Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions
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Math 651 Homework 1 - Algebras and Groups Due 2/22/2013
Math 651 Homework 1 - Algebras and Groups Due 2/22/2013 1) Consider the Lie Group SU(2), the group of 2 × 2 complex matrices A T with A A = I and det(A) = 1. The underlying set is z −w jzj2 + jwj2 = 1 (1) w z with the standard S3 topology. The usual basis for su(2) is 0 i 0 −1 i 0 X = Y = Z = (2) i 0 1 0 0 −i (which are each i times the Pauli matrices: X = iσx, etc.). a) Show this is the algebra of purely imaginary quaternions, under the commutator bracket. b) Extend X to a left-invariant field and Y to a right-invariant field, and show by computation that the Lie bracket between them is zero. c) Extending X, Y , Z to global left-invariant vector fields, give SU(2) the metric g(X; X) = g(Y; Y ) = g(Z; Z) = 1 and all other inner products zero. Show this is a bi-invariant metric. d) Pick > 0 and set g(X; X) = 2, leaving g(Y; Y ) = g(Z; Z) = 1. Show this is left-invariant but not bi-invariant. p 2) The realification of an n × n complex matrix A + −1B is its assignment it to the 2n × 2n matrix A −B (3) BA Any n × n quaternionic matrix can be written A + Bk where A and B are complex matrices. Its complexification is the 2n × 2n complex matrix A −B (4) B A a) Show that the realification of complex matrices and complexifica- tion of quaternionic matrices are algebra homomorphisms. -
Models of 2-Dimensional Hyperbolic Space and Relations Among Them; Hyperbolic Length, Lines, and Distances
Models of 2-dimensional hyperbolic space and relations among them; Hyperbolic length, lines, and distances Cheng Ka Long, Hui Kam Tong 1155109623, 1155109049 Course Teacher: Prof. Yi-Jen LEE Department of Mathematics, The Chinese University of Hong Kong MATH4900E Presentation 2, 5th October 2020 Outline Upper half-plane Model (Cheng) A Model for the Hyperbolic Plane The Riemann Sphere C Poincar´eDisc Model D (Hui) Basic properties of Poincar´eDisc Model Relation between D and other models Length and distance in the upper half-plane model (Cheng) Path integrals Distance in hyperbolic geometry Measurements in the Poincar´eDisc Model (Hui) M¨obiustransformations of D Hyperbolic length and distance in D Conclusion Boundary, Length, Orientation-preserving isometries, Geodesics and Angles Reference Upper half-plane model H Introduction to Upper half-plane model - continued Hyperbolic geometry Five Postulates of Hyperbolic geometry: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. A circle may be described with any given point as its center and any distance as its radius. 4. All right angles are congruent. 5. For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Some interesting facts about hyperbolic geometry 1. Rectangles don't exist in hyperbolic geometry. 2. In hyperbolic geometry, all triangles have angle sum < π 3. In hyperbolic geometry if two triangles are similar, they are congruent. -
CLIFFORD ALGEBRAS and THEIR REPRESENTATIONS Introduction
CLIFFORD ALGEBRAS AND THEIR REPRESENTATIONS Andrzej Trautman, Uniwersytet Warszawski, Warszawa, Poland Article accepted for publication in the Encyclopedia of Mathematical Physics c 2005 Elsevier Science Ltd ! Introduction Introductory and historical remarks Clifford (1878) introduced his ‘geometric algebras’ as a generalization of Grassmann alge- bras, complex numbers and quaternions. Lipschitz (1886) was the first to define groups constructed from ‘Clifford numbers’ and use them to represent rotations in a Euclidean ´ space. E. Cartan discovered representations of the Lie algebras son(C) and son(R), n > 2, that do not lift to representations of the orthogonal groups. In physics, Clifford algebras and spinors appear for the first time in Pauli’s nonrelativistic theory of the ‘magnetic elec- tron’. Dirac (1928), in his work on the relativistic wave equation of the electron, introduced matrices that provide a representation of the Clifford algebra of Minkowski space. Brauer and Weyl (1935) connected the Clifford and Dirac ideas with Cartan’s spinorial represen- tations of Lie algebras; they found, in any number of dimensions, the spinorial, projective representations of the orthogonal groups. Clifford algebras and spinors are implicit in Euclid’s solution of the Pythagorean equation x2 y2 + z2 = 0 which is equivalent to − y x z p = 2 p q (1) −z y + x q ! " ! " # $ so that x = q2 p2, y = p2 + q2, z = 2pq. If the numbers appearing in (1) are real, then − this equation can be interpreted as providing a representation of a vector (x, y, z) R3, null ∈ with respect to a quadratic form of signature (1, 2), as the ‘square’ of a spinor (p, q) R2. -
On the Complexification of the Classical Geometries And
On the complexication of the classical geometries and exceptional numb ers April Intro duction The classical groups On R Spn R and GLn C app ear as the isometry groups of sp ecial geome tries on a real vector space In fact the orthogonal group On R represents the linear isomorphisms of arealvector space V of dimension nleaving invariant a p ositive denite and symmetric bilinear form g on V The symplectic group Spn R represents the isometry group of a real vector space of dimension n leaving invariant a nondegenerate skewsymmetric bilinear form on V Finally GLn C represents the linear isomorphisms of a real vector space V of dimension nleaving invariant a complex structure on V ie an endomorphism J V V satisfying J These three geometries On R Spn R and GLn C in GLn Rintersect even pairwise in the unitary group Un C Considering now the relativeversions of these geometries on a real manifold of dimension n leads to the notions of a Riemannian manifold an almostsymplectic manifold and an almostcomplex manifold A symplectic manifold however is an almostsymplectic manifold X ie X is a manifold and is a nondegenerate form on X so that the form is closed Similarly an almostcomplex manifold X J is called complex if the torsion tensor N J vanishes These three geometries intersect in the notion of a Kahler manifold In view of the imp ortance of the complexication of the real Lie groupsfor instance in the structure theory and representation theory of semisimple real Lie groupswe consider here the question on the underlying geometrical -
Old-Fashioned Relativity & Relativistic Space-Time Coordinates
Relativistic Coordinates-Classic Approach 4.A.1 Appendix 4.A Relativistic Space-time Coordinates The nature of space-time coordinate transformation will be described here using a fictional spaceship traveling at half the speed of light past two lighthouses. In Fig. 4.A.1 the ship is just passing the Main Lighthouse as it blinks in response to a signal from the North lighthouse located at one light second (about 186,000 miles or EXACTLY 299,792,458 meters) above Main. (Such exactitude is the result of 1970-80 work by Ken Evenson's lab at NIST (National Institute of Standards and Technology in Boulder) and adopted by International Standards Committee in 1984.) Now the speed of light c is a constant by civil law as well as physical law! This came about because time and frequency measurement became so much more precise than distance measurement that it was decided to define the meter in terms of c. Fig. 4.A.1 Ship passing Main Lighthouse as it blinks at t=0. This arrangement is a simplified model for a 1Hz laser resonator. The two lighthouses use each other to maintain a strict one-second time period between blinks. And, strict it must be to do relativistic timing. (Even stricter than NIST is the universal agency BIGANN or Bureau of Intergalactic Aids to Navigation at Night.) The simulations shown here are done using RelativIt. Relativistic Coordinates-Classic Approach 4.A.2 Fig. 4.A.2 Main and North Lighthouses blink each other at precisely t=1. At p recisel y t=1 sec. -
Patterns of Maximally Entangled States Within the Algebra of Biquaternions
J. Phys. Commun. 4 (2020) 055018 https://doi.org/10.1088/2399-6528/ab9506 PAPER Patterns of maximally entangled states within the algebra of OPEN ACCESS biquaternions RECEIVED 21 March 2020 Lidia Obojska REVISED 16 May 2020 Siedlce University of Natural Sciences and Humanities, ul. 3 Maja 54, 08-110 Siedlce, Poland ACCEPTED FOR PUBLICATION E-mail: [email protected] 20 May 2020 Keywords: bipartite entanglement, biquaternions, density matrix, mereology PUBLISHED 28 May 2020 Original content from this Abstract work may be used under This paper proposes a description of maximally entangled bipartite states within the algebra of the terms of the Creative Commons Attribution 4.0 biquaternions–Ä . We assume that a bipartite entanglement is created in a process of splitting licence. one particle into a pair of two indiscernible, yet not identical particles. As a result, we describe an Any further distribution of this work must maintain entangled correlation by the use of a division relation and obtain twelve forms of biquaternions, attribution to the author(s) and the title of representing pure maximally entangled states. Additionally, we obtain other patterns, describing the work, journal citation mixed entangled states. Finally, we show that there are no other maximally entangled states in Ä and DOI. than those presented in this work. 1. Introduction In 1935, Einstein, Podolski and Rosen described a strange phenomenon, in which in its pure state, one particle behaves like a pair of two indistinguishable, yet not identical, particles [1]. This phenomenon, called by Einstein ’a spooky action at a distance’, has been investigated by many authors, who tried to explain this strange correlation [2–9]. -
Majorana Spinors
MAJORANA SPINORS JOSE´ FIGUEROA-O'FARRILL Contents 1. Complex, real and quaternionic representations 2 2. Some basis-dependent formulae 5 3. Clifford algebras and their spinors 6 4. Complex Clifford algebras and the Majorana condition 10 5. Examples 13 One dimension 13 Two dimensions 13 Three dimensions 14 Four dimensions 14 Six dimensions 15 Ten dimensions 16 Eleven dimensions 16 Twelve dimensions 16 ...and back! 16 Summary 19 References 19 These notes arose as an attempt to conceptualise the `symplectic Majorana{Weyl condition' in 5+1 dimensions; but have turned into a general discussion of spinors. Spinors play a crucial role in supersymmetry. Part of their versatility is that they come in many guises: `Dirac', `Majorana', `Weyl', `Majorana{Weyl', `symplectic Majorana', `symplectic Majorana{Weyl', and their `pseudo' counterparts. The tra- ditional physics approach to this topic is a mixed bag of tricks using disparate aspects of representation theory of finite groups. In these notes we will attempt to provide a uniform treatment based on the classification of Clifford algebras, a work dating back to the early 60s and all but ignored by the theoretical physics com- munity. Recent developments in superstring theory have made us re-examine the conditions for the existence of different kinds of spinors in spacetimes of arbitrary signature, and we believe that a discussion of this more uniform approach is timely and could be useful to the student meeting this topic for the first time or to the practitioner who has difficulty remembering the answer to questions like \when do symplectic Majorana{Weyl spinors exist?" The notes are organised as follows. -
1 Euclidean Geometry
Dr. Anna Felikson, Durham University Geometry, 26.02.2015 Outline 1 Euclidean geometry 1.1 Isometry group of Euclidean plane, Isom(E2). A distance on a space X is a function d : X × X ! R,(A; B) 7! d(A; B) for A; B 2 X satisfying 1. d(A; B) ≥ 0 (d(A; B) = 0 , A = B); 2. d(A; B) = d(B; A); 3. d(A; C) ≤ d(A; B) + d(B; C) (triangle inequality). We will use two models of Euclidean plane: p 2 2 a Cartesian plane: f(x; y) j x; y 2 Rg with the distance d(A1;A2) = (x1 − x2) + (y1 − y2) ; a Gaussian plane: fz j z 2 Cg, with the distance d(u; v) = ju − vj. Definition 1.1. A Euclidean isometry is a distance-preserving transformation of E2, i.e. a map f : E2 ! E2 satisfying d(f(A); f(B)) = d(A; B). Corollary 1.2. (a) Every isometry of E2 is a one-to-one map. (b) The composition of any two isometries is an isometry. (c) Isometries of E2 form a group (denoted Isom(E2)). Example 1.3: Translation, rotation, reflection in a line are isometries. Theorem 1.4. Let ABC and A0B0C0 be two congruent triangles. Then there exists a unique isometry sending A to A0, B to B0 and C to C0. Corollary 1.5. Every isometry of E2 is a composition of at most 3 reflections. (In particular, the group Isom(E2) is generated by reflections). Remark: the way to write an isometry as a composition of reflection is not unique. -
Catoni F., Et Al. the Mathematics of Minkowski Space-Time.. With
Frontiers in Mathematics Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta, USA) Benoît Perthame (Ecole Normale Supérieure, Paris, France) Laurent Saloff-Coste (Cornell University, Rhodes Hall, USA) Igor Shparlinski (Macquarie University, New South Wales, Australia) Wolfgang Sprössig (TU Bergakademie, Freiberg, Germany) Cédric Villani (Ecole Normale Supérieure, Lyon, France) Francesco Catoni Dino Boccaletti Roberto Cannata Vincenzo Catoni Enrico Nichelatti Paolo Zampetti The Mathematics of Minkowski Space-Time With an Introduction to Commutative Hypercomplex Numbers Birkhäuser Verlag Basel . Boston . Berlin $XWKRUV )UDQFHVFR&DWRQL LQFHQ]R&DWRQL LDHJOLD LDHJOLD 5RPD 5RPD Italy Italy HPDLOYMQFHQ]R#\DKRRLW Dino Boccaletti 'LSDUWLPHQWRGL0DWHPDWLFD (QULFR1LFKHODWWL QLYHUVLWjGL5RPD²/D6DSLHQ]D³ (1($&5&DVDFFLD 3LD]]DOH$OGR0RUR LD$QJXLOODUHVH 5RPD 5RPD Italy Italy HPDLOERFFDOHWWL#XQLURPDLW HPDLOQLFKHODWWL#FDVDFFLDHQHDLW 5REHUWR&DQQDWD 3DROR=DPSHWWL (1($&5&DVDFFLD (1($&5&DVDFFLD LD$QJXLOODUHVH LD$QJXLOODUHVH 5RPD 5RPD Italy Italy HPDLOFDQQDWD#FDVDFFLDHQHDLW HPDLO]DPSHWWL#FDVDFFLDHQHDLW 0DWKHPDWLFDO6XEMHFW&ODVVL½FDWLRQ*(*4$$ % /LEUDU\RI&RQJUHVV&RQWURO1XPEHU Bibliographic information published by Die Deutsche Bibliothek 'LH'HXWVFKH%LEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD½H GHWDLOHGELEOLRJUDSKLFGDWDLVDYDLODEOHLQWKH,QWHUQHWDWKWWSGQEGGEGH! ,6%1%LUNKlXVHUHUODJ$*%DVHOÀ%RVWRQÀ% HUOLQ 7KLVZRUNLVVXEMHFWWRFRS\ULJKW$OOULJKWVDUHUHVHUYHGZKHWKHUWKHZKROHRUSDUWRIWKH PDWHULDOLVFRQFHUQHGVSHFL½FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRILOOXVWUD -
Maxwell's Equations
Maxwell’s equations Daniel Henry Gottlieb August 1, 2004 Abstract We express Maxwell’s equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current, and then the divergence of a special matrix to obtain the Electromagnetic field. These two equations give rise to a remarkable dual set of equations in which the operators become the matrices and the vectors become the fields. The decoupling of the equations into the wave equation is very simple and natural. The divergence of the stress energy tensor gives the Lorentz Law in a very natural way. We compare this approach to the related descriptions of Maxwell’s equations by biquaternions and Clifford algebras. 1 Introduction Maxwell’s equations have been expressed in many forms in the century and a half since their dis- covery. The original equations were 16 in number. The vector forms, written below, consist of 4 equations. The differential form versions consists of two equations; see [Misner, Thorne and Wheeler(1973); see equations 4.10, 4.11]. See also [Steven Parrott(1987) page 98 -100 ] The ap- plication of quaternions, and their complexification, the biquaternions, results in a version of one equation. William E. Baylis (1999) equation 3.8, is an example. In this work, we obtain one Maxwell equation, (10), representing the electromagnetic field as a matrix and the divergence as a vector multiplying the field matrix. But we also obtain a remarkable dual formulation of Maxwell’s equation, (15), wherein the operator is now the matrix and the field is now the vector. -
Hyperbolic Geometry on a Hyperboloid, the American Mathematical Monthly, 100:5, 442-455, DOI: 10.1080/00029890.1993.11990430
William F. Reynolds (1993) Hyperbolic Geometry on a Hyperboloid, The American Mathematical Monthly, 100:5, 442-455, DOI: 10.1080/00029890.1993.11990430. (c) 1993 The American Mathematical Monthly 1985 Mathematics Subject Classification 51 M 10 HYPERBOLIC GEOMETRY ON A HYPERBOLOID William F. Reynolds Department of Mathematics, Tufts University, Medford, MA 02155 E-mail: [email protected] 1. Introduction. Hardly anyone would maintain that it is better to begin to learn geography from flat maps than from a globe. But almost all introductions to hyperbolic non-Euclidean geometry, except [6], present plane models, such as the projective and conformal disk models, without even mentioning that there exists a model that has the same relation to plane models that a globe has to flat maps. This model, which is on one sheet of a hyperboloid of two sheets in Minkowski 3-space and which I shall call H2, is over a hundred years old; Killing and Poincar´eboth described it in the 1880's (see Section 14). It is used by differential geometers [29, p. 4] and physicists (see [21, pp. 724-725] and [23, p. 113]). Nevertheless it is not nearly so well known as it should be, probably because, like a globe, it requires three dimensions. The main advantages of this model are its naturalness and its symmetry. Being embeddable (distance function and all) in flat space-time, it is close to our picture of physical reality, and all its points are treated alike in this embedding. Once the strangeness of the Minkowski metric is accepted, it has the familiar geometry of a sphere in Euclidean 3-space E3 as a guide to definitions and arguments. -
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry.