The Four Dimensional Dirac Equation in Five Dimensions
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Quaternions and Cli Ord Geometric Algebras
Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a . -
Hypercomplex Algebras and Their Application to the Mathematical
Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory Torsten Hertig I1, Philip H¨ohmann II2, Ralf Otte I3 I tecData AG Bahnhofsstrasse 114, CH-9240 Uzwil, Schweiz 1 [email protected] 3 [email protected] II info-key GmbH & Co. KG Heinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland 2 [email protected] March 31, 2014 Abstract Quantum theory (QT) which is one of the basic theories of physics, namely in terms of ERWIN SCHRODINGER¨ ’s 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating EINSTEIN’s theory of Special Relativity (SR) (SCHRODINGER¨ , OSKAR KLEIN, WALTER GORDON, 1926, PAUL DIRAC 1928) leads to an equation which requires some coefficients which can neither be real nor complex but rather must be hypercomplex. It is conventional to write down the DIRAC equation using pairwise anti-commuting matrices. However, a unitary ring of square matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic properties of the elements and their relations to one another, rather than their precise form as matrices which is important. This encourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinations of some basis elements. In the case of the DIRAC equation, these elements are called biquaternions, also known as quaternions over the complex numbers. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutative in contrast to H. -
Codimension One Isometric Immersions Betweenlorentz Spaces by L
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 252, August 1979 CODIMENSION ONE ISOMETRIC IMMERSIONS BETWEENLORENTZ SPACES BY L. K. GRAVES In memory of my father, Lucius Kingman Graves Abstract. The theorem of Hartman and Nirenberg classifies codimension one isometric immersions between Euclidean spaces as cylinders over plane curves. Corresponding results are given here for Lorentz spaces, which are Euclidean spaces with one negative-definite direction (also known as Minkowski spaces). The pivotal result involves the completeness of the relative nullity foliation of such an immersion. When this foliation carries a nondegenerate metric, results analogous to the Hartman-Nirenberg theorem obtain. Otherwise, a new description, based on particular surfaces in the three-dimensional Lorentz space, is required. The theorem of Hartman and Nirenberg [HN] says that, up to a proper motion of E"+', all isometric immersions of E" into E"+ ' have the form id X c: E"-1 xEUr'x E2 (0.1) where c: E1 ->E2 is a unit speed plane curve and the factors in the product are orthogonal. A proof of the Hartman-Nirenberg result also appears in [N,]. The major step in proving the theorem is showing that the relative nullity foliation, which is spanned by those tangent directions in which a local unit normal field is parallel, has complete leaves. These leaves yield the E"_1 factors in (0.1). The one-dimensional complement of the relative nullity foliation gives rise to the curve c. This paper studies isometric immersions of L" into Ln+1, where L" denotes the «-dimensional Lorentz (or Minkowski) space. Its chief goal is the classifi- cation, up to a proper motion of L"+1, of all such immersions. -
K-Theory and Algebraic Geometry
http://dx.doi.org/10.1090/pspum/058.2 Recent Titles in This Series 58 Bill Jacob and Alex Rosenberg, editors, ^-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara) 57 Michael C. Cranston and Mark A. Pinsky, editors, Stochastic analysis (Cornell University, Ithaca) 56 William J. Haboush and Brian J. Parshall, editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E. -
Embeddings of Integral Quadratic Forms Rick Miranda Colorado State
Embeddings of Integral Quadratic Forms Rick Miranda Colorado State University David R. Morrison University of California, Santa Barbara Copyright c 2009, Rick Miranda and David R. Morrison Preface The authors ran a seminar on Integral Quadratic Forms at the Institute for Advanced Study in the Spring of 1982, and worked on a book-length manuscript reporting on the topic throughout the 1980’s and early 1990’s. Some new results which are proved in the manuscript were announced in two brief papers in the Proceedings of the Japan Academy of Sciences in 1985 and 1986. We are making this preliminary version of the manuscript available at this time in the hope that it will be useful. Still to do before the manuscript is in final form: final editing of some portions, completion of the bibliography, and the addition of a chapter on the application to K3 surfaces. Rick Miranda David R. Morrison Fort Collins and Santa Barbara November, 2009 iii Contents Preface iii Chapter I. Quadratic Forms and Orthogonal Groups 1 1. Symmetric Bilinear Forms 1 2. Quadratic Forms 2 3. Quadratic Modules 4 4. Torsion Forms over Integral Domains 7 5. Orthogonality and Splitting 9 6. Homomorphisms 11 7. Examples 13 8. Change of Rings 22 9. Isometries 25 10. The Spinor Norm 29 11. Sign Structures and Orientations 31 Chapter II. Quadratic Forms over Integral Domains 35 1. Torsion Modules over a Principal Ideal Domain 35 2. The Functors ρk 37 3. The Discriminant of a Torsion Bilinear Form 40 4. The Discriminant of a Torsion Quadratic Form 45 5. -
Hyperbolicity of Hermitian Forms Over Biquaternion Algebras
HYPERBOLICITY OF HERMITIAN FORMS OVER BIQUATERNION ALGEBRAS NIKITA A. KARPENKO Abstract. We show that a non-hyperbolic hermitian form over a biquaternion algebra over a field of characteristic 6= 2 remains non-hyperbolic over a generic splitting field of the algebra. Contents 1. Introduction 1 2. Notation 2 3. Krull-Schmidt principle 3 4. Splitting off a motivic summand 5 5. Rost correspondences 7 6. Motivic decompositions of some isotropic varieties 12 7. Proof of Main Theorem 14 References 16 1. Introduction Throughout this note (besides of x3 and x4) F is a field of characteristic 6= 2. The basic reference for the staff related to involutions on central simple algebras is [12].p The degree deg A of a (finite dimensional) central simple F -algebra A is the integer dimF A; the index ind A of A is the degree of a central division algebra Brauer-equivalent to A. Conjecture 1.1. Let A be a central simple F -algebra endowed with an orthogonal invo- lution σ. If σ becomes hyperbolic over the function field of the Severi-Brauer variety of A, then σ is hyperbolic (over F ). In a stronger version of Conjecture 1.1, each of two words \hyperbolic" is replaced by \isotropic", cf. [10, Conjecture 5.2]. Here is the complete list of indices ind A and coindices coind A = deg A= ind A of A for which Conjecture 1.1 is known (over an arbitrary field of characteristic 6= 2), given in the chronological order: • ind A = 1 | trivial; Date: January 2008. Key words and phrases. -
Introduction to Clifford's Geometric Algebra
SICE Journal of Control, Measurement, and System Integration, Vol. 4, No. 1, pp. 001–011, January 2011 Introduction to Clifford’s Geometric Algebra ∗ Eckhard HITZER Abstract : Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branchesof physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows, e.g., to optimize learning algorithms of Clifford neurons, etc. Key Words: Hypercomplex algebra, hypercomplex analysis, geometry, science, engineering. 1. Introduction unique associative and multilinear algebra with this property. W.K. Clifford (1845-1879), a young English Goldsmid pro- Thus if we wish to generalize methods from algebra, analysis, ff fessor of applied mathematics at the University College of Lon- calculus, di erential geometry (etc.) of real numbers, complex don, published in 1878 in the American Journal of Mathemat- numbers, and quaternion algebra to vector spaces and multivec- ics Pure and Applied a nine page long paper on Applications of tor spaces (which include additional elements representing 2D Grassmann’s Extensive Algebra. In this paper, the young ge- up to nD subspaces, i.e. plane elements up to hypervolume el- ff nius Clifford, standing on the shoulders of two giants of alge- ements), the study of Cli ord algebras becomes unavoidable. -
Zeros of Conformal Fields in Any Metric Signature
Zeros of conformal fields in any metric signature Andrzej Derdzinski Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA E-mail: [email protected] Abstract. The connected components of the zero set of any conformal vector field, in a pseudo-Riemannian manifold of arbitrary signature, are shown to be totally umbilical conifold varieties, that is, smooth submanifolds except possibly for some quadric singularities. The singularities occur only when the metric is indefinite, including the Lorentzian case. This generalizes an analogous result for Riemannian manifolds, due to Belgun, Moroianu and Ornea (2010). AMS classification scheme numbers: 53B30 1. Introduction A vector field v on a pseudo-Riemannian manifold (M; g) of dimension n ≥ 2 is called conformal if, for some function φ : M ! IR, $vg = φg; that is; in coordinates; vj;k + vk;j = φgjk : (1) One then obviously has div v = nφ/2. The class of conformal vector fields on (M; g) includes Killing fields v, characterized by (1) with φ = 0. Kobayashi [15] showed that, for any Killing vector field v on a Riemannian manifold (M; g), the connected components of the zero set of v are mutually isolated totally geodesic submanifolds of even codimensions. Assuming compactness of M, Blair [5] established an analogue of Kobayashi's theorem for conformal vector fields, in which the word `geodesic' is replaced by `umbilical' and the codimension clause is relaxed for one-point connected components. Very recently, Belgun, Moroianu and Ornea [4] proved that Blair's conclusion is valid in the noncompact case as well. The last result is also a direct consequence of a theorem of Frances [12] stating that a conformal field on a Riemannian manifold is linearizable at any zero z unless some neighborhood of z is conformally flat. -
Issue 2006 Volume 4
ISSUE 2006 PROGRESS IN PHYSICS VOLUME 4 ISSN 1555-5534 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS A quarterly issue scientific journal, registered with the Library of Congress (DC, USA). This journal is peer reviewed and includedinthe abstracting and indexing coverage of: Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sweden), Zentralblatt MATH (Germany), Referativnyi Zhurnal VINITI (Russia), etc. Electronic version of this journal: OCTOBER 2006 VOLUME 4 http://www.ptep-online.com http://www.geocities.com/ptep_online CONTENTS To order printed issues of this journal, con- tact the Editor in Chief. D. Rabounski New Effect of General Relativity: Thomson Dispersion of Light in Stars as a Machine Producing Stellar Energy. .3 Chief Editor A. N. Mina and A. H. Phillips Frequency Resolved Detection over a Large Dmitri Rabounski Frequency Range of the Fluctuations in an Array of Quantum Dots . 11 [email protected] C. Y. Lo Completing Einstein’s Proof of E = mc2 ..........................14 Associate Editors Prof. Florentin Smarandache D. Rabounski A Source of Energy for Any Kind of Star . 19 [email protected] Dr. Larissa Borissova J. Dunning-Davies The Thermodynamics Associated with Santilli’s Hadronic [email protected] Mechanics . 24 Stephen J. Crothers F. Smarandache and V.Christianto A Note on Geometric and Information [email protected] Fusion Interpretation of Bell’s Theorem and Quantum Measurement . 27 Department of Mathematics, University of J. X. Zheng-Johansson and P.-I. Johansson Developing de Broglie Wave . 32 New Mexico, 200 College Road, Gallup, NM 87301, USA The Classical Theory of Fields Revision Project (CTFRP): Collected Papers Treating of Corrections to the Book “The Classical Theory of Fields” by L. -
In Praise of Quaternions
IN PRAISE OF QUATERNIONS JOACHIM LAMBEK With an appendix on the algebra of biquaternions Michael Barr Abstract. This is a survey of some of the applications of quaternions to physics in the 20th century. In the first half century, an elegant presentation of Maxwell's equations and special relativity was achieved with the help of biquaternions, that is, quaternions with complex coefficients. However, a quaternionic derivation of Dirac's celebrated equation of the electron depended on the observation that all 4 × 4 real matrices can be generated by quaternions and their duals. On examine quelques applications des quaternions `ala physique du vingti`emesi`ecle.Le premier moiti´edu si`ecleavait vu une pr´esentation ´el´egantes des equations de Maxwell et de la relativit´e specialle par les quaternions avec des coefficients complexes. Cependant, une d´erivation de l'´equationc´el`ebrede Dirac d´ependait sur l'observation que toutes les matrices 4 × 4 r´eelles peuvent ^etre gener´eespar les representations reguli`eresdes quaternions. 1. Prologue. This is an expository article attempting to acquaint algebraically inclined readers with some basic notions of modern physics, making use of Hamilton's quaternions rather than the more sophisticated spinor calculus. While quaternions play almost no r^olein mainstream physics, they afford a quick entry into the world of special relativity and allow one to formulate the Maxwell-Lorentz theory of electro-magnetism and the Dirac equation of the electron with a minimum of mathematical prerequisites. Marginally, quaternions even give us a glimpse of the Feynman diagrams appearing in the standard model. As everyone knows, quaternions were invented (discovered?) by William Rowan Hamilton. -
Geometric Algebra Techniques
Geometric Algebra Techniques in Mathematics, Physics and Engineering S. Xamb´o UPC IMUVA 16-19 Nov 2015 · · S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 1 / 47 Index Main points Background. Lorentz transformations. Introduction. Definitions, notations and conventions. The Dirac representation of . D + Lorentzian GA. = 1;3 = ΛηE. Involutions and . Complex structure. G G G The nature of the even subalgebra. Inner products in and . Relative P D involutions. Space-time kinematics. Space-time paths. Timelike paths and proper time. Instantaneous rest frame. Relative velocity. Lorentz transformations. Lorentz spinors. Rotations. Lorentz boosts. Relativistic addition of velocities. General boosts. Lorentz group. Appendix A. Appendix B. EM field. Maxwell's equations. Gradient operator. Riesz form of Maxwell's equations. GA form of Dirac's equation. References. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 2 / 47 Background Lorentz transformations The Lorentz boost for speed v is given by t0 = γ(t βx); x 0 = γ(x βt) − p − where β = v=c and γ = 1= 1 β2. − This transformation leaves invariant the Lorentz quadratic form (ct)2 x 2 y 2 z 2. − − − S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 3 / 47 Background Lorentz transformations Let γx ; γy ; γz ; γt be the unit vectors on the axes of the reference S 0 0 0 0 0 and γx ; γy ; γz ; γt be the unit vectors on the axes of the reference S . Then the above Lorentz boost is equivalent to the linear transformation 0 0 0 0 γt = γ(γt + βγx ); γx = γ(γx + βγt ); γy = γy ; γz = γz ; which is a Lorentz isometry. -
Notes on Geometry and Spacetime Version 2.7, November 2009
Notes on Geometry and Spacetime Version 2.7, November 2009 David B. Malament Department of Logic and Philosophy of Science University of California, Irvine [email protected] Contents 1 Preface 2 2 Preliminary Mathematics 2 2.1 Vector Spaces . 3 2.2 Affine Spaces . 6 2.3 Metric Affine Spaces . 15 2.4 Euclidean Geometry . 21 3 Minkowskian Geometry and Its Physical Significance 28 3.1 Minkowskian Geometry { Formal Development . 28 3.2 Minkowskian Geometry { Physical Interpretation . 39 3.3 Uniqueness Result for Minkowskian Angular Measure . 51 3.4 Uniqueness Results for the Relative Simultaneity Relation . 56 4 From Minkowskian Geometry to Hyperbolic Plane Geometry 64 4.1 Tarski's Axioms for first-order Euclidean and Hyperbolic Plane Geometry . 64 4.2 The Hyperboloid Model of Hyperbolic Plane Geometry . 69 These notes have not yet been fully de-bugged. Please read them with caution. (Corrections will be much appreciated.) 1 1 Preface The notes that follow bring together a somewhat unusual collection of topics. In section 3, I discuss the foundations of \special relativity". I emphasize the invariant, \geometrical approach" to the theory, and spend a fair bit of time on one special topic: the status of the relative simultaneity relation within the theory. At issue is whether the standard relation, the one picked out by Einstein's \definition" of simultaneity, is conventional in character, or is rather in some significant sense forced on us. Section 2 is preparatory. When the time comes, I take \Minkowski space- time" to be a four-dimensional affine space endowed with a Lorentzian inner product.