Introduction to Quantum Field Theory for Mathematicians
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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. -
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 ........................... -
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. -
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). -
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. -
The Quaternionic Commutator Bracket and Its Implications
S S symmetry Article The Quaternionic Commutator Bracket and Its Implications Arbab I. Arbab 1,† and Mudhahir Al Ajmi 2,* 1 Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khartoum 11115, Sudan; [email protected] 2 Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, P.C. 123, Muscat 999046, Sultanate of Oman * Correspondence: [email protected] † Current address: Department of Physics, College of Science, Qassim University, Qassim 51452, Saudi Arabia. Received: 11 August 2018; Accepted: 9 October 2018; Published: 16 October 2018 Abstract: A quaternionic commutator bracket for position and momentum shows that the i ~ quaternionic wave function, viz. ye = ( c y0 , y), represents a state of a particle with orbital angular momentum, L = 3 h¯ , resulting from the internal structure of the particle. This angular momentum can be attributed to spin of the particle. The vector y~ , points in an opposite direction of~L. When a charged particle is placed in an electromagnetic field, the interaction energy reveals that the magnetic moments interact with the electric and magnetic fields giving rise to terms similar to Aharonov–Bohm and Aharonov–Casher effects. Keywords: commutator bracket; quaternions; magnetic moments; angular momentum; quantum mechanics 1. Introduction In quantum mechanics, particles are described by relativistic or non-relativistic wave equations. Each equation associates a spin state of the particle to its wave equation. For instance, the Schrödinger equation applies to the spinless particles in the non-relativistic domain, while the appropriate relativistic equation for spin-0 particles is the Klein–Gordon equation. -
Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions
PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 05 Tuesday, January 14, 2020 Topics: Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions. Feynman Diagrams We can use the Wick’s theorem to express the n-point function h0jT fφ(x1)φ(x2) ··· φ(xn)gj0i as a sum of products of Feynman propagators. We will be interested in developing a diagrammatic interpretation of such expressions. Let us look at the expression containing four fields. We have h0jT fφ1φ2φ3φ4g j0i = DF (x1 − x2)DF (x3 − x4) +DF (x1 − x3)DF (x2 − x4) +DF (x1 − x4)DF (x2 − x3): (1) We can interpret Eq. (1) as the sum of three diagrams, if we represent each of the points, x1 through x4 by a dot, and each factor DF (x − y) by a line joining x to y. These diagrams are called Feynman diagrams. In Fig. 1 we provide the diagrammatic interpretation of Eq. (1). The interpretation of the diagrams is the following: particles are created at two spacetime points, each propagates to one of the other points, and then they are annihilated. This process can happen in three different ways, and they correspond to the three ways to connect the points in pairs, as shown in the three diagrams in Fig. 1. Thus the total amplitude for the process is the sum of these three diagrams. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 ⟨0|T {ϕ1ϕ2ϕ3ϕ4}|0⟩ = DF(x1 − x2)DF(x3 − x4) + DF(x1 − x3)DF(x2 − x4) +DF(x1 − x4)DF(x2 − x3) 1 2 1 2 1 2 + + 3 4 3 4 3 4 Figure 1: Diagrammatic interpretation of Eq. -
Commutator Groups of Monomial Groups
Pacific Journal of Mathematics COMMUTATOR GROUPS OF MONOMIAL GROUPS CALVIN VIRGIL HOLMES Vol. 10, No. 4 December 1960 COMMUTATOR GROUPS OF MONOMIAL GROUPS C. V. HOLMES This paper is a study of the commutator groups of certain general- ized permutation groups called complete monomial groups. In [2] Ore has shown that every element of the infinite permutation group is itsself a commutator of this group. Here it is shown that every element of the infinite complete monomial group is the product of at most two commutators of the infinite complete monomial group. The commutator subgroup of the infinite complete monomial group is itself, as is the case in the infinite symmetric group, [2]. The derived series is determined for a wide class of monomial groups. Let H be an arbitrary group, and S a set of order B, B ^ d, cZ = ^0. Then one obtains a monomial group after the manner described in [1], A monomial substitution over H is a linear transformation mapping each element x of S in a one-to-one manner onto some element of S multi- plied by an element h of H, the multiplication being formal. The ele- ment h is termed a factor of the substitution. If substitution u maps xi into hjXj, while substitution v maps xό into htxt, then the substitution uv maps xt into hόhtxt. A substitution all of whose factor are the iden- tity β of H is called a permutation and the set of all permutations is a subgroup which is isomorphic to the symmetric group on B objects. -
Commutator Theory for Congruence Modular Varieties Ralph Freese
Commutator Theory for Congruence Modular Varieties Ralph Freese and Ralph McKenzie Contents Introduction 1 Chapter 1. The Commutator in Groups and Rings 7 Exercise 10 Chapter 2. Universal Algebra 11 Exercises 19 Chapter 3. Several Commutators 21 Exercises 22 Chapter 4. One Commutator in Modular Varieties;Its Basic Properties 25 Exercises 33 Chapter 5. The Fundamental Theorem on Abelian Algebras 35 Exercises 43 Chapter 6. Permutability and a Characterization ofModular Varieties 47 Exercises 49 Chapter 7. The Center and Nilpotent Algebras 53 Exercises 57 Chapter 8. Congruence Identities 59 Exercises 68 Chapter 9. Rings Associated With Modular Varieties: Abelian Varieties 71 Exercises 87 Chapter 10. Structure and Representationin Modular Varieties 89 1. Birkhoff-J´onsson Type Theorems For Modular Varieties 89 2. Subdirectly Irreducible Algebras inFinitely Generated Varieties 92 3. Residually Small Varieties 97 4. Chief Factors and Simple Algebras 102 Exercises 103 Chapter 11. Joins and Products of Modular Varieties 105 Chapter 12. Strictly Simple Algebras 109 iii iv CONTENTS Chapter 13. Mal’cev Conditions for Lattice Equations 115 Exercises 120 Chapter 14. A Finite Basis Result 121 Chapter 15. Pure Lattice Congruence Identities 135 1. The Arguesian Equation 139 Related Literature 141 Solutions To The Exercises 147 Chapter 1 147 Chapter 2 147 Chapter 4 148 Chapter 5 150 Chapter 6 152 Chapter 7 156 Chapter 8 158 Chapter 9 161 Chapter 10 165 Chapter 13 165 Bibliography 169 Index 173 Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz- ers, are all defined from the binary operation [x, y]= x−1y−1xy. -
Chapter 4 Introduction to Many-Body Quantum Mechanics
Chapter 4 Introduction to many-body quantum mechanics 4.1 The complexity of the quantum many-body prob- lem After learning how to solve the 1-body Schr¨odinger equation, let us next generalize to more particles. If a single body quantum problem is described by a Hilbert space of dimension dim = d then N distinguishable quantum particles are described by theH H tensor product of N Hilbert spaces N (N) N ⊗ (4.1) H ≡ H ≡ H i=1 O with dimension dN . As a first example, a single spin-1/2 has a Hilbert space = C2 of dimension 2, but N spin-1/2 have a Hilbert space (N) = C2N of dimensionH 2N . Similarly, a single H particle in three dimensional space is described by a complex-valued wave function ψ(~x) of the position ~x of the particle, while N distinguishable particles are described by a complex-valued wave function ψ(~x1,...,~xN ) of the positions ~x1,...,~xN of the particles. Approximating the Hilbert space of the single particle by a finite basis set with d basis functions, the N-particle basisH approximated by the same finite basis set for single particles needs dN basis functions. This exponential scaling of the Hilbert space dimension with the number of particles is a big challenge. Even in the simplest case – a spin-1/2 with d = 2, the basis for N = 30 spins is already of of size 230 109. A single complex vector needs 16 GByte of memory and will not fit into the memory≈ of your personal computer anymore. -
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). -
On the Definition of the Renormalization Constants in Quantum Electrodynamics
On the Definition of the Renormalization Constants in Quantum Electrodynamics Autor(en): Källén, Gunnar Objekttyp: Article Zeitschrift: Helvetica Physica Acta Band(Jahr): 25(1952) Heft IV Erstellt am: Mar 18, 2014 Persistenter Link: http://dx.doi.org/10.5169/seals-112316 Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag. Ein Dienst der ETH-Bibliothek Rämistrasse 101, 8092 Zürich, Schweiz [email protected] http://retro.seals.ch On the Definition of the Renormalization Constants in Quantum Electrodynamics by Gunnar Källen.*) Swiss Federal Institute of Technology, Zürich. (14.11.1952.) Summary. A formulation of quantum electrodynamics in terms of the renor- malized Heisenberg operators and the experimental mass and charge of the electron is given. The renormalization constants are implicitly defined and ex¬ pressed as integrals over finite functions in momentum space. No discussion of the convergence of these integrals or of the existence of rigorous solutions is given. Introduction. The renormalization method in quantum electrodynamics has been investigated by many authors, and it has been proved by Dyson1) that every term in a formal expansion in powers of the coupling constant of various expressions is a finite quantity.