#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 "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

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 : "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 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/