A (Mostly) Linear Algebraic Introduction to Quaternions
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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 ........................... -
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. -
Chapter 1 the Field of Reals and Beyond
Chapter 1 The Field of Reals and Beyond Our goal with this section is to develop (review) the basic structure that character- izes the set of real numbers. Much of the material in the ¿rst section is a review of properties that were studied in MAT108 however, there are a few slight differ- ences in the de¿nitions for some of the terms. Rather than prove that we can get from the presentation given by the author of our MAT127A textbook to the previous set of properties, with one exception, we will base our discussion and derivations on the new set. As a general rule the de¿nitions offered in this set of Compan- ion Notes will be stated in symbolic form this is done to reinforce the language of mathematics and to give the statements in a form that clari¿es how one might prove satisfaction or lack of satisfaction of the properties. YOUR GLOSSARIES ALWAYS SHOULD CONTAIN THE (IN SYMBOLIC FORM) DEFINITION AS GIVEN IN OUR NOTES because that is the form that will be required for suc- cessful completion of literacy quizzes and exams where such statements may be requested. 1.1 Fields Recall the following DEFINITIONS: The Cartesian product of two sets A and B, denoted by A B,is a b : a + A F b + B . 1 2 CHAPTER 1. THE FIELD OF REALS AND BEYOND A function h from A into B is a subset of A B such that (i) 1a [a + A " 2bb + B F a b + h] i.e., dom h A,and (ii) 1a1b1c [a b + h F a c + h " b c] i.e., h is single-valued. -
Real Numbers and Their Properties
Real Numbers and their Properties Types of Numbers + • Z Natural numbers - counting numbers - 1, 2, 3,... The textbook uses the notation N. • Z Integers - 0, ±1, ±2, ±3,... The textbook uses the notation J. • Q Rationals - quotients (ratios) of integers. • R Reals - may be visualized as correspond- ing to all points on a number line. The reals which are not rational are called ir- rational. + Z ⊂ Z ⊂ Q ⊂ R. R ⊂ C, the field of complex numbers, but in this course we will only consider real numbers. Properties of Real Numbers There are four binary operations which take a pair of real numbers and result in another real number: Addition (+), Subtraction (−), Multiplication (× or ·), Division (÷ or /). These operations satisfy a number of rules. In the following, we assume a, b, c ∈ R. (In other words, a, b and c are all real numbers.) • Closure: a + b ∈ R, a · b ∈ R. This means we can add and multiply real num- bers. We can also subtract real numbers and we can divide as long as the denominator is not 0. • Commutative Law: a + b = b + a, a · b = b · a. This means when we add or multiply real num- bers, the order doesn’t matter. • Associative Law: (a + b) + c = a + (b + c), (a · b) · c = a · (b · c). We can thus write a + b + c or a · b · c without having to worry that different people will get different results. • Distributive Law: a · (b + c) = a · b + a · c, (a + b) · c = a · c + b · c. The distributive law is the one law which in- volves both addition and multiplication. -
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. -
Commutator Formulas
Commutator formulas Jack Schmidt This expository note mentions some interesting formulas using commutators. It touches on Hall's collection process and the associated Hall polynomials. It gives an alternative expression that is linear in the number of commutators and shows how to find such a formula using staircase diagrams. It also shows the shortest possible such expression. Future versions could touch on isoperimetric inequalities in geometric group theory, powers of commutators and Culler's identity as well as its effect on Schur's inequality between [G : Z(G)] and jG0j. 1 Powers of products versus products of powers In an abelian group one has (xy)n = xnyn so in a general group one has (xy)n = n n x y dn(x; y) for some product of commutators dn(x; y). This section explores formulas for dn(x; y). 1.1 A nice formula in a special case is given by certain binomial coefficients: n (n) (n) (n) (n) (n) (n) (xy) = x 1 y 1 [y; x] 2 [[y; x]; x] 3 [[[y; x]; x]; x] 4 ··· [y; n−1x] n The special case is G0 is abelian and commutes with y. The commutators involved are built inductively: From y and x, one gets [y; x]. From [y; x] and x, one gets [[y; x]; x]. From [y; n−2x] and x, one gets [y; n−1; x]. In general, one would also need to consider [y; x] and [[y; x]; x], but the special case assumes commutators commute, so [[y; x]; [[y; x]; x]] = 1. In general, one would also need to consider [y; x] and y, but the special case assumes commutators commute with y, so [[y; x]; y] = 1. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Hamilton Equations, Commutator, and Energy Conservation †
quantum reports Article Hamilton Equations, Commutator, and Energy Conservation † Weng Cho Chew 1,* , Aiyin Y. Liu 2 , Carlos Salazar-Lazaro 3 , Dong-Yeop Na 1 and Wei E. I. Sha 4 1 College of Engineering, Purdue University, West Lafayette, IN 47907, USA; [email protected] 2 College of Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA; [email protected] 3 Physics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA; [email protected] 4 College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310058, China; [email protected] * Correspondence: [email protected] † Based on the talk presented at the 40th Progress In Electromagnetics Research Symposium (PIERS, Toyama, Japan, 1–4 August 2018). Received: 12 September 2019; Accepted: 3 December 2019; Published: 9 December 2019 Abstract: We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations. Keywords: quantum mechanics; commutator relations; Heisenberg picture 1. Introduction In quantum theory, a classical observable, which is modeled by a real scalar variable, is replaced by a quantum operator, which is analogous to an infinite-dimensional matrix operator. -
Universit¨At Stuttgart Fachbereich Mathematik
Universitat¨ Fachbereich Stuttgart Mathematik Heisenberg Groups, Semifields, and Translation Planes Norbert Knarr, Markus J. Stroppel Preprint 2013/006 Universitat¨ Fachbereich Stuttgart Mathematik Heisenberg Groups, Semifields, and Translation Planes Norbert Knarr, Markus J. Stroppel Preprint 2013/006 Fachbereich Mathematik Fakultat¨ Mathematik und Physik Universitat¨ Stuttgart Pfaffenwaldring 57 D-70 569 Stuttgart E-Mail: [email protected] WWW: http://www.mathematik.uni-stuttgart.de/preprints ISSN 1613-8309 c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors. LATEX-Style: Winfried Geis, Thomas Merkle Heisenberg Groups, Semifields, and Translation Planes Norbert Knarr, Markus J. Stroppel Abstract For Heisenberg groups constructed over semifields (i.e., not neccessarily associative divi- sion rings), we solve the isomorphism problem and determine the automorphism groups. We show that two Heisenberg groups over semifields are isomorphic precisely if the semi- fields are isotopic or anti-isotopic to each other. This condition means that the corre- sponding translation are isomorphic or dual to each other. Mathematics Subject Classification: 12K10 17A35 20D15 20F28 51A35 51A10 Keywords: Heisenberg group, nilpotent group, automorphism, translation plane, semi- field, division algebra, isotopism, autotopism In [7], Heisenberg’s example of a step 2 nilpotent group has been generalized, replacing the ground field by an arbitrary associative ring S with 2 2 S×. The main focus of [7] was on the group of automorphisms of such a Heisenberg group. In the present note we extend this further (see 1.2 below), dropping the restriction on invertibility of 2 and allowing the multiplication in S to be non-associative. The main results of the present paper require that S is a semifield. -
Math 11200/20 Lectures Outline I Will Update This
Math 11200/20 lectures outline I will update this document after every lecture to keep track of what we covered. \Textbook" refers to the book by Diane Herrmann and Paul Sally. \Intro to Cryptography" refers to: • Introduction to Cryptography with Coding Theory, Second Edition, by Wade Trappe and Lawrence Washington. • http://calclab.math.tamu.edu/~rahe/2014a_673_700720/textbooks.html. week 1 9/26/16. Reference: Textbook, Chapter 0 (1) introductions (2) syllabus (boring stuff) (3) Triangle game (a) some ideas: parity, symmetry, balancing large/small, largest possible sum (b) There are 6 \symmetries of the triangle." They form a \group." (just for fun) 9/28/16. Reference: Textbook, Chapter 1, pages 11{19 (1) Russell's paradox (just for fun) (2) Number systems: N; Z; Q; R. (a) π 2 R but π 62 Q. (this is actually hard to prove) (3) Set theory (a) unions, intersections. (remember: \A or B" means \at least one of A, B is true") (b) drawing venn diagrams is very useful! (c) empty set. \vacuously true" (d) commutative property, associative, identity (the empty set is the identity for union), inverse, (distributive?) (i) how to show that there is no such thing as \distributive property of addi- tion over multiplication" (a + (b · c) = (a · b) + (a · c)) (4) Functions 9/30/16. Reference: Textbook, Chapter 1, pages 20{32 (1) functions (recap). f : A ! B. A is domain, B is range. (2) binary operations are functions of the form f : S × S ! S. (a) \+ is a function R × R ! R." Same thing as \+ is a binary operation on R." Same thing as \R is closed under +." (closed because you cannot escape!) a c ad+bc (b) N; Z are also closed under addition. -
1 Vectors & Tensors
1 Vectors & Tensors The mathematical modeling of the physical world requires knowledge of quite a few different mathematics subjects, such as Calculus, Differential Equations and Linear Algebra. These topics are usually encountered in fundamental mathematics courses. However, in a more thorough and in-depth treatment of mechanics, it is essential to describe the physical world using the concept of the tensor, and so we begin this book with a comprehensive chapter on the tensor. The chapter is divided into three parts. The first part covers vectors (§1.1-1.7). The second part is concerned with second, and higher-order, tensors (§1.8-1.15). The second part covers much of the same ground as done in the first part, mainly generalizing the vector concepts and expressions to tensors. The final part (§1.16-1.19) (not required in the vast majority of applications) is concerned with generalizing the earlier work to curvilinear coordinate systems. The first part comprises basic vector algebra, such as the dot product and the cross product; the mathematics of how the components of a vector transform between different coordinate systems; the symbolic, index and matrix notations for vectors; the differentiation of vectors, including the gradient, the divergence and the curl; the integration of vectors, including line, double, surface and volume integrals, and the integral theorems. The second part comprises the definition of the tensor (and a re-definition of the vector); dyads and dyadics; the manipulation of tensors; properties of tensors, such as the trace, transpose, norm, determinant and principal values; special tensors, such as the spherical, identity and orthogonal tensors; the transformation of tensor components between different coordinate systems; the calculus of tensors, including the gradient of vectors and higher order tensors and the divergence of higher order tensors and special fourth order tensors.