Lounesto, Pertti

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Cayley transform, outer exponential and spinor norm

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WSGP 7Pertti LounestoCayley transform, outer exponential and spinor normIn: Jarolím Bureš and Vladimír Souček (eds.): Proceedings of the Winter School "Geometry andPhysics". Circolo Matematico di Palermo, Palermo, 1987. Rendiconti del Circolo Matematico diPalermo, Serie II, Supplemento No. 16. pp. [191]--198.Persistent URL: http://dml.cz/dmlcz/701421Terms of use:© Circolo Matematico di Palermo, 1987Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czCAYLEY TRANSFORM, OUTER EXPONENTIAL AND SPINOR NORM Pertti Lounesto Abstract. Cayley transform of an antisymmetric nxn-matrix A is the rotation matrix £/ = (I + A)(I-A)_1 in SO(n). In the Clifford algebra the matrices A and U correspond to the bivector B in R^, Ax = x • B, and to its outer exponential defined by the finite sum eAB = 1 + B + ^BAB + *BABAB + ... The outer exponential 8 = eAR of B is the unique element in the group Tn, with real part 1, inducing the rotation U, t/x = 8~1x8. This representation of rotations was first invented by R. Lipschitz. In this paper the above known result is given a new proof, which does not rely on indices and is therefore independent of the coordinates. The proof employes the outer and inner products only and is based on the formula x8 = (x + x • B)A8. The absolute value 181 of 8, or the spinor norm of £/, is the square root of 8*8 = detC-^—) , where 8 -»8* is the reversion of the Clifford algebra. This paper is in final form and no version of it will be submitted for publication elsewhere 1 9 2 PERTTI LOUNESTO 1. Properties of Clifford algebras The Clifford algebra Rn shall be the associative algebra over the reals R generated by the elements ei,e2,..,en subject to the relations e f-2=l and eiei---ejei,i*j. In order to guarantee the universal property we must also require e\e^^en ^ ±1. Rn is a linear space of dimension 2n. It is a sum of the spaces R* each having basis elements e/i ; = e;e: ...e:, 1< ii <i2< ... </^<n where k = 0,l,...,n is fixed. More precicely, the basis elements are k ЄҺҺ-iк 0 1 2 1 eij~ei j n e12...n 1 < i < n l<i<j<n R* shall be identified with the euclidean space Rn. The sum of the R* with even k will be denoted by R ^ , while R ^ refers to odd k. R ^ is a subalgebra of Rn. For more information about the Clifford algebras see Refs. [1], [6], [9], [10], [11]. Involutions. The Clifford algebra Rn has three important involutions, similar to complex conjugation. The first, called main involution, is the isomorphism a -»a' obtained by replacing each e,- by - ê , thereby replacing each a in R* by a' = (-l)*a. By definition (ab)' =a'b'. The second involution, called reversion, is an anti-isomorphism a -> a* obtained by reversing the order of factors et in each e ^ 2 ^ , thereby replacing each a in R* by a* = (-l)^/2]a. By definition (ab)* = b*a*. The third involution, called conjugation, is a combination of the two others 5 = a*' = a'*. Absolute Value. The euclidean square norm on Rn extends to the whole Clifford algebra Rn by defining rj \a\2 = l a i i i i i k for ^ = ^ %i2.jk%i2.jk (%i2.Jk r e a l > where the sum ranges over all ordered multi-indices iW-'ifc such that 1< i\<ii< ... < i/c < n. This gives the absolute value \a\ of a, also obtained as the square root of the real scalar part of a*a; as an equation Id2 = Re(a*a). CAYLEY TRANSFORM, OUTER EXPONENTIAL AND SPINOR NORM 193 2. Unit products and spin group Products of vectors in Rn are called products in short. The invertible products in Rn form ihcLipschitz group Tn. If a is in Tn then a*a is real and so \a\2 = a*a, from which it follows that \ab\ = \a\ \b\. If x is in Rn and a is in Tn, then a'-^xa is again a vector in Rn. Furthermore, the transformation x->a'~*xa is a euclidean isometry. In other words, for every a in Tn there is a matrix Ua in 0(n) such that a'_1xa = Ua(x). Conversely, every orthogonal matrix can be represented in this way. The main involution a-* a' is included here in the map x -> a!'^xa in order to guarantee a coherent treatment of even-dimensional and odd-dimensional spaces. The Lipschitz group Tn splits in even and odd parts Tn = r ^ u l ^1 ) , where r ® = R$nTn. The even part T^ covers the rotation group SO(n) so that the unit products a, ld = l,in T^) form a two-fold covering group Spin(n) of SO(n). Example. Forabivector B in R2, n<6, (1+B)(1-B)_1 is in Spin(n). Exercise. Prove that if s is in Spin(/i) so that 1+8 is invertible, then Re-j—= \. 3. Outer and inner product If two elements a in R* and b in RI are multiplied, then their product ab is in the direct sum RI+> + Rl+-^2 + ... + R,j[-A The component in Rl+I is called the outer product a A b and the component in R -̂I1 the inner product a • b. Both products can be extended by linearity to all of Rn. The outer product is associative (aAb)Ac = aA(bAc). In the graded sense the outer product is also commutative, that is, (1) aAb = (A)%Aa for a in R(p and b in R<{). Example. If x is a vector and B a bivector, then xB = x*B + xAB = ^(xB - Bx) + £(xB + Bx). Also (x-B)AB = £((x-B)B + B(x-B)) = ±(xB2 - BxB + BxB - B2x) = \(xB2 - B2x). On the other hand x(BAB) = ^(x(BAB) - (BAB)x) = \(xB2 - B2x). Therefore £x-(BAB) = (x-B)AB. !94 PERTTI LOUNESTO Exercise. If a l 5a2 , a3, a4 and x are vectors, then x • (aiAa2Aa3Aa4) = (x • ai)(a2Aa3Aa4) - (x • a2)(ajAa3Aa4) + (x • a3)(aiAa2Aa4) - (x • a4)(a1Aa2Aa3). Denote Bx = ai Aa2, B2 = a3Aa4 and prove x • ( B ^ B ^ = (x • B ^ ^ + (x • B2)ABV 4. Outer exponential The outer exponential of a bivector B in R2 is the exponential series with outer product as multiplication [7] (2) eAB = 1 + B + 1BAB + ±BABAB + ... This series is finite. The bivector B can be written as a sum B = Bj + B2 + ... + B^ of at most £ = [n/2] mutually orthogonal plain bivectors B,-, B,AB,- = 0, where the completely orthogonal planes Bi have only one point in common. This decomposi-tion is unique unless B,-2 = By2. Notwithstanding, the product ( l + B ^ l + B ^ . . ACl+Bj) = (1+B1)(1+B2)...(1+BI) depends only on B and equals the outer exponential eAB of B [4], [5]. The reversion of s = eAB is 8* = eA(~B). Since 8A8* = 1, one can say that the outer inverse 8A(_1) of s equals 8*. The ordinary inverse of s is given by 8"1 =8*/l8l2. 5. Cayley transform An antisymmetric nxn-matrix A is sent by the Cayley transform to the rotation matrix [/ = (/+AXI-A)"1 in SO(n). There corresponds a bivector B in R2 to A so that Ax = x • B for all vectors x in R". If y = t/x, then y - Ay = x + Ax, or equivalently (3) y + B • y = x + x • B. Next, compute 8A(y + B-y) = 8Ay + 8A(B*y) when s = eAB. Sum up i j (BABA.. AB)A (B • y) = ̂ j (BABA...AB) • y K /C+l CAYLEY TRANSFORM, OUTER EXPONENTIAL AND SPINOR NORM 195 for k = 0,1,2,... to obtain 8A(B-y) = (8 - 1) • y. Since 8Ay + (8-1 )Ay = 8y, it follows that 8A(y + B-y) = 8y. Similarly, 8A(x + x-B) = 8Ax - (8-l)Ax = xA8 +x • (8-1) = x8. Therefore, the equation (3) is equivalent to (4) •sy = xí' or Ux = 8_1x8. This representation of rotations was first invented by R. Lipschitz. To pay hommage to him we have denoted the Lipschitz group by T, a mirror image of L. All told we have sketched a novel proof for a previously known result [5], [9], [13]. Theorem. An antisymmetric nxn-matrix A and the rotation matrix U = (I + A)(I-A)_1 e SO(n) correspond, respectively, to the bivector B e R^, Ax = x • B, and to its outer exponential 8 = eAB e T$\ which is the unique element of r ^ ) , with real part 1, inducing the rotation £/, Ux = 8_1x8. Conversely, every rotation U in SO(rc), with eigenvalues different from -1, is uniquely obtained in this way. The following table gives two different kinds of connections between the rotation and spin groups SO(n) Spin(n) I + A I - A + eB/2 e л в leлB| Absolute value Of OUter exponential. If a rotation matrix U in SO(n) does not rotate any plane by a half-turn, then there is a unique element 8 in T$\ with real part 1, so that f/x = 8"1x8. The absolute value 181 of 8 is the square root of 8*8, which equals [12], [13] (5) det(I-A) = d e t ( - ^ ) " where A = ([/ - I)(U + I)"1. The absolute value is also the square root of 8*8 = (1 - B^Xl - B 22 ) . . . (1 -B^2). i n , PERTTI LOUNESTO lvo Combined rotations. Take two antisymmetric matrices -4i,A2 and the corre-sponding rotations C/i, U2 as well as bivectors Bi, B2 and their outer exponentials 8! = eAfi1,s2 = eAR2. Then also 8iA82 = eA(Bi+B2) is in T^\ If the combined rota-tion U2U\ does not have -1 as its eigenvalue, or equivalendy, is represented by such an element 8182 in T ^ that Re(si 82) * 0, then the matrix identity [12] U2UX+I U2+I Ux + I o = —~—(Z + A^! )—*— shows that X = 1/Re(8is2) is a solution of the quadratic equation (6) ?i2 = det(/ + A2A1)-1. In other words, when multiplied by the scalar X the product s^s2 is sent to the outer exponential of a unique bivector Xs^s2 = eAR. This is the reason why the spinor norm was introduced in the first place [2], [12], [13]. See also Refs. [3], [8], [9]. Remark. It is important to observe that the set of rotations £/, det(£7 +1) * 0, repre-sented by the products s = eAB, which are expressed in terms of the outer product only, does not depend on the scalar product of the underlying vector space. How-ever, when writing down the actual rotation Ux = s~lxs, the inner product is also employed. Therefore, it might be interesting to know the effect of the quadratic form on the rotation and spin groups. 6. Indefinite quadratic forms The Clifford algebra Rpq shall be the associative algebra over the reals R gener-ated by4he elements ei ,e2,...,ert subject to the relations Є ; 2 = l l < i < p e,-2 = -1 p+1 < / < p+q = n P P . =r —P .P. 1 J j l i<j. In order to guarantee the universal property we must also require eie2...e„ * ±1. The main differences between the positive definite case Rn = RnQ and the other Clifford algebras Rpq will be reviewed in the following. In the Clifford algebra Rpq the quadratic forms a -> Re(a*a) and a -» Re(5a) are usually non-definite. The Lipschitz group Yp „ consists of all elements in Rp„ which can be written as products of non-isotropic vectors x, x2 * 0, in R* = RP>4. For a product u in CAYLEY TRANSFORM, OUTER EXPONENTIAL AND SPINOR NORM 1 9 7 Гpq the expression ӣu is always real. A unit product u satisfies 5м = ± l . The unit products forma subgroup Pin(p,q) of Гpq. Theeven Lipschitz group Г ^ has a subgroup of even unit products Spin(p,q) = Rj^nPin^ q, which further has a subgroup Spin+(p,q) where Ö Í І = 1 . Since R^)~R(Ф we also have the ІSO-moфhisms Spin(p,q) ~ Spin(q,p). Thegroups Pin(p,q), Spin(p,q) and Spin+(p,q) are two-fold coverings of the matrix groups 0»(p,q), SO(p,q) and SO+(p,q), which is the identity component of SO(p,q). Also the group Spin+(p,q) is connected with the following exceptions Spin+(0,0) = Spin+(1,0) = Spin+(0,1) = ±1 and Spin+(1,1) = {x + ye12lx2-y2=l} [9,p.427]. Every linear isometry L of RPЯ, corшected with the identity of SO(p,q), is the exponential of an antisymmetric transformation A of RP>Q, L = И, if and only if RP>Ч isoneofthefollowing R* = R">°, R0^, RP>1 or R 1 ^ [10, pp. 150-152]. In the same orthogonal spaces there is a bivector B є Rp a such that Lx = e~вxeв foranyvector x є R M [10,p. 160]. Finally, given a bivector B in R 2 o n e c a n> i*1 geneгal, find other bivectors F such that eв = - eғ, and hence e~вxeв = e~ғxeғ for all vectors x in RP4. The only exceptions concern the following cases [10, p. 172]: R1*1 forałl B R 1 ' 2 and R 2 1 forall B * 0 suchthat B 2 > 0 R 1 ' 3 and R 3 ' 1 forall B * 0 suchthat B 2 = 0. The outer exponential eлß of the bivector B in Rpq need not be invertible. that is, it does not necessarily belong to the Lipschitz group Гn a. However, an invertible 8 = e ь is in Г„ ø. If the mutually commuting plain bivectors in the orthogonal decomposition B = B^ + B2 + ... + B^ satisfy B ř2 Ф1, ťhen ss = (1-B12)(1-B22)...(1-BІ2) Ф 0, and the product 8 = (1+B1)(1+B2)...(1+BJ-) is invertible. In the orthogonal space RP>Q an antisymmetric transformation A, det(/-A)?fc0, corresponds to the rotation U = (I + A)(I -. A)-1 є SO(p,q), det(t/ +1) # 0. 198 PERTTI LOUNESTO REFERENCES [I] L. Ahlfors: Mobius transformations and Clifford numbers. I. Chavel, H. M. Farkas (ed.): Differential Geometry and Complex Analysis. Dedicated to H. E. Rauch. Springer, Berlin, 1985, pp. 65-75. [2] E. Artin: Geometric Algebra. Interscience, New York, 1957. [3] H. Bass: Clifford algebras and spinor norms over a commutative ring. Amer. J. Math. 96 (1974), 156-206. [4] C Chevalley: The Algebraic Theory ofSpinors. Columbia University Press, New York, 1954. [5] J. Helmstetter: Groupe de Chfford pour des formes quadratiques de rang quelconque. C. R. Acad. ScL Paris Sir. A 285 (1977), 175-177. [6] D. Hestenes: New Foundations for Classical Mechanics. Reidel, Dordrecht, 1986. [7] B. Magneron: Une realisation des groupes de Clifford par fibration en droites des groupes orthogonaux. C. R. Acad. ScL Paris Sir. 1293 (1981), 451-454. [8] M. Perroud: A new formula for the spinor norm. J. Math. Phys. 19 (1978), 1265-1266. [9] I. Porteous: Topological Geometry. Van Nostrand Reinhold, London, 1969. Cambridge University Press, Cambridge, 1981. [10] M. Riesz: Clifford Numbers and Spinors. University of Maryland, College Park, MD, 1958. [II] B. L. van der Waerden: A History of Algebra. Springer, Berlin, 1985. [12] H. Zassenhaus: On the spinor norm. Arch. Math. (Basel) 13 (1962), 434-451. [13] Correspondence under the pseudonym R. Lipschitz. Ann. Math. 69 (1959), 247-251. INSTITUTE OF MATHEMATICS HELSINKI UNIVERSITY OF TECHNOLOGY SF-02150 ESPOO, FINLAND 

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Cayley transform, outer exponential and spinor norm

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Institute of Mathematics AS CR, v. v. i.