DOCSLIB.ORG

CLASSICAL FIELD THEORY - a Quick Guide

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CLASSICAL FIELD THEORY - a Quick Guide CLASSICAL FIELD THEORY - A Quick Guide - Huan Q. Bui Colby College PHYSICS & MATHEMATICS Statistics Class of 2021 February 25, 2020 Preface Greetings, Classical Field Theory, A Quick Guide to is compiled based on my inde- pendent study PH491/2: Topics in Classical Field Theory notes with professor Robert Bluhm. Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity, along with other resources, serves as the main guiding text. This text also references a number of other texts such as Quantum Field The- ory by Ryder, Quantum Field Theory by Mandl and Shaw, Gauge Theories Of The Strong, Weak, and Electromagnetic Interactions by Quigg, A First Book of Quantum Field Theory by A. Lahiri and P. B. Pal, Quantum Field Theory in a Nutshell by Zee. This text is a continuation of General Relativity and Cosmology, A Quick Guide to. Familiarity with classical mechanics, linear algebra, vector calculus, and especially general relativity is expected. There will be a quick review of general relativity where important concepts are revisited and derivations high- lighted, but familiarity with basic notions such as geodesics, Christoffel symbols, the Riemann curvature tensors, etc. is assumed. Note: As a consequence of being developed from a variety of sources, there will be some overlapping among the sections. However, the objective of each section is unique. Enjoy! Contents Preface................................2 1 Introduction to the Lagrangian and the Principle of Least Ac- tion7 1.1 A Classical-Mechanical Example..................7 2 Group Theory: a quick guide in a quick guide9 3 Introduction to Classical Field Theory 11 3.1 Relativistic Notation......................... 11 3.2 Classical Lagrangian Field Theory................. 12 3.3 Quantized Lagrangian Field Theory (primer)........... 13 3.4 Symmetries and Conservation Laws (primer)........... 15 4 Gauge Invariance 21 4.1 Introduction.............................. 21 4.2 Gauge Invariance in Classical Electrodynamics.......... 21 4.3 Phase Invariance in Quantum Mechanics.............. 24 4.4 Significance of Potentials in Quantum Theory........... 26 4.4.1 The Aharonov-Bohm Effect & The Physical Vector Potential 26 4.4.2 Path-dependent Phase Factors............... 28 4.5 Phase Invariance in Field Theory.................. 28 5 Lagrangian Field Theory in Flat Spacetime 31 5.1 Real Scalar Fields.......................... 31 5.2 Complex Scalar Fields and Electromagnetism........... 35 5.2.1 Gauge transformation of the first kind - Global Symmetry 35 5.2.2 Gauge transformation of the second kind - Local Symmetry 36 5.2.3 Motivations in the derivation of the E&M Lagrangian.. 37 5.2.4 A few remarks........................ 41 5.3 Vector Fields and Photons...................... 43 6 Symmetries and Conservation Laws in Field Theory 51 6.1 Hamiltonian formalism (primer)................... 51 3 4 CONTENTS 6.2 Overview of Noether's Theorem: A Consequence of Variational Principle................................ 51 6.3 Noether's Theorem on Symmetries and Conservation Laws.... 56 6.3.1 Space-time translations................... 60 6.3.2 Lorentz transformations................... 61 6.3.3 Internal symmetries..................... 61 7 Spontaneous Symmetry Breaking 63 7.1 Introduction.............................. 63 7.2 Continuous Global Symmetry.................... 69 7.3 An O(2) example........................... 71 7.4 Introduction to the Higgs Mechanism................ 75 8 Gravitation and Lagrangian Formulation of General Relativity 79 8.1 Review of General Relativity & Curved Spacetime........ 79 8.1.1 General Relativity...................... 79 8.1.2 Curved Spacetime...................... 79 8.2 Lagrangian Formulation....................... 80 8.3 Overview............................... 80 8.3.1 Introduction to the Lagrangian Formulation of General Relativity........................... 81 8.3.2 Variation on the metric tensor, without matter: Tµν = 0. 83 8.3.3 Variations on the metric tensor, with matter: Tµν 6= 0.. 88 8.4 Properties of Einstein Equations.................. 91 9 Diffeomorphism, Vierbein Formalism, and general spacetime symmetries 93 9.1 Overview of Diffeomorphisms and the Lie Derivative....... 93 9.2 Spacetime Symmetry......................... 97 9.2.1 Global Lorentz Transformations in Minkowski Spacetime 97 9.2.2 Diffeomorphism in Curved Spacetime........... 100 9.2.3 Local Lorentz Transformation & Vierbein Formalism... 102 9.2.4 Riemann-Cartan spacetime................. 106 10 Linearized Gravity 109 10.1 The metric.............................. 109 10.2 The Christoffel symbols and Einstein tensor............ 110 10.3 Massive Gravity........................... 123 10.3.1 A Brief History........................ 123 10.3.2 Quick review of Field Dimensions.............. 124 10.3.3 Fierz-Pauli Massive Gravity................. 126 10.3.4 Fierz-Pauli Massive Gravity with Source.......... 144 10.3.5 The St¨uckelberg Trick.................... 160 10.3.6 Nonlinear Massive Gravity................. 171 10.3.7 The Nonlinear St¨uckelberg Formalism........... 202 10.3.8 St¨uckelberg Analysis of Interacting Massive Gravity... 211 CONTENTS 5 10.4 The Λ3 theory............................ 215 10.4.1 Tuning interactions to raise the cutoff........... 215 10.4.2 The appearance of Galileons and the absence of ghosts.. 216 10.4.3 The Vainshtein radius.................... 217 10.4.4 Th Vainshtein mechanism in the Λ3 theory........ 218 10.4.5 Quantum corrections in the Λ3 theory........... 219 10.5 xACT Tutorial............................ 220 10.5.1 Importing packages...................... 220 10.5.2 xTensor Basics........................ 222 10.5.3 xPert Basics......................... 228 10.5.4 xTras Basics: Metric Variations............... 236 10.5.5 Undefining Basics: Playtime's Over!............ 241 6 CONTENTS Part 1 Introduction to the Lagrangian and the Principle of Least Action Proposition 1.0.1. All fundamental physics obeys least action principles. The action S is defined as b S = L dt: (1.1) ˆa where L is called the Lagrangian. Refer for Farlow's Partial Differential Equation, page 353, for detailed ex- planation of Lagrange's calculus of variations. I will derive the Euler-Lagrange equation(s) here, but we are not going to use it in the following subsection for the introduction to field theory for now. @L @L − @µ = 0: (1.2) @φ @(@µφ) 1.1 A Classical-Mechanical Example In this subsection we take a look at how the Lagrangian formulation of classical mechanics can give rise to Newton's second law of motion. In mechanics, the Lagrangian often takes the form: L = K − V; (1.3) 7 8PART 1. INTRODUCTION TO THE LAGRANGIAN AND THE PRINCIPLE OF LEAST ACTION where K is the kinetic energy, and V is the potential energy. Let us consider a simple example where 1 K = mx_ 2 (1.4) 2 V = V (x): (1.5) Variations on the Lagrangian gives 1 δL = δ mx_ 2 − V (x) (1.6) 2 dV = mxδ_ x_ − δx (1.7) dx dV = mx_δx_ − δx (1.8) dx d dV = m −xδx¨ + xδx_ − δx (1.9) dt dx d dV = −mxδx¨ − m xδx_ − δx: (1.10) dt dx It follows that the variations on the action gives b b dV δS = δL dt = − mx¨ + δx dt: (1.11) ˆa ˆa dx The principle of least action requires δS = 0 for all δx. Therefore it follows that dV mx¨ + = 0; (1.12) dx which is simply Newton's second law of motion in disguise. Before we move on, we should note that in order for the Lagrangian formu- lation to work in electromagnetism or in general relativity, we need to promote the Lagrangian to its relativistic version where the Lagrangian is given by b L = L d3x: (1.13) ˆa L is called the Lagrangian density, but we can colloquially refer to it as \the Lagrangian." The relativistic action hence takes the form S = L d4x; (1.14) ˆ where d4x implies integrating over all spacetime. Part 2 Group Theory: a quick guide in a quick guide 1. Consider an N-dimensional vector with complex elements. A gauge trans- formation that takes jzj2 ! jzj2 (modulus preserving) is call a U(N) gauge transformation. The letter \U" denotes \unitary." These kinds of trans- formations can be represented by a unitary matrix, which is defined as a matrix whose conjugate transpose is the same as its inverse. If z ! Uz, then jzj2 ! (Uz)y(Uz) = zyU yUz = zyz = jzj2; (2.1) hence modulus preserving. Note that the transformation eiα is unitary. It is simply a 1×1 unitary matrix. This immediately implies that complex scalars have a U(1) gauge invariance. The special group of U(N) transformations is denoted SU(N), which rep- resents those with det U = 1. 2. Consider another N-dimensional real-valued-element vector. We denote the group of orthogonal transformations O(N). Orthogonal transforma- tions are orthogonal, i.e., length-preserving. (Ox)>(Ox) = x>O>Ox = x>x = ~x2 (2.2) Once again, we denote the special group of the O(N) the SO(N) group. This group represents transformations with det O = 1. The Lorentz group is a sub-group of SO(N), as the norm is defined differently, nevertheless it is still length-preserving. We call the Lorentz group SO(3,1), signifying that one sign is different from the other three. 9 10 PART 2. GROUP THEORY: A QUICK GUIDE IN A QUICK GUIDE Part 3 Introduction to Classical Field Theory 3.1 Relativistic Notation Throughout this text, we will use the particle physics' Minkowski spacetime metric tensor: 01 0 0 0 1 B0 −1 0 0 C ηµν = B C : (3.1) @0 0 −1 0 A 0 0 0 −1 We shall assume knowledge of inner products and index-lowering/raising oper- ations (please refer to General Relativity & Cosmology: A Quick Guide to) for more details about this \indexing" business. Quick notes: the four-dimensional generalization of the gradient operator r transforms like a four-vector. If φ(x) is a scalar function, so is @φ δφ = δxµ (3.2) @xµ and so @φ ≡ @ φ ≡ φ, : (3.3) @xµ µ µ Similarly, we also have something for the contravariant four-vector: @φ ≡ @µφ ≡ φ,µ : (3.4) @xµ Lastly, we define the d'Alembertian as: @2 ≡ − @µ@ : (3.5) @t2 µ 11 12 PART 3.
Load more

Recommended publications
  • Chapter 2 Introduction to Electrostatics
    Chapter 2 Introduction to electrostatics 2.1 Coulomb and Gauss’ Laws We will restrict our discussion to the case of static electric and magnetic fields in a homogeneous, isotropic medium. In this case the electric field satisfies the two equations, Eq. 1.59a with a time independent charge density and Eq. 1.77 with a time independent magnetic flux density, D (r)= ρ (r) , (1.59a) ∇ · 0 E (r)=0. (1.77) ∇ × Because we are working with static fields in a homogeneous, isotropic medium the constituent equation is D (r)=εE (r) . (1.78) Note : D is sometimes written : (1.78b) D = ²oE + P .... SI units D = E +4πP in Gaussian units in these cases ε = [1+4πP/E] Gaussian The solution of Eq. 1.59 is 1 ρ0 (r0)(r r0) 3 D (r)= − d r0 + D0 (r) , SI units (1.79) 4π r r 3 ZZZ | − 0| with D0 (r)=0 ∇ · If we are seeking the contribution of the charge density, ρ0 (r) , to the electric displacement vector then D0 (r)=0. The given charge density generates the electric field 1 ρ0 (r0)(r r0) 3 E (r)= − d r0 SI units (1.80) 4πε r r 3 ZZZ | − 0| 18 Section 2.2 The electric or scalar potential 2.2 TheelectricorscalarpotentialFaraday’s law with static fields, Eq. 1.77, is automatically satisfied by any electric field E(r) which is given by E (r)= φ (r) (1.81) −∇ The function φ (r) is the scalar potential for the electric field. It is also possible to obtain the difference in the values of the scalar potential at two points by integrating the tangent component of the electric field along any path connecting the two points E (r) d` = φ (r) d` (1.82) − path · path ∇ · ra rb ra rb Z → Z → ∂φ(r) ∂φ(r) ∂φ(r) = dx + dy + dz path ∂x ∂y ∂z ra rb Z → · ¸ = dφ (r)=φ (rb) φ (ra) path − ra rb Z → The result obtained in Eq.
    [Show full text]
  • Quantum Field Theory*
    Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
    [Show full text]
  • Introductory Lectures on Quantum Field Theory
    Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties.
    [Show full text]
  • Effective Field Theories, Reductionism and Scientific Explanation Stephan
    To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics.
    [Show full text]
  • Electromagnetic Field Theory
    Electromagnetic Field Theory BO THIDÉ Υ UPSILON BOOKS ELECTROMAGNETIC FIELD THEORY Electromagnetic Field Theory BO THIDÉ Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden and School of Mathematics and Systems Engineering Växjö University, Sweden Υ UPSILON BOOKS COMMUNA AB UPPSALA SWEDEN · · · Also available ELECTROMAGNETIC FIELD THEORY EXERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik Freely downloadable from www.plasma.uu.se/CED This book was typeset in LATEX 2" (based on TEX 3.14159 and Web2C 7.4.2) on an HP Visualize 9000⁄360 workstation running HP-UX 11.11. Copyright c 1997, 1998, 1999, 2000, 2001, 2002, 2003 and 2004 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. Preface The current book is an outgrowth of the lecture notes that I prepared for the four-credit course Electrodynamics that was introduced in the Uppsala University curriculum in 1992, to become the five-credit course Classical Electrodynamics in 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty. Intended primarily as a textbook for physics students at the advanced undergradu- ate or beginning graduate level, it is hoped that the present book may be useful for research workers
    [Show full text]
  • An Introduction to Supersymmetry
    An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems.
    [Show full text]
  • A Guiding Vector Field Algorithm for Path Following Control Of
    1 A guiding vector field algorithm for path following control of nonholonomic mobile robots Yuri A. Kapitanyuk, Anton V. Proskurnikov and Ming Cao Abstract—In this paper we propose an algorithm for path- the integral tracking error is uniformly positive independent following control of the nonholonomic mobile robot based on the of the controller design. Furthermore, in practice the robot’s idea of the guiding vector field (GVF). The desired path may be motion along the trajectory can be quite “irregular” due to an arbitrary smooth curve in its implicit form, that is, a level set of a predefined smooth function. Using this function and the its oscillations around the reference point. For instance, it is robot’s kinematic model, we design a GVF, whose integral curves impossible to guarantee either path following at a precisely converge to the trajectory. A nonlinear motion controller is then constant speed, or even the motion with a perfectly fixed proposed which steers the robot along such an integral curve, direction. Attempting to keep close to the reference point, the bringing it to the desired path. We establish global convergence robot may “overtake” it due to unpredictable disturbances. conditions for our algorithm and demonstrate its applicability and performance by experiments with real wheeled robots. As has been clearly illustrated in the influential paper [11], these performance limitations of trajectory tracking can be Index Terms—Path following, vector field guidance, mobile removed by carefully designed path following algorithms. robot, motion control, nonlinear systems Unlike the tracking approach, path following control treats the path as a geometric curve rather than a function of I.
    [Show full text]
  • Branched Hamiltonians and Supersymmetry
    Branched Hamiltonians and Supersymmetry Thomas Curtright, University of Miami Wigner 111 seminar, 12 November 2013 Some examples of branched Hamiltonians are explored, as recently advo- cated by Shapere and Wilczek. These are actually cases of switchback poten- tials, albeit in momentum space, as previously analyzed for quasi-Hamiltonian dynamical systems in a classical context. A basic model, with a pair of Hamiltonian branches related by supersymmetry, is considered as an inter- esting illustration, and as stimulation. “It is quite possible ... we may discover that in nature the relation of past and future is so intimate ... that no simple representation of a present may exist.” – R P Feynman Based on work with Cosmas Zachos, Argonne National Laboratory Introduction to the problem In quantum mechanics H = p2 + V (x) (1) is neither more nor less difficult than H = x2 + V (p) (2) by reason of x, p duality, i.e. the Fourier transform: ψ (x) φ (p) ⎫ ⎧ x ⎪ ⎪ +i∂/∂p ⎪ ⇐⇒ ⎪ ⎬⎪ ⎨⎪ i∂/∂x p − ⎪ ⎪ ⎪ ⎪ ⎭⎪ ⎩⎪ This equivalence of (1) and (2) is manifest in the QMPS formalism, as initiated by Wigner (1932), 1 2ipy/ f (x, p)= dy x + y ρ x y e− π | | − 1 = dk p + k ρ p k e2ixk/ π | | − where x and p are on an equal footing, and where even more general H (x, p) can be considered. See CZ to follow, and other talks at this conference. Or even better, in addition to the excellent books cited at the conclusion of Professor Schleich’s talk yesterday morning, please see our new book on the subject ... Even in classical Hamiltonian mechanics, (1) and (2) are equivalent under a classical canonical transformation on phase space: (x, p) (p, x) ⇐⇒ − But upon transitioning to Lagrangian mechanics, the equivalence between the two theories becomes obscure.
    [Show full text]
  • Lo Algebras for Extended Geometry from Borcherds Superalgebras
    Commun. Math. Phys. 369, 721–760 (2019) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-019-03451-2 Mathematical Physics L∞ Algebras for Extended Geometry from Borcherds Superalgebras Martin Cederwall, Jakob Palmkvist Division for Theoretical Physics, Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden. E-mail: [email protected]; [email protected] Received: 24 May 2018 / Accepted: 15 March 2019 Published online: 26 April 2019 – © The Author(s) 2019 Abstract: We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin–Vilkovisky framework, or equivalently, an L∞ algebra. The L∞ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra B(gr+1), which is a double extension of the structure algebra gr . The construction includes a set of “ancillary” ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra. Contents 1. Introduction ................................. 722 2. The Borcherds Superalgebra ......................... 723 3. Section Constraint and Generalised Lie Derivatives ............. 728 4. Derivatives, Generalised Lie Derivatives and Other Operators ....... 730 4.1 The derivative .............................. 730 4.2 Generalised Lie derivative from “almost derivation” .......... 731 4.3 “Almost covariance” and related operators ..............
    [Show full text]
  • Towards a Glossary of Activities in the Ontology Engineering Field
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid Towards a Glossary of Activities in the Ontology Engineering Field Mari Carmen Suárez-Figueroa, Asunción Gómez-Pérez Ontology Engineering Group, Universidad Politécnica de Madrid Campus de Montegancedo, Avda. Montepríncipe s/n, Boadilla del Monte, Madrid, Spain E-mail: [email protected], [email protected] Abstract The Semantic Web of the future will be characterized by using a very large number of ontologies embedded in ontology networks. It is important to provide strong methodological support for collaborative and context-sensitive development of networks of ontologies. This methodological support includes the identification and definition of which activities should be carried out when ontology networks are collaboratively built. In this paper we present the consensus reaching process followed within the NeOn consortium for the identification and definition of the activities involved in the ontology network development process. The consensus reaching process here presented produces as a result the NeOn Glossary of Activities. This work was conceived due to the lack of standardization in the Ontology Engineering terminology, which clearly contrasts with the Software Engineering field. Our future aim is to standardize the NeOn Glossary of Activities. field of knowledge. In the case of IEEE Standard Glossary 1. Introduction of Software Engineering Terminology (IEEE, 1990), this The Semantic Web of the future will be characterized by glossary identifies terms in use in the field of Software using a very large number of ontologies embedded in Engineering and establishes standard definitions for those ontology networks built by distributed teams (NeOn terms.
    [Show full text]
  • Perturbative Algebraic Quantum Field Theory
    Perturbative algebraic quantum field theory Klaus Fredenhagen Katarzyna Rejzner II Inst. f. Theoretische Physik, Department of Mathematics, Universität Hamburg, University of Rome Tor Vergata Luruper Chaussee 149, Via della Ricerca Scientifica 1, D-22761 Hamburg, Germany I-00133, Rome, Italy [email protected] [email protected] arXiv:1208.1428v2 [math-ph] 27 Feb 2013 2012 These notes are based on the course given by Klaus Fredenhagen at the Les Houches Win- ter School in Mathematical Physics (January 29 - February 3, 2012) and the course QFT for mathematicians given by Katarzyna Rejzner in Hamburg for the Research Training Group 1670 (February 6 -11, 2012). Both courses were meant as an introduction to mod- ern approach to perturbative quantum field theory and are aimed both at mathematicians and physicists. Contents 1 Introduction 3 2 Algebraic quantum mechanics 5 3 Locally covariant field theory 9 4 Classical field theory 14 5 Deformation quantization of free field theories 21 6 Interacting theories and the time ordered product 26 7 Renormalization 26 A Distributions and wavefront sets 35 1 Introduction Quantum field theory (QFT) is at present, by far, the most successful description of fundamental physics. Elementary physics , to a large extent, explained by a specific quantum field theory, the so-called Standard Model. All the essential structures of the standard model are nowadays experimentally verified. Outside of particle physics, quan- tum field theoretical concepts have been successfully applied also to condensed matter physics. In spite of its great achievements, quantum field theory also suffers from several longstanding open problems. The most serious problem is the incorporation of gravity.
    [Show full text]
  • (Aka Second Quantization) 1 Quantum Field Theory
    221B Lecture Notes Quantum Field Theory (a.k.a. Second Quantization) 1 Quantum Field Theory Why quantum field theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum field theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content. 1.1 Limitations of Multi-body Schr¨odinger Wave Func- tion We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) difference. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The final state contains yet another particle, photon in this case. But the Schr¨odinger equation is a differential equation acting on the arguments of the Schr¨odingerwave function, and can never change the number of arguments.
    [Show full text]