DOCSLIB.ORG

The Conceptual Foundations of Contemporary Gauge Theories

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Conceptual Foundations of Contemporary Gauge Theories Gauging What’s Real This page intentionally left blank Gauging What’s Real The Conceptual Foundations of Contemporary Gauge Theories Richard Healey 1 1 Great Clarendon Street, Oxford 26 Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York Richard Healey 2007 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2007 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Laserwords Private Limited, Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd, King’s Lynn, Norfolk ISBN 978–0–19–928796–3 10987654321 Contents Preface ix Acknowledgements xiii Introduction xv 1. What is a gauge theory? 1 1.1 Classical electromagnetism: a paradigm gauge theory 3 1.2 A fiber bundle formulation 7 1.2.1 Electromagnetic interactions of quantum particles 14 1.2.2 Electromagnetic interactions of matter fields 18 2. The Aharonov–Bohm effect 21 2.1 Fiber bundles 26 2.2 A gauge-invariant, local explanation? 31 2.3 Geometry and topology in the Aharonov–Bohm effect 40 2.4 Locality in the Aharonov–Bohm effect 44 2.5 Lessons for classical electromagnetism 54 3. Classical gauge theories 58 3.1 Non-Abelian Yang–Mills theories 58 3.1.1 The fiber bundle formulation 64 3.1.2 Loops, groups, and hoops 70 3.1.3 Topological issues 74 3.2 A fiber bundle formulation of general relativity 77 3.2.1 A gravitational analog to the Aharonov–Bohm effect 78 4. Interpreting classical gauge theories 82 4.1 The no gauge potential properties view 83 4.2 The localized gauge potential properties view 85 4.2.1 Problems defining theoretical terms 91 4.2.2 Leeds’s view 99 4.2.3 Maudlin’s interpretation 102 4.3 The non-localized gauge potential properties view 104 4.4 A holonomy interpretation 111 4.4.1 Epistemological considerations 112 4.4.2 Objections considered 119 4.4.3 Semantic considerations 122 4.5 Metaphysical implications: non-separability and holism 123 vi contents 5. Quantized Yang–Mills gauge theories 129 5.1 How to quantize a classical field 131 5.2 Coulomb gauge quantization 133 5.3 Lorenz gauge quantization 135 5.4 Classical electromagnetism as a constrained Hamiltonian system 136 5.5 The free Maxwell field as a Hamiltonian system 139 5.6 Path-integral quantization 141 5.7 Canonical quantization of non-Abelian fields 143 5.8 Path-integral quantization of non-Abelian fields 145 5.9 Interacting fields in the Lagrangian formulation 146 6. The empirical import of gauge symmetry 149 6.1 Two kinds of symmetry 150 6.2 Observing gauge symmetry? 155 6.3 The gauge argument 159 6.4 Ghost fields 167 6.5 Spontaneous symmetry-breaking 169 6.6 The θ-vacuum 175 6.7 Anomalies 182 7. Loop representations 184 7.1 The significance of loop representations 185 7.2 Loop representations of the free Maxwell field 186 7.3 Loop representations of other free Yang–Mills fields 192 7.4 Interacting fields in loop representations 195 7.5 The θ-vacuum in a loop representation 197 7.6 Conclusion 198 8. Interpreting quantized Yang–Mills gauge theories 200 8.1 Auyang’s event ontology 200 8.2 Problems of interpreting a quantum field theory 203 8.2.1 Particle interpretations 205 8.2.2 Bohmian interpretations 209 8.2.3 Copenhagen interpretations 212 8.2.4 Everettian interpretations 215 8.2.5 Modal interpretations 218 9. Conclusions 220 A. Electromagnetism and its generalizations 229 B. Fiber bundles 233 C. The constrained Hamiltonian formalism 248 contents vii D. Alternative quantum representations 257 E. Algebraic quantum field theory 265 F. Interpretations of quantum mechanics 272 F.1 The Copenhagen interpretation 272 F.2 Bohmian mechanics 274 F.3 Everettian interpretations 276 F.4 Modal interpretations 278 Bibliography 280 Index 287 This page intentionally left blank Preface The quest for the deep structure of the world, begun by the pre-Socratic philosophers some two and a half thousand years ago, was pursued with great success by physicists during the 20th century. The revolutions wrought by relativity and quantum theory in the first quarter century created the scaffolding around which theories of the fundamental forces of nature were then constructed. After some mid-century struggles, by the end of the century these theories had been consolidated into two ‘‘packages.’’ Gravity has now been successfully described up to cosmological distances by Einstein’s general theory of relativity, while what has come to be known as the Standard Model has been successfully confirmed in its predictions of the behavior of the so- called elementary particles and their interactions down to the tiny subnuclear distance scales explored by high energy accelerators. While physicists keep trying to combine these into a single overarching theory, now is an appropriate time to pause and take stock of what has already been achieved. By ‘‘taking stock,’’ I don’t mean offering a detailed historical account lauding scientists and their achievements, though the history is fascinating and there is plenty of praise to spread around. Nor do I mean popularizing contemporary theories of fundamental forces to engage the interest of the casual reader by sub- stituting metaphor for faithful exposition and anecdote for critical analysis. What is needed is a careful logical and philosophical examination of the conceptual structure and broader implications of contemporary theories of fundamental forces like that which followed the rise of relativity and quantum theory. Logical positivists like Rudolf Carnap and Moritz Schlick, along with many other philosophers including Bertrand Russell and (especially) Hans Reichenbach, eagerly undertook such an examination of the physical theories of their day. Karl Popper acknowledged Einstein’s relativity as a major stimulus to the development of his own critical realist view of scientific investigation, while rejecting what he saw as the instrumentalism infecting Bohr and Heisenberg’s understanding of quantum theory. The philosophy of physics has since become a flourishing branch of the philosophy of science, raising the level of foundational discussion of relativity and quantum theory within this small interdisciplinary community of philosophers, physicists, and mathematicians. But science has moved on, and there are as yet few (if any) sustained investigations into the conceptual foundations of the theories of fundamental forces that emerged as an important culmination of twentieth century physics. I offer this book as both a contribution and a stimulus to the study of the conceptual foundations of gauge theories of fundamental interactions. I have x preface changed my mind and corrected my own misunderstandings often enough while working toward it to realize quite clearly that the book does not represent the last word—or even my last word—on this topic. If I have conveyed a sense of its intellectual interest and importance, while provoking some reader to do a better job, then I will be satisfied. To whom is this book addressed? To my fellow philosophers of physics, certainly—but I set out to reach a far wider class of readers. There is much food for thought here for other philosophers; not only for philosophers of science, but also for general epistemologists and metaphysicians. We have here a paradigm of reliable, objective knowledge of matters very distantly related to human sensory experience. But what are the scope and limits of this knowledge, and by what processes and what reasoning have we come by it? Indeed, how exactly should we understand the content of theoretically mediated knowledge claims in this domain? Do fundamental forces act at a distance, and if not how (and on what) do they act? Do traditional distinctions between matter and force even make sense in these theories, in the absence of any clear ontology of localized particles or fields? I write also for students and practitioners of physics and other sciences. The distinction between philosophy and science may be convenient for librarians and educational administrators, but it marks no true intellectual boundary. Physics is a source of both novel concepts and intractable conceptual problems. Understanding these concepts and dealing with these problems is the common concern of the physicist and the philosopher. Each can learn from the other, and both can learn from the mathematician. Professional commitments are only a potential hindrance to what is best viewed as an essentially collaborative enterprise. Students of physics typically have little time to reflect on the foundations of their discipline, and even find themselves discouraged from such reflections by their teachers.
Load more

Recommended publications
  • Jhep01(2020)007
    Published for SISSA by Springer Received: March 27, 2019 Revised: November 15, 2019 Accepted: December 9, 2019 Published: January 2, 2020 Deformed graded Poisson structures, generalized geometry and supergravity JHEP01(2020)007 Eugenia Boffo and Peter Schupp Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany E-mail: [email protected], [email protected] Abstract: In recent years, a close connection between supergravity, string effective ac- tions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded ge- ometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the 2-graded symplectic manifold T ∗[2]T [1]M that are based on a metric g, a closed Neveu-Schwarz 3-form H (locally expressed in terms of a Kalb-Ramond 2-form B) and a scalar dilaton φ. The derived bracket formalism relates this structure to the generalized differential geometry of a Courant algebroid, which has the appropriate stringy symme- tries, and yields a connection with non-trivial curvature and torsion on the generalized “doubled” tangent bundle E =∼ TM ⊕ T ∗M. Projecting onto TM with the help of a natural non-isotropic splitting of E, we obtain a connection and curvature invariants that reproduce the NS-NS sector of supergravity in 10 dimensions. Further results include a fully generalized Dorfman bracket, a generalized Lie bracket and new formulas for torsion and curvature tensors associated to generalized tangent bundles.
    [Show full text]
  • Natural Operators in the View of Cartan Geometries
    NATURAL OPERATORS IN THE VIEW OF CARTAN GEOMETRIES Martin Panak´ Abstract. We prove, that r-th order gauge natural operators on the bundle of Cartan connections with a target in the gauge natural bundles of the order (1, 0) (”tensor bundles”) factorize through the curvature and its invariant derivatives up to order r−1. On the course to this result we also prove that the invariant derivations (a generalization of the covariant derivation for Cartan geometries) of the curvature function of a Cartan connection have the tensor character. A modification of the theorem is given for the reductive and torsion free geometries. In [P] we have shown that Cartan connections on principal fibered bundles with a given structure group, say H, with values in g (H ⊂ G Lie groups, h, g their Lie alge- bras) are (all) sections of a gauge natural bundle which we call the bundle of Cartan connections and we will write C for it. In fact it is a bundle of elements of Car- tan connections. It is a functor on the category PBm(H) of principal bundles with a structure group H and principal bundle morphisms with local diffeomorphisms as base maps. For each principal bundle P the bundle CP can be viewed as a subbundle of the bundle of principal connections on the associated bundle P ×H G. We use the terms gauge natural bundle and gauge natural operator in the sense of [KMS]. We will study r-th order gauge natural operators on the bundle of Cartan connec- tions with gauge natural bundles of the order (1, 0) as target spaces.
    [Show full text]
  • (TEGR) As a Gauge Theory: Translation Or Cartan Connection? M Fontanini, E
    Teleparallel gravity (TEGR) as a gauge theory: Translation or Cartan connection? M Fontanini, E. Huguet, M. Le Delliou To cite this version: M Fontanini, E. Huguet, M. Le Delliou. Teleparallel gravity (TEGR) as a gauge theory: Transla- tion or Cartan connection?. Physical Review D, American Physical Society, 2019, 99, pp.064006. 10.1103/PhysRevD.99.064006. hal-01915045 HAL Id: hal-01915045 https://hal.archives-ouvertes.fr/hal-01915045 Submitted on 7 Nov 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Teleparallel gravity (TEGR) as a gauge theory: Translation or Cartan connection? M. Fontanini1, E. Huguet1, and M. Le Delliou2 1 - Universit´eParis Diderot-Paris 7, APC-Astroparticule et Cosmologie (UMR-CNRS 7164), Batiment Condorcet, 10 rue Alice Domon et L´eonieDuquet, F-75205 Paris Cedex 13, France.∗ and 2 - Institute of Theoretical Physics, Physics Department, Lanzhou University, No.222, South Tianshui Road, Lanzhou, Gansu 730000, P R China y (Dated: November 7, 2018) In this paper we question the status of TEGR, the Teleparallel Equivalent of General Relativity, as a gauge theory of translations. We observe that TEGR (in its usual translation-gauge view) does not seem to realize the generally admitted requirements for a gauge theory for some symmetry group G: namely it does not present a mathematical structure underlying the theory which relates to a principal G-bundle and the choice of a connection on it (the gauge field).
    [Show full text]
  • G -Metrics Arising from Non-Integrable Special Lagrangian Brations
    Complex Manifolds 2019; 6:348–365 Research Article Open Access Ryohei Chihara* G2-metrics arising from non-integrable special Lagrangian brations https://doi.org/10.1515/coma-2019-0019 Received March 29, 2019; accepted July 18, 2019 Abstract: We study special Lagrangian brations of SU(3)-manifolds, not necessarily torsion-free. In the case where the ber is a unimodular Lie group G, we decompose such SU(3)-structures into triples of sol- der 1-forms, connection 1-forms and equivariant 3 × 3 positive-denite symmetric matrix-valued functions on principal G-bundles over 3-manifolds. As applications, we describe regular parts of G2-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of G = T3 and SO(3). Keywords: G-structures, SU(3)-structures, G2-structures, Lagrangian brations, Einstein metrics MSC: 53C10, 53C25, 53C38 1 Introduction The geometry of G2-structures on 7-manifolds is closely related to that of SU(3)-structures on 6-manifolds. For a one-parameter family (ω(t), ψ(t)) of SU(3)-structures on a 6-manifold X, the 3-form ω(t) ^ dt + ψ(t) is a G2-structure on X × (t1, t2). Here ω(t) and ψ(t) denote the 2- and 3-form on X dening an SU(3)-structure for each t 2 (t1, t2). Conversely, any G2-structure on Y is locally described by one-parameter families of SU(3)- structures on 6-dimensional hypersurfaces in Y as above. This viewpoints has been studied by many authors [2, 4, 9, 15].
    [Show full text]
  • Why Is Quantum Gravity So Difficult (Compared to QCD)?
    OU-HET-990 Why is quantum gravity so difficult (compared to QCD)? Hidenori Fukaya∗ Department of Physics, Osaka University, Toyonaka 560-0043, Japan Abstract Gravity is difficult to quantize. This is a well-known fact but its reason is given simply by non-renormalizability of the Newton constant and little is discussed why among many quantum gauge theories, gravity is special. In this essay1, we try to treat the gravity as one of many gauge theories, and discuss how it is special and why it is difficult to quantize. 1 Introduction The author of this article is mainly working on numerical simulations of lattice QCD. His ordinary target is a many-body system of quarks and gluons at energy of a few GeV, and there is no chance for gravity to appear. But he was asked to teach general relativity to senior students in a seminar class in 2014 and he needed a lot of study to recover what he all forgot. In fact, it turned out that no student entered the high-energy theory lab, which had never happened in his lab for more than 80 years, and the class was not opened. Since the class was not opened, the author could have stopped his study on gravity and forget all of them again. But he took this one hour and half in a week as a good chance to compare it to QCD and understand how gravity is different and why it is difficult to quantize. The difficulty of quantization of gravity is a well-known fact and it is well arXiv:1811.11577v3 [hep-th] 17 Mar 2020 explained by the negative mass dimension of the Newton constant.
    [Show full text]
  • Synthetic Geometry of Manifolds Beta Version August 7, 2009 Alpha Version to Appear As Cambridge Tracts in Mathematics, Vol
    Synthetic Geometry of Manifolds beta version August 7, 2009 alpha version to appear as Cambridge Tracts in Mathematics, Vol. 180 Anders Kock University of Aarhus Contents Preface page 6 1 Calculus and linear algebra 11 1.1 The number line R 11 1.2 The basic infinitesimal spaces 13 1.3 The KL axiom scheme 21 1.4 Calculus 26 1.5 Affine combinations of mutual neighbour points 34 2 Geometry of the neighbour relation 37 2.1 Manifolds 37 2.2 Framings and 1-forms 47 2.3 Affine connections 52 2.4 Affine connections from framings 61 2.5 Bundle connections 66 2.6 Geometric distributions 70 2.7 Jets and jet bundles 81 2.8 Infinitesimal simplicial and cubical complex of a manifold 89 3 Combinatorial differential forms 92 3.1 Simplicial, whisker, and cubical forms 92 3.2 Coboundary/exterior derivative 101 3.3 Integration of forms 106 3.4 Uniqueness of observables 115 3.5 Wedge/cup product 119 3.6 Involutive distributions and differential forms 123 3.7 Non-abelian theory of 1-forms 125 3.8 Differential forms with values in a vector bundle 130 3.9 Crossed modules and non-abelian 2-forms 132 3 4 Contents 4 The tangent bundle 135 4.1 Tangent vectors and vector fields 135 4.2 Addition of tangent vectors 137 4.3 The log-exp bijection 139 4.4 Tangent vectors as differential operators 144 4.5 Cotangents, and the cotangent bundle 146 4.6 The differential operator of a linear connection 148 4.7 Classical differential forms 150 4.8 Differential forms with values in TM ! M 154 4.9 Lie bracket of vector fields 157 4.10 Further aspects of the tangent bundle 161 5 Groupoids
    [Show full text]
  • Connections and Frame Bundle Reductions
    Connections and Frame Bundle Reductions M. Lachi`eze-Rey APC-Astroparticule et Cosmologie (UMR-CNRS 7164), Universit´eParis Diderot-Paris 7, 10, rue Alice Domon et L´eonie Duquet F-75205 Paris Cedex 13, France [email protected] February 5, 2020 Abstract In general relativity, the gravitational potential is represented by the Levi-Civita connection, the only symmetric connection preserving the metric. On a differentiable manifold, a metric identifies with an orthog- onal structure, defined as a Lorentz reduction of the frame bundle. The Levi-Civita connection appears as the only symmetric connection preserv- ing the reduction. This paper presents generalization of this process to other aproaches of gravitation: Weyl structure with Weyl connections, teleparallel structures with Weitzenb¨ock connections, unimodular struc- ture, similarly appear as frame bundle reductions, with preserving con- nections. To each subgroup H of the linear group GL correspond reduced struc- tures, or H-structures. They are subbundles of the frame bundle (with GL as principal group), with H as principal group. A linear connection in a manifold M is a principal connection on the frame bundle. Given a reduction, the corresponding preserving connections on M are the linear connections which preserve it. I also show that the time gauge used in the 3+1 formalism for general relativity similarly appears as the result of a bundle reduction. PACS numbers: 11.15.-q, 04.50.-h, 02.40.-k, 11.30.-j, arXiv:2002.01410v1 [math-ph] 4 Feb 2020 1 Introduction Principal bundle reductions are the mathematical expression of a physi- cal process of symmetry reduction, from a principal group to one of its subgroups.
    [Show full text]
  • Fiber Bundles, Yang and the Geometry of Spacetime
    Fiber bundles, Yang and the geometry of spacetime. A bachelor research in theoretical physics Federico Pasinato Univeristy of Groningen E-mail: [email protected] First supervisor: P rof: Dr: Elisabetta P allante Second supervisor: P rof: Dr: Holger W aalkens Last updated on July 24, 2018 Acknowledgments To my family, nothing would have been possible without their sacrifices and love. A special thanks to C.N.Yang public lectures, Kenneth Young’s lectures on thematic melodies of 20th century and a particular thanks to Simon Rea’s lecture notes of Frederic Schuller’s course on the “Geometric anatomy of theoretical physics”, taught in the academic year 2013/14 at the Friedrich-Alexander-Universität Erlangen-Nürnberg. The entire course is hosted on YouTube at the following address: www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic We will be mainly interested in the fiber bundle formalism introduced. Federico Pasinato Contents Acknowledgements1 Introduction1 1 The first field theory2 1.1 James Clerk Maxwell2 1.2 Herman Weyl4 1.3 Yang and Mills, the gauge symmetry5 1.4 The emergence of a central mathematical construct7 2 Abelian case 11 2.1 Magnetostatics 11 2.2 Adding Quantum Mechanics 12 2.3 Differential geometry for physicists 14 2.4 Back to EM 18 3 Non-Abelian case 20 3.1 Differential geometry for physicists - continued 21 3.2 General Relativity 23 3.3 Concluding remarks, a path to unification 24 4 Fiber bundle 25 4.1 Topological manifolds and bundles 25 4.1.1 Bundles 25 4.1.2 Product bundles 26 4.1.3 Fiber Bundles 26 4.1.4 Application
    [Show full text]
  • Arxiv:0904.4616V1 [Math-Ph] 29 Apr 2009 Manifolds Obtained by Soldering Together Points, Lines, Etc
    Manifolds obtained by soldering together points, lines, etc. Fr´ed´eric Helein´ ∗ 1 The solder form 1.1 A physical motivation : ideas from the general relativity One fundamental concept in the theory of general relativity is that the gravitation should not be considered as a force which takes place in some a priori given space-time but that, instead, the space-time is built out of the gravitational field. A mathematical achievement of this idea is the Einstein equation for a (pseudo-)metric tensor gij on a manifold M (the space-time): 1 R − Rg = T , (1) ij 2 ij ij where Rij is the Ricci curvature tensor of the metric and R is its scalar curvature. This equation tells us how the metric tensor gij (i.e. the field mediating the gravitation) is obeying to constraints imposed by the energy-momentum tensor Tij, which encodes informations about the distribution of energy and momentum in spacetime. For simplicity we will assume in the following that Tij = 0, i.e. we consider the Einstein equation in the arXiv:0904.4616v1 [math-ph] 29 Apr 2009 vacuum. Note that if we work with equation (1) in the same way as for any other partial differential equation, then it is very easy to forget about the original intuition of Einstein (which is that the space-time is built out of the gravitational field gij) and erroneously to implicitely assume that we are given some manifold M, and we are looking for an unknown field gij satisfying (1). We want to take this physical intuition seriously into account and to look for some mathematical framework which would help us to keep in mind that the manifold M and the metric should be built simultaneously when solving equation (1).
    [Show full text]
  • Natural Operators on the Bundle of Cartan Connections
    NATURAL OPERATORS ON THE BUNDLE OF CARTAN CONNECTIONS Ph.D. dissertation Martin Pan´ak Brno 2001 Thesis advisor: prof. RNDr. Jan Slov´ak,DrSc. Department of Algebra and Geometry Faculty of Science, Masaryk University Brno Jan´aˇckovo n´am.2a 662 95 Brno Czech Republic Ackowledgement. I would like to thank my thesis advisor prof. Jan Slov´akfor numerous productive discussions on my thesis. I also thank him for his moral support, which has helped me a lot. Contents Introduction ............................................................... 2 Episode I: Cartan Geometries .............................................. 5 Curvature ............................................................... 7 The Invariant Differentiation.............................................8 Reductive geometries ................................................... 10 Episode IV: Natural Operations ........................................... 15 Gauge Natural Bundle of Cartan Connections ........................... 19 Natural Sheafs..........................................................21 Curvature spaces ....................................................... 23 Reductive geometries ................................................... 25 Episode V: The Algebra Strikes Back ...................................... 28 Reductive geometries ................................................... 32 List of Symbols ........................................................... 40 2 Introduction There are several purposes of this PhD thesis. At first I want my PhD degree. At second
    [Show full text]
  • The Versatile Preform
    As originally published in the SMTA Proceedings THE VERSATILE PREFORM Ed Briggs Indium Corporation Clinton, NY, USA [email protected] ABSTRACT powder, which creates a significant amount of flux Solder preforms are composed of solder alloys and are outgassing. The shapes and thicknesses of solder preforms designed and manufactured into various shapes and can be adapted to a wide variety of processes, working well configurations. The solder preform concept is quite simple; with component placement equipment such as vibratory however, the applications are very diverse. Solder preforms feeders, vacuum pick-up manifolds, and tube-type are used in numerous ways, e.g., from simple PCB rework dispensers. In addition, preforms work well with a variety of to high-reliability applications, and from package level to reflow processes from simple hot plates to convection board level assembly. They are used in low-volume ovens. applications, as in military electronics as well as in high throughput, high-volume applications, such as smartphones. PREFORM APPLICATIONS Preforms are also used in robotics and automated processes Low-Temperature Soldering and to optimize labor-intensive steps in manufacturing One advantage of solder preforms formed from a low- processes such as hand soldering. Solder preforms and melting temperature alloy is the ability to allow for a true solder-cored wire offer an opportunity to increase low-temperature reflow process. In applications that require consistency and decrease cycle times in many reflow soldering below 170°C, solder pastes generally are manufacturing environments. not a viable option. Regardless of how low the liquidus temperature of the alloy is within the solder paste, a peak This paper will discuss attributes associated within the temperature of 180°C is most often required to provide solder families that the preforms are composed of, offer adequate wetting (see Figure 1).
    [Show full text]
  • Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results
    Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results Adam Marsh February 27, 2019 Abstract A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. The present paper is best read in conjunction with the similar paper on Riemannian geometry cited herein. Contents 1 Introduction 2 2 Gauge theory 2 2.1 Matter fields and gauges . 2 2.2 The gauge potential and field strength . 3 2.3 Spinor fields . 4 3 Defining bundles 7 3.1 Fiber bundles . 7 3.2 G-bundles . 9 3.3 Principal bundles . 10 arXiv:1607.03089v2 [math.DG] 27 Feb 2019 4 Generalizing tangent spaces 13 4.1 Associated bundles . 13 4.2 Vector bundles . 15 4.3 Frame bundles . 17 4.4 Gauge transformations on frame bundles . 20 4.5 Smooth bundles and jets . 23 4.6 Vertical tangents and horizontal equivariant forms . 24 1 5 Generalizing connections 28 5.1 Connections on bundles . 28 5.2 Parallel transport on the frame bundle . 28 5.3 The exterior covariant derivative on bundles . 31 5.4 Curvature on principal bundles . 32 5.5 The tangent bundle and solder form .
    [Show full text]