We review several Grassmann techniques applied to
the transport of itinerant and spinless fermions moving as defects in dilute two-dimensional magnetic media, where they occupy site vacancies. 
Grassmann non-commuting variables are intrinsically related to classical spins, fermions, and field theories. Exact results are provided when a correspondence is made between partition functions of solvable problems and Grassmann integrals which can be expressed as determinants or Pfaffians. These techniques can be applied to both classical models of spins and fermionic problems. In this manuscript, we consider as an example the problem of fermions interacting indirectly via local classical spins in a magnetic medium, and we address the computation of thermal quantities from the Grassmann algebraic tools. The presence of strong non-local interactions between the fluctuations of the magnetic medium and the itinerant fermions can be analyzed via coupled self-consistent mean-field equations, after expressing the spins and fermion Hamiltonian in term of a Grassmannian effective action. We also apply these methods to the electronic transport at finite temperature, and show that Grassmann and Matsubara formalisms are an alternative solution to compute the Kubo conductivities. We also compare the results with the standard technique relying on the spectral decomposition of the Hamiltonian.

Amid deepening China‑Africa cooperation, social media discussions of Africans in Guangzhou have generated extensive textual data reflecting public opinion. Although these discourses reveal collective sentiment, they also expose persistent communication challenges, highlighting the need for constructive dialogue and the reduction of cross‑cultural barriers. Drawing on field theory and big data–assisted online content analysis, this study examines 62,877 valid data posts from Sina Weibo, WeChat, and Zhihu (2009–2024) to investigate Chinese social media users’ discursive marginalization of Africans in Guangzhou. Findings indicate that the topic is framed within a multithematic, negatively skewed discourse that fosters a negative emotional community. Within this space, patriotism intertwines with racism, and cultural conflict converges with identity negotiation, generating complex transmission dynamics. Communication challenges—namely, obstructed information flow, generalization of noise flow, and turbulence of influence flow—intensify the uncertainty and complexity of public discourse on China‑Africa relations. These challenges arise from the consolidation of negative habitus and the structural asymmetry of field power. Issue-specific habitus does not emerge spontaneously but arises from the asymmetrical coupling of multiple capitals in the field. Unlike classical field theory, which emphasizes habitus as stable dispositional tendencies formed through long-term individual socialization, social media fields generate short-term, affect-driven, and collective issue-specific habitus.

Abstract Within density-functional theory (DFT), Moreau–Yosida regularization enables both a reformulation of the theory and a mathematically well-defined definition of the Kohn–Sham approach. It is further employed in density–potential inversion schemes and, through the choice of topology for the density and potential space, can be directly linked to classical field theories. This perspective collects various appearances of the regularization technique within DFT alongside possibilities for their future development.

This work presents a comprehensive overview of three recently developed geometric frameworks for the study of classical action-dependent field theories. Specifically, the three underlying geometric structures — namely, k-contact, k-cocontact, and multicontact — are first introduced, and then used to develop the Lagrangian and Hamiltonian formalisms of the aforementioned theories. Finally, the relationship among these three types of structures is analyzed in the case of trivial bundles; as well as the comparison with other alternative definitions of multicontact structure presented in the literature.

This paper establishes profound connections between metallic ratios, Diophantine equations, and their underlying symmetry groups. Building on recent work on Diophantine equations and their solutions, we develop a comprehensive theory revealing the algebraic and geometric structures governing families of Diophantine equations associated with metallic ratios. Through detailed investigations of automorphism groups, continued fractions, and arithmetic geometry, we provide complete proofs and explicit examples that illuminate the deep arithmetic properties of metallic ratios. Our work offers a unified framework bridging number theory, group theory, and algebraic geometry, with applications to class field theory and computational number theory.

Abstract This article presents a comprehensive exposition on deriving the interaction potential between point-like particles within the framework of a scalar field. The Yukawa interaction energy, as well as the Coulomb interaction as a particular case, is obtained purely through classical arguments, without invoking field quantization, and directly from the field sources, with no need for artifacts such as the work-on-sources concept. Furthermore, we adopt an approach grounded in the language of classical field theory and based solely on dynamical equations, avoiding the use of analytic tools such as the Lagrangian and Hamiltonian density formalisms, in the hope that this exposition will help beginners appreciate the role of fields in mediating interactions and spark their interest in classical field theory. We also hope this approach is accessible, comprehensive, and self-contained for undergraduate physics students in the final year of a typical program. We recover a general formula that enables the calculation of the interaction energy between arbitrary stationary source distributions, not necessarily point-like, whose interaction is mediated by the scalar field. As an exercise for interested readers, we suggest investigating additional physical systems using the same framework employed in this work.

The (emergent) symmetry of a critical point constitutes fundamental pieces of information for determining the universality class and effective field theory. However, the underlying symmetry thus far can be conjectured only indirectly from the dimension of the order parameters in symmetry-breaking phases, and its correctness requires further verification to avoid overlooking hidden order parameters, which by itself is also a difficult task. In this Letter, we introduce an unbiased numerical approach to identify the underlying (emergent) symmetry of a critical point in quantum many-body systems without prior knowledge about the corresponding low-energy effective field theory. By numerically calculating the reduced density matrix in a very small subsystem of the total system, the Anderson tower of states in the entanglement spectrum can be obtained, clearly reflecting the underlying (emergent) symmetry of criticality. This is attributed to the fact that the entanglement spectrum can reveal the broken symmetry of the ground state of entanglement Hamiltonian after cooling from the critical point along an extra entanglement-temperature axis.

Abstract The uniqueness and rigidity of black holes remain central themes in gravitational research. In this work, we investigate the construction of all extremal black hole solutions to the Einstein equation for a given near-horizon geometry, employing the homotopy algebraic perspective, a powerful and increasingly influential framework in both classical and quantum field theory. Utilising Gaußian null coordinates, we recast the deformation problem as an analysis of the homotopy Maurer–Cartan equation associated with an L ∞ -algebra. Through homological perturbation theory, we systematically solve this equation order by order in directions transverse to the near-horizon geometry. As a concrete application of this formalism, we examine the deformations of the extremal Kerr horizon. Notably, this homotopy-theoretic approach enables us to characterise the moduli space of deformations by studying only the lowest-order solutions, offering a systematic way to understand the landscape of extremal black hole geometries.

We present a geometric framework for the regularization of the three-dimensional incompressible Navier-Stokes equations (NSE) based on the coupling between internal frequency fields and the pressure gradient. The model introduces an effective graduated viscosity μeff(p, Ω) and two anisotropic constitutive stresses, σ(Ω) and σ[Ω], linked to an internal frequency field Ω(x, t). The constitutive law ∇Ω ∝ −∇p establishes a coherent alignment between frequency and pressure, generating an intrinsic geometric damping in the direction most susceptible to vortex stretching. Within this setting, the extended Navier-Stokes system preserves Galilean invariance and classical energy dissipation while producing a coercive enstrophy inequality of the form d/ dt ‖ ω ‖ 2 +2 μ min ‖ ∇ω ‖ 2 +2c ‖ ( b⋅∇ )ω ‖ 2 ≤nonlinear terms , where b = ∇Ω/|∇Ω| and c_ represents the anisotropic damping amplitude. This new term acts as a directional energy sink, dynamically aligned with ∇p, and effectively suppresses local vorticity amplification even when c ∗ →0 . A global existence theorem (Theorem 8.1) is established for fixed constitutive parameters, followed by a vanishing-regularization program (Theorem 8.2) showing that if uniform Beale-Kato-Majda (BKM) bounds persist as (αμ, χ, η) → 0, then the limit solution satisfies the classical 3D NSE smoothly for all finite times. The key analytical challenge is maintaining a uniform BKM bound independent of the anisotropic regularization coefficients—a property conjectured to hold due to the intrinsic geometric coupling ∇Ω ∝ −∇p, which remains non-vanishing even in the limit. Physically, the mechanism can be interpreted as an internal redistribution of pressure energy along geometric flow directions. In hydrostatic or stratified configurations, this coupling enforces vertical coherence and horizontal energy dispersion, mirroring the natural stabilization observed in buoyant fluids and microgravity water spheres. The framework thus bridges classical fluid dynamics and geometric field theory, revealing how the internal geometry of pressure and frequency ensures smoothness and finite enstrophy in three-dimensional incompressible flows.

Gérard and Grellier proposed, under the name of the cubic Szegő equation, a remarkable classical field theory on a circle with a quartic Hamiltonian. The Lax integrability structure that emerges from their definition is so constraining that it allows for writing down an explicit general solution for prescribed initial data, and at the same time, the dynamics is highly nontrivial and involves turbulent energy transfer to arbitrarily short wavelengths. The quantum version of the same Hamiltonian is even more striking: not only the Hamiltonian itself, but also its associated conserved hierarchies display purely integer spectra, indicating a structure beyond ordinary quantum integrability. Here, we initiate a systematic study of this quantum system by presenting a mixture of analytic results and empirical observations on the structure of its eigenvalues and eigenvectors, conservation laws, ladder operators, etc.

Quantum-like modeling (QLM)—quantum theory applications outside of physics—are intensively developed with applications in biology, cognition, psychology, and decision-making. For cognition, QLM should be distinguished from quantum reductionist models in the spirit of Hameroff and Penrose, as well as Umezawa and Vitiello. QLM is not only concerned with just quantum physical processes in the brain but also with QL information processing by macroscopic neuronal structures. Although QLM of cognition and decision-making has seen some success, it suffers from a knowledge gap that exists between oscillatory neuronal network functioning in the brain and QL behavioral patterns. Recently, steps toward closing this gap have been taken using the generalized probability theory and prequantum classical statistical field theory (PCSFT)—a random field model beyond the complex Hilbert space formalism. PCSFT is used to move from the classical “oscillatory cognition” of the neuronal networks to QLM for decision-making. In this study, we addressed the most difficult problem within this construction: QLM for entanglement generation by classical networks, that is, “mental entanglement.” We started with the observational approach to entanglement based on operator algebras describing “local observables” and bringing into being the tensor product structure in the space of QL states. Moreover, we applied the standard states entanglement approach: entanglement generation by spatially separated networks in the brain. Finally, we discussed possible future experiments on “mental entanglement” detection using the EEG/MEG technique.

Let G G be a finite abelian p p -group. We count étale G G -extensions of global rational function fields F q ( T ) \mathbb F_q(T) of characteristic p p by the degree of what we call their Artin–Schreier conductor. The corresponding (ordinary) generating function turns out to be rational. This gives an exact answer to the counting problem, and seems to beg for a geometric interpretation. This is in contrast with the generating functions for the ordinary conductor (from class field theory) and the discriminant, which in general have no meromorphic continuation to the entire complex plane.

The aim of this paper is to formulate a non-commutative geometrical version of the classical electromagnetic field theory in the vacuum with the Moyal–Weyl algebra as the spacetime by using the theory of quantum principal bundles and quantum principal connections. As a result we will present the correct Maxwell equations in the vacuum of the model, in which we can appreciate the existence of electric and magnetic charges and currents. Finally, in Sec. [4], we present a mathematical model for which there are instantons that are not solutions of the corresponding Yang–Mills equation.

Topological defects occurring in nonlinear classical field theories are described by a system of second-order differential equations. A breakthrough was made in 1976 by E. B. Bogomoln’yi who demonstrated that in several field theories these equations can be reduced to first-order provided the coupling constants take on particular values. One of the examples involved a string in the Abelian Higgs model which is equivalent to the Abrikosov flux line of the Ginzburg-Landau theory of superconductivity. In a similar vein, in the 1966 textbook Superconductivity of Metals and Alloys P. G. de Gennes explained how to reduce the second-order Ginzburg-Landau equations to first-order at a particular value of the Ginzburg-Landau parameter by a method due to G. Sarma. We analyze the two ways of arriving at the first-order Sarma-Bogomol’nyi equations and conclude that while they both rely on the same operator identity, Sarma’s method is free of the assumption that there is a topological defect. The implication is that Bogomol’nyi equations found in other field theories may be a source of a wider range of solutions beyond topological defects.

Hard-magnetic soft materials (HMSMs) are susceptible to wrinkling instability, especially under compressive loading, which can significantly affect their functional performance. In this study, we report a theoretical framework based on continuum mechanics principles and nonlinear field theory of HMSMs to analyze both equal and unequal bi-axial deformation in planar hard-magnetic soft plates. The constitutive behavior of the HMSMs is modeled using a hyperelastic incompressible neo-Hookean model in conjunction with the ideal HMSM model. The presented theoretical framework incorporates classical tension field theory for predicting the onset and evolution of stable states in HMSM plates. A parametric study is conducted to highlight the effect of applied magnetic flux density, remanent magnetization, and bi-axiality ratio on the deformation behavior and stability of the HMSM plate. The insights gained from this study contribute toward the design and development of remotely controlled next-generation magneto-active polymer (MAP)-based soft robotic systems.

Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, we can try to determine them by using class field theory. For this, it is necessary to know the ramified primes. We show that the ramified primes of the subfield can be computed efficiently. Using this information, we give algorithms to determine all quadratic and cyclic cubic subfields of the initial field. The approach generalises to cyclic subfields of prime degree. In the case of quadratic subfields, our approach is much faster than other methods.

We review the construction of classical elliptic integrable systems on the $$\mathrm{SL}(N,\mathbb C)$$ Higgs bundles with non-trivial characteristic classes. The set of integrable models includes the Calogero–Moser system, its spin generalization, the Euler–Arnold top, and the Gaudin models. In some particular cases, mixed-type models known as the generalized models of interacting tops appear. Then we also review the field generalizations for these models on the $$(1\,{+}\,1)$$ -dimensional space–time. Finally, we propose a Maillet-type classical $$r$$ -matrix structure for the field analog of the generalized model of interacting tops corresponding to $$\mathrm{SL}$$ -bundles with non-trivial characteristic classes.

Coherent States in Classical Field Theory

The Theory of new Axioms and Laws contains 2 new Axioms and 8 Laws and it is by the same author. The classical axiom (Maxuell 1864) states that one closed vortex has constant velocity and it explains the Classical Field Theory. In contrast, the first new Axiom1 asserts that an open vortex has always a changing speed and it explains the new Theory of Open vortices. The Law1 describes that the electron is generated by a self -decelerating transverse open vortex in direction from out to in that is like open input. It generates inward (in2D) in its Gravity center a perpendicular self-accelerating longitudinal vortex upward (in 3D). Because generating transverse vortex is strongly self-decelerating the electron becomes to eccentric toroid. The Geometric center of body moves from to a new Gravity center that is in second quadrant. Thus the distance between these centers forms a Vector of Eccentricity. The Vector of Eccentricity lies on diameter of this toroid. Along this diameter on side of Gravity center, the distance between the transverse windings is minimal, and on the opposite side of diameter – the distance is maximal. Pulsating in time the electron emits a larger electric wave from place of the maximum distance, but a smaller one -from the side with minimum distance, Therefore, the resultant Electric wave moves in direction of towards the maximal distance that coincides open input of the electron. If pulsating electrons are phased with their open inputs towards one and the same end of a Conductor then their predominant radiation towards that end. It cause the movement of Electric wave to the this end. Thus the Electric wave flows inside to this end and Electric current – on surface of Conductor to its opposite end. The author proposes the Induction to be achieved without movement of the Conductor or the Magnetic field. The scientists can implant on the surface of Conductor a dense nano-grid emitting impulse Magnetic field. The pulsating lines of force of the Magnetic field hit the pulsating electrons and thus electrons rotate. The rotating continues until most of the electrons in the conductor order and phased their perpendicular vectors parallel and unidirectional to the external magnetic lines. Thus open inputs most likely point towards one and the same end of the Conductor. Therefore the phenomenon Induction of Electric current is connected with phasing of the electrons along the 3 axes. The existence of the phenomena Induction are direct evidence of exactly this structure of the electron according Theory of new Axioms and Laws. Ecxactly the inner structure of the electron is the reason it to react the outer impact of the lines of external Magnetic field in this kind. Therefore electron reacts like a particle or cell possessing some internal sensitivity and external reflex.

According to classical electromagnetic theory, quantum mechanics, and quantum field theory, particles are often associated with self-energy flow. In electromagnetic theory, the so-called self-energy flow corresponds to the Poynting vector of energy flow, and in quantum mechanics, it represents the probability current density. The author believes this is likely incorrect. The author finds that self-energy flow can be replaced by mutual energy flow. Mutual energy flow consists of retarded waves emitted by the source and advanced waves emitted by the sink. The source includes the primary coil of a transformer, the transmitting antenna, radiation atoms, and the cathode emitting electrons. The sink includes the secondary coil of the transformer, the receiving antenna, absorbing atoms, and the anode receiving electrons. Originally, all energy flows are generated by the source. If we assume energy flow is generated jointly by the source and the sink, we first need to compress the field produced by the source to half its original size. In this way, the source produces half of the field, and the sink produces the other half. The mutual energy flow generated by the source and sink together will exactly match the original self-energy flow produced by the source. However, there is still a problem. If the energy flow is mutual energy flow rather than self-energy flow, then self-energy flow should not transfer energy. However, if we do not alter the original electromagnetic theory or quantum theory, self-energy flow still transmits energy. This creates an absurd situation where mutual energy flow particles and self-energy flow particles are two distinct particles, both transferring energy. The author finds that both self-energy flows in electromagnetic theory and quantum theory should be reactive power and therefore do not transfer energy. The author adds a magnetic field, source, and sink to the Schrödinger equation and extends the mutual energy flow law from electromagnetic theory to quantum theory. The author then corrects the phase of the magnetic field in Maxwell’s electromagnetic theory and quantum theory by 90 degrees. After the correction, both self-energy flows in the two theories become reactive power. Thus, particles, including photons, electrons, and all other particles, are mutual energy flows, not self-energy flows. For electromagnetic theory, the author also proves that the composite energy flow of many photon mutual energy flows coincides with the self-energy flow corresponding to the Poynting vector calculated from Maxwell’s electromagnetic theory. Thus, electromagnetic waves have three different modes: (1) broadcasting mode, where the source and sink simultaneously emit spherical waves at random, which decay with propagation distance. (2) Photon mode, where the retarded wave emitted by the source and the advanced wave emitted by the sink are perfectly synchronized. At this point, the advanced wave forms the waveguide of the retarded wave, and the retarded wave forms the waveguide of the advanced wave. Due to the interference between the retarded and advanced waves, the two waves interfere constructively along the line connecting the source and the sink and weaken in other directions, ultimately turning into quasi-planar waves along the connection line between the source and sink. These retarded and advanced waves constitute the mutual energy flow, which is the photon. (3) The average energy flow consisting of countless photons is consistent with the energy flow calculated according to the Poynting vector. Therefore, Maxwell’s electromagnetic theory is still valid for calculating average radiation electromagnetic energy flow. However, calculating photons requires the author’s electromagnetic theory. The situation for electrons or other particles is roughly similar.