Classical Field Theory Research Articles

On the automorphic side of the K-theoretic Artin symbol

Clausen has constructed a homotopical enrichment of the Artin reciprocity symbol in class field theory. On the Galois side, Selmer K-homology replaces the abelianized Galois group, while on the automorphic side the K-theory of locally compact vector spaces replaces classical idelic objects. We supply proofs for some predictions of Clausen regarding the automorphic side.

Emergent holographic description for the Kondo effect: Comparison with Bethe ansatz

Implementing Wilsonian renormalization group transformations in an iterative way, we develop a non-perturbative field theoretical framework, which takes into account all-loop quantum corrections organized in the $1/N$ expansion, where $N$ represents the flavor number of quantum fields. The resulting classical field theory is given by an effective Landau-Ginzburg theory for a local order parameter field, which appears in one-dimensional higher spacetime. We claim that such all-loop quantum corrections are introduced into an equation of motion for the order parameter field through the evolution in the emergent extra dimension. Based on this non-perturbative theoretical framework, we solve the Kondo effect, where the quantum mechanics problem in the projective formulation is mapped into a Landau-Ginzburg field theory for the hybridization order parameter field with an emergent extra dimension. We confirm the non-perturbative nature of this field theoretical framework. Intriguingly, we show that the Wilsonian renormalization group method can explain non-perturbative thermodynamic properties of an impurity consistent with the Bethe ansatz solutions. Finally, we speculate how our non-perturbative field theoretical framework can be connected with the AdS$_{d+2}$/CFT$_{d+1}$ duality conjecture.

Tribonacci numbers and primes of the form p = x 2 + 11y 2

Abstract In this paper we show that for any prime number p not equal to 11 or 19, the Tribonacci number T p−1 is divisible by p if and only if p is of the form x 2 + 11y 2. We first use class field theory on the Galois closure of the number field corresponding to the polynomial x 3 − x 2 − x − 1 to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for T p−1. We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight 2 and level 11.

Thermodensity coupling in phase-field-crystal-type models for the study of rapid crystallization

We self-consistently derive a formalism that couples a Phase Field Crystal (PFC) density field to thermal transport. It yields a theory for non-uniform transient temperature and density evolution, and includes local latent heat release during atomic rearrangements of the PFC density field. The basic formalism is applied to the basic PFC model, demonstrating the approach's capacity to capture solidification and recalescence. With an aim towards linking physical temperature to PFC temperature, a new classical density field theory for solid/liquid/vapor systems is then derived. It presents a different approach to those traditionally used in the PFC literature while still retaining the major advantages that have become the hallmark of PFC modelling. The new model is based entirely on physical density and temperature scales. We end the paper by applying the thermal-density coupling formalism to this new multi-phase density functional theory/PFC model.

Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields

We describe the elements of a novel structural approach to classical field theory, inspired by recent developments in perturbative algebraic quantum field theory. This approach is local and focuses mainly on the observables over field configurations, given by certain spaces of functionals which are studied here in depth. The analysis of such functionals is characterized by a combination of geometric, analytic and algebraic elements which (1) make our approach closer to quantum field theory, (2) allow for a rigorous analytic refinement of many computational formulae from the functional formulation of classical field theory and (3) provide a new pathway towards understanding dynamics. Particular attention will be paid to aspects related to nonlinear hyperbolic partial differential equations and their linearizations.

Mechanisms of Calcium Leak from Cardiac Sarcoplasmic Reticulum Revealed by Statistical Mechanics

Heart muscle contraction is normally activated by a synchronized Ca release from sarcoplasmic reticulum (SR), a major intracellular Ca store. However, under abnormal conditions, Ca leaks from the SR, decreasing heart contraction amplitude and increasing risk of life-threatening arrhythmia. The mechanisms and regimes of SR operation generating the abnormal Ca leak remain unclear. Here, we employed both numerical and analytical modeling to get mechanistic insights into the emergent Ca leak phenomenon. Our numerical simulations using a detailed realistic model of the Ca release unit reveal sharp transitions resulting in Ca leak. The emergence of leak is closely mapped mathematically to the Ising model from statistical mechanics. The system steady-state behavior is determined by two aggregate parameters: the analogs of magnetic field (h) and the inverse temperature (β) in the Ising model, for which we have explicit formulas in terms of SR [Ca] and release channel opening and closing rates. The classification of leak regimes takes the shape of a phase β-h diagram, with the regime boundaries occurring at h = 0 and a critical value of β (β∗) that we estimate using a classical Ising model and mean field theory. Our theory predicts that a synchronized Ca leak will occur when h > 0 and β >β∗, and a disordered leak occurs when β <β∗ and h is not too negative. The disorder leak is distinguished from synchronized leak (in long-lasting sparks) by larger Peierls contour lengths, an output parameter reflecting degree of disorder. Thus, in addition to our detailed numerical model approach, we also offer an instantaneous computational tool using analytical formulas of the Ising model for respective ryanodine receptor parameters and SR Ca load that describe and classify phase transitions and leak emergence.

Hertz\u2019s Viewpoint on Quantum Theory

In the nineteenth century (in the process of transition from mechanics to the field theory), the German school of theoretical physics confronted problems similar to the basic problems in the foundations of quantum mechanics (QM). Hertz tried to resolve such problem through analysis of the notion of a scientific theory and interrelation of theory and experiment. This analysis led him to the Bild (image) conception of theory (which was later essentially developed, but also modified by Boltzmann). In this paper, we claim that to resolve the basic foundational problems of QM, one has to use the Bild conception and reject the observational viewpoint on physical theory. As an example of a Bild theory underlying QM (treated as an observational theory), we consider prequantum classical statistical field theory (PCSFT): theory of random subquantum fields.

Nonlinear Schrödinger equation and semiclassical description of the microwave-to-optical frequency conversion based on the Lamb–Retherford experiment

It is shown that Lamb–Retherford experiment can be completely described within the framework of classical field theory without using such concepts as discrete states of the atom and jump-like electron transitions between them. The rate of stimulated decay of the metastable state of a hydrogen atom in an external periodic electric field is determined. The dependence of this rate on the frequency and amplitude of the external electric field, as well as on the parameters of the atom, is obtained. It is shown that the maximum value of the stimulated decay rate of the metastable state of a hydrogen atom is achieved at a frequency of an external electric field equal to the frequency shift corresponding to either the fine structure of the hydrogen atom or the Lamb shift. Using the Lamb–Retherford idea, a direct method of the microwave-to-optical frequency conversion is suggested. A theory of this conversion is developed, and the intensity of optical emission induced by microwave radiation is calculated.

On computing integral points of a Mordell curve – the method of Wildanger revisited

We develop a new efficient algorithm for solving Mordell’s equation. Our method is based on Wildanger’s geometry of number approach. Major new ingredients come from Kummer theory and class field theory. This allows to enlarge the range of computations considerably. For explicit calculations, we used Magma and KANT.

Decay of I-ball/oscillon in classical field theory

I-balls/oscillons are long-lived and spatially localized solutions of real scalar fields. They are produced in various contexts of the early universe in, such as, the inflaton evolution and the axion evolution. However, their decay process has long been unclear. In this paper, we derive an analytic formula of the decay rate of the I-balls/oscillons within the classical field theory. In our approach, we calculate the Poynting vector of the perturbation around the I-ball/oscillon profile by solving a relativistic field equation, with which the decay rate of the I-ball/oscillon is obtained. We also perform a classical lattice simulation and confirm the validity of our analytical formula of the decay rate numerically.

Covariant Canonical Gauge Gravitation and Cosmology

The covariant canonical transformation theory applied to the relativistic Hamiltonian theory of classical matter fields in dynamical space-time yields a novel (first order) gauge field theory of gravitation. The emerging field equations necessarily embrace a quadratic Riemann term added to Einstein’s linear equation. The quadratic term endows space-time with inertia generating a dynamic response of the space-time geometry to deformations relative to (Anti) de Sitter geometry. A “deformation parameter” is identified, the inverse dimensionless coupling constant governing the relative strength of the quadratic invariant in the Hamiltonian, and directly observable via the deceleration parameter q0. The quadratic invariant makes the system inconsistent with Einstein’s constant cosmological term, Λ = const. In the Friedman model this inconsistency is resolved with the scaling ansatz of a “cosmological function”, Λ(a), where a is the scale parameter of the FLRW metric. The cosmological function can be normalized such that with the A CDM parameter set the present-day observables, the Hubble constant and the deceleration parameter, can be reproduced. With this parameter set we recover the dark energy scenario in the late epoch. The proof that inflation in the early phase is caused by the “geometrical fluid” representing the inertia of space-time is yet pending, though. Nevertheless, as according to the CCGG theory the present-day cosmological function, identified with the currently observed Λobs, is a balanced mix of two contributions. These are the (A)dS curvature plus the residual vacuum energy of space-time and matter. The curvature term is proportional to the deformation parameter given by the coupling strength of the quadratic Riemann term. This allows for a fresh look at the Cosmological Constant Problem that plagues the standard Einstein-Friedman cosmology.

An invitation to multisymplectic geometry

In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we reformulate the latter in “multisymplectic terms”. Furthermore, we investigate basic questions on normal forms of multisymplectic manifolds, notably the questions whether and when Darboux-type theorems hold, and “how many” diffeomorphisms certain, important classes of multisymplectic manifolds possess. Finally, we survey recent advances in the area of symmetries and conserved quantities on multisymplectic manifolds.

Diophantine definability of nonnorms of cyclic extensions of global fields

We show that, for any square-free natural number n n and any global field K K with ( char ( K ) , n ) = 1 (\textrm {char}(K), n)=1 containing a primitive n n th root of unity, the pairs ( x , y ) ∈ K × × K × (x,y)\in K^{\times }\times K^{\times } such that x x is not a relative norm of K ( y n ) / K K(\sqrt [n]{y})/K form a diophantine set over K K . We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that, for any n ∈ N n\in {\mathbb {N}} and any global field K K with char ( K ) ≠ n \textrm {char}(K)\neq n , K × ∖ K × n K^{\times }\setminus K^{\times n} is diophantine over K K . For a number field K K , this is a result of Colliot-Thélène and Van Geel, proved using results on the Brauer–Manin obstruction. Additionally, we prove a variation of our main theorem for global fields K K without the n n th roots of unity, where we parametrize varieties arising from norm forms of cyclic extensions of K K without any rational points by a diophantine set.

Three-Dimensional Reference Interaction Site Model Self-Consistent Field Study on the Coordination Structure and Excitation Spectra of Cu(II)-Water Complexes in Aqueous Solution.

The molecular and solvation structures of the hydrated Cu2+ ions and their excitation spectra were investigated using the Kohn-Sham density functional theory (DFT) and the three-dimensional reference interaction site model (3D-RISM) self-consistent field method. Five stable geometrical structures were found to exist in aqueous solution: the distorted octahedral [Cu(H2O)6]2+ in C i and D2 h symmetries, the square pyramidal and trigonal bipyramidal [Cu(H2O)5]2+, and the square planar [Cu(H2O)4]2+. The distorted octahedral structure in the C i symmetry is preferred in [Cu(H2O)6]2+, and the square pyramidal and trigonal bipyramidal [Cu(H2O)5]2+ show almost the same stability. Among these geometries, the six-coordinate complex [Cu(H2O)6]2+ in the C i symmetry had the lowest Helmholtz energy. [Cu(H2O)6]2+ had a distorted octahedral structure, that is, two long axial bonds and four short equatorial bonds. The spatial and radial distribution function analyses for [Cu(H2O)5]2+ and [Cu(H2O)4]2+ showed that [Cu(H2O)5]2+ and [Cu(H2O)4]2+ had one and two solvent water molecules that constituted a distorted octahedron with ligand water distribution. The coordination numbers (CNs) derived from the distribution functions were 5.2-5.4 for [Cu(H2O)5]2+ and 5.3 for [Cu(H2O)4]2+. These results indicated that the Cu2+ ion in an aqueous solution had 5-6 coordination water molecules in the first hydration shell and some structures with different CNs may interchange in the solution. The excitation energies and electronic configurations of low-lying d-d excited states were calculated using the time-dependent DFT with the electric field generated by 3D-RISM. The orbital energies and electronic configurations were in a similar picture to those of the classical crystal field theory because of the highly symmetrical features of all structures. In [Cu(H2O)6]2+, the degeneracies of orbitals were resolved, whereas in [Cu(H2O)5]2+ and [Cu(H2O)4]2+, weak and strong quasi-degeneracies remained. As a result, only the four-coordinate complex generated third and fourth excited states, whereas in other complexes, there were no obvious characters of degeneracies. The resulting excitation energies were in good agreement with the absorption spectra.

Idèlic class field theory for 3-manifolds and very admissible links

We study a topological analogue of id\`elic class field theory for 3-manifolds, in the spirit of arithmetic topology. We firstly introduce the notion of a very admissible link $\mathcal{K}$ in a 3-manifold $M$, which plays a role analogous to the set of primes of a number field. For such a pair $(M,\mathcal{K})$, we introduce the notion of id\`eles and define the id\`ele class group. Then, getting the local class field theory for each knot in $\mathcal{K}$ together, we establish analogues of the global reciprocity law and the existence theorem of id\`elic class field theory.

Attractive interactions between like-oriented surface steps from an ab initio perspective: Role of the elastic and electrostatic contributions

At low temperatures the interactions between like-oriented steps on a surface are believed to be dominated by elastic repulsions. This belief is based on the results of the classical continuum field theories of elasticity. Electrostatic interactions are also believed to have a role but no theory has yet been developed for their description. In our paper we critically revisit the assumptions of these theories, and found that the ab initio approaches, based on the Born-Oppenheimer distinction between the valence electrons and the ions degrees of freedom, allow to identify distinctly both elastic and electrostatic contributions to the step energy and the step interaction energy. By using ab-initio calculations and specifying to the case of the technologically important (001)surface of GaAs, we show that the elastic interactions are not always repulsive and attractive step interactions between like-oriented steps are also possible. Furthermore, the repulsiveness and attractiveness of the step interactions depend on the step atomic structure.

Scientific Life and Works of Walter Noll

Walter Noll (1925–2017) was an American mathematician of German birth who made lasting contributions to the foundations of continuum physics and the classical non-linear field theory. This essay is an attempt to put in a broader perspective Noll’s methods and achievements in the hope that young generations of researchers may find their inspiration in the talent and depth of the old. By no means should this be considered as a historical account on the development of continuum mechanics through the second half of the twentieth century. We are content to illuminate Noll’s precious legacy.

Reinterpretation of the Mean Field Hypothesis in Analytical Models of Ostwald Ripening and Grain Growth

A long right tail is a common feature of experimental quasi-stationary size distributions of particles and grains that is not explained by the classical theories based on the mean field hypothesis. In this work, it is shown that the “pairwise interaction” approach, here presented in a comprehensive exposition involving both Ostwald ripening and grain growth, is a valid alternative to classical mean field theories since it produces more realistic predictions of the distribution shapes. The new analytical models are based on the mean field concept but rely on a detailed physical description of the elementary interactions responsible for the exchange of matter. They are jointly reviewed and compared with the corresponding classical Lifshitz–Slyozov–Wagner and Hillert models. The interactions are treated as a sum of elementary and specific contributions rather than as a generalized exchange with the mean field. The framework is complemented by the introduction of the “interaction volume” in Ostwald ripening and of the “local grain boundary curvature” in grain growth which are both size-dependent and permit to represent more precisely the local physics of the exchanges. The excellent results obtained in reproducing the experiments without any ad hoc parameters suggest that the mean field hypothesis adopted in the classical theories to describe the environment of a growing particle or grain represents a too drastic approximation. Therefore, it is proposed to replace the classical mean field by a “local mean field,” i.e., the ensemble of actual mean environments interacting with any single element of a given size. This alternative assumption induces a higher growth rate for large particles or grains compared with their respective mean field theories, thus producing right-skewed asymptotic distributions. For particles at small volume fraction the stationary distribution resembles a lognormal function, whereas for grains in normal grain growth regime the Rayleigh distribution is found as solution.

Canonical transformation path to gauge theories of gravity-II: Space-time coupling of spin-0 and spin-1 particle fields

The generic form of space-time dynamics as a classical gauge field theory has recently been derived, based on only the action principle and on the principle of general relativity. It was thus shown that Einstein’s general relativity is the special case where (i) the Hilbert Lagrangian (essentially the Ricci scalar) is supposed to describe the dynamics of the “free” (uncoupled) gravitational field, and (ii) the energy–momentum tensor is that of scalar fields representing real or complex structureless (spin-[Formula: see text]) particles. It followed that all other source fields — such as vector fields representing massive and nonmassive spin-[Formula: see text] particles — need careful scrutiny of the appropriate source tensor. This is the subject of our actual paper: we discuss in detail the coupling of the gravitational field with (i) a massive complex scalar field, (ii) a massive real vector field, and (iii) a massless vector field. We show that different couplings emerge for massive and nonmassive vector fields. The massive vector field has the canonical energy–momentum tensor as the appropriate source term — which embraces also the energy density furnished by the internal spin. In this case, the vector fields are shown to generate a torsion of space-time. In contrast, the system of a massless and charged vector field is associated with the metric (Hilbert) energy–momentum tensor due to its additional [Formula: see text] symmetry. Moreover, such vector fields do not generate a torsion of space-time. The respective sources of gravitation apply for all models of the dynamics of the “free” (uncoupled) gravitational field — which do not follow from the gauge formalism but must be specified based on separate physical reasoning.

Integrable Coupled σ Models.

A systematic procedure for constructing classical integrable field theories with arbitrarily many free parameters is outlined. It is based on the recent interpretation of integrable field theories as realizations of affine Gaudin models. In this language, one can associate integrable field theories with affine Gaudin models having arbitrarily many sites. We present the result of applying this general procedure to couple together an arbitrary number of principal chiral model fields on the same Lie group, each with a Wess-Zumino term.