Of connections and fields—II (original) (raw)
2005
We describe some instances of the appearance of Chern's mathematical ideas in physics. By means of simple examples, we bring out the geometric and topological ideas which have found application in describing the physical world. These applications range from magnetic monopoles in electrodynamics to instantons in quantum chromodynamics to the geometric phase of quantum mechanics. The first part of this article is elementary and addressed to a general reader. The second part is somewhat more demanding and is addressed to advanced students of mathematics and physics.
Recent Developments in Classical and Quantum Theories of Connections, Including General Relativity
General relativity can be recast as a theory of connections by performing a canonical transformation on its phase space. In this form, its (kinematical) structure is closely related to that of Yang-Mills theory and topological field theories. Over the past few years, a variety of techniques have been developed to quantize all these theories non-perturbatively. These developments are summarized with special emphasis on loop space methods and their applications to quantum gravity.
The Tensor Theory of Connections
2018
This paper extends the univariate Theory of Connections, introduced in (Mortari,2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons,1984), by providing analytical expressions, called "constrained expressions," representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, g(x,y), satisfy all constraints no matter what the g(x,y) is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the "Tensor Theory of Connections," validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order deri...
Generalised connections and curvature
Mathematical Proceedings of the Cambridge Philosophical Society, 2005
The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang-Mills theory are given.
Differential pseudoconnections and field theories
Annales De L Institut Henri Poincare-physique Theorique, 1981
2014 We define a « differential pseudoconnection of order k » on a bundle p : E -~ M as a translation morphism r : V T* (8) VE . k on the affine bundle J~ -1 E. Such concept is a generalization of usual connections. Then we study in the framework of jet spaces several important differential operators used in physics. In this context an interest arises naturally for the second order affine differential equations, called « special », given by E2 == ker (G o H) 4 J2E, where H : V T* (8) VE 2 is a differential pseudoconnection and G : V T* (8) VE -~ VE is the linear 2 submersion induced by a metric g : V T* -~ ~. Particular cases of special 2 equations are both the geodesics equation (an ordinary equation) and any kind of Laplace equation (a partial equation) even modified by the addition of physical terms. So special equations are candidate to fit a lot of fundamental physical fields. At the present state of the theory we can emphasize several common features of physical fields. Furthe...
Geometry, Topology and Physics
Geometry, Topology and Physics, 1990
Preface to the First Edition xvii Preface to the Second Edition xix How to Read this Book xxi Notation and Conventions xxii Problems xvi CONTENTS 13 Anomalies in Gauge Field Theories 13.1 Introduction 13.2 Abelian anomalies 13.2.1 Fujikawa's method 13.3 Non-Abelian anomalies 13.4 The Wess-Zumino consistency conditions 13.4.1 The Becchi-Rouet-Stora operator and the Faddeev-Popov ghost 13.4.2 The BRS operator, FP ghost and moduli space 13.4.3 The Wess-Zumino conditions 13.4.4 Descent equations and solutions of WZ conditions 13.5 Abelian anomalies versus non-Abelian anomalies 13.5.1 m dimensions versus m + 2 dimensions 13.6 The parity anomaly in odd-dimensional spaces 13.6.1 The parity anomaly 13.6.2 The dimensional ladder: 4-3-2
Introduction to Differential Geometry
2011
Classical differential geometry is often considered as an “art of manipulating with indices”. In these lectures we develop a more geometric approach by explaining the true mathematical meaning of all introduced notions. Where possible, we try to avoid coordinates totally. But the correspondence to the traditional coordinate presentation is also explained. The usage of invariant language not only simplifies many arguments but also reduces the amount of computations in particular problems. The best example is a simple formula for the Gaussian curvature of a surface based on the concept of connection 1-form.
Geometry, Topology and Quantum Field Theory
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