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Associative Space - Time Sedenions Associative space - time sedenions Victor L. Mironov, Sergey V. Mironov Institute for physics of microstructures RAS, 603950, Nizhniy Novgorod, GSP-105, Russia e-mail: [email protected] (Submitted 15 January 2014) We present an alternative type of sixteen-component hypercomplex scalar-vector values named “space-time sedenions”, generating associative noncommutative space-time Clifford algebra. 1. Introduction The eight-component hypercomplex numbers such as biquaternions and octonions [1-6] enclosing scalar, pseudoscalar, vector and pseudovector components are widely used for the generalization of equations of quantum mechanics and field theory. These structures take into account the space symmetry with respect to the spatial inversion. However, a consistent relativistic approach requires taking into consideration full time and space symmetries that leads to the sixteen- component space-time algebras. The well known sixteen-component hypercomplex numbers, sedenions, are obtained from octonions by the Cayley-Dickson extension procedure [7,8]. In this case the sedenion is defined as SOO1 2e , (1) 2 where Oi is an octonion and the parameter of duplication e is similar to imaginary unit e 1. The algebra of sedenions has the specific rules of multiplication. The product of two sedenions SOO1 11 12e , SOO2 21 22e , is defined as SSOOOOOOOOOOOO12 11 12e 21 22 e 1121 2212 2211 1221 e , (2) where Oi is conjugated octonion. The sedenionic multiplication (2) allows one to introduce a well defined norm of sedenion. However such procedure of constructing the higher hypercomplex numbers leads to the fact that the sedenions as well as octonions generate normed but nonassociative algebra [9-11]. This greatly complicates the use of the Cayley-Dickson sedenions in the physical applications. Recently we have developed an alternative approach to constructing the multicomponent values based on our scalar-vector conception realized in associative eight-component octons [12-14] and sixteen-component sedeons [15-19]. In this paper we present an alternative version of the sixteen- component associative space-time algebra. 2. Sedenionic space-time algebra It is known, the quaternion is a four-component object q q0a0 q 1 a 1 q 2 a 2 q 3 a 3 , (3) where components q (Greek indexes 0,1, 2, 3 ) are numbers (complex in general), a0 1 is scalar units and values am (Latin indexes m 1, 2, 3) are quaternionic units, which are interpreted 1 as unit vectors. The rules of multiplication and commutation for am are presented in Table 1. We introduce also the space-time basis et,, e r e tr , which is responsible for the space-time inversions. The indexes t and r indicate the transformations (t for time inversion and r for spatial inversion), which change the corresponding values. The value e0 1 is a scalar unit. For convenience we introduce numerical designations e1 e t (time scalar unit); e2 e r (space scalar unit) and e3 e tr (space-time scalar unit). The rules of multiplication and commutation for this basis we choose similar to the rules for quaternionic units (see Table 2). Table 1. Table 2. a1 a2 a3 e1 e2 e3 a1 1 a3 a2 e1 1 e3 e2 a a a e e e 2 3 1 1 2 3 1 1 a3 a2 a1 1 e3 e2 e1 1 Note that the unit vectors a1,, a 2 a 3 and the space-time units e1,, e 2 e 3 generate the anticommutative algebras: an a m a m a n , (4) en e m e m e n , for n m , but e1,, e 2 e 3 commute with a1,, a 2 a 3 : en a m a m e n , (5) for any n and m . Then we can introduce the sixteen-component space-time sedenion V in the following form: V e0VVVVVVVV00 a 0 01 a 1 02 a 2 03 a 3 e 1 10 a 0 11 a 1 12 a 2 13 a 3 (6) e2VVVVVVVV20 a 0 21 a 1 22 a 2 23 a 3 e 3 30 a 0 31 a 1 32 a 2 33 a 3 . The sedenionic components V are numbers (complex in general). Introducing designation of scalar and vector values in accordance with the following relations VV e a , 000 0 VVVVe0 01 a 1 02 a 2 03 a 3 , VVV e a , t 1 110 0 VVVVVt 1 e 1 11 a 1 12 a 2 13 a 3 , (7) VVV e a , r 2 220 0 VVVVVr 2 e 2 21 a 1 22 a 2 23 a 3 , VVV e a , tr 3 330 0 VVVVVtr 3 e 3 31 a 1 32 a 2 33 a 3 , We can represent the sedenion in the following scalar-vector form: 2 V VVVVVVVV t t r r tr tr . (8) Thus, the sedenionic algebra encloses four groups of values, which are differed with respect to spatial and time inversion. (1) Absolute scalars (V ) and absolute vectors (V ) are not transformed under spatial and time inversion. (2) Time scalars (Vt ) and time vectors (Vt ) are changed (in sign) under time inversion and are not transformed under spatial inversion. (3) Space scalars (Vr ) and space vectors (Vr ) are changed under spatial inversion and are not transformed under time inversion. (4) Space-time scalars (Vtr ) and space-time vectors (Vtr ) are changed under spatial and time inversion. Further we will use the symbol 1 instead units a0 and e0 for simplicity. Introducing the designations of scalar-vector values V0VVVV 00 01a1 02 a 2 03 a 3 , V1VVVV 10 11a1 12 a 2 13 a 3 , (9) V2VVVV 20 21a1 22 a 2 23 a 3 , V3VVVV 30 31a1 32 a 2 33 a 3 , we can write the sedenion (6) in the following compact form V = V0 +e1 V 1 e 2 V 2 e 3 V 3 . (10) On the other hand, introducing designations of space-time sedenion-scalars V0()VVVV 00 e1 10 e 2 20 e 3 30 , V1()VVVV 01 e1 11 e 2 21 e 3 31 , V2()VVVV 02 e1 12 e 2 22 e 3 32 , (11) V3()VVVV 03 e1 13 e 2 23 e 3 33 , we can write the sedenion (6) as VVVVV0 1a1 2 a 2 3 a 3 , (12) or introducing the sedenion-vector V =VVVVt r tr = V1a 1 V 2 a 2 V 3 a 3 , (13) we can rewrite the sedenion in following compact form VVV0 . (14) Further we will indicate sedenion-scalars and sedenion-vectors with the bold capital letters. Let us consider the sedenionic multiplication in detail. The sedenionic product of two sedenions A and B can be represented in the following form AB A AB B AB ABAB AB AB (15) 0 0 0 0 0 0 Here we denoted the sedenionic scalar multiplication of two sedenion-vectors (internal product) by symbol “ ” and round brackets 3 ABABABAB 1 1 2 2 3 3 , (16) and sedenionic vector multiplication (external product) by symbol “” and square brackets, AB AB ABa +AB AB a +AB AB a . (17) 2332 1 3113 2 1221 3 In (16) and (17) the multiplication of sedenionic components is performed in accordance with (11) and Table 2. Thus the sedenionic product F = AB F0 + F (18) has the following components: F0 = A 0 B 0 A 1 B 1 A 2 B 2 A 3 B 3 , F=AB+AB+1 1 0 01 AB 2 3 AB 3 2 , (19) F=AB+AB2 2 0 0 2 AB 3 1 AB 1 3 , F=AB+AB3 3 0 0 3 AB 1 2 AB 2 1 . Note that in the sedenionic algebra the square of vector is defined as 2 2 2 2 AAAAAA 1 2 3 , (20) and the square of modulus of vector is 2 2 2 2 A A A A1 + A 2 + A 3 . (21) 3. Spatial rotation and space-time inversion The rotation of sedenion V on the angle around the absolute unit vector n is realized by uncompleted sedenion U cos / 2 n sin / 2 (22) and by conjugated sedenion U * : U * cos / 2 n sin / 2 (23) with UU ** = U U 1. (24) The transformed sedenion V is defined as sedenionic product V U * V U , (25) Thus the transformed sedenion V can be written as VVVcos /2 n sin /2 0 cos /2 n sin /2 VVVV cos n n 1 cos n sin . (26) 0 It is clearly seen that rotation does not transform the sedenion-scalar part, but the sedenionic vector V is rotated on the angle around n . 4 ˆ ˆ ˆ The operations of time inversion ( Rt ), space inversion ( Rr ) and space-time inversion ( Rtr ) are connected with transformations in e1 , e2 , e 3 basis and can be presented as ˆ RtV e 2V e 2 V0 e 1 V 1 e 2 V 2 e 3 V 3 , ˆ RrVVVVVV e 1 e 1 0 e 1 1 e 2 2 e 3 3 , (27) ˆ RtrVVVVVV e 3 e 3 0 e 1 1 e 2 2 e 3 3 . 4. Sedenionic Lorentz transformations The relativistic event four-vector can be represented in the follow sedenionic form: S e1ct+ e 2 r . (28) The square of this value is the Lorentz invariant SS c2 t 2 + x 2 + y 2 + z 2 . (29) The Lorentz transformation of event four-vector is realized by uncompleted sedenions L ch e3m sh , (30) * L ch e3m sh , where th 2 v / c , v is velocity of motion along the absolute unit vector m . Note that LLLL 1.
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