DOCSLIB.ORG

Geometric Algebra Techniques

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. One basic results of elementary relativity theory is that any proper Lorentz isometry that preserves the temporal orientation is, up to a roation of the x; y; z axis and a rotation of the x 0; y 0; z 0 axis, a Lorentz boost. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 4 / 47 Introduction Definitions, notations and conventions Historically, the spacetime algebra was the first modern implementation of geometric algebra. This is because it provides a sythetic framework for studying spacetime physics. Doran-Lasenby-2003, Ch. 5. A Lorentzian spacetime is a real quadratic space (E; η) of signature (1; 3). We express this by writing E1;3 instead of E. We also write a b to mean η(a; b). The elements a E are called events. We will use· the customary terms time-like, space-like2 and light-light to refer to vectors such that η(a; a) is positive, negative or null, respectively. An (inertial) frame of E1;3 is an orthonormal basis γ0; γ1; γ2; γ3: γ0 γ0 = 1; γ0 γj = 0; γj γk = δj;k (j; k 1; 2; 3 : · · · − 2 f g Or, in the familiar relativistic notations, γµ γν = ηµν (µ, ν 0; 1; 2; 3 ): · 2 f g S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 5 / 47 Introduction Definitions, notations and conventions Instead of the indices 0; 1; 2; 3 , we will also use the labels t; x; y; z . The componentsf ofg an event a in the frame γ are f gµ µ denoted a , and so a = a γµ (Einstein's summation convention). Instead of a0 = at we often write t (actually ct, but we can choose k units for which c = 1), so that a = tγ0 + a γk . Remark. In Dirac's theory, the γµ are certain 4 4 matrices (Dirac matrices), but here they are vectors. The Dirac× matrices yield (see next slide) a representation of the space-time algebra = 1;3 = ΛηE1;3, and so is Dirac's algebra without matrices. The beautyD G and usefulness of Dwill be apparent along the way, much as it happened with the treatmentD of quaternions by geometric algebra. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 6 / 47 Introduction The Dirac representation of D Proposition. Define σ0 σk Γ0 = and Γk = − , k = 1; 2; 3. σ0 σk − Then there exists an algebra isomorphism R(4) such that G! γµ Γµ. 7! Proof. The Γµ satisfy the Clifford relations ΓµΓν + ΓνΓµ = 2ηµνI4. This follows from the Clifford relations σj σk + σk σj = 2δjk satisfied by the Pauli matrices σ1; σ2; σ3 and straightfoward matrix computations. So there is an algebra homomorphism R(4) such G! that γµ Γµ. Finally, this homomorphism is an isomorphism: the 7! images ΓI of the γI (I running over the multiindeces of 0; 1; 2; 3) turn out to be lineraly independent. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 7 / 47 Lorentzian GA D = G1;3 = Λη E In terms of the frame γ0; γ1; γ2; γ3, the basic rule in is D γµγν + γνγµ = 2ηµν: Let i be the pseudo-scalar unit associated to that frame: i = γ0γ1γ2γ3: By GAT-03.24, Theorem 5, i anticommutes with vectors and trivectors and commutes with scalars, bivectors and pseudo-scalars. In a compact form, xi = ( 1)r ix for x r . Moreover, we have (see also the Remark after its proof):− 2 G Proposition 1) i2 = Q(i) = det(diag(+; ; ; )) = 1: 2) The Hodge duality map −r − − 4−r , x− x ∗ = xi = ( 1)r ix is an antiisometry for r = 0D; 1;!D2; 3. This implies7! that the− signatures of 2, 3 and 4 are ( 3; 3), (3; 1) and (0; 1) = 1.¯ D D D − S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 8 / 47 Lorentzian GA D = G1;3 = Λη E Let σ1 = γ1γ0 = γ10, σ2 = γ2γ0 = γ20, σ3 = γ3γ0 = γ30. Then a ∗ short computation shows that σj = σj i = γk γl = γkl (j; k; l a cyclic permutation of (1; 2; 3)).1 Explicitely,− − ∗ ∗ ∗ σ = γ23; σ = γ31; σ = γ12: 1 − 2 − 3 − ∗ The σj and σj have signatures 1 and +1, respectively, and together form a basis of 2. − D ∗ 3 The γµ = γµi form a basis of and they have signatures 1 for µ = 0 and +1 otherwise. NoteD that − ∗ ∗ ∗ ∗ γ0 = γ123; γ1 = γ023; γ2 = γ031; γ3 = γ012: With these notations, we finally have i = γ0123: 1 ∗ This is often condensed as σj = jkl γk γl , where jkl denotes the sign of the permutation jkl of 123 (Levi-Civita symbol).− S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 9 / 47 Lorentzian GA D = G1;3 = Λη E The preceding facts are summarized in the following table: 1 γ0 γ1 γ2 γ3 x∗ = x i σ1 σ2 σ3 σ1∗ σ2∗ σ3∗ Q(x)=+1 γ0∗ γ1∗ γ2∗ γ3∗ Q(x)= 1 − 1∗ S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 10 / 47 Lorentzian GA D = G1;3 = Λη E ∗ ∗ ∗ ∗ ∗ ∗ ∗ G γ0 γ1 γ2 γ3 σ1 σ2 σ3 σ1 σ2 σ3 γ0 γ1 γ2 γ3 i ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ0 1σ ¯1 σ¯2 σ¯3 γ¯1 γ¯2 γ¯3 γ¯1 γ¯2 γ¯3 i σ¯1 σ¯2 σ¯3 γ0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ1 σ1 -1σ ¯3 σ2 γ¯0 γ3 γ¯2 γ¯0 γ¯3 γ2 σ1 -i σ3 σ¯2 γ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ2 σ2 σ3 -1σ ¯1 γ¯3 γ¯0 γ1 γ3 γ¯0 γ¯1 σ2 σ¯3 -i σ1 γ2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ3 σ3 σ¯2 σ1 -1 γ2 γ¯1 γ¯0 γ¯2 γ1 γ¯0 σ3 σ2 σ¯1 -i γ3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ1 γ1 γ0 γ¯3 γ2 1 σ3 σ¯2 i σ¯3 σ2 γ1 γ0 γ3 γ¯2 σ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ2 γ2 γ3 γ0 γ¯1 σ¯3 1 σ1 σ3 i σ¯1 γ2 γ¯3 γ0 γ1 σ2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ3 γ3 γ¯2 γ1 γ0 σ2 σ¯1 1σ ¯2 σ1 i γ3 γ2 γ¯1 γ0 σ3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ1 γ¯1 γ¯0 γ¯3 γ2 i σ¯3 σ2 -1σ ¯3 σ2 γ1 γ0 γ¯3 γ2 σ¯1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ2 γ¯2 γ3 γ¯0 γ¯1 σ3 i σ¯1 σ3 -1σ ¯1 γ2 γ3 γ0 γ¯1 σ¯2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ3 γ¯3 γ¯2 γ1 γ¯0 σ¯2 σ1 i σ¯2 σ1 -1 γ3 γ¯2 γ1 γ0 σ¯3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ0 -i σ1 σ2 σ3 γ¯1 γ¯2 γ¯3 γ1 γ2 γ3 1σ ¯1 σ¯2 σ¯3 γ¯0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ1 σ¯1 i σ¯3 σ2 γ¯0 γ¯3 γ2 γ0 γ¯3 γ2 σ1 -1σ ¯3 σ2 γ¯1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ2 σ¯2 σ3 i σ¯1 γ3 γ¯0 γ¯1 γ3 γ0 γ¯1 σ2 σ3 -1σ ¯1 γ¯2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ3 σ¯3 σ¯2 σ1 i γ¯2 γ1 γ¯0 γ¯2 γ1 γ0 σ3 σ¯2 σ1 -1 γ¯3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ i γ¯0 γ¯1 γ¯2 γ¯3 σ1 σ2 σ3 σ¯1 σ¯2 σ¯3 γ0 γ1 γ2 γ3 -1 This is the multiplication table of in terms of the described basis. The bar over D symbols indicates minus sign, not Clifford conjugation. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 11 / 47 Lorentzian GA D = G1;3 = Λη E A product γI γJ is determined by the rule GAT-03.25, Corollary 1. In this case it yields γK , K = I M J (symmetric difference) and 1 = ( 1)ν, where± ν is the sum of the number of elements in ±1; 2; 3− I J and the number of inversions in the sequence I ; J. f g \ \ By GAT-03.25, Corollary 2, the table is symmetric up to sign, c rs because γJ γI = ( 1) ( 1) γI γJ , where r = I , s = J and − − j j j j c = I J .
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • The Group Sk 2 of a Biquaternion Algebra
    THE GROUP SK2 OF A BIQUATERNION ALGEBRA BAPTISTE CALMES` Abstract. M. Rost has proved the existence of an exact sequence relating the group SK1 - kernel of the reduced norm for K1 - of a biquaternion algebra D whose center is a field F with Galois cohomology groups of F . In this paper, we relate the group SK2 - kernel of the reduced norm for K2 - of D with Galois cohomology of F through an exact sequence. Introduction To understand the K-theory of central simple algebras, one of the most useful tools is the reduced norm. It is defined for K0, K1 and K2. It has been proved (see [14, Proposition 4]) that it cannot be defined for K3 and satisfy reasonable properties. The definition of the reduced norm is trivial for K0, elementary for K1, but much less elementary for K2. A definition for K2 was given by Suslin in [26, Corollary 5.7], which uses the highly non trivial result that the K-cohomology 0 group H (X, K2) is isomorphic to K2F when X is a complete smooth rational va- riety over the field F (see [26, Corollary 5.6]). The kernel of the reduced norm for Ki, i = 0, 1, 2, is denoted by SKi and is difficult to compute for i = 1, 2 (it is always zero for i = 0). The first result on SK1 was obtained by Wang in 1949. He proved in [34] that SK1A is zero when the index of A is a product of different prime num- bers. Whether SK1 was always zero or not was then known as the Tannaka-Artin problem.
    [Show full text]
  • The Descent of Biquaternion Algebras in Characteristic Two
    The descent of biquaternion algebras in characteristic two Demba Barrya,b, Adam Chapmanc, Ahmed Laghribid aFacult´edes Sciences et Techniques de Bamako, BP: E3206 Bamako, Mali bDepartement Wiskunde–Informatica, Universiteit Antwerpen, Belgium cDepartment of Computer Science, Tel-Hai College, Upper Galilee, 12208 Israel dUniversit´ed’Artois, Facult´edes Sciences Jean Perrin, Laboratoire de math´ematiquesde Lens EA 2462, rue Jean Souvraz - SP18, 62307 Lens, France Abstract In this paper we associate an invariant to a biquaternion algebra B over a field K with a subfield F such that K=F is a quadratic separable extension and char(F) = 2. We show that this invariant is trivial exactly when B B0 ⊗ K for some biquaternion algebra B0 over F. We also study the behavior of this invariant under certain field extensions and provide several interesting examples. Keywords: Kato-Milne Cohomology, Cohomological invariants, Algebras with involution, Biquaternion algebras 2010 MSC: primary 11E81; secondary 11E04, 16K20, 19D45 1. Introduction Given a central simple algebra C over a field K with a subfield F, we say that C has a descent to F if there exists a central simple algebra C0 over F such that C C0 ⊗ K. When [K : F] = exp(C), one necessary condition for C to have a descent to F is that corK=F(C) ∼Br F, because if C C0⊗K ⊗ ⊗[K:F] ∼ ∼ then corK=F(C) = corK=F(C0 K) = C0 Br F, where Br denotes the Brauer equivalence. If K=F is a separable quadratic extension and Q is a quaternion algebra over K, then corK=F(Q) ∼Br F is a necessary and sufficient condition for Q to have a descent to F by [1, Chapter X, Theorem 21].
    [Show full text]
  • The Four Dimensional Dirac Equation in Five Dimensions
    The Four Dimensional Dirac Equation in Five Dimensions Romulus Breban Institut Pasteur, Paris, France Abstract The Dirac equation may be thought as originating from a theory of five-dimensional (5D) space- time. We define a special 5D Clifford algebra and introduce a spin-1/2 constraint equation to describe null propagation in a 5D space-time manifold. We explain how the 5D null formalism breaks down to four dimensions to recover two single-particle theories. Namely, we obtain Dirac’s relativistic quantum mechanics and a formulation of statistical mechanics. Exploring the non-relativistic limit in five and four dimensions, we identify a new spin-electric interaction with possible applications to magnetic resonance spectroscopy (within quantum mechanics) and superconductivity (within statistical mechanics). 1 Introduction The Dirac equation [1, 2] established itself as a fundamental tool in four dimensional (4D) physics. With increasing popularity of 5D physics, many authors generalized it from four to five dimensions [3–12]. A different approach was that of Barut [13] who recognized that the Dirac matrices αj (j, k, ... =1, 2, 3) and β close under commutation to provide a representation of so(1,4), thus establishing an intrinsic 5D symmetry of the Dirac equation. Later, Bracken and Cohen [14] introduced an algebraic formalism, for the Dirac equation, based on so(1,4). In this paper, we present a curved geometry for 5D space-times compatible to the Dirac equation, where the electromagnetic field enters according to the minimal coupling recipe. Furthermore, we reconsider the non-relativistic limit of the Dirac equation using 5D-based approximations.
    [Show full text]
  • Determination of the Biquaternion Divisors of Zero, Including
    Determination of the biquaternion divisors of zero, including the idempotents and nilpotents Stephen J. Sangwine∗ Daniel Alfsmann†† November 7, 2018 Abstract The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the present paper. Using several different representations for biquaternions, we present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares that vanishes, and that this condition is equivalent to two conditions on the inner product of the real and imaginary parts of the biquaternion, and the equality of the norms of the real and imaginary parts. We give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement about the composition of the set of biquaternion divisors of zero, and its subsets, the idempotents and the nilpotents. 1 Introduction The complexified quaternions, or biquaternions1, were discovered by Hamilton himself and published within ten years of the discovery of quaternions [1]. Hamilton discovered that a subset of the biquater- nions were divisors of zero [2, Lecture VII, 672, p 669]2.
    [Show full text]
  • Robust Tracking of Dynamic Targets with Aerial Vehicles Using Quaternion-Based Techniques Hernan Abaunza Gonzalez
    Robust tracking of dynamic targets with aerial vehicles using quaternion-based techniques Hernan Abaunza Gonzalez To cite this version: Hernan Abaunza Gonzalez. Robust tracking of dynamic targets with aerial vehicles using quaternion- based techniques. Automatic. Université de Technologie de Compiègne, 2019. English. NNT : 2019COMP2480. tel-02155857 HAL Id: tel-02155857 https://tel.archives-ouvertes.fr/tel-02155857 Submitted on 14 Jun 2019 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. Par Hernán ABAUNZA GONZALEZ Robust tracking of dynamic targets with aerial vehicles using quaternion-based techniques Thèse présentée pour l’obtention du grade de Docteur de l’UTC Soutenue le 26 avril 2019 Spécialité : Automatique et Robotique : Unité de recherche Heudyasic (UMR-7253) D2480 SORBONNE UNIVERSITES, UNIVERSITE DE TECHNOLOGIE DE COMPIEGNE HEUDIASYC, UMR 7253 CNRS Ecole Doctorale “Sciences Pour l’Ingénieur” ROBUST TRACKING OF DYNAMIC TARGETS WITH AERIAL VEHICLES USING QUATERNION‐BASED TECHNIQUES T H E S I S presented by Hernan ABAUNZA GONZALEZ´ to obtain
    [Show full text]
  • Differential Geometry Revisited by Biquaternion Clifford Algebra Patrick Girard, Patrick Clarysse, Romaric Pujol, Liang Wang, Philippe Delachartre
    Differential Geometry Revisited by Biquaternion Clifford Algebra Patrick Girard, Patrick Clarysse, Romaric Pujol, Liang Wang, Philippe Delachartre To cite this version: Patrick Girard, Patrick Clarysse, Romaric Pujol, Liang Wang, Philippe Delachartre. Differential Ge- ometry Revisited by Biquaternion Clifford Algebra. Lecture Notes in Computer Science - Curves and Surfaces, 9213, pp.216-242, 2015, 978-3-319-22804-4. 10.1007/978-3-319-22804-4_17. hal-01622682 HAL Id: hal-01622682 https://hal.archives-ouvertes.fr/hal-01622682 Submitted on 24 Oct 2017 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. DIFFERENTIAL GEOMETRY REVISITED BY BIQUATERNION CLIFFORD ALGEBRA Patrick R. Girard1, Patrick Clarysse1, Romaric Pujol2, Liang Wang1, and Philippe Delachartre1 1 Universit´ede Lyon, CREATIS; CNRS UMR 5220; Inserm U1044; INSA-Lyon; Universit´eLYON 1, France, B^at.Blaise Pascal, 7 avenue Jean Capelle, F-69621 Villeurbanne, France 2 Universit´ede Lyon, P^olede Math´ematiques,INSA-Lyon, B^at.L´eonardde Vinci, 21 avenue Jean Capelle, F-69621 Villeurbanne, France Abstract. In the last century, differential geometry has been expressed within various calculi: vectors, tensors, spinors, exterior differential forms and recently Clifford algebras. Clifford algebras yield an excellent repre- sentation of the rotation group and of the Lorentz group which are the cornerstones of the theory of moving frames.
    [Show full text]
  • Neutrosophic Physics: More Problems, More Solutions
    University of New Mexico UNM Digital Repository Faculty and Staff Publications Mathematics Spring 5-2010 Neutrosophic Physics: More Problems, More Solutions Florentin Smarandache University of New Mexico, [email protected] Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Applied Mathematics Commons, Astrophysics and Astronomy Commons, and the Mathematics Commons Recommended Citation Florentin Smarandache (ed.), Neutrosophic Physics: More Problems, More Solutions. Hanko: North- European Scientific Publishers, 2010 This Book is brought to you for free and open access by the Mathematics at UNM Digital Repository. It has been accepted for inclusion in Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. Neutrosophic Physics - Cover - 2.15.10:Layout 1 2/16/10 3:29 PM Page 1 In this concern, neutrosophic logics and neutrosophy in general, Neutrosophic Physics: established by Prof. Smarandache, is one of the promising research Neutrosophic Physics: More Problems, More Solutions (Collected Papers) Editor: Prof. Florentin Smarandache More Problems, More Solutions instruments, which could be successfully applied by a theoretical physicist. Collected Papers Naturally, neutrosophic logics, being a part of modern logics, states that neutralities may be between any physical states, or states of space-time. In particular, this leads, sometimes, to paradoxist situations, when two opposite states are known in physics, while the neutral state between them seems absolutely impossible from a physical viewpoint! Meanwhile, when considering the theoretically possible neutralities in detail, we see that these neutral states indicate new phenomena which were just discovered by the experimentalists in the last decade, or shows a new field for further experimental studies, as for example unmatter which is a state between matter and antimatter.
    [Show full text]