Classical Gauge Field Theories Research Articles

Finite-group gauge theories on lattices as Hamiltonian systems with constraints

In this work, we present a brief but insightful overview of the gauge theories, which are defined on n-dimensional lattices by using finite gauge groups, in order to show how they can be interpreted as a Hamiltonian system with constraints, analogous to what happens with the classical (continuous) gauge (field) theories. As this interpretation is not usually explored in the literature that discusses/introduces the concept of lattice gauge theory, but some recent works have been exploring Hamiltonian models in order to support some kind of quantum computation, we use this interpretation to, for example, present a brief geometric view of one class of these models: the Kitaev Quantum Double Models.

Edge modes as reference frames and boundary actions from post-selection

We introduce a general framework realizing edge modes in (classical) gauge field theory as dynamical reference frames, an often suggested interpretation that we make entirely explicit. We focus on a bounded region M with a co-dimension one time-like boundary Γ, which we embed in a global spacetime. Taking as input a variational principle at the global level, we develop a systematic formalism inducing consistent variational principles (and in particular, boundary actions) for the subregion M. This relies on a post-selection procedure on Γ, which isolates the subsector of the global theory compatible with a general choice of gauge-invariant boundary conditions for the dynamics in M. Crucially, the latter relate the configuration fields on Γ to a dynamical frame field carrying information about the spacetime complement of M; as such, they may be equivalently interpreted as frame-dressed or relational observables. Generically, the external frame field keeps an imprint on the ensuing dynamics for subregion M, where it materializes itself as a local field on the time-like boundary Γ; in other words, an edge mode. We identify boundary symmetries as frame reorientations and show that they divide into three types, depending on the boundary conditions, that affect the physical status of the edge modes. Our construction relies on the covariant phase space formalism, and is in principle applicable to any gauge (field) theory. We illustrate it on three standard examples: Maxwell, Abelian Chern-Simons and non-Abelian Yang-Mills theories. In complement, we also analyze a mechanical toy-model to connect our work with recent efforts on (quantum) reference frames.

Lie algebroids and the geometry of off-shell BRST

It is well-known that principal bundles and associated bundles underlie the geometric structure of classical gauge field theories. In this paper, we explore the reformulation of gauge theories in terms of Lie algebroids and their associated bundles. This turns out to be a simple but elegant change, mathematically involving a quotient that removes spurious structure. The payoff is that the entire geometric structure involves only vector bundles over space-time, and we emphasize that familiar concepts such as BRST are built into the geometry, rather than appearing as adjunct structure. Thus the formulation of gauge theories in terms of Lie algebroids provides a fully off-shell account of the BRST complex. We expect that this formulation will have appealing impacts on the geometric understanding of quantization and anomalies, as well as entanglement in gauge theories. The formalism covers all gauge theories, and we discuss Yang-Mills theories with matter as well as gravitational theories explicitly.

Domain Wall Solitons Arising in Classical Gauge Field Theories

Domain wall solitons are basic constructs realizing phase transitions in various field-theoretical models and are solutions to some nonlinear ordinary differential equations descending from the corresponding full sets of governing equations in higher dimensions. In this paper, we present a series of domain wall solitons arising in several classical gauge field theory models. In the context of the Abelian gauge field theory, we unveil the surprising result that the solutions may explicitly be constructed, which enriches our knowledge on integrability of the planar Liouville type equations in their one-dimensional limits. In the context of the non-Abelian gauge field theory, we obtain some existence theorems for domain wall solutions arising in the electroweak type theories by developing some methods of calculus of variations formulated as direct and constrained minimization problems over a weighted Sobolev space.

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.

Physics architecture. Part two

Physics architecture. Part two. The talk is devoted to the jubilee of academician M.A. Markov, who was born in 1908. Now we have also the 100th anniversary of the gauge invariance principle proposed by Weyl in 1918. During the period from 1967 to 1988 M.A. Markov was Academician Secretary of Nuclear Physics Department of AN USSR. He organized the series of conferences dedicated to the topical problems of theoretical, nuclear and elementary particle physics. One of them was international seminar “Vector mesons and electromagnetic interactions” under A.M. Baldin direct leadership. It went off in JINR in September of 1969. This very that seminar gave birth to the conferences series named “Baldin autumn”. Thanks to the discussions at the A.M. Baldin seminar many critical situations in theoretical and experimental physics were subsequently settled. The first plenary session of the seminar included L.D. Faddeev talk about gauge field quantization, which cleared the way to develop the renormalized quantum gauge field theory. My talk was dedicated to the transformation of the classical gauge field theory into geometrical one. As a result the unified geometrical theory was obtained, which could include any interactions together with Einsteinian gravity. In this connection both formulations of Weyl’s local gauge invariance principle (of 1918 and 1929) as well as Einsteinian principle of general relativity were used. This approach was based on the fibre bundle geometry, which was developed by mathematicians at that time. This theory gives a realizable classification of interactions according to the local Lie’s symmetry groups associated with them. This approach leads to the VIth Hilbert’s problem solution. To obtain further development of the gauge field theory, it is necessary to decide the question of the gauge field mass origin. It is the key moment both in quantum and geometrical classical theories of the gauge fields. Therefore it is very important to analyze the experiments, where massive and massless particles are converted in each other (similar Baldin experiments in JINR in 1967).

The energy–momentum tensor(s) in classical gauge theories

We give an introduction to, and review of, the energy–momentum tensors in classical gauge field theories in Minkowski space, and to some extent also in curved space–time. For the canonical energy–momentum tensor of non-Abelian gauge fields and of matter fields coupled to such fields, we present a new and simple improvement procedure based on gauge invariance for constructing a gauge invariant, symmetric energy–momentum tensor. The relationship with the Einstein–Hilbert tensor following from the coupling to a gravitational field is also discussed.

Modified Proca equation and modified dispersion relation from a power-law Lagrangian functional

This work discusses some developments in classical gauge field theory which have grown out of the ideas of non-standard Lagrangians in the framework of the calculus of variations. A substantial part of this work is concerned with the modification of some basic equations of classical field theories starting from a power-law Lagrangian functional with time-dependent power and which belongs to the class of non-standard Lagrangians. Many equations principally the modified Proca equation and some modified dispersion relations for the consequent field equations are derived.

Proof of the Julia–Zee Theorem

It is a well accepted principle that finite-energy static solutions in the classical relativistic gauge field theory over the (2 + 1)-dimensional Minkowski spacetime must be electrically neutral. We call such a statement the Julia–Zee theorem. In this paper, we present a mathematical proof of this fundamental structural property.

Lectures on Classical Integrable Systems and Gauge Field Theories

In these lectures we consider Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimension. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL(N, C) and explain the Symplectic Hecke correspondence for these systems.

Projective geometry and special relativity

In this paper we present the Koopman-von Neumann (KvN) formulation of classical non-Abelian gauge field theories. In particular we shall explore the functional (or classical path integral) counterpart of the KvN method. In the quantum path integral quantization of Yang-Mills theories concepts like gauge-fixing and Faddeev-Popov determinant appear in a quite natural way. We will prove that these same objects are needed also in this classical path integral formulation for Yang-Mills theories. We shall also explore the classical path integral counterpart of the BFV formalism and build all the associated universal and gauge charges. These last are quite different from the analog quantum ones and we shall show the relation between the two. This paper lays the foundation of this formalism which, due to the many auxiliary fields present, is rather heavy. Applications to specific topics outlined in the paper will appear in later publications.

Exact solutions of classical electrodynamics and the Yang-Mills-Wong theory in even-dimensional space-time

Exact solutions of classical gauge theories in even-dimensional (D=2n) space-time are discussed. Common and specific properties of these solutions are analyzed for the particular dimensions D=2, D=4, and D=6. The consistent formulation of classical gauge field theories with pointlike charged or colored particles is proposed for D=6. The particle Lagrangian must then depend on the acceleration. In D=2, radiation is absent and all processes are invertible w.r.t. time. In D=6, the expression for the radiation intensity, as well as the equation of motion of a self-interacting particle, is obtained; trembling always leads to radiation. Non-Abelian solutions are absent for any D≠4, and only Coulomb-like solutions, which correspond to the Abelian limit of the D-dimensional Yang-Mills-Wong theory, are admitted.

Kaluza-Klein or fiber bundles?

Two basic geometric approaches to the modern theory of gauge fields are analyzed and compared. The first approach is an extension of the Kaluza-Klein unified theory of gravity and electromagnetism (1921). The second approach generalizes Cartan's formulation (1925) of Riemannian geometry and GR which now is transformed to fiber bundle theory. The goal of this paper is to show that the above-mentioned geometric approaches to the classical gauge field theory are nonequivalent and lead to different forms of quantum gauge field theory. Bibliography:12 titles.

Nöther formalism for conserved quantities in classical gauge field theories. II. The arbitrary Bosonic matter case

A formalism suited to deal with Bosonic matter represented by tensor densities coupled with a gauge field and gravitational field is developed. Nöther conserved currents and superpotentials are found and the relation between the canonical and the Hilbert stress tensors are investigated in the general case. The particular cases of gauge theories with scalar matter fields and of purely geometrical theories with arbitrary matter are recovered from this general formalism; as an example of application we consider scalar densities and the vector matter fields.

Erratum: Nöther formalism for conserved quantities in classical gauge field theories [J. Math. Phys. 35, 1644–1657 (1994)

Erratum: Nöther formalism for conserved quantities in classical gauge field theories [J. Math. Phys. <b>35</b>, 1644–1657 (1994)

Nöther formalism for conserved quantities in classical gauge field theories

Gauge theories coupled with the gravitational field and external matter fields are considered and Nötherian techniques are used to build the canonical conserved quantities. Their superpotentials are found and the relation between the canonical stress tensor and the Hilbert tensor is determined. The case of Yang–Mills theories coupled with gravity and scalar matter fields is considered in detail and it is shown that in this case the canonical stress tensor and the Hilbert tensor coincide.

Pure gauge theory of the cosserat surface

Within the framework of classical gauge field theory we present the Cosserat model of a surface described by two-dimensional gauge continuum with T(3) > SO(3) symmetry group. The essential result of this paper is the identification of several basic objects of gauge theory, such as the pure gauge potential, the strength field tensor and the Yang-Mills equations with the basic objects of the Cosserat surface theory. This allows to derive several new results both within the Cosserat surface theory and the gauge field theory itself.

An application of functional equations to the analysis of the invariance identities of classical gauge field theory

The equations of motion for a particle in a classical gauge field are derived from the invariance identities 2 and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions Eα. The functions Eα are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang-Mills theory. 3 These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner.

A new gauge field theory of a continuum with dislocations and disclinations

Combining the non-Riemannian geometric continuum theory of defects with the classical gauge field theory, we establish a new gauge field theory of a continuum with dislocations and disclinations. the governing equations are derived on the basis of a variational principle. The varied physics quantities such as dislocation density tensor, disclination density tensor, strain tensors and strain density tensors and their reciprocal relations are specifically expressed. An intuitive explanation of why the Lagrangian density should include the SO(3) gauge field is given in the paper. The difference between this paper and [9] is finally discussed and it shows that this paper is more reasonable.

Higgs fields from symmetric connections—The bundle picture

The derivation (Forgács and Manton) of a classical gauge plus Higgs field theory from a purely symmetric gauge theory by dimensional reduction is translated into the pictorial language of symmetric connections on principal bundles. The bundle description provides a better understanding of symmetric gauge theories; its global nature excludes some gauge group-symmetry group combinations, and shows that the reduced gauge group is larger than the centralizer CH as previously thought. In general it is a certain factor group, NK/K, involving both the gauge and symmetry groups, and contains CH as a normal subgroup. The commonly used F2 Lagrangian of symmetric gauge theories will not prescribe the dynamics of fields associated with this larger group, unless these fields are suitably constrained. It is shown that a constrained Kaluza–Klein metric is required to construct the F2 Lagrangian, thus pointing the way to include dynamics for the NK/K group.