View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server August, 1995 hep-th/95mmnnn Quaternions and Sp ecial Relativity a) Stefano De Leo UniversitadiLecce, Dipartimento di Fisica Instituto di Fisica Nucleare, Sezione di Lecce Lecce, 73100, ITALY Abstract We reformulate Sp ecial Relativityby a quaternionic algebra on reals. Using real linear quaternions,we show that previous diculties, concerning the appropriate transformations on the 3 + 1 space-time, maybeovercome. This implies that a complexi ed quaternionic version of Sp ecial Relativityisachoice and not a necessity. a) e-mail: [email protected] 1 Intro duction \The most remarkable formula in mathematics is: i e = cos + i sin : (1) This is our jewel. Wemay relate the geometry to the algebra by representing complex numb ers in a plane i x + iy = re : This is the uni cation of algebra and geometry."-Feynman [1]. i We know that a rotation of -angle around the z -axis, can b e represented by e , in fact i i(+ ) e (x + iy )=re : In 1843, Hamilton in the attempt to generalize the complex eld in order to describ e 1 the rotation in the three-dimensional space, discovered quaternions . Today a rotation ab out an axis passing trough the origin and parallel to a given uni- tary vector ~u (u ;u;u)by an angle , can b e obtained taking the transformation x y z (iu +ju +ku ) (iu +ju +ku ) x y z x y z 2 2 e (ix + jy + kz)e : (2) Therefore if we wish to represent rotations in the three-dimensional space and com- plete \the uni cation of algebra and geometry"we need quaternions. The quaternionic algebra has b een exp ounded in a series of pap ers [2] and b o oks [3] with particular reference to quantum mechanics, the reader may refer to these for further details. For convenience we rep eat and develop the relevant p oints in the following section, where the terminology is also de ned. Nothing that U (1;q) is algebraically isomorphic to SU (2;c), the imaginary units i; j; k can b e realized by means of the 2 2Pauli matrices through (i; j; k ) $ (i ; i ; i ) : 3 2 1 (this particular representation of the imaginary units i; j; k has b een intro duced in ref. [4]). So a quaternion q can b e represented bya 22 complex matrix ! z z 1 2 q $ Q = ; (3) z z 2 1 1 Quaternions, as used in this pap er, will always mean \real quaternions" q = a + ib + jc + kd ; a; b; c; d 2R: 1 where z = a + ib ; z = c id 2 C (1;i) : 1 2 It follows that a quaternion with unitary norm is identi ed by a unitary 2 2 matrix 2 with unit determinant, this gives the corresp ondence b etween unitary quaternions U (1;q) and SU (2;c). Let us consider the transformation law of a spinor (two-dimensional representations of the rotation group) 0 = U ; (4) where ! z 1 = ; U2SU (2;c) : z 2 We can immediately verify that ! z 2 ~ = z 1 transforms as follows 0 ~ ~ = U ; (5) so ! ! 0 z z z z 1 1 2 2 = U z z z z 2 2 1 1 represents again the transformation law of a spinor. Thanks to the identi cation (3) we can write the previous transformations by real quaternions as follows 0 q = U q; + with q = z + jz and U quaternion with unitary norm (N (U )=U U =1). Note that 1 2 we don't need right op erators to indicate the transformation law of a spinor. Nowwe can obtain the transformation law of a three-dimensional vector ~r (x; y ; z )by pro duct of spinors, in fact if we consider the purely imaginary quaternion + ! = qiq = ix + jy + kz or the corresp onding traceless 2 2 complex matrix ! ix y iz + = i = ; y iz ix 3 a rotation in the three-dimensional space can b e written as follows 2 In a recent pap er [5] the representation theory of the group U (1;q) has b een discussed in detail. 3 \No such `trick' works to relate the full four-vector (ct; x; y ; z ) with real quaternions." - Penrose [6]. 2 0 + ! = U ! U (quaternions) , 0 + = U U (2 2 complex matrices) . ~ ~ For in nitesimal transformations, U =1+Q,we nd 0 ~ ~ ~ ~ ~ ~ ~ ~ Q ~r = Q ~r + Q Q ~r Q ~r Q ; where ~ ~ Q (i; j; k ) ; ( ; ; ) : 4 If we rewrite the mentioned ab ove transformation in the following form 0 ~ ~ ~ ~ ~ Q ~r =[1+(Q1 j Q)] Q ~r; (6) we identify i 1 j i j 1 j j k 1 j k ; ; ; 2 2 2 5 as the generators for rotations in the three-dimensional space . Up till now, wehave considered only particular op erations on quaternions. A quaternion q can also b e multiplied by unitary quaternions V from the right. A p ossible transformation which preserves the norm is given by 0 q = U q V ; (7) + + ( U U = V V =1) : Since left and rightmultiplications commute, the group is lo cally isomorphic to SU (2) SU (2), and so to O (4), the four-dimensional Euclidean rotation group. As far as here we can recognize only particular real linear quaternions, namely 1 ; i; j; k; 1ji; 1jj; 1jk: Wehaven't hop e to describ e the Lorentz group if we use only previous ob jects. An- alyzing the most general transformation on quaternions (see section 4), weintro duce new real linear quaternions which allow ustoovercome the ab ove diculty and so obtain a quaternionic version of the Lorentz group, without the use of complexi ed 4 Barred op erators Ojq act on quaternionic ob jects as in (Ojq) = O q: 1 5 The factor guarantees that our generators satisfy the usual algebra: 2 [A ;A ]= A m; n; p =1; 2; 3 : m n mnp p 3 quaternions. This result app ears, to the b est of our knowledge, for the rst time in print. First of that we brie y recall the standard way to rewrite sp ecial relativitybya quaternionic algebra on complex (see section 3). In section 5, we present a quaternionic version of the sp ecial group SL(2;c), which is as well-known collected to the Lorentz group. Our conclusions are drawn in the nal section. 2 Quaternionic algebras A quaternionic algebra over a eld F is a set H = f + i + j + k j ; ; ; 2Fg with multiplication op erations de ned by following rules for imaginary units i; j; k 2 2 2 i = j = k = 1 ; jk = kj = i; ki = ik = j; ij = ji = k: In our pap er we will work with quaternionic algebras de ned on reals and complex, so in this section we give a panoramic review of such algebras. We start with a quaternionic algebra on reals H = f + i + j + k j ; ; ; 2Rg: R + Weintro duce the quaternion conjugation denoted by and de ned by + q = i j k : The previous de nition implies + + + ( ') = ' ; for , ' quaternionic functions. A conjugation op eration which do es not reverse the order of , ' factors is given, for example, by q~ = i + j k : An imp ortant di erence b etween quaternions and complexi ed quaternions is based on the concept of division algebra, which is a nite dimensional algebra for which a 6=0,b6= 0 implies ab 6= 0, in others words, which has not nonzero divisors of zero. A classical theorem [7] states that the only division algebras over the reals are 4 algebras of dimension 1, 2, 4 and 8; the only asso ciative algebras over the reals are R, C and H (Frob enius [8]); the nonasso ciative division algebras include the o ctonions R O (but there are others as well; see Okub o [9]). A simple example of a nondivision algebra is provided by the algebra of complexi ed quaternions H = f + i + j + k j ; ; ; 2C(1; I )g ; C [I ;i]= [I;j]= [I;k]=0 : In fact since (1 + iI )(1 iI )=0 ; there are nonzero divisors of zero. For complexi ed quaternions wehave di erent opp ortunities to de ne conjugation op erations, we shall use the following terminology: 1. The complex conjugate of q is C q = + i + j + k : C Under this op eration (I ;i;j;k) ! (I ;i;j;k) ; and (q p ) = q p : C C C C 2. The quaternion conjugate of q is C ? q = i j k : C Here (I ;i;j;k) ! (I;i; j; k ) ; and ? ? ? (q p ) = p q : C C C C 3. In the absence of standard terminology,we call that formed by combining these op erations, the full conjugate + = i j k : q C Under this op eration (I ;i;j;k) ! (I;i;j;k) ; and + + + (q p ) = p q : C C C C 5 3 Complexi ed Quaternions and Sp ecial Relativi- ty We b egin this section by recalling a sentence of Anderson and Joshi [10] ab out the quaternionic reformulation of sp ecial relativity: \There has b een a long tradition of using quaternions for Sp ecial Relativity ... The use of quaternions in sp ecial relativity,however, is not entirely straightforward. Since the eld of quaternions is a four-dimensional Euclidean space, complex comp onents for the quaternions are required for the 3+1 space-time of sp ecial relativity." In the following section, we will demonstrate that a reformulation of sp ecial rela- tivityby a quaternionic algebra on reals is p ossible.