#quantumfieldtheory — Public Fediverse posts

The #paperOfTheDay is "how to resum Feynman graphs" from 2013. In #quantumFieldTheory , #FeynmanIntegral s are the individual constituents of a perturbation expansion with respect to a small coupling parameter. Since the number of graphs grows factorially with their number of vertices, and (in Bosonic theories) these graphs all have the same sign, this perturbation series is divergent.
One can in some cases improve the situation by a Hubbard-Stratonovich transformation, or loop-vertex expansion. This is a rearrangement of the terms of the perturbation series with respect to a different parameter, such as the parameter 1/N (where N is large) in an O(N) symmetric theory. Morally, this amounts to a "perpendicular" summation: Fix a certain order in 1/N, and sum all orders in a small coupling.
The present article goes even further: The Feynman integral of a graph with E many edges consists of E! many "Hepp sectors", each given by a spanning tree of the graph. The proposal is to use these spanning trees as fundamental object, i.e. first fix a tree and compute the sum of all graphs where that tree contributes, and in a second step sum over all trees.
In the paper, this is envisoned somewhat schematically as an alternative way to compute the exact perturbation series. Today, #tropicalFieldTheory morally realizes this approach as a numerical sampling algorithm, where a Hepp sector is constructed as a primary object, and then any of the corresponding graphs is taken as a numerical sample. #physics
https://link.springer.com/article/10.1007/s00023-013-0299-8

The #paperOfTheDay is a classic in #mathematics from 1909: "Der Eulersche Dilogarithmus Und Seine Verallgemeinerungen" by Niels Nielsen. This article concerns generalizations of logarithms, in particular the dilogarithm. Starting from the fundamental identity
log(x) = integral from 1 to x of 1/t dt,
various authors had thought of "generalized logarithms" by integrating another time. One defines the dilogarithm
Li(x) = - integral from 0 to x of log(1-t)/t dt,
but many other generalizations are conceivable, for example why not allow some other rational function in place of 1/t in the integrand? This was answered by Hill and Kummer, who showed that if F and G are arbitrary rational functions, then any integral of F(t) log(G(t)) dt evaluates to a finite linear combination of Li(x) plus logarithms.
Nielsen gives a comprehensive review of known results about dilogarithms, remarking that notation and definitions in the existing literature are inconsistent and unnecessarily complicated.
In the second part of the article, one class of further generalizations is proposed, today known as "Nielsen polylogarithms", essentially integrals of log^(m-1)(t) log^p(1-t) for arbitrary positive integers m and p. These are truly more general than the dilogarithm.
Funnily, Nielsen starts his work by remarking that dilogarithms appear to be a rather unimportant class of functions, compared to elliptic integrals and hypergeometric functions which have plenty application in #physics . Of course, this changed 30 years later with the advent of #QuantumFieldTheory . Today, generalized polylogarithms are the most common class of special functions encountered in #FeynmanIntegral s.
https://katalog.bibliothek.kit.edu/bib/516451

АНТРОПОЛОГИЯ, ANTHROPOLOGY, АНТРОПОЛОГІЯ
#АНТРОПОЛОГИЯ, #ANTHROPOLOGY, #АНТРОПОЛОГІЯ
t.me/scilib_yura15cbx/542

Thermodynamics, statistical physics
Термодинамика, статистическая физика
Термодинаміка, статистична фізика
#Thermodynamics, #statistical physics
#Термодинамика, #статистическаяфизика
#Термодинаміка, #статистичнафізика
t.me/scilib_yura15cbx/541

PQm Quantum mechanics
Квантовая механика
Квантова механіка
#Quantum mechanics
#Квантоваямеханика
#Квантовамеханіка
t.me/scilib_yura15cbx/540

PQft Quantum field theory
Квантовая теория поля
Квантова теорія поля
#Quantum field theory
t.me/scilib_yura15cbx/539

Фазовые переходы
Phase_transitions, Фазовіпереходи
#Фазовые переходы
#Phase transitions, #Фазовіпереходи
t.me/scilib_yura15cbx/538

Пиротехника, Піротехніка, Pyrotechnics
#Пиротехника, #Піротехніка, #Pyrotechnics
t.me/scilib_yura15cbx/537

Астрономия, Астрономія, Astronomy
#Астрономия, #Астрономія, #Astronomy
t.me/scilib_yura15cbx/536

PPop Popular-level
Популярная физика
Популярна Фізика
t.me/scilib_yura15cbx/535

PG General courses
Общие курсы
Загальні курси
t.me/scilib_yura15cbx/534

PPl Plasma Плазма
Физика плазмы
#Plasma #Плазма
#Физика плазмы
t.me/scilib_yura15cbx/533

PPh Philosophy of Physics
Философия физики,
Філософія фізики
t.me/scilib_yura15cbx/532

POs Oscillations and waves
Колебания и волны
Коливання і хвилі
t.me/scilib_yura15cbx/531

PNu Nuclear Physics
Ядерна фізика
Ядерная физика
#Nuclear Physics
#Ядернафізика
#Ядернаяфизика
t.me/scilib_yura15cbx/530

PNc Nonlinear chaos
Нелинейный хаос
Нелінійний хаос
#Nonlinear chaos
#Нелинейныйхаос
#Нелінійнийхаос
t.me/scilib_yura15cbx/529

PM Atomic Molecular and Optical Physics
Атомна молекулярна та оптична Фізика
Атомная молекулярная и оптическая физика
t.me/scilib_yura15cbx/528

PGrc Cosmology
Космология
Космологія
#Cosmology
#Космология
#Космологія
t.me/scilib_yura15cbx/527

PGr Gravitation
Гравитация
Гравітація
#Gravitation
#Гравитация
#Гравітація
t.me/scilib_yura15cbx/526

PGe Encyclopaediae physics
Енциклопедія
Энциклопедии
t.me/scilib_yura15cbx/525

PE Electromagnetism
Електромагнетизм
Электромагнетизм
#Electromagnetism
#Електромагнетизм
#Электромагнетизм
t.me/scilib_yura15cbx/523

PD Dynamical systems
Динамические системы
Динамічна система
t.me/scilib_yura15cbx/522

PCh Chemical physics
Хімічна фізика
Химическая физика
#Chemical physics
#Хімічнафізика
#Химическаяфизика
t.me/scilib_yura15cbx/521

PCtm Theoretical mechanics
Теоретическая механика
Теоретична механіка
t.me/scilib_yura15cbx/520

PCstr Special relativity
Спеціальна теорія відносності
Специальная теория относительности
t.me/scilib_yura15cbx/519

PCft Classical fields
Классические поля, классическая теория поля, класична теорія поля
t.me/scilib_yura15cbx/518

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 "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 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

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 "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 "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 "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 "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