitudes, so that they are actually integrals. It then turns out that the individual trees correspond to different domains of integration, a so-called fan (i.e. a sum of wedges in coordinate space). Crucially, this fan can be described with a rather simple formula, thus defining the full (tree-level) amplitude without ever generating all tree graphs.
#mathematics
https://arxiv.org/abs/2602.21194

#paperOfTheDay is "Non-perturbative renormalization of the energy-momentum tensor in the 2d O(3) nonlinear sigma model" from 2026. The nonlinear sigma model is a popular model #quantumFieldTheory where the field variable is constrained to live in some manifold. Here, the O(3) signifies that the field ls a 3-component vector, confined to a sphere, that is, the field is a unit vector in 3 dimensional space. The constraint of being a unit vector is obviously non-linear, this implies that the model is an interacting theory.
The present paper studies it numerically on a lattice. In that setting, one has to make choices, obviously the size of the lattice, but also which particular discretization of e.g. the action functional one wants. All of these correspond to the same continuum limit, but for finite lattices, they are distinct, and can give rise to discretization artifacts. The paper compares a few such choices, and in particular analyzes the computation of the energy-momentum tensor of the field.
#physics #latticeQFT
https://arxiv.org/abs/2602.23078

#paperOfTheDay is "Non-perturbative renormalization of the energy-momentum tensor in the 2d O(3) nonlinear sigma model" from 2026. The nonlinear sigma model is a popular model #quantumFieldTheory where the field variable is constrained to live in some manifold. Here, the O(3) signifies that the field ls a 3-component vector, confined to a sphere, that is, the field is a unit vector in 3 dimensional space. The constraint of being a unit vector is obviously non-linear, this implies that the model is an interacting theory.
The present paper studies it numerically on a lattice. In that setting, one has to make choices, obviously the size of the lattice, but also which particular discretization of e.g. the action functional one wants. All of these correspond to the same continuum limit, but for finite lattices, they are distinct, and can give rise to discretization artifacts. The paper compares a few such choices, and in particular analyzes the computation of the energy-momentum tensor of the field.
#physics #latticeQFT
https://arxiv.org/abs/2602.23078

#paperOfTheDay is "Perturbative renormalization and ifrared finiteness in the Wilson renormalization group: the massless scalar case" from 1993. The point of this paper is to prove renormalizability of massless scalar #quantumFieldTheory (which had been known for decades at that point) , but from a new perspective. Namely, instead of discussing the properties of #FeynmanIntegral s, the authors set up a set of integral equations for the quantum effective action in presence of UV and IR momentum cutoffs, and then show that the renormalized versions of these equations stay finite as either of the two cutoffs is removed, thus proving UV and IR finiteness of the renormalized theory.
Notice that this paper appeared in the early 1990s, at the same time as many foundational articles of the #functionalRenormalizationGroup , but the present article uses a custom derivation and a version of functional renormalization group that is not obviously equal to e.g. the Wetterich equation (although, as the authors discuss, it is a version of Polchinski's equation, and my impression is that all these functional renormalization group equations are to some extent equivalent up to changes of variables).
Regardless of whether renormalizability had been known, it is of course very important to check if an emergent new formulation of quantum field theory reproduces this result, or perhaps leads to new insights (or difficulties).
https://www.sciencedirect.com/science/article/abs/pii/S0370157321000156

#paperOfTheDay is "Perturbative renormalization and ifrared finiteness in the Wilson renormalization group: the massless scalar case" from 1993. The point of this paper is to prove renormalizability of massless scalar #quantumFieldTheory (which had been known for decades at that point) , but from a new perspective. Namely, instead of discussing the properties of #FeynmanIntegral s, the authors set up a set of integral equations for the quantum effective action in presence of UV and IR momentum cutoffs, and then show that the renormalized versions of these equations stay finite as either of the two cutoffs is removed, thus proving UV and IR finiteness of the renormalized theory.
Notice that this paper appeared in the early 1990s, at the same time as many foundational articles of the #functionalRenormalizationGroup , but the present article uses a custom derivation and a version of functional renormalization group that is not obviously equal to e.g. the Wetterich equation (although, as the authors discuss, it is a version of Polchinski's equation, and my impression is that all these functional renormalization group equations are to some extent equivalent up to changes of variables).
Regardless of whether renormalizability had been known, it is of course very important to check if an emergent new formulation of quantum field theory reproduces this result, or perhaps leads to new insights (or difficulties).
https://www.sciencedirect.com/science/article/abs/pii/S0370157321000156

The #paperOfTheDay : "Nonperturbative study of the fermion propagator in quenched QED in covariant gauges using a renormalizable truncation of the Schwinger-Dyson equation" from 1993 does what the title says.
Concretely, the Dyson-Schwinger equations are an infinite set of integral equations between all correlation functions of a #quantumFieldTheory . These equations are believed to contain all information about the theory in question, but they can not be solved exactly. In practice, one has to first truncate the system to a finite number of equations, and secondly also make some assumptions about the solutions of the remaining equation (e.g. expand in power series or solve numerically to finite accuracy).
One issue with this is that the truncations and assumptions can easily be inconsistent, for example break gauge symmetry or be non-renormalizable. This is easily understood from the perspective of #FeynmanIntegral s: The sum of all integrals has the desired properties, but these rely on identities or cancellations. A random subset, in general, will break the symmetries. The point of the present article is to set up a more consistent truncation than what had been used before. They use it to examine whether the fermion in QED dynamically acquires a mass. The result is that indeed, if the coupling is strong enough, a QED-type theory can produce a mass by itself, even if the input Lagrangian was massless. The threshold is alpha~1, which, if I understand the conventions correctly, is much larger than the physical value, but such things are always tricky because the computation still is only a coarse approximation.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.48.4933

The #paperOfTheDay : "Nonperturbative study of the fermion propagator in quenched QED in covariant gauges using a renormalizable truncation of the Schwinger-Dyson equation" from 1993 does what the title says.
Concretely, the Dyson-Schwinger equations are an infinite set of integral equations between all correlation functions of a #quantumFieldTheory . These equations are believed to contain all information about the theory in question, but they can not be solved exactly. In practice, one has to first truncate the system to a finite number of equations, and secondly also make some assumptions about the solutions of the remaining equation (e.g. expand in power series or solve numerically to finite accuracy).
One issue with this is that the truncations and assumptions can easily be inconsistent, for example break gauge symmetry or be non-renormalizable. This is easily understood from the perspective of #FeynmanIntegral s: The sum of all integrals has the desired properties, but these rely on identities or cancellations. A random subset, in general, will break the symmetries. The point of the present article is to set up a more consistent truncation than what had been used before. They use it to examine whether the fermion in QED dynamically acquires a mass. The result is that indeed, if the coupling is strong enough, a QED-type theory can produce a mass by itself, even if the input Lagrangian was massless. The threshold is alpha~1, which, if I understand the conventions correctly, is much larger than the physical value, but such things are always tricky because the computation still is only a coarse approximation.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.48.4933

#paperOfTheDay : "Fixed-point structure and effective fractional dimensionality for O(N) models with long-range interactions" from 2015.
This paper considers scalar lattice models in statistical #physics (which, in the appropriate limit, are equivalent to scalar Euclidean #quantumFieldTheory ). The "classic" such model, the Ising model, has interactions only between nearest neighbours, but many relevant physical systems allow for long-range interactions that decay with a power law. In that case, there is a "crossover": As one might guess, when the long-range interaction decays slowly, it dominates the behaviour of the system at large scales. However, if it decays fast enough, the system is effectively equivalent to having only shoart-range interaction. In particular, there is a parameter range where the interaction Hamiltonian/ the action contains a long-range term, but the resulting system is equivalent to not having any long-range term.
It has long been known that this crossover happens exactly at the critical dimension of the short-range theory: Basically, the local interactions give rise to an effective long-range behaviour, and when this one decays slower than the manually inserted long-range interaction, the short-range one dominates.
The present paper uses methods of the functional #renormalization group to confirm this picture. Concretely, they study a local potential approximation of the Wetterich equation, and find that indeed above the crossover, the presence of a quickly decaying long-range term does not alter the results, while below, when it decays less quickly, it gives rise to a different solution.
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.052113

#paperOfTheDay : "Fixed-point structure and effective fractional dimensionality for O(N) models with long-range interactions" from 2015.
This paper considers scalar lattice models in statistical #physics (which, in the appropriate limit, are equivalent to scalar Euclidean #quantumFieldTheory ). The "classic" such model, the Ising model, has interactions only between nearest neighbours, but many relevant physical systems allow for long-range interactions that decay with a power law. In that case, there is a "crossover": As one might guess, when the long-range interaction decays slowly, it dominates the behaviour of the system at large scales. However, if it decays fast enough, the system is effectively equivalent to having only shoart-range interaction. In particular, there is a parameter range where the interaction Hamiltonian/ the action contains a long-range term, but the resulting system is equivalent to not having any long-range term.
It has long been known that this crossover happens exactly at the critical dimension of the short-range theory: Basically, the local interactions give rise to an effective long-range behaviour, and when this one decays slower than the manually inserted long-range interaction, the short-range one dominates.
The present paper uses methods of the functional #renormalization group to confirm this picture. Concretely, they study a local potential approximation of the Wetterich equation, and find that indeed above the crossover, the presence of a quickly decaying long-range term does not alter the results, while below, when it decays less quickly, it gives rise to a different solution.
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.052113

#paperOfTheDay : "From nondegenerate conducting polymers to dense matter in the massive Gross-Neveu model" from 2005.
I shared a few papers about the Gross-Neveu model before: It is a renormalizable #quantumFieldTheory in 1+1 dimensions consisting of fermions with a 4-fermion-interaction vertex (which would not be renormalizable in 4 dimension). It is asymptotically free at high energies, and has a discrete chiral symmetry psi -> gamma_5 psi (where psi is the fermion field) when it is massless. However, under certain conditions a mass is dynamically generated, which leads to a quite interesting phase diagram: There is a massive and a massless phase, and also an intermediate non-homogeneous "crystal" phase.
This all is about the GN model as a relativistic field theory. However, the same (i.e. mathematically equivalent) model arises in condensed matter #physics as an effective model for the behaviour of e.g. polymers. A simple example are polymers consisting of long chains of carbon with alternating single and double bonds, C-C=C-C=... (with appropriate hydrogen atoms attached). The discrete chiral symmetry corresponds to flipping the location of the bonds, which might or might not yield an equivalent molecule (which, in field theory language, means that the mass is intact or broken). The present paper re-derives the ground state of the massive GN model from this polymer perspective, by solving the corresponding Schrödinger equations and finding the minimal energy solution. The results are fully compatible with field theory. As the authors put it, they foster the relation between "Phys Rev D" (fields) and "Phys Rev B" (condensed matter) communities. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.72.105008