Classical Field Theory Research Articles

A kinetic generator for classical field theories with conservation laws

In this study, we demonstrate that, by suitably modifying the equilibrium distribution, the Bhatnagar-Gross-Krook form of the Boltzmann equation can serve as a generator of a broad class of nonlinear partial differential equations via a suitable generalization of the Chapman-Enskog method. We achieve this by properly choosing the relaxation time and moments of the equilibrium state in such a way as to suppress the dynamics of the zeroth moment, density, at increasingly faster time scales. As concrete examples, explicit analytical formulations of the Burgers', Korteweg-de Vries and Kuramoto-Sivashinsky equations are presented, along with a discussion of the associated conservation laws of the corresponding moments. The concept is numerically demonstrated by solving the BGK Boltzmann equation with the finite-element method. Because the Boltzmann-BGK equation can be equipped with a H-theorem, the framework developed in this work indicates a path towards the development of new algorithms for the numerical solution of the aforementioned equations with enhanced stability.

Fractional values of orbital angular momentum in problems of classical physics

Classical field theory problems for which the presence of nontrivial topological Pauli phase is essential (i.e. fractional values of the orbital angular momentum are possible in two-dimensional case) are discussed within the two-dimensional Helmholtz equation. As examples, we consider the “wedge problem” — determination of the field created by a point charge placed between two conducting half-planes — and the Fresnel diffraction from knife edge.

Iterated Laurent series over rings and the Contou-Carrère symbol

This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol. The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described. As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol. Bibliography: 67 titles.

Coulomb law in the nonuniform Euler–Heisenberg theory

We consider the nonlinear classical field theory which results from adding to the Maxwell’s Lagrangian the contributions from the weak-field Euler–Heisenberg Lagrangian and a nonuniform part which involves derivatives of the electric and magnetic fields. We focus on the electrostatic case where the magnetic field is set to zero, and we derive the modified Gauss law, resulting in a higher-order differential equation. This equation gives the electric field produced by stationary charges in the higher-order nonlinear electrodynamics. Specializing for the case of a point charge, we investigate the solutions of the modified Gauss law and calculate the correction to the Coulomb law.

Integrable defects at junctions within a network

The purpose of this article is to explore the properties of integrable, purely transmitting, defects placed at the junctions of several one-dimensional domains within a network. The defect sewing conditions turn out to be quite restrictive—for example, requiring the number of domains meeting at a junction to be even—and there is a clear distinction between the behaviour of conformal and massive integrable models. The ideas are mainly developed within classical field theory and illustrated using a variety of field theory models defined on the branches of the network, including both linear and nonlinear examples.

Orbit-like trajectory of the vortex core in ferrimagnetic dots close to the compensation point

In physics, conserved quantities are key to understand and describe physical phenomena. These conserved quantities are related to Noether’s theorem and Lagrangian description in both classical mechanics and field theory. In this article we have found the equation of the vortex core trajectory in ferrimagnets close to the compensation point in terms of two conserved physical quantities, namely the energy, E, and a vector perpendicular to the orbit plane, A→=−L→+G→|r→c|2∕2 where G→, L→ and r→c are the topological gyrovector, the angular momentum and the position of the vortex core, respectively. We found that in the absence of a dissipative term, for small deviations of the vortex core, the trajectory is bounded between two concentric circles. On the contrary, under the action of a dissipative term proportional to the damping coefficient, A→ is no longer conservative and the vortex core moves either towards the center or out of the cylinder, depending on the circularity of the magnetic vortex and the intensity of the magnetic field applied in the plane of the cylinder.

The relation between quantum and classical field theory with a classical source.

Classical field theory is about fields and how they behave in space-time. Quantum field theory, in practice, usually seems to be about particles and how they scatter. Nevertheless, classical fields must emerge from quantum field theory in appropriate limits, and Michael Duff showed how this happens for the Schwarzschild solution in perturbative quantum gravity. In a series of papers, we and others have shown how classical radiation from an accelerated charge emerges from quantum field theory when the Unruh thermal effect is taken into account. Here, we sharpen those conclusions by showing that, even at finite times, the quantum picture is meaningful and is in close agreement with the classical picture.

Mesoscopic quantum superposition states of weakly-coupled matter-wave solitons

The Josephson junctions (JJs) are at the heart of modern quantum technologies and metrology. In this work we establish quantum features of an atomic soliton Josephson junction (SJJ) device, which consists of two weakly-coupled condensates with negative scattering length. The condensates are trapped in a double-well potential and elongated in one dimension. Starting with classical field theory we map for the first time a two-soliton problem onto the effective two-mode Hamiltonian and perform a second quantization procedure. Compared to the conventional bosonic Josephson junction condensate system, we show that the SJJ-model in quantum domain exhibits unusual features due to its effective nonlinear strength proportional to the square of total particle number, N2. A novel self-tuning effect for the effective tunneling parameter is also demonstrated in the SJJ-model, which depends on the particle number and rapidly vanishes as the JJ population imbalance increases. The formation of entangled Fock state superposition is predicted for the quantum SJJ-model, revealing dominant N00N-state components at the ‘edges’ for n = 0, N particle number. We have shown that the obtained quantum state is more resistant to few particle losses from the condensates if tiny components of entangled Fock states are present in the vicinity of the major N00N-state component. This peculiarity of the quantum SJJ-model establishes an important difference from its semiclassical analogue obtained in the framework of Hartree approach. Our results are confirmed by studying the first and N-order Hillery–Zubairy criteria applied for studying multiparticle entanglement and planar spin squeezing. The Einstein–Podolsky–Rosen quantum steering represents an important prerequisite for the crossover to the mesoscopic superposition Schrödinger-cat and/or N00N-states. The feasibility in observation for these predicted states of the SJJ-model in the experiments is also discussed by taking into account one- and three-body losses for lithium condensates.

K-spin Hamiltonian for quantum-resolvable Markov decision processes

The Markov decision process is the mathematical formalization underlying the modern field of reinforcement learning when transition and reward functions are unknown. We derive a pseudo-Boolean cost function that is equivalent to a K-spin Hamiltonian representation of the discrete, finite, discounted Markov decision process with infinite horizon. This K-spin Hamiltonian furnishes a starting point from which to solve for an optimal policy using heuristic quantum algorithms such as adiabatic quantum annealing and the quantum approximate optimization algorithm on near-term quantum hardware. In arguing that the variational minimization of our Hamiltonian is approximately equivalent to the Bellman optimality condition for a prevalent class of environments we establish an interesting analogy with classical field theory. Along with proof-of-concept calculations to corroborate our formulation by simulated and quantum annealing against classical Q-Learning, we analyze the scaling of physical resources required to solve our Hamiltonian on quantum hardware.

Schmid’s Formula for Higher Local Fields

In local class field theory, the Schmid-Witt symbol encodes interesting data about the ramification theory of $p$-extensi-ons of $K$ and can, for example, be used to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid-Witt symbol of Artin-Schreier extensions of local fields. Later, his formula was generalized to Artin-Schreier-Witt extensions, but still over a local field. In this paper we generalize Schmid's formula to compute the Artin-Schreier-Witt-Parshin symbol for Artin-Schreier-Witt extensions of two-dimensional local fields of positive characteristic.

Emergent dual holographic description for interacting Dirac fermions in the large N limit

We derive an effective dual holographic Einstein-Maxwell theory, applying renormalization group transformations to interacting Dirac fermions in a recursive way. In particular, we show how both background metric tensor and U(1) gauge fields become dynamical to describe the renormalization group flows of coupling functions and order parameter fields of the corresponding quantum field theory through natural emergence of an extra-dimensional space, respectively. Finally, we propose a prescription on how to calculate correlation functions in a non-perturbative way, where quantum fluctuations of both metric and gauge fields are frozen to cause a classical field theory of the Einstein-Maxwell type in the large $N$ limit. Here, $N$ is the flavor degeneracy of Dirac fermions. This prescription turns out to coincide with that of the holographic duality conjecture.

Power-law Genesis: Strong coupling and Galileon-like vector fields

A simple way to construct models with early cosmological Genesis epoch is to employ bosonic fields whose Lagrangians transform homogeneously under scaling transformation. We show that in these theories, for a range of parameters defining the Lagrangian, there exists a homogeneous power-law solution in flat space-time, whose energy density vanishes, while pressure is negative (power-law Genesis). We find the condition for the legitimacy of the classical field theory description of such a situation. We note that this condition does not hold for our earlier Genesis model with vector field. We construct another model with vector field and power-law background solution in flat space-time, which is legitimately treated within classical field theory, violates the Null Energy Condition (NEC) and is stable. Upon turning on gravity, this model describes the early Genesis stage.

New Axioms and Laws

This report contains 2 new axioms and 8 laws. It predicts to include except the Electromagnetic Field and other unknown and unexplored fields, for example the Gravity Field. As it is well known the Classic Field Theory is described mostly by Theory of Electromagnetic Field. The Electromagnetic Field is described by Maxwell’s laws (1864). The Maxwell’s laws are certified by a single axiom which claims that the movement of a vector E in a closed loop (div rot E = 0) is evenly. The author replaces this axiom with a new one, according to which the movement of a vector E in an open loop (div rot E ≠ 0) or an open vortex (div Vor E ≠ 0) is unevenly. If the vortex is in plane (2D), it is named a cross vortex. If the vortex is in volume (3D), it is named a longitudinal vortex. Something more- each vortex can be decelerating (div (VorE) <0) or accelerating (div (VorE)> 0). After the first axiom are obtained immediately 4 types of movements – cross vortex, which can be accelerating or decelerating and longitudinal vortex, which can also be accelerating or decelerating. The following results are obtained : evenly movement is replaced with unevenly movement (decelerating or accelerating); movement in a closed loop is replaced with movement in an open loop or vortex ; a cross vortex in 2D generates a longitudinal vortex in 3D through a special transformation and vice versa- the longitudinal vortex in 3D through another special transformation generates a cross vortex in 2D ; the decelerating vortex emits primary vortices to environment, but the accelerating vortex sucks into the same primary vortices from environment; the accelerating longitudinal vortices are attracted to one another, as the faster vortex is inserted into the slower and thus form a funnel, which is the model of gravity funnel. It should be noted, in particular, the results of the second axiom. It claims that two complex (cross-longitudinal) vortex objects in 3D that work in one direction as one complementary pair, are existed simultaneously. This way they are obtained 2 pairs of complementary objects in both directions. As a final result are received many models with similar shapes and content. For example, the pair in one direction of complex complementary vortex objects is a model of the electron-proton chain, and in the opposite direction is an antiproton-positron chain model.

Lagrangian constraint analysis of first-order classical field theories with an application to gravity

We present a method that is optimized to explicitly obtain all the constraints and thereby count the propagating degrees of freedom in (almost all) manifestly first order classical field theories. Our proposal uses as its only inputs a Lagrangian density and the identification of the a priori independent field variables it depends on. This coordinate-dependent, purely Lagrangian approach is complementary to and in perfect agreement with the related vast literature. Besides, generally overlooked technical challenges and problems derived from an incomplete analysis are addressed in detail. The theoretical framework is minutely illustrated in the Maxwell, Proca and Palatini theories for all finite $d\geq 2$ spacetime dimensions. Our novel analysis of Palatini gravity constitutes a noteworthy set of results on its own. In particular, its computational simplicity is visible, as compared to previous Hamiltonian studies. We argue for the potential value of both the method and the given examples in the context of generalized Proca and their coupling to gravity. The possibilities of the method are not exhausted by this concrete proposal.

The Einstein and Landau‐Lifshitz pseudotensors—A mathematical note on existence

For a closed system, the conservation of energy and momentum has been affirmed through a vast array of experiments. In an attempt to reconcile the General Theory of Relativity with these findings, Einstein constructed, ad hoc, his so-called pseudotensor [A. Einstein, Ann. Phys. 49, 769 (1916)]. Yet this solution fell outside the tensorial mathematical structure of his theory. Landau and Lifshitz also constructed, ad hoc, an even more complex pseudotensor, as a proposed improvement upon the work of Einstein [The Classical Theory of Fields (Addison-Wesley Press, Inc., Cambridge, MA, 1951)]. Their pseudotensor is symmetric, unlike that proposed by Einstein. They advance that their pseudotensor yields a conservation law which also included angular momentum. However, once again, this approach leads to a mathematical construct which is not a tensor and thereby falls outside the very mathematical structure of Einstein’s theory. Both pseudotensors, whether that advanced by Einstein or by Landau and Lifshitz, violate the rules of pure mathematics and therefore can hold no place in physics.

Constraining Images of Quadratic Arboreal Representations

Abstract In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First, we show that, over global fields, quadratic post-critically finite (PCF) polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes’ conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily PCF, and we combine our results with local class field theory to classify quadratic pairs over ${ {\mathbb{Q}}}$ with abelian dynamical Galois group, improving on recent results of Andrews and Petsche. On the other hand, we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence toward Jones’ finite index conjecture.

The maximum number of points on a curve of genus eight over the field of four elements

The Oesterlé bound shows that a curve of genus 8 over the finite field F4 can have at most 24 rational points, and Niederreiter and Xing used class field theory to show that there exists such a curve with 21 points. We improve both of these results: We show that a genus-8 curve over F4 can have at most 23 rational points, and we provide an example of such a curve with 22 points, namely the curve defined by the two equations y2+(x3+x+1)y=x6+x5+x4+x2 and z3=(x+1)y+x2.

Antifungal activity of etomidate against growing biofilms of fluconazole-resistant Candida spp. strains, binding to mannoproteins and molecular docking with the ALS3 protein

This study evaluated the effect of etomidate against biofilms of Candida spp. and analysed through molecular docking the interaction of this drug with ALS3, an important protein for fungal adhesion. Three fluconazole-resistant fungi were used: Candida albicans, Candida parapsilosis and Candida tropicalis. Growing biofilms were exposed to etomidate at 31.25-500 µg ml-1. Then, an ALS3 adhesive protein from C. albicans was analysed through a molecular mapping technique, composed of a sequence of algorithms to perform molecular mapping simulation based on classic force field theory. Etomidate showed antifungal activity against growing biofilms of resistant C. albicans, C. parapsilosis and C. tropicalis at all concentrations used in the study. The etomidate coupling analysis revealed three interactions with the residues of interest compared to hepta-threonine, which remained at the ALS3 site. In addition, etomidate decreased the expression of mannoproteins on the surface of C. albicans. These results revealed that etomidate inhibited the growth of biofilms.

Particles, fields, and the measurement of electron spin

This article compares treatments of the Stern-Gerlach experiment across different physical theories, building up to a novel analysis of electron spin measurement in the context of classical Dirac field theory. Modeling the electron as a classical rigid body or point particle, we can explain why the entire electron is always found at just one location on the detector (uniqueness) but we cannot explain why there are only two locations where the electron is ever found (discreteness). Using non-relativistic or relativistic quantum mechanics, we can explain both uniqueness and discreteness. Moving to more fundamental physics, both features can be explained within a quantum theory of the Dirac field. In a classical theory of the Dirac field, the rotating charge of the electron can split into two pieces that each hit the detector at a different location. In this classical context, we can explain a feature of electron spin that is often described as distinctively quantum (discreteness) but we cannot explain another feature that could be explained within any of the other theories (uniqueness).

Witnessing the nonclassical nature of gravity in the presence of unknown interactions

General relativity as a classical field theory does not predict gravitationally induced entanglement, as such, recent proposals seek an empirical demonstration of this feature which would represent a significant milestone for physics. We introduce improvements to a spin witness protocol that reduce the highly challenging experimental requirements. After rigorously assessing approximations from the original proposal [S. Bose et al. Phys. Rev. Lett. 119, 240401 (2017)], we focus on entanglement witnessing. We propose a new witness which greatly reduces the required interaction time, thereby making the experiment feasible for higher decoherence rates, and we show how statistical analysis can separate the gravitational contribution from other possibly dominant and ill-known interactions. We point out a potential loophole and show how it can be closed using state tomography.