Tracking down gauge: an ode to the constrained Hamiltonian formalism

It is essentially known that useful gauge field Lagrangians arise as Weil polynomials of the curvature of the gauge connection. The deeper implications and details of this fact are worked out in two widely differing cases. The Glashow–Weinberg–Salam gauge field Lagrangian for electroweak theory and the Townsend–Zardecki action for gravitation are obtained from the same type of ‘‘Yang–Mills’’ Weil form on a principal fiber bundle over space-time, with symmetry group U(2) and SO(2,3), respectively. The unified geometrical approach given here shows that fiber bundle reduction and symmetry breaking are essential not only in electroweak theory but also in the SO(2,3) gauge theory for gravitation. In fact, the process of symmetry breaking in electroweak theory and the soldering of the anti-de Sitter bundle, essential in the interpretation of SO(2,3) gauge theory as a theory for gravitation, are corresponding geometrical concepts.

We review how gauge theories, initial introduced for classical mechanics and quantum led theory, seem to apply to many fields of research such as information theory, computer science, economy, biology. This non-exhaustive list raises natural question on future developments of this theory. Short Communication How gauge theories appear in information theory and economics Basically, the mathematical structure of groups is among the best adapted for describing transformations and moves. A path on a group can encode the evolution of a dynamical system, or the moves of an exterior observer with respect to a given system. In the theory of quantum fields, more general objects called principal bundles, which consist in a total space P with fibers isomorphic to a (Lie) group G over a base (simplicial complex or manifold). M For one of the most simple settings, one can see when the base is a manifold (we call them continuum gauge theories) [1-5]. When the base is a simplicial complex (or a lattice), the models are called discrete gauge theories. They stand formally as an integrated version of the previous ones which (more or less formally again). The physical study consists in solving equations on P or after quantization into fields, in minimizing so-called action functionals. These action functionals are in fact acting on connections, which are infinitesimal expressions of local slices. They can be understood also as differential operators. These delicate settings coming from physics have, in the last twenty years, found two applications to our knowledge: In so-called geometric arbitrage theory» for managing portfolios in financial markets [2,6-10], where fiber bundles or principal bundles describe, along their fibers and on the base, the external and internal parameters for the prediction of the evolution of the portfolio. Fiber bundles over a manifold described in [9] are related to principal bundles in a way described in [10]. In so-called approximate reasoning in pairwise comparisons where complete studies can be performed when * G + =  and with normalized objects, with very recent axiomatization [10-19]. However, the need of so-called non-numerical ranking appeals the use of non-abelian groups G and of mathematical tools coming from discrete gauge theories [12,18]. Gauge wraith in biology and neurology The aim of this short article, giving the opinion of the sole author, is to mention that there are some hidden aspects of neurology and biology which reveal, in some kind of matching mirror process, the presence of gauge effects. This aspect is not actually truly confirmed, it is more-or-less at a step of conceptualization, and that is the reason why we talk of gauge wraith. Other authors have already mentioned the similarities of models coming from biology and neuronal science [11,15,16], each time with very mild interpretations of classical facts. From a heuristic viewpoint, this is not so surprising since «physics of life» deal with entities interacting with each other, exactly like elementary particles of nuclear physics interact. At the microscopic scale, a simplicial structure arise naturally by linking an entity (neuron, molecule, cell) with each of its neighbours making discrete gauge theories appropriate, and at macroscopic level, a first approximation of phenomena can rise form continuum gauge theories. However, as a last remark, the mathematical theory of discretization of gauge theories has to be questioned very seriously, because the classical scheme based on leads to mathematical problems while other still non studied ways to discretize a continuum model can be proposed [17].

A basic problem of classical field theory, which has attracted growing attention over the past decade, is to find and classify all nonlinear deformations of linear abelian gauge theories. The physical interest in studying deformations is to address uniqueness of known nonlinear interactions of gauge fields and to look systematically for theoretical possibilities for new interactions. Mathematically, the study of deformations aims to understand the rigidity of the nonlinear structure of gauge field theories and to uncover new types of nonlinear geometrical structures. The first part of this paper summarizes and significantly elaborates a field-theoretic deformation method developed in earlier work. Some key contributions presented here are, firstly, that the determining equations for deformation terms are shown to have an elegant formulation using Lie derivatives in the jet space associated with the gauge field variables. Secondly, the obstructions (integrability conditions) that must be satisfied by lowest-order deformations terms for existence of a deformation to higher orders are explicitly identified. Most importantly, a universal geometrical structure common to a large class of nonlinear gauge theory examples is uncovered. This structure is derived geometrically from the deformed gauge symmetry and is characterized by a covariant derivative operator plus a nonlinear field strength, related through the curvature of the covariant derivative. The scope of these results encompasses Yang–Mills theory, Freedman–Townsend theory, and Einstein gravity theory, in addition to their many interesting types of novel generalizations that have been found in the past several years. The second part of the paper presents a new geometrical type of Yang–Mills generalization in three dimensions motivated from considering torsion in the context of nonlinear sigma models with Lie group targets (chiral theories). The generalization is derived by a deformation analysis of linear abelian Yang–Mills Chern–Simons gauge theory. Torsion is introduced geometrically through a duality with chiral models obtained from the chiral field form of self-dual (2+2) dimensional Yang–Mills theory under reduction to (2+1) dimensions. Field-theoretic and geometric features of the resulting nonlinear gauge theories with torsion are discussed.

We derive an explicit formula for factorizing an $n$-point closed string amplitude into open string amplitudes. Our results are phrased in terms of a momentum kernel which in the limit of infinite string tension reduces to the corresponding field theory kernel. The same momentum kernel encodes the monodromy relations which lead to the minimal basis of color-ordered amplitudes in Yang-Mills theory. There are interesting consequences of the momentum kernel pertaining to soft limits of amplitudes. We also comment on surprising links between gravity and certain combinations of kinematic and color factors in gauge theory.

The global feature of the spontaneously broken gauge theory is discussed in the context of fiber bundle. With the example of 'tHooft-Polyakov monopole, we show how the Higgs mechanism makes a nontrivial fiber bundle. The general relation of the topology of Higgs field ancl the structure of fiber bundle is presented, and extension of the argument to a general gauge theory with Higgs mechanism is also macle.

Note on the gauge fixing in gauge theory

We review the Dirac formalism for dealing with constraints in a canonical Hamiltonian formulation, discuss gauge freedom, and display constraints for gauge theories in a general context. We introduce the Dirac bracket and show that it provides a consistent method to remove any gauge freedom present. We discuss stability in evolution of gauge theories and show that fixing all gauge freedom is sufficient to ensure well-posedness for a large class of gauge theories. Electrodynamics provides examples of the methods outlined for general gauge theories. Future work will apply the formalism, and results derived here, to General Relativity.

We write a gravity theory with Yang–Mills-type action using the biconformal gauging of the conformal group. We show that the resulting biconformal Yang–Mills gravity theories describe 4-dim, scale-invariant general relativity in the case of slowly changing fields. In addition, we systematically extend arbitrary 4-dim Yang–Mills theories to biconformal space, providing a new arena for studying flat-space Yang–Mills theories. By applying the biconformal extension to a 4-dim pure Yang–Mills theory with conformal symmetry, we establish a 1-1, onto mapping between a set of gravitational gauge theories and 4-dim, flat-space gauge theories.

A gauge invariant Wilsonian effective action is constructed for pure SU(N) Yang-Mills theory by formulating the corresponding flow equation. Manifestly gauge invariant calculations can be performed i.e. without gauge fixing or ghosts. Regularisation is implemented in a novel way which realises a spontaneously broken SU(N|N) supergauge theory. As an example we sketch the computation of the one-loop β function, performed for the first time without any gauge fixing.

SU( N)-gauge theories in Polyakov gauge on the torus

Dirac introduced the theory of the magnetic monopole which bears his name in 1931 in [9], quite a few years before the advent of either the Yang–Mills theory or the theory of fiber bundles. However, by 1975, Wu and Yang had learned enough about fiber bundles to reformulate Dirac’s theory in terms of a connection on a principal bundle. See [52].

We present a method to quantize SU([ital N]) gauge theories which does not use gauge fixing, and therefore circumvents the problematics around gauge fixing and Gribov copying. The method is illustrated in the Abelian theory where no Gribov ambiguity is present, and it is shown to be equivalent to Landau gauge fixing in this case. It is shown that there are fundamental differences between Abelian and non-Abelian theories. These differences imply that in Abelian theories the gauge particle must be massless, but this is not necessarily true in non-Abelian theories.

Part 1 Anomalies and fractional charge: non-canonical behaviour in canonical theories quantum mechanical symmetry breaking delta function potentials in two- and three-dimensional quantum mechanics update on anomalous theories effects of Dirac's negative energy sea on quantum numbers Fermion fractionization in physics the chiral anomaly. Part 2 Gauge theories and gravity: Yang-Mills vacuum as a bloch wave bifurcation and stability in Yang-Mills theory with sources Gauge theories in three dimensions (= at high temperature) planar gravity time travel? Gauge theories for gravity on a line. Part 3 Symmetry behavior: introducing scale symmetry hidden symmetry of the point magnetic monopole invariance, symmetry and periodicity in Gauge theories symmetry restoration at finite temperature mean field theory for non-equilibrium quantum fields. Part 4 Approaches to quantum theory following Dirac: canonical light-cone commutators and their applications invariant quantization, scale symmetry and Euclidean field theory (constrained) quantization without tears analysis on infinite-dimensional manifolds - Schrodinger representation for quantized fields. Part 5 Solitons, instantons and semi-classical quantum field theory: non-perturbative and topological methods in quantum field theory self-dual Chern-Simons solitions.

In the preceding chapter we introduced the concept of the geometric phase and showed its natural emergence from the basic principles of quantum mechanics. We emphasized the geometric nature of the phase and briefly outlined a holonomy interpretation of the geometric phase using the Aharonov-Anandan principal fiber bundle. This involved several differential geometric concepts such as those of a fiber bundle, a horizontal lift, a connection, a parallel transport and a holonomy.

Gribov's observation that global gauge fixing is impossible has led to suggestions that there may be a deep connection between gauge fixing and confinement. We find an unexpected relation between the topological nontriviality of the gauge bundle and colored states in SU (N) Yang–Mills theory, and show that such states are necessarily impure. We approximate QCD by a rectangular matrix model that captures the essential topological features of the gauge bundle, and demonstrate the impure nature of colored states explicitly. Our matrix model also allows the inclusion of the QCD θ-term, as well as to perform explicit computations of low-lying glueball masses. This mass spectrum is gapped. Since an impure state cannot evolve to a pure one by a unitary transformation, our result shows that the solution to the confinement problem in pure QCD is fundamentally quantum information-theoretic.