Paul Balduf

Mathematical physicist.
Postdoc at the University of Waterloo and Perimeter Institute, Canada.
I work with Feynman integrals, renormalization, combinatorics, large-order asymptotics of perturbation theory, and resummation.
My goal is to understand the physical properties of renormalized Green functions in quantum field theory.
Has anyone seen an instanton around here lately?
#quantum #physics
#math

Website: https://paulbalduf.com

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Last Notes

2026-10-20 15:57:16 GMT

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Das ist sehr interessant! Wie misst man denn Bodenfeuchte? Irgendwie über Leitfähigkeit? Oder wird ein Hohlraum gemacht und darin Luftfeuchtigkeit gemessen?

2026-10-18 16:57:52 GMT

The #paperOfTheDay is "Infinite conformal symmetry in two-dimensional quantum field theory" from 1984, probably one of the most cited works in theoretical #physics.
The authors study CFTs, i.e. field theories that are invariant under conformal transformations. Analogous to rotation symmetry (a physical theory should be the same if you rotate the coordinate system), conformal symmetry is a restriction to how a theory behaves: It says that a theory stays the same under all transformations that change the metric tensor by only a (position-dependent) scalar factor. Such transformations locally leave angles unchanged, but they can warp the spacetime globally (e.g. two straight lines crossing at a right angle will turn into two other lines, which still cross at a right angle, but might be curved otherwise). Generically, a real-world #quantumFieldTheory does not have conformal symmetry, but it is interesting to study from a conceptual perspective.
In our usual 4-dimensional spacetime, conformal symmetry gives "just" a few extra constraints, similar to what rotation symmetry implies. The situation is much different in a 2-dimensional spacetime, because there, a metric tensor is only a small (2x2) symmetric matrix, and the position-dependent scaling is in fact the only non-trivial degree of freedom. Hence, conformal symmetry in 2 dimensions is a much stronger constraint than in higher dimension.
The present paper classifies the quantities that behave consistently in a CFT: There are "primary operators", and each of them has an infinite chain of "secondary operators". Their correlation functions are heavily constrained and can be computed in terms of "conformal blocks". https://www.sciencedirect.com/science/article/pii/055032138490052X

2026-09-25 10:18:39 GMT

Many people visit #Leipzig for the Thomanerchor, the historic significance, or a scientific workshop at the uni or Max Planck Institute. Today I learned that it's also worth checking out the zoo with its newly built underwater tunnels. The seals are incredibly playful and active, it feels like they are watching the visitors more than the other way around. (Germans will know that there is also a weekly TV soap on MDR about this zoo).
https://media.mathstodon.xyz/media_attachments/files/117/331/138/241/050/235/original/e2c39583742a994e.mp4

2026-07-08 10:03:15 GMT

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

2026-06-30 11:12:53 GMT

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If one is mostly interested in (approximate) numerical values rather than proofs, one can sometimes use such information to improve resummation (secretly assuming that the singularities that might obstruct resummability are so far away that they don't matter numericall). E.g. if one knows that the leading singularity in the Borel plane is at a certain point, one can make a rational ansatz that has a singularity at that point. Effectively, this allows to determine one additional coefficient in the ansatz, hopefully making it more accurate. People have done many different such procedures in the resummation of e.g. critical exponents, but I am not aware of any systematic study if, and how much, this really increases the accuracy. The problem is that in practice one maybe knows 5 or 6 terms in the perturbation series, so that a rational approximant is e.g. of order [3/2] or [2/3], and these low-order rational functions are qualitatively very different depending on the particular degree one picks, and the resulting resummation might depend on this choice more than on knowing one additional coefficient.

2026-06-23 13:30:07 GMT

#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

2026-06-08 09:46:16 GMT

#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

2026-05-27 16:46:03 GMT

#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

2026-05-12 11:50:00 GMT

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This paper is by Gregor Wentzel, but I believe that other authors worked on related questions, too, at that time.
Often I only mention the authors when the thing they describe got named after them, or when I know of a particular personal circumstance that is relevant to the work. Especially the modern papers often have so many authors that I don't want to write a list of 6 names (and I want to avoid the implications that come with whether an author is famous or not, or who is the first author).
Even beyond posting, I usually immediately forget the authors of a paper I have read, but I remember "there is a paper where someone showed ..."

2026-05-11 07:40:56 GMT

#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

2026-05-07 19:46:19 GMT

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Yes.

2026-05-07 07:27:14 GMT

#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

2025-10-25 14:44:49 GMT

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good explanation. And beyond the original amplituhedron story, by now, there is a whole synergy grant full of people working on various aspects of positive geometry (https://positive-geometry.com ). They have literally written a paper titled "What is positive geometry?" (https://arxiv.org/abs/2502.12815v2 ) to answer this question for interested mathematicians and physicists.

2025-10-07 14:40:00 GMT

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personally, I would prefer category theory, not so much the mathematical definitions (which are reasonably clear), but rather an explicit discussion with examples of how it is relevant and useful for physics. I have the impression that category theory is mentioned at every single mathematical physics research talk these days, and in many cases, it doesn't become clear to me what the underlying physical intuition really is. Clearly, I must be missing a part of the story. But maybe I'm not your target audience, and I should rather read a review 😁

2025-08-27 12:10:45 GMT

#Perimeter Institute has many objects that are both for decoration and #physics education. One of them is an inflatable semi-spherical moon. Overnight, it lost its pressure, but not its educational value: it is now an illustration of spherical #geometry , showing that the surface of a sphere can not easily be mapped onto a plane without wrinkles.
https://media.mathstodon.xyz/media_attachments/files/115/100/640/780/519/361/original/e334271622b2d605.jpg

2025-07-03 07:26:00 GMT

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This also has a deep relation to renormalization in quantum field theory: The Green functions are formal power series, and renormalization is reversion and insertion of these series, hence its coefficients are enumerated by associahedra as you explained. On the other hand, the coefficients are finite sums of Feynman graphs, and the subgraphs of a given graph naturally have the structure of rooted trees. It turns out that the contribution of every individual tree is combinatorially described by the sum of all tubings of that tree. Now those tubings, for a fixed tree, themselves furnish the vertices of associahedra. Physically, this means that the renormalization of individual Feynman graphs has the same combinatorial structure as the renormalization of the full theory.
https://doi.org/10.1112/jlms.70006
https://media.mathstodon.xyz/media_attachments/files/114/788/076/255/755/804/original/b40101ccee0012be.jpg

2025-05-04 12:33:26 GMT

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actually, I found the article itself quite interesting. It contains an informative historical review of solving algebraic equations and a nice geometric interpretation of generalisations of Catalan numbers (where at least the basic case had long been known).
Formal power series can also be indexed by rooted trees (in the obvious way where concatenation is attaching subtrees). In the (Connes-Kreimer) Hopf algebra of rooted trees, there is a Hochschild 1-cocycle B, which acts by merging multiple trees to a new common root, "grafting". From an analysis perspective, iterating this operation solves a certain class of differential equations (which correspond to Dyson-Schwinger equations if one interprets the power series as a renormalized coupling in a QFT). The operation \nabla introduced in the paper appears to me to be (a special case of?) the grafting operator B, if one maps the triangulations to rooted trees. So this paper is actually interesting to me from the combinatorial perspective of giving another instance/interpretation of the cocycle (it is known that the cocycle B is universal among all graded... Hopf algebras, so if this new thing is a cocycle, it is necessarily a special case of B).
But somehow, the popular science headlines made it sound like "This guy disproved Galois theory," which is not doing it justice. Nevertheless, I wouldn't necessarily agree that infinite formal power series are more "existing" than irrational numbers.

2025-03-16 00:10:45 GMT

I almost overlooked that I've been on #Mastodon for a full year now. I must say, I was sceptical in the beginning, but I feel that we have been making continuous progress in terms of reach and interesting content, while at the same time not falling for too excessive spam. Keep it up 👍!
Since many scientific institutions have joined in recent months, and with them hopefully many more scientists, it's maybe time for an updated #introduction
I'm a theoretical physicist who works as a postdoc at the Math Institute in Oxford 🐂. My research is about #quantum field theory, with an emphasis on combinatorics. Some aspects are rather "hands on", such as the numerical values and distribution of Feynman integrals, which is input for new numerical methods to compute scattering cross sections. Other aspects of my work are rather mathematical, such as the relationship between topological field theory and the cohomology of the odd graph complex. I have a website that contains a summary of my research and a sporadic blog about my postdoc life and what to do with physics in real life. https://paulbalduf.com

2025-02-08 00:23:54 GMT

Currently, I'm working on a problem in #quantum field theory where we use #FeynmanIntegral s. These integrals are depicted by graphs, and they can be divergent when a graph has too many edges for a given number of vertices. The task is to identify all subgraphs that are divergent. This is a coproduct: It produces multiple terms, and each term is a list of 2 elements. The first element is one or multiple divergent subgraphs, and the second element is the remainder. It is surprising how many terms the coproduct has even for small graphs. For my example, even if the red graph is rather small, there are already 15 combinations of divergent subgraphs. To compute a physically sensible result, one needs to sum over all original graphs, and subtract all these combinations of subgraphs. #physics #research
https://media.mathstodon.xyz/media_attachments/files/113/965/379/239/924/337/original/01f2e030937b4833.jpg