Geometric Field Theory Research Articles

A Pure Geometric Field Theory with Parameterized Canonical W-tensor

A Pure Geometric Field Theory with Parameterized Canonical W-tensor

Generalized Amit-Roginsky model from perturbations of 3D quantum gravity

A generalized Amit-Roginsky vector model in flat space is obtained as the effective dynamics of pertubations around a classical solution of the Boulatov group field theory for 3D Euclidean quantum gravity, extended to include additional matter degrees of freedom. By further restricting the type of perturbations, the original Amit-Roginsky model can be obtained. This result suggests a general link (and possibly a unified framework) between two types of tensorial quantum field theories; on one hand quantum geometric group field theories and tensorial models for random geometry, and on the other hand melonic-dominated vector and tensorial models in flat space, such as the Amit-Roginsky model and the Sachdev-Ye-Kitaev model. Published by the American Physical Society 2024

High-order field theory and a weak Euler–Lagrange–Barut equation for classical relativistic particle-field systems

In both quantum and classical field systems, conservation laws such as the conservation of energy and momentum are widely regarded as fundamental properties. A broadly accepted approach to deriving conservation laws is built using Noether’s method. However, this procedure is still unclear for relativistic particle-field systems where particles are regarded as classical world lines. In the present study, we establish a general manifestly covariant or geometric field theory for classical relativistic particle-field systems. In contrast to quantum systems, where particles are viewed as quantum fields, classical relativistic particle-field systems present specific challenges. These challenges arise from two sides. The first comes from the mass-shell constraint. To deal with the mass-shell constraint, the Euler–Lagrange–Barut (ELB) equation is used to determine the particle’s world lines in the four-dimensional (4D) Minkowski space. Besides, the infinitesimal criterion, which is a differential equation in formal field theory, is reconstructed by an integro-differential form. The other difficulty is that fields and particles depend on heterogeneous manifolds. To overcome this challenge, we propose using a weak version of the ELB equation that allows us to connect local conservation laws and continuous symmetries in classical relativistic particle-field systems. By applying a weak ELB equation to classical relativistic particle-field systems, we can systematically derive local conservation laws by examining the underlying symmetries of the system. Our proposed approach provides a new perspective on understanding conservation laws in classical relativistic particle-field systems.

Ramification theory of reciprocity sheaves, I: Zariski–Nagata purity

Abstract We prove a Zariski–Nagata purity theorem for the motivic ramification filtration of a reciprocity sheaf. An important tool in the proof is a generalization of the Kato-Saito reciprocity map from geometric global class field theory to all reciprocity sheaves. As a corollary we obtain cut-by-curves and cut-by-surfaces criteria for various ramification filtrations. In some cases this reproves known theorems, in some cases we obtain new results.

Power operations in the Stolz–Teichner program

The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant $K$-theory. Further, we use the geometric approach to construct power operations for complexified equivariant elliptic cohomology.

The symplectic geometrical formulation of quantum hydrodynamics

Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.

QFT, EFT and GFT

We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences. We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.

P-torsion étale sheaves on the Jacobian of a curve

Suppose X is a smooth, proper, geometrically connected curve over Fq with an Fq-rational point x0. For any Fq×-character σ of π1(X) trivial on x0, we construct a functor Lnσ from the derived category of coherent sheaves on the moduli space of deformations of σ over the Witt ring Wn(Fq) to the derived category of constructible Wn(Fq)-sheaves on the Jacobian of X. The functors Lnσ categorify the Artin reciprocity map for geometric class field theory with p-torsion coefficients. We then give a criterion for the fully faithfulness of (an enhanced version of) Lnσ in terms of the Hasse–Witt matrix of X.

A Framework for Geometric Field Theories and their Classification in Dimension One

In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with geometric structures. We take particular care to establish a notion of smooth variation of such geometric structures, so that it makes sense to require the output of our field theory to depend smoothly on the input. We then test our framework on the case of 1-dimensional field theories (with or without orientation) over a manifold M. Here the expectation is that such a field theory is equivalent to the data of a vector bundle over M with connection and, in the nonoriented case, the additional data of a nondegenerate bilinear pairing; we prove that this is indeed the case in our framework.

Albert Einstein’s Epistemic Virtues and Vices

The article is based on the concepts of epistemic virtues and epistemic vices and explores A. Einstein’s contribution to the creation of fundamental physical theories, namely the special theory of relativity and general theory of relativity, as well as to the development of a unified field theory on the basis of the geometric field program, which never led to success. Among the main epistemic virtues that led Einstein to success in the construction of the special theory of relativity are the following: a unique physical intuition based on the method of thought experiment and the need for an experimental justification of space-time concepts; striving for simplicity and elegance of theory; scientific courage, rebelliousness, signifying the readiness to engage in confrontation with scientific conventional dogmas and authorities. In the creation of general theory of relativity, another intellectual virtue was added to these virtues: the belief in the heuristic power of the mathematical aspect of physics. At the same time, he had to overcome his initial underestimation of the H. Minkowski’s four-dimensional concept of space and time, which has manifested in a distinctive flexibility of thinking typical for Einstein in his early years. The creative role of Einstein’s mistakes on the way to general relativity was emphasized. These mistakes were mostly related to the difficulties of harmonizing the mathematical and physical aspects of theory, less so to epistemic vices. The ambivalence of the concept of epistemic virtues, which can be transformed into epistemic vices, is noted. This transformation happened in the second half of Einstein’s life, when he for more than thirty years unsuccessfully tried to build a unified geometric field theory and to find an alternative to quantum mechanics with their probabilistic and Copenhagen interpretation In this case, we can talk about the following epistemic vices: the revaluation of mathematical aspect and underestimation of experimentally – empirical aspect of the theory; adopting the concepts general relativity is based on (continualism, classical causality, geometric nature of fundamental interactions) as fundamental; unprecedented persistence in defending the GFP (geometrical field program), despite its failures, and a certain loss of the flexibility of thinking. A cosmological history that is associated both with the application of GTR (general theory of relativity) to the structure of the Universe, and with the missed possibility of discovering the theory of the expanding Universe is intermediate in relation to Einstein’s epistemic virtues and vices. This opportunity was realized by A.A. Friedmann, who defeated Einstein in the dispute about if the Universe was stationary or nonstationary. In this dispute some of Einstein’s vices were revealed, which Friedman did not have. The connection between epistemic virtues and the methodological principles of physics and also with the “fallibilist” concept of scientific knowledge development has been noted.

Development of singularities of the Skyrme model

The Skyrme model is a geometric field theory and a quasilinear modification of the Nonlinear Sigma Model (Wave Maps). In this paper, we study the development of singularities for the equivariant Skyrme Model, in the strong-field limit, where the restoration of scale invariance allows us to look for self-similar blow-up behavior. After introducing the Skyrme Model and reviewing what’s known about formation of singularities in equivariant Wave Maps, we prove the existence of smooth self-similar solutions to the [Formula: see text]-dimensional Skyrme Model in the strong-field limit, and use that to conclude that the solution to the corresponding Cauchy problem blows-up in finite time, starting from a particular class of everywhere smooth initial data.

Generalized Jacobians and explicit descents

We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $n$ dividing the degree of some reduced effective divisor $\mathfrak{m}$ on a curve $C$, we show that multiplication by $n$ on the generalized Jacobian $J_\frak{m}$ factors through an isogeny $\varphi:A_{\mathfrak{m}} \rightarrow J_{\mathfrak{m}}$ whose kernel is naturally the dual of the Galois module $(\operatorname{Pic}(C_{\overline{k}})/\mathfrak{m})[n]$. By geometric class field theory, this corresponds to an abelian covering of $C_{\overline{k}} := C \times_{\operatorname{Spec}{k}} \operatorname{Spec}(\overline{k})$ of exponent $n$ unramified outside $\mathfrak{m}$. The $n$-coverings of $C$ parameterized by explicit descents are the maximal unramified subcoverings of the $k$-forms of this ramified covering. We present applications of this to the computation of Mordell-Weil groups of Jacobians.

Blow-ups and class field theory for curves

We propose another proof of the geometric class field theory for curves by considering blow-ups of symmetric products of curves.

On the ramified class field theory of relative curves

We generalize Deligne's approach to tame geometric class field theory to the case of a relative curve, with arbitrary ramification.

Unification Principle and a Cosmological Model

We present a cosmological model constructed using a pure geometric field theory. The unification principle implies defining any physical object in the model using the building blocks of the geometry used, the Absolute Parallelism (AP) geometry. The type of AP geometry used has simultaneously nonvanishing curvature and torsion. The AP geometric structure, used for this application, satisfies the cosmological principle and switches off the electromagnetic sector of the theory automatically. The model obtained is found to be free from particle horizons and flatness problems. This model can be considered as representing a transition phase between a decelerated and accelerated Universe. Conservation in the model is guaranteed by the theory used.

Teleparallel gravity with non-vanishing curvature

Guided by the rules of Einstein’s geometrization philosophy, a pure geometric field theory is constructed. The Lagrangian used to derive the field equations of the theory is a curvature scalar of a version of absolute parallelism (AP) geometry known in the literature as the parameterized absolute parallelism (PAP) geometry. The linear connection of this version has simultaneously non-vanishing curvature and torsion. Analysis of the theory obtained shows clearly that it is a pure gravity theory. The theory is a teleparallel one, since the building blocks of both PAP and AP geometries are the same. It is shown analytically that the theory has a trivial version in the AP-geometry, if gravity is attributed to curvature not to torsion. In the case of spherical symmetry, solutions of the field equations give rise to the Schwarzschild exterior field. The theory depends on two principles: covariance and unification. The weak equivalence principle is satisfied under a certain condition. The work preserves Einstein’s main idea that gravity is just space–time curvature, although it is not a metric theory. It is shown that the theory reduces to vacuum general relativity upon taking the parameter of the geometry b = 0.

A pure geometric approach to stellar structure: Mass–radius relation

This paper represents the second step towards understanding stellar structure using pure geometric tools. It is an attempt to get a theoretical expression for a mass–radius relation. The stellar model used has been obtained as an analytic solution of the field equations of a pure geometric field theory. The method suggested to get this relation is very simple. It depends mainly on a set of differential equations implying the vanishing of all components of a geometric material-energy tensor on a boundary of stellar configuration. The theoretical relation obtained is a linear one [Formula: see text] with one free parameter [Formula: see text] Comparison with observation, using a sample of lower main-sequence stars, members of binary systems, is given. For the primary members [Formula: see text], we get [Formula: see text]. It is worthy of mention that the model obtained is a simple one. Rotation, magnetic field, etc. are not considered in the present treatment. So, the model is far from being complete. It is just a step to show that pure geometric consideration objects can be used to treat problems of stellar structure.

Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems

A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is developed. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space-time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles’ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler–Lagrange (EL) equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.

Initial singularity and pure geometric field theories

In the present article we use a modified version of the geodesic equation, together with a modified version of the Raychaudhuri equation, to study initial singularities. These modified equations are used to account for the effect of the spin-torsion interaction on the existence of initial singularities in cosmological models. Such models are the results of solutions of the field equations of a class of field theories termed pure geometric. The geometric structure used in this study is an absolute parallelism structure satisfying the cosmological principle. It is shown that the existence of initial singularities is subject to some mathematical (geometric) conditions. The scheme suggested for this study can be easily generalized.

On the concept of a filtered bundle

We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalizations of double vector bundles and double affine bundles. Furthermore, we show that the linearization of a filtered bundle — which can be seen as a partial polarization of the admissible changes of local coordinates — is well defined.