Quaternions, Octonions, and the Cauchy-Riemann Equations in Higher Dimensions

Transcription

1 Quaternions, Octonions, and the Cauchy-Riemann Equations in Higher Dimensions James Emery version 5/31/2018 Contents 1 This Document gives some definitions, Simple Properties, and a Bibliography Concerning the Title Subjects. 2 2 The Quaternions 2 3 The Quaternion Group 4 4 Matrix Representation of Quaternions 5 5 The Quaternion Basis Vectors are Linearly Independent 6 6 The Use of Quaternions for Representing Rotations 7 7 Octonions 9 8 Bibliography 10 9 Index 13 1

2 1 This Document gives some definitions, Simple Properties, and a Bibliography Concerning the Title Subjects. I think to make progress in using the paper: Stiefel E., On Cauchy-Riemann Equations in Higher Dimensions, Journal of Research of the National Bureau of Standards, Vol. 48, No. 5, May 1952, Research Paper 2328, for whatever purpose, one must make a rather intense study of several items listed in this bibliography, as well as others. It would be very useful to find out if this paper made a significant contribution to follow on papers that have lead to significant new results and contributions to knowledge either in applied or theoretical fields. If so what are these significant contributions? Because this paper was written so long ago, if no important knowledge has flowed, perhaps the paper is not really very important and thus perhaps extensive new work should not be undertaken. On the other hand if such follow on papers can be located, they would probably serve to clarify this rather obscure and opaque publication. For my part these areas are not of great interest to me, and so I do not plan to spend much, or any more, time on them, because it would take much effort, and take away from practical things in my life that I must do, as well as time I plan to devote to other studies, and interests. I had planned to locate the German references in the above paper and translate these to English. I did find them at Linda Hall Library and started to translate the paper by Adolph Hurwitz. However, I soon realized the limitations of my knowledge of german, and conseqently the great amount of time it would take for me to complete this translation, and so I have given up on it at this time. Please take a look at the bibliography. 2 The Quaternions The quaternions form a four dimensional vector space over the reals, with basis vectors 1,i,j,k. Thus a general element is x = x 0 +x 1 i+x 2 j +x 3 k, 2

3 where x 0,x 1,x 2,x 3 are real numbers. Multiplication is defined by i 2 = 1,j 2 = 1,k 2 = 1,ij = ji = k,jk = kj = i,ki = ik = j If α,β are real numbers, and x,y are quaternions, then define So if and then Define the norm of x as (αx)(βy) = (αβ)(xy). x = x 0 +x 1 i+x 2 j +x 3 k, y = y 0 +y 1 i+y 2 j +y 3 k, xy = x 0 y 0 x 1 y 1 x 2 y 2 x 3 y 3 +(x 0 y 1 +x 1 y 0 +x 2 y 3 x 3 y 2 )i +(x 0 y 2 +x 2 y 0 +x 3 y 1 x 1 y 3 )j +(x 0 y 3 +x 3 y 0 +x 1 y 2 x 2 y 1 )k x = Define the conjugate of x as Then So x 2 0 +x 2 1 +x 2 2 +x 2 3 x = x 0 x 1 i x 2 j x 3 k. xx = x x = x 2. x x 2 is the inverse of x, if x is not zero. Thus every nonzero element has an inverse, that is every nonzero element in the ring is a unit, so the quaternions form a division ring. However it is not a field because multiplication is not commutative, that is in general xy yx. Because it is also a vector space it is an algebra, and hence a division algebra. Complex numbers form a two dimensional vector space over the reals and can be used as vectors in physics for two dimensional problems. Hamilton 3

4 invented quaternions to extend complex numbers to three dimensions to be used for three dimensional vector analysis in physics. They were used widely for this purpose in the nineteenth century until J. Willard Gibbs introduced his simplified vector analysis. Notice that Gibbs took from quaternions the unit i, j, k vectors. The two dimensional subspace of the quaternions generated by bases vectors 1,i is the complex numbers. A pure quaternion is one of the form x = x 1 i+x 2 j +x 3 k. If y also is a pure quaternion, then one finds that the product is xy = x y x y, that is the cross product (outer product, vector product) minus the dot product (inner product). So we can begin to see the connection between Hamilton s quaternions and Gibb s vector analysis. The quaternions are associative, but the vector product is not. For example (i i) j = 0, but i (i j) = i k = j. 3 The Quaternion Group The elements{1, i, j, k, 1, i, j, k}, under quaternion multiplication, form a non-abelian group of order eight. The pairs of products selected from the three quaternions i,j,k behave like the corresponding i,j,k unit vectors of vector analysis under the cross product. And ii = 1, jj = 1, kk = 1. And i is the inverse of i, j is the inverse of j, and k is the inverse of k. Thus the quaternions are not communitive, although they are associative. 4

5 The Quaternion Multiplicative Group Table 1 i j k 1 i j k 1 1 i j k 1 i j k i i 1 k j i 1 k j j j k 1 i j k 1 i k k j i 1 k j i i j k 1 i j k i i 1 k j i 1 k j j j k 1 i j k 1 i k k j i 1 k j i 1 4 Matrix Representation of Quaternions The four basis vectors 1,i,j,k can be represented by 2 2 complex matrices, or by 4 4 real matrices, which gives a way of computing with quaternions. See 1 Hungerford Thomas H., Abstract Algebra: An Introduction, 2nd edition, Saunders College Publishing, Page 54, Exercise = i = i 0 0 i j = k = The set H of real quaternions is 0 i i 0 5

6 a1+bi+cj+dk, where a,b,c,d are real numbers. Then each element of H is of the form a+bi c+di c+di a di Exercise. Solve the problems (a),(b),(c),(d) on page 55 of Hungerford. It is easy to verify by doing some multiplications that these four matrices satisfy the defined properties of the quaternions. Here is a script used to load the basis into octave.. % name of script is c:\je\om\quat.m, called an m script % The percentage sign is a comment % quaternion matrices as defined in octave by % 1=q1, i=qi, j= qj, k=qk % q1=1 0; 0 1; q1 qi=i 0; 0 -i; qi qj=0 1 ; -1 0; qj qk=0 i; i 0; qk % First we change the working directory to c:\je\om\quat.m by typing cd \je\om % Typing quat without the extension loads the script into octave. % The m scripts (octave scripts) are located in local directory \je\om % The directory for matlab m scripts is \je\m % These two different directories are used because there is some incompatibility % between matlab and octave scripts, because there can be differences, but not in simple % computations. % Once the quat.m script is loaded into octave, we can proceed to compute with % these basis elements, namely the complex matrix representations of the % quaternions 1,i,j,k 5 The Quaternion Basis Vectors are Linearly Independent Let x,y,z,w satisfy 1 0 x +y 0 1 i 0 0 i +z w 0 i i 0 = 0

7 Then the following four equations must be satisfied, x+yi = 0, z +wi = 0, z +wi = 0, x yi = 0. These have the unique solution x = y = z = w = 0, so the four vectors 1, i, j, k are linearly independent. These four vectors span the quaternions by definition, and so are a basis of the vector space over the real numbers. 6 The Use of Quaternions for Representing Rotations Let x be a unit quaternion, meaning its norm equals 1. We claim that x represents a rotation. Let Let the vector part be then is a unit vector and Then So for some angle θ and So x = x 0 +x 1 i+x 2 j +x 3 k y = x 1 i+x 2 j +x 3 k u = y y x = x 0 + y u 1 = x 2 = x y 2 u 2 = x y 2 x 0 = cos(θ) y = sin(θ). x = cos(θ)+sin(θ)u 7

8 Then given a vector v, the vector v = xvx 1 = xvx, where the multiplication is quaternion multiplication, is the vector obtained by rotating vector v about the u axis by angle 2θ (proof omitted). Conversely, given a rotation axis specified by a unit vector u, and a rotation about this axis by angle 2θ, then the rotation is given by the quaternion x = cos(θ)+sin(θ)u. Rotations specified by matrices are 3 by 3 orthogonal matrices, that is the column vectors are unit vectors and the columns are orthogonal to each other. One can compute the rotation matrix from the quaternion, or the quaternion from the matrix. The quaternion is specified with just four numbers, and a nine number matrix need not be specified to do calculations. 8

9 7 Octonions Theoctonionswere discovered in1843by JohnT. Graves, a friendofwilliam Hamilton, who called them octaves. They were discovered independently by Arthur Cayley (1845). They are sometimes referred to as Cayley numbers or the Cayley algebra. Octonions are not communitive or associative. The octonions are an eight dimensional vector space, with basis vectors 1,i,j,k,l,il,jl,kl with multiplication table: 1 i j k l il jl kl i 1 k j il l kl jl j k 1 i jl kl l il k j i 1 kl jl il l l il jl kl 1 i j k il l kl jl i 1 k j jl kl l il j k 1 i kl jl il l k j i 1 The octonions are the largest of the four normed division algebras, which are, the Real Numbers, the Complex Numbers, the Quaternions, and the Octonions. The Octonians bare a relation to the spinors of quantum mechanics and Pauli s spin matrices. See Penrose in the bibliography. 9

10 8 Bibliography 1 Altmann Simon L., Rotations, Quaternions, and Double Groups, clarendon press, oxford, 1986, (linda hall QC R65A48). 2 Baez, John, The Octonions, 2002, Bulletin of the American Mathematical Society 39: pp 145 to Online HTML versions at Baez s site or see lanl.arxiv.org copy. Theorem 1., p. 150, The real numbers, the complex numbers, the quaternions, and the octonions are the only normed division algebras. See the 1898 paper by Adolf Hurwitz ( ). For a modern proof see Shafer. 3 Birkhoff Garret, Mac Lane Saunders, (Maclane), A Survey of Modern Algebra, 1941, revised edition 1953, Macmillan. 4 Birkhoff Garret, Mac Lane Saunders, (Maclane), A Survey of Modern Algebra, 4th edition, 1977, Macmillan. 5 Cayley, Arthur, On Jacobi s Elliptic Functions and on quaternions, (in reply to... ), Philos. Mag. 26: , reprinted in The Collected Mathematical Papers, 1885 Johnson Reprint Co., New York, 1963, p Conway, John Horton, Smith, Derek A., On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, 2003, A. K. Peters, Ltd., ISBN Dickson L. E., Modern Algebraic Theories, Dover New York, (1936). (Algebraic Theories, 1959). 8 Dickson, Leonard Eugene (1975), Albert, A. Adrian, ed., The collected mathematical papers of Leonard Eugene Dickson, IVI, New York: AMS Chelsea Publishing, ISBN , MR MR Dummit David S., Foote Richard M., Abstract Algebra, Prentice Hall,

11 10 Eckmann B. (Beno), Gruppentheoretischer Beweis des Satzes von Hurwitz-Radon uber die Komposition quatratisher Formen, Comment Math. Helv. 15, (1942), LHL. 11 Herstein I. N., Topics in Algebra, Blaisdell, 1964., (Frobenius Theorem on Division Algebras, Quaternions). 12 Hungerford Thomas H., Abstract Algebra: An Introduction, 2nd edition, Saunders College Publishing, Hurwitz A. (Adolph), Uber die Komposition der Quadratishen Formen, Collected Papers 2, (1933), LHL. 14 Hurwitz Adolph, Uber die Composition der quadratishen Formen von Beliebig vielen Variabeln. Nachr. Ges. Wiss. Göttingen (1898), (On the Composition of Quadratic Forms in Many (any number of) Variables) 15 Murnaghan Francis D., The Theory of Group Representations, 1938, Dover Penrose Roger, The Road to Reality, Alfred A. Knopf, Shafer Richard D., Introduction to Non-Associative Algebras, Dover, New York, 1995, MR 96j: Stiefel E., On Cauchy-Riemann Equations in Higher Dimensions, Journal of Research of the National Bureau of Standards, Vol. 48, No. 5, May 1952, Research Paper Taussky O.(Olga Taussky-Todd), An Algebraic Property of Laplace s Differential Equation, Quart. J. Math. 10, 99, (1939). 20 Ward J. P., Quaternions and Cayley Numbers, 1997, Kluwer Academic Publishers. (linda hall library QA196.w37), (an interesting book, with a good bibliography.) 11

12 21 Weyl Herman, The Classical Groups, their Invarients and Representations, Princeton University Press (1938). 22 Weyl Herman, The Theory of Groups and Quantum Mechanics, E. P. Dutton, New York, (1931), (Dover). 12

13 9 Index Cayley algebra 9 Cayley numbers 9 linear independence 6 pure quaternions 4 quaternion, matrix representation 5 real quaternions 5 13