# Quantum Field Theory a Tourist Guide for Mathematicians

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• Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay.
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• Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation speciﬁes how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases.
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• An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
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• Perturbative Versus Non-Perturbative Quantum Field Theory: Tao’S Method, the Casimir Effect, and Interacting Wightman Theories
universe Review Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories Walter Felipe Wreszinski Instituto de Física, Universidade de São Paulo, São Paulo 05508-090, SP, Brazil; [email protected] Abstract: We dwell upon certain points concerning the meaning of quantum ﬁeld theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-deﬁned (rigorous) mathematical sense, as well as some of the few existent mathematically precise results on fully quantized ﬁeld theories. Emphasis is brought on how the mathematical contributions help to elucidate or illuminate certain conceptual aspects of the theory when applied to real physical phenomena, in particular, the singular nature of quantum ﬁelds. In a ﬁrst part, we present a comprehensive review of divergent versus asymptotic series, with qed as background example, as well as a method due to Terence Tao which conveys mathematical sense to divergent series. In a second part, we apply Tao’s method to the Casimir effect in its simplest form, consisting of perfectly conducting parallel plates, arguing that the usual theory, which makes use of the Euler-MacLaurin formula, still contains a residual inﬁnity, which is eliminated in our approach. In the third part, we revisit the general theory of nonperturbative quantum ﬁelds, in the form of newly proposed (with Christian Jaekel) Wightman axioms for interacting ﬁeld theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as Citation: Wreszinski, W.F. unstable particles. Various problems (mostly open) are ﬁnally discussed in connection with concrete Perturbative versus Non-Perturbative models.
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• Research Statement
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• The Dirac Equation
The Dirac Equation Asaf Pe’er1 February 11, 2014 This part of the course is based on Refs. [1], [2] and [3]. 1. Introduction So far we have only discussed scalar ﬁelds, such that under a Lorentz transformation ′ ′ xµ xµ = λµ xν the ﬁeld transforms as → ν φ(x) φ′(x)= φ(Λ−1x). (1) → We have seen that quantization of such ﬁelds gives rise to spin 0 particles. However, in the framework of the standard model there is only one spin-0 particle known in nature: the Higgs particle. Most particles in nature have an intrinsic angular momentum, or spin. To describe particles with spin, we are going to look at ﬁelds which themselves transform non-trivially under the Lorentz group. In this chapter we will describe the Dirac equation, whose quantization gives rise to Fermionic spin 1/2 particles. To motivate the Dirac equation, we will start by studying the appropriate representation of the Lorentz group. 2. The Lorentz group, its representations and generators The ﬁrst thing to realize is that the set of all possible Lorentz transformations form what is known in mathematics as group. By deﬁnition, a group G is deﬁned as a set of objects, or operators (known as the elements of G) that may be combined, or multiplied, to form a well-deﬁned product in G which satisfy the following conditions: 1. Closure: If a and b are two elements of G, then the product ab is also an element of G. 1Physics Dep., University College Cork –2– 2. The multiplication is associative: (ab)c = a(bc).
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• Quantum Optics: an Introduction
The Himalayan Physics, Vol.1, No.1, May 2010 Quantum Optics: An Introduction Min Raj Lamsal Prithwi Narayan Campus, Pokhara Email: [email protected] Optics is the physics, which deals with the study of in electrodynamics gave beautiful and satisfactory nature of light, its propagation in different media description of the electromagnetic phenomena. At (including vacuum) & its interaction with different the same time, the kinetic theory of gases provided materials. Quantum is a packet of energy absorbed or a microscopic basis for the thermo dynamical emitted in the form of tiny packets called quanta. It properties of matter. Up to the end of the nineteenth means the amount of energy released or absorbed in century, the classical physics was so successful a physical process is always an integral multiple of and impressive in explaining physical phenomena a discrete unit of energy known as quantum which is that the scientists of that time absolutely believed also known as photon. Quantum Optics is a ﬁ eld of that they were potentially capable of explaining all research in physics, dealing with the application of physical phenomena. quantum mechanics to phenomena involving light and its interactions with matter. However, the ﬁ rst indication of inadequacy of the Physics in which majority of physical phenomena classical physics was seen in the beginning of the can be successfully described by using Newton’s twentieth century where it could not explain the laws of motion is called classical physics. In fact, the experimentally observed spectra of a blackbody classical physics includes the classical mechanics and radiation. In addition, the laws in classical physics the electromagnetic theory.
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• Mathematical Methods of Classical Mechanics-Arnold V.I..Djvu
V.I. Arnold Mathematical Methods of Classical Mechanics Second Edition Translated by K. Vogtmann and A. Weinstein With 269 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest V. I. Arnold K. Vogtmann A. Weinstein Department of Department of Department of Mathematics Mathematics Mathematics Steklov Mathematical Cornell University University of California Institute Ithaca, NY 14853 at Berkeley Russian Academy of U.S.A. Berkeley, CA 94720 Sciences U.S.A. Moscow 117966 GSP-1 Russia Editorial Board J. H. Ewing F. W. Gehring P.R. Halmos Department of Department of Department of Mathematics Mathematics Mathematics Indiana University University of Michigan Santa Clara University Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053 U.S.A. U.S.A. U.S.A. Mathematics Subject Classifications (1991): 70HXX, 70005, 58-XX Library of Congress Cataloging-in-Publication Data Amol 'd, V.I. (Vladimir Igorevich), 1937- [Matematicheskie melody klassicheskoi mekhaniki. English] Mathematical methods of classical mechanics I V.I. Amol 'd; translated by K. Vogtmann and A. Weinstein.-2nd ed. p. cm.-(Graduate texts in mathematics ; 60) Translation of: Mathematicheskie metody klassicheskoi mekhaniki. Bibliography: p. Includes index. ISBN 0-387-96890-3 I. Mechanics, Analytic. I. Title. II. Series. QA805.A6813 1989 531 '.01 '515-dcl9 88-39823 Title of the Russian Original Edition: M atematicheskie metody klassicheskoi mekhaniki. Nauka, Moscow, 1974. Printed on acid-free paper © 1978, 1989 by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis.
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• 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.
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• The Concept of Quantum State : New Views on Old Phenomena Michel Paty
The concept of quantum state : new views on old phenomena Michel Paty To cite this version: Michel Paty. The concept of quantum state : new views on old phenomena. Ashtekar, Abhay, Cohen, Robert S., Howard, Don, Renn, Jürgen, Sarkar, Sahotra & Shimony, Abner. Revisiting the Founda- tions of Relativistic Physics : Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht: Kluwer Academic Publishers, p. 451-478, 2003. halshs-00189410 HAL Id: halshs-00189410 https://halshs.archives-ouvertes.fr/halshs-00189410 Submitted on 20 Nov 2007 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. « The concept of quantum state: new views on old phenomena », in Ashtekar, Abhay, Cohen, Robert S., Howard, Don, Renn, Jürgen, Sarkar, Sahotra & Shimony, Abner (eds.), Revisiting the Foundations of Relativistic Physics : Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht: Kluwer Academic Publishers, 451-478. , 2003 The concept of quantum state : new views on old phenomena par Michel PATY* ABSTRACT. Recent developments in the area of the knowledge of quantum systems have led to consider as physical facts statements that appeared formerly to be more related to interpretation, with free options.
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• On the “Equivalence” of the Maxwell and Dirac Equations
On the “equivalence” of the Maxwell and Dirac equations Andre´ Gsponer1 Document ISRI-01-07 published in Int. J. Theor. Phys. 41 (2002) 689–694 It is shown that Maxwell’s equation cannot be put into a spinor form that is equivalent to Dirac’s equation. First of all, the spinor ψ in the representation F~ = ψψ~uψ of the electromagnetic ﬁeld bivector depends on only three independent complex components whereas the Dirac spinor depends on four. Second, Dirac’s equation implies a complex structure speciﬁc to spin 1/2 particles that has no counterpart in Maxwell’s equation. This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxwell’s and Dirac’s equations into each other. Key words: Maxwell equation; Dirac equation; Lanczos equation; fermion; boson. 1. INTRODUCTION The conventional view is that spin 1 and spin 1/2 particles belong to distinct irreducible representations of the Poincaré group, so that there should be no connection between the Maxwell and Dirac equations describing the dynamics of arXiv:math-ph/0201053v2 4 Apr 2002 these particles. However, it is well known that Maxwell’s and Dirac’s equations can be written in a number of different forms, and that in some of them these equations look very similar (e.g., Fushchich and Nikitin, 1987; Good, 1957; Kobe 1999; Moses, 1959; Rodrigues and Capelas de Oliviera, 1990; Sachs and Schwebel, 1962). This has lead to speculations on the possibility that these similarities could stem from a relationship that would be not merely formal but more profound (Campolattaro, 1990, 1997), or that in some sense Maxwell’s and Dirac’s equations could even be “equivalent” (Rodrigues and Vaz, 1998; Vaz and Rodrigues, 1993, 1997).
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• Mathematical Foundations of Supersymmetry
MATHEMATICAL FOUNDATIONS OF SUPERSYMMETRY Lauren Caston and Rita Fioresi 1 Dipartimento di Matematica, Universit`adi Bologna Piazza di Porta S. Donato, 5 40126 Bologna, Italia e-mail: [email protected], ﬁ[email protected] 1Investigation supported by the University of Bologna 1 Introduction Supersymmetry (SUSY) is the machinery mathematicians and physicists have developed to treat two types of elementary particles, bosons and fermions, on the same footing. Supergeometry is the geometric basis for supersymmetry; it was ﬁrst discovered and studied by physicists Wess, Zumino [25], Salam and Strathde [20] (among others) in the early 1970’s. Today supergeometry plays an important role in high energy physics. The objects in super geometry generalize the concept of smooth manifolds and algebraic schemes to include anticommuting coordinates. As a result, we employ the techniques from algebraic geometry to study such objects, namely A. Grothendiek’s theory of schemes. Fermions include all of the material world; they are the building blocks of atoms. Fermions do not like each other. This is in essence the Pauli exclusion principle which states that two electrons cannot occupy the same quantum me- chanical state at the same time. Bosons, on the other hand, can occupy the same state at the same time. Instead of looking at equations which just describe either bosons or fermions separately, supersymmetry seeks out a description of both simultaneously. Tran- sitions between fermions and bosons require that we allow transformations be- tween the commuting and anticommuting coordinates. Such transitions are called supersymmetries. In classical Minkowski space, physicists classify elementary particles by their mass and spin.
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