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04.50.+h; numbers: PACS approach. our in naturally fundamen appear of coordi theories, space appearance physical of non-commutativity the a no and to elements of correspond Grassmann definiteness could positive of octonions condition of the uncer from tu and appears It constant) and Planck space-time. and light of of phen (velocity coordinatization elemen constants physical the (the picture raise divisors give zero a while norms) such algebra, chosen In of elements split-octonions. of algebra hsclapiaiiyo omdsltagba,sc as such split-algebras, normed of applicability physical A .INTRODUCTION I. aaahiiSr,bls 807 Georgia 380077, Str.,Tbilisi Tamarashvili 6 nrnksvl nttt fPhysics of Institute Andronikashvili -al [email protected] E-mail: bevbeAlgebra Observable ea Gogberashvili Merab Otbr2,2018) 24, (October 1 h qaeo h ntelement unit the of square The elui element unit real complex of [2]. algebras octonions the and quaternions are the numbers, These theorem, real satisfy that Hurwitz of conditions. algebras the required algebra unique to usual and extraordinary the according three of are, Besides there over numbers element numbers. a unit need real the we with the world real algebra the composition of geometry normed the element. describe unit to a that an with as algebra them means an of need always one we distance using Thus objects a etalon. two between of of distance comparison Introduction a some introduce to objects. able two be must one nals), h qae ftehprcmlxunits hyper-complex the of squares the n ye-ope units hyper-complex hsproew att hoetepstv ini (1.1), in sign positive For the choose distances). or to of want we measurements purpose for this used signals unit be contain the must can to it (which corresponding that on elements is of requirement apply class to physical special like new would we one the algebra of the So result and direct the signals. of norm, backward multiplication the the actually a that is have assume measurement, to We any is requirement signals to norm. important some real correspond The of can number. exchange of signals the kind One-way by only them. done between is be algebra can the jects of norm case this In negative defined. used. the positively is mainly (1.1) algebras in division sign of applications In rnegative or ,frquaternions for 1, = seta o l hs lersi h xsec fthe of existence the is algebras these all for Essential mean requirements these language, algebraic the In ent hti elwrdaycmaio ftoob- two of comparison any world real in that note We anypicpehv emtia meaning geometrical have principle tainty so pi-lerscrepnigt zero to corresponding split-algebras of ts n ob osbeta w fundamental two that possible be to rns begoer a edsrbdb the by described be can geometry able m.Tepoet fnon-associativity of property The rms. yeblcnmes split-quaternions numbers, hyperbolic mn r ecie yteordinary the by described are omena ae,wihaewdl sdi various in used widely are which nates, a rbblte nfu dimensions. four in probabilities tal e 0 e e n ieetnme fadjoined of number different a and n 2 n o h aeo ope numbers complex of case the For . = n ± n o octonions for and 3 = e 0 . e 0 sawy oiieand positive always is e n a epositive be can n (1.1) 7. = 2 en = e0 . (1.2) where εnmk is fully anti-symmetric tensor. Or vice versa, we can say that from the postulated algebra (1.6) for This leads to so-called split algebras with the equal num- basis elements one recovers the observed multiplication ber of terms with the positive and negative signs in the laws (1.4) and (1.5) of physical vectors. definition of their norms. Adjoining to a real unit ele- Because of the ambiguity in choosing of direction of ∗ ment (which could be connected with time), any invert- basis vectors, one must introduce conjugate elements en, ible hyper-complex units with negative norms could be which differ from en by the sign understood as an appearance of dynamics. The critical ∗ e = en , (1.7) elements of split-algebras, so-called zero divisors, corre- n − sponding to zero norms can be used as the unit signals what physically can be understood as a reflection. to measure distances between physical objects, wile or- The algebra of the vector basis elements (1.6) and (1.7) dinary physical objects are assumed to be described by is exactly the same as we have for basis units of split alge- regular elements (with division properties) of the algebra. bras (with the positive sign (1.2)), which we had chosen We believe that the dimension of the proper algebra from some general requirements. Since only vector prod- and the properties of its basis elements also can be ex- ucts of different basis vectors and also scalar product of tracted from physical considerations. a basis vector with itself are non-zero, we do not need to It is known that observed physical quantities are di- write different kind of brackets to express their multipli- vided into three main classes, scalars, vectors, and cations. spinors. We need to have a unique way to describe these It is known that the notions of vector products and objects in an appropriate algebra. composition algebras in fact are equivalent [3]. However, Scalar quantities have magnitude only, and do not in- pure quaternions (with the negative sign in (1.1)) can volve directions. The complete specification of a scalar not produce the usual 3-dimensional vectors, since with quantity requires a unit of the same kind and a number the respect of reflections quaternions basis units behave counting how many times the unit is contained in the like a pseudo-vectors [4]. Generalization of the standard quantity. Scalar quantities are manipulated by applying vector products is only possible in seven dimensions and the rules of ordinary algebra or real numbers. is generated by octonions [3,5]. We shall show below that A vast range of physical phenomena finds its most exactly three orthogonal vector-like objects are elements natural description as vector quantities, being a way to of the split-octonion algebra. describe the size and orientation of something. A vec- Restrictions on the appropriate algebra can be imposed tor quantity requires for its specification a real number, also from the properties of spinors, another important which gives the magnitude to the quantity in terms of class of observed physical quantities. Spinors as vectors the unit (as for scalars) and an assignment of direction. can be characterized by some magnitude and direction, By revising the process of addition of physical vectors but they changes sign after a 2π rotation in 3-space. Us- it was found that using unit orthogonal basis vectors en ing a multi-dimensional picture, when we transform ordi- any vector could be decomposed into components nary space coordinates, spinors rotate also in some extra space. It is well known that there exists an equivalence a = anen . (1.3) of vectors and spinors in eight dimensions [6], which ren- ders the 8-dimensional theory (connected with octonions) The ordinary idea of a product in scalar algebra can- a very special case. In eight dimensions a vector and a not apply to vectors because of their directional proper- left and right spinor all have an equal number of compo- ties. Since vectors have their origin in physical problems, nents (namely, eight) and their invariant forms all look the definition of the products of vectors historically was the same. One may consider the vector’s rotation as the obtained from the way in which such products occur in primary rotation that induces the transformations of two physical applications. Two types of multiplication of vec- kinds of spinors, or equivalently starts from the spinor tors were introduced - the scalar product rotation, which induces a corresponding rotation of the (ab)= ab cos θ , (1.4) vector and the second kind of spinor. This property is referred in the literature as ”principle of triality”. The and the vector product physical result of this principle can be the observed fact that interaction in particle physics exhibit by the vertex [ab]= n ab sin θ , (1.5) containing a left spinor, a right spinor and a vector [7]. To summarize our general considerations, we conclude where n is a unit vector perpendicular to the plane of a that the proper algebra needed to describe the real world and b. must be the 8-dimensional algebra of split-octonions over Formulae (1.3) - (1.5) result in the following properties the real numbers. Geometry can be constructed by the of unit basis vectors exchanging of octonionic signals. The assumption that our Universe is built from pairs (enem)=0 , enem = emen , of octonions is now an important idea of string the- 2 − (enen)= en =1 , [enem]= εnmkek , (1.6) ory also (for example, see some recent publications [8]).
2 2 String theory (where octonions are used only for extra Gn =0 , GnGm =0 . (2.6) dimensions), however, in difference with our approach is an attempt to marry classical considerations about 4- Since the square of hyper-complex basis units of a split- algebra is +1 or 1, Grassmann numbers can be con- dimensional space-time to the small-scale uncertainties − inherent in quantum theory. structed by coupling of two basis elements (except of In the next section the properties of zero divisors, unity) with the opposite choice of the sign in (1.1). For which appear in split-algebras, are considered. The fol- example, primitive Grassmann numbers are the sums of lowing three sections are devoted to geometrical appli- any two basis elements e1 and e2 cations of hyper-numbers, split-quaternions and split- ± 1 octonions, respectively. Since physical applications of G = (e1 e2) (2.7) 2 ± split-algebras over the real numbers are purely studied main considerations of all sections are original. with the properties
e2 =1 , e2 = 1 , e∗ = e , e∗ = e . (2.8) 1 2 − 1 − 1 2 − 2 II. ZERO DIVISORS A Grassmann number does not contain the unit ele- ment 1 as the one of its terms (the structure with the The division and split algebras have some differences unit element forms projection operator) and can be rep- following from the fact that in split-algebras special ob- resented as jects, called zero divisors, can be constructed [2,9]. We assume that these objects (we shall call them critical el- G = e1D, (2.9) ements of the algebra) could be serve as the unit sig- nals characterizing physical events. Sommerfeld specially where D is a projection operator and e1 is a vector-like 2 notes physical importance of zero divisors of split alge- basis element of the algebra with positive square e1 = 1. bras in his book [10]. For the completeness we note that the Grassmann The first types of zero divisors we want to study are numbers G± and projection operators D± obey the fol- projection operators with the property lowing algebra: ± ∓ ± ± ± D2 = D, (2.1) D D =0 ,D D = D , G±G± =0 , G±G∓ = D∓ , for any non-zero D. In division algebras only projection D±G± =0 ,D±G∓ = G∓ , (2.10) operator is the identity 1 = e . o ± ∓ ± ± ± The most general form of projection operator in split- G D =0 , G D = G . algebra is From this relations we see that, separately, the quanti- 1 ties G+ and G− are Grassmann numbers, but they do D = + a e , (2.2) 2 n n not commute with each other (in contrast with the pro- jection operators D+ and D−) and thus do not obey the + with anan =1/4. In (2.2) index n runs over the number full Grassmann algebra (2.6). Instead the quantities G − of the orthogonal hyper-complex basis elements en. and G are the elements of so-called algebra of Fermi In general, a zero divisor is called primitive if it cannot operators with the anti-commutator be expressed as the sum of two other commuting zero − − divisors. For simplicity below we shall use only primitive G+G + G G+ =1 . (2.11) projection operators. Commuting primitive projection operators can be con- The algebra of Fermi operators is some syntheses of the structed by decomposition of the identity element Grassmann and Clifford algebras. Commuting zero divisors correspond to simultaneously + − measurable signals and can serve as the geometrical ba- 1= Dn + Dn . (2.3) sis for the physical events. Ordinary physical events are with the particular choice describing by the regular elements of the algebra.
+ 1 − 1 D = (1 + en) ,D = (1 en) . (2.4) n 2 n 2 − III. HYPER-NUMBERS The elements D+ and D− commute, since n n There are certain important types of physical phe- + − − + Dn Dn = Dn Dn =0 . (2.5) nomenon that are intrinsically 2-dimensional in nature. Many physical functions occur naturally in pairs and are Another type of zero divisors in split-algebras are expressed by 2-dimensional real numbers that satisfies Grassmann numbers defined as the set of anti-commuting a different type of algebra [11]. The most common 2- numbers G1, G2, ..., Gn with the properties dimensional algebra is the algebra of complex numbers,
3 however, there is no special reason to prefer one algebra The main reason why complex numbers are popular is to the other [12]. Euler’s formula. As it was mentioned by Feynman [15], Historically, the motivation to introduce complex num- ”the most remarkable formula in mathematics is: bers was mathematical rather than physical. This num- iθ bers made sense of the solution of algebraic equations, of e = cos θ + i sin θ . (3.4) the convergence of series, of formulae for trigonometric functions, differential equations, and many other things. This is our jewel. We may relate the geometry to the This was quite unlike the initial motivation for using real algebra by representing complex numbers in a plane numbers, which came about as an idealization of the kind iθ of quantity that directly arose from physical measure- x + iy = ρe . (3.5) ments. This is the unification of algebra and geometry.” Essential for the complex numbers Using (3.5) the rule for complex multiplication looks z = x + iy , (3.1) almost obvious as a consequence of the behavior of ro- tations in plane. We know that a rotation of α-angle where x and y are some real numbers, is existence of around the z-axis, can be represented by the unit element and one imaginary element i, with the iα i(θ+α) property i2 = 1. Later it was found that if we introduce e (x + iy)= ρe . (3.6) the conjugation− of complex numbers Another possible 2-dimensional normed algebra is the z∗ = x iy , (3.2) algebra of hyper-numbers − there algebra is division. A regular complex number can z = T + hx . (3.7) be represented geometrically by the amplitude and polar angle This has also a long history but is rarely used in physics y [16]. In (3.7) the quantities T, x are the real numbers and ρ2 = zz∗ = x2 + y2 , θ = arctan . (3.3) hyper-unit h has the properties similar to ordinary unit x vector The amplitude ρ is multiplicative and the polar angle θ 2 is additive upon the multiplication of complex numbers. h =1 . (3.8) Complex number have found many uses in physics, but these were considered as a ”mathematical tricks”, like Conjugation of h can be understand as reflection when h changes its direction on the opposite h∗ = h. So, the employment of complex numbers in 2-dimensional − hydrodynamics, electrical circuit theory, or the theory of in contrast with ordinary complex numbers, the norm vibrations. of a hyper-number is invariant under of rotation with In quantum mechanics, complex numbers enter at the reflection. foundation of the theory from probability amplitudes and Hyper-numbers constitute a commutative ring, but not the superposition principle. To pass from a quantum a field, since norm of a nonzero hyper-number information link, which does not respect the rules of N = zz∗ = T 2 x2 , (3.9) ordinary classical causality (as in the case of Einstein- − Podolsky-Rosen phenomenon) to a classical information can be zero. link (which necessarily propagates causally into the fu- A hyper-number does not have an inverse when its ture) and to obtain real number corresponding to the norm is zero, i. e., when x = T , or, alternatively, when probability, one must multiply the complex amplitude z = T (1 h). These two lines± whose elements have no by its complex conjugate. In fact, the complex conju- inverses,± play the same role as the point z = 0 does for gated amplitude may be thought of as applying to the the complexes, and provide the essential property of the quantum-information link in the reverse direction in time light cone, which makes the hyper-numbers relevant for [13]. the representation of relativistic coordinate transforma- Up to now it is admitted without justification that tions. Lorentz boosts can be succinctly expressed as two-valuedness of physical quantities naturally appear- ing in quantum mechanics is to be described in terms (T + hx) = (T ′ + hx′)ehθ , (3.10) of complex numbers. Complex numbers achieve a par- ticular representation of quantum mechanics in term where tanh θ = v. of which the fundamental equations take their sim- Any hyper-number (3.7) can be expressed in terms of plest form. Other choices for the representation of light cone coordinates the two-valuedness also possible, but would give to the Schrodinger equation a more complicated form (involving z = T + hx = (T + x)D+ + (T x)D− , (3.11) additional terms), although its physical meaning would − be unchanged [14]. where we introduced projection operators
4 1 ∗ ∗ 2 2 2 2 D± = (1 h) (3.12) N = qq = q q = a b c + d , (4.4) 2 ± − − has (2+2)-signature and in general is not positively de- corresponding to the critical signals. fined. The unit of the algebra of hyper-numbers is connected We see that the third unit element (ij) of split- with the critical signals and is universal for all physical quaternions is the vector product of the unit vectors i quantities (which are described by the ordinary elements and j and thus is a pseudo-vector. It is important to re- of the algebra). Thus it is just the hyper-unit is respon- alize that a pseudo-vector is not a geometrical object in sible for the signature of the space-time metric and the the usual sense. In particular, it should not be considered velocity of light c is related with it. Thus the norm (3.9) as a real physical arrow in space. can be understand as the distance and the constant c The pseudo-vector (ij) differs from the other two explicitly extracted from the unit element of the algebra hyper-complex units of split-quaternions i and j and be- haves like a pure imaginary object N = c2t2 x2 , (3.13) − (ij)∗ = (ij) , (ij)(ij)= i2j2 = 1 . (4.5) where quantity t now has the dimension of time. So − − − we can say that hyper-numbers are useful to describe In this fashion one can justify the origins of complex num- dynamics in (1+1)-space. bers
z = x + (ij)y , (4.6) IV. QUATERNIONS without introducing them ad-hoc. In this sense we can William Hamilton’s discovery of quaternions in 1843 say that split-quaternions are more rich in structure com- was the first time in history when the concept of 2- pared to Hamilton’s quaternions, since they contain com- dimensional numbers was successfully generalized. Many plex and hyper-complex numbers as particular cases. authors have proposed applications of quaternions in In the algebra of split quaternions two classes (totally physics [12,17]. four) projection operators General element of the quaternion algebra can be writ- 1 1 ten by using only the two basis elements i and j in the D± = (1 i) ,D± = (1 j) , (4.7) form i 2 ± j 2 ±
q = a + bi + (c + di)j , (4.1) can be introduced. The two classes does not commute with each other. The commuting projection operators where a,b,c and d are some real numbers. The third basis with the properties element of the quaternion (ij) is possible to obtained by + − + + + − − − composition of the first two. [D D ]=0 ,D D = D ,D D = D , (4.8) The quaternion reverse to (4.1), the conjugated quater- ± ± + ∗ are only the pairs Di , or Dj . The operators D and nion q , can be constructed using the properties of the − basis units under the conjugation (reflection) D differ from each other by reflection of the one basis element and thus correspond to the direct and reverse i∗ = i , j∗ = j , (ij)∗ = (ij) . (4.2) critical signals along one of the two real directions. − − − In the algebra we have also two classes of Grassmann A physically observable quantity is the norm of a quater- numbers nion N = qq∗ corresponding to the multiplication of the ± 1 ± 1 direct and reverse signals. Gi = (1 i)j , Gj = (1 j)i . (4.9) When the basis elements i and j are imaginary i2 = 2 ± 2 ± j2 = 1 (similar to ordinary complex unit) we have From the properties of zero divisors Hamilton’s− quaternion with the positively defined norm D±G± = G± ,D±G± = G± , ∗ ∗ 2 2 2 2 i i i j j j N = qq = q q = a + b + c + d . (4.3) ± ∓ ± ∓ Di Gi =0 ,Dj Gj =0 , (4.10) In this case the third orthogonal basis unit (ij) has prop- erties analogous to i and j (they all are similar to pseudo- we see that the Grassmann numbers (4.9) have pairwise vectors [4]). Because of this fact it is impossible to iden- commuting relations with the projection operators (4.7) tify pure quaternion with a vector [4,17]. ± ∓ ± ∓ For the positive squares i2 = j2 = 1 we have the al- [Di Gi ]=0 , [Dj Gj ]=0 . (4.11) gebra of split-quaternions. The unit elements i, j have Using commuting zero divisors any quaternion can be the properties of real unit vectors. The norm of a split- written in the form quaternion
5 (a + b) (c + d) q = a + ib + (c + id)j = q = (4.19) + + (c d) (a b) = D [a + b + (c + d)G ] + (4.12) − − +D−[a b + (c d)G−] , − − and the norm (4.4) is the determinant of this matrix ± ± where D and G are the projection operators and detq = (a2 b2) (c2 d2) . (4.20) Grassmann elements belonged to the one of the classes − − − from (4.7) and (4.9) labelled by i, or by j. If we treat the unit element of the algebra as the time, The quaternion algebra is associative and therefore can then split-quaternions could be used for describing of dy- be represented by matrices. We get the simplest non- namics in 2-dimensional space, just as hyper-numbers are trivial representation of the split-quaternion basis units used to study dynamics in 1-space. Here we have two if we choose the real Pauli matrices accompanied by the independent rotations in the (t i) and (t j) planes unit matrix. Note that for a real matrix representation described by D±, which similar− to hyper-numbers− case of Hamilton’s quaternions one needs 4-dimensional ma- (3.13) introduces fundamental constant c. trices. But what is the geometrical meaning of the Grassmann The independent unit basis elements of split quater- numbers and the fourth orthogonal element in the split- nions i and j have the following matrix representation quaternion algebra? There was no similar element in the algebra of hyper-numbers. 1 0 0 1 i = , j = . (4.13) It can be shown, that pseudo-vector (ij) is connected 0 1 1 0 − with the rotation in the (i j) plane. Indeed, the oper- − The third unit basis element (ij) is formed by multipli- ators cation of i and j and has the representation 1 R± = [1 (ij)] , (4.21) √ 0 1 2 ± (ij)= . (4.14) 1 0 − (which are ordinary complex numbers with the property R+R− = 1) represent the rotation by π/2 in (i j) plane. It is easy to noticed that, in contrast with the complex − case, the three real Pauli matrices have different proper- Thus they can be used to transform i into j ties, since their squares give the unit matrix R+iR− = j , R+jR− = i . (4.22) − 1 0 (1) = (4.15) Thus the extra pseudo-direction (ij) is similar to the an- 0 1 gular momentum and it is natural to assume that it can with different signs be connected with the property of the matter such as spin. i2 = j2 = (1) , (ij)2 = (1) . (4.16) In the algebra there is a symmetry in choice of the in- − ± ± dependent projection operators Di or Dj for the quater- Conjugation of the unit elements i and j means chang- nion decomposition (4.12). This means that we have two ing of signs of matrices i, j and (ij), thus independent space directions i and j, where in principle ii∗ = jj∗ = (1) , (ij)(ij)∗ = (1) . (4.17) signals could be exchanged. However, we can simultane- − ously measure signals only from the one direction (since ± ± Matrix representation of the independent projection the operators Di and Dj do not commute). This prop- operators and Grassmann elements from (4.7) and (4.9) erty is equivalent to the non-commutativity of coordi- labelled by i are nates [18]. The constant c, as for hyper-numbers (3.13), can be 1 1 0 extracted explicitly from the unit element of any quater- D+ = (1 + i)= , i 2 0 0 nion 1 0 0 D− = (1 i)= , q = ct + xi + yj + λ(ij) , (4.23) i 2 − 0 1 1 0 1 where λ is some quantity with the dimension of length. G+ = (j + ij)= , (4.18) i 2 0 0 Projection operators correspond to critical rotations for the (1 i) and (1 j) planes and introduce maxi- 1 0 0 G− = (j ij)= . mal velocity,− following from− the requirement to have the i 2 − 1 0 positive norm for the quaternion (4.23) It is easy to find also matrix representation of similar ∆x ∆y zero divisors labelled by the second index j. < 1 , < 1 . (4.24) The decomposition (4.12) of a quaternion now can be c∆t c∆t written in the form
6 Analogously Grassmann numbers correspond to critical As we have seen in previous sections, hyper-numbers rotations in the (i ij) and (j ij) planes. When con- are useful to describe dynamics in 1-dimensional space, ditions (4.24) are satisfied− for the− positive definiteness of and split-quaternions, in 2-dimensional space. According the norm, now we must write also the extra relations to the Hurwitz theorem there is only one further compos- ite division algebra, the octonion algebra [2]. We want to ∆x ∆y > 1 , > 1 . (4.25) show that split-octonions can describe dynamics in the ∆λ ∆λ usual 3-dimensional space. For the construction of the octonion algebra with the This means that there must exist the second fundamental general element constant (which can be extracted from λ) characterizing this critical property of the algebra. Since the pseudo- O = a0e0 + anen , n =1, 2, ..., 7 (5.1) direction (ij) is similar to angular momentum it is natu- ral to identify new constant with Plank’s constant ¯h and where e0 is the unit element and a0,an are real num- write λ in the form bers, the multiplication law of its eight basis units e0,en ¯h usually is given. For the case of ordinary octonions λ = , (4.26) 2 2 ∗ ∗ P e = e , e = e , e = e , e = en , (5.2) 0 0 n − 0 0 0 n − where quantity P has the dimension of the momentum. and norms are positively defined Inserting (4.26) into (4.25) we can conclude that uncer- tainty principle N = OO∗ = O∗O = 2 2 2 2 2 2 2 2 ∆x∆P > ¯h , (4.27) = a0 + a1 + a2 + a3 + a4 + a5 + a6 + a7 . (5.3) probably has same geometrical meaning as the existence The multiplication table for the basis elements (5.2) can of the maximal velocity. be written in the form To summarize, split quaternions with the norm (enem)= δnme + ǫnmkǫk , (5.4) − 0 ¯h2 2 2 2 2 where δnm is the Kronecker symbol and ǫnmk is the fully N = c t + 2 x y , (4.28) P − − anti-symmetric tensor with the postulated value ǫnmk = could be used to describe dynamics of a particle with spin +1 for the following values of indices in a 2-dimensional space. The two fundamental constants nmk = 123, 145, 176, 246, 257, 347, 365 . c and ¯h needed for this case have the geometrical origin and correspond to two kinds of critical signals in (2+2)- This definition of structure constants used by Cayley in space. The Lorentz factor his original paper but is not unique. There are 16 differ- ent possibilities of definition of ǫnmk leading to equivalent v2 ¯h2 F 2 algebras. Sometimes for visualization of the multiplica- γ = 1 + , (4.29) s − c2 c2 P 2 tion law of the octonion basis units the geometrical pic- ture of the Fano plane (a little gadget with 7 points and corresponding to a general boost in the space (4.28) con- 7 lines) also used [7]. Unfortunately both of these meth- tains an extra positive term, where F is some kind of ods are almost unenlightening. We want to consider a force. Thus, the dispersion relation in the (2+2)-space more obvious picture using properties of usual vectors (4.28) has a form similar to that of double-special rela- products. tivity models [19]. For the case of quaternions we had shown that not all basis elements of the algebra are independent. One eas- ily recover the full algebra from two fundamental basis V. OCTONIONS elements without consideration of multiplication tables or graphics. The same can be done with the octonions. Despite the fascination of octonions for over a century Now beside of unit element 1 we have the three funda- since their discovery in 1844-1845 by Graves and Cayley, mental basis elements i, j and l of split-octonions with it is fair to say that they still await universal acceptance. the properties of ordinary unit vectors The octonions have remained in the shadow for a long i2 = j2 = l2 =1 . (5.5) time. This is not to say that there have not been vari- ous attempts to find appropriate uses for them in physics Other four basis elements (ij), (il), (jl) and ((ij)l) can (see, for example [9,17,20,21]). Recently we can say that be constructed by vector multiplications of i, j and l. the octonions have revealed themselves as the most im- All brackets denote vector product and they show only portant number system of all, since they are crucial to the order of multiplication. Here we have some ambi- string theory [8]. guity of the placing of brackets for the eights element
7 ((ij)l). However, we can choose any convenient form, all dependent real Grassmann algebras with the elements ± ± ± the other possibilities are defined by the ordinary laws of Gn1, Gn2 and Gn3, where n runs over i,j,l and ((ij)l). triple vector multiplication. As it was mentioned in the In contrast with quaternions, octonions cannot be rep- introduction to denote scalar and vector multiplications resented by matrices with the usual multiplication lows of basis units we does not need to use different kind of [2]. The reason is the non-associativity, leading to the brackets, since the product of the different basis vectors is different rules for the left and right multiplications. always the vector product, and the product of two same Using commuting projection operators and Grassmann basis vectors is always the scalar product. numbers any split octonion Using properties of standard scalar and vector prod- ucts of i, j and l we can write the properties of other O = A + Bi + Cj + D(ij)+ four, not fundamental, basis elements of the octonion al- +El + F (il)+ G(jl)+ H(i(jl)) , (5.9) gebra where A,B,C,D,E,F,G and H are some real numbers, (i(jl)) = ((ij)l)= ((li)j) , − − can be written in the form (ij)= (ji) , (il)= (li) , (jl)= (lj) , − − − + + (ij)2 = (il)2 = (jl)2 = 1 , (5.6) O = D [(A + B) + (C + D)G1 + ∗ ∗ ∗ − + + (ij)(ij) = (il)(il) = (jl)(jl) =1 , +(E + F )G2 + (G + H)G3 ]+ − − 2 ∗ +D [(A B) + (C D)G + (5.10) ((ij)l) =1 , ((ij)l)((ij)l) = 1 . − − 1 − +(E F )G− + (G H)G−] . Conjugation as for the quaternion case means reflection − 2 − 3 of the basis units, or changing of there signs (except of Because of the symmetry between i, j and l using (5.7) unit element). These obvious properties of the scalar and and (5.8) we can find three different representation of vector products of basis elements are hidden if we write (5.10). the abstract octonion algebra (5.4). Two terms in formula (5.10) with the opposite signs in In the algebra of split-octonions it is possible to intro- the labels of zero divisors correspond to the direct and duce four classes (totally eight) of projection operators backward signals for the one of three real directions i, j, or l. However, as for quaternions, the different classes ± 1 ± 1 D = (1 i) ,D = (1 j) , of projection operators D ,D and D do not commute. i 2 ± j 2 ± i j l This is the analog of non-commutativity of coordinates ± 1 ± 1 Dl = (1 l) ,D(ij)l = (1 (ij)l) , (5.7) [18]. 2 ± 2 ± As in the case with quaternions, here we also have corresponding to the critical unit signal along i,j,l and the two fundamental constants c and ¯h corresponding to ((ij)l) directions. These four classes do not commute the two types of critical signals D± and G±. Thus any with each other. Only one class, i.e. any pair D± with octonion can be written in the form the same label from i,j,l and ((ij)l), is independent. In the algebra we have also the four classes (totally O = ct + xi + yj + zl + twenty four) of Grassmann numbers labelled by i,j,l and ¯h ¯h ¯h ((ij)l) + (ij)+ (il)+ (jl)+ ω(i(jl)) , (5.11) Pz Py Px
± 1 ± 1 Gi1 = (1 i)j , Gi2 = (1 i)l , where ω is some quantity with the dimension of length. 2 ± 2 ± We had used expressions similar to (3.13) and (4.28) to 1 G± = (1 i)(jl) , extract c andh ¯ from the components of (5.9) and to in- i3 2 ± troduce the quantities Px, Py and Pz with the dimensions ± 1 ± 1 of momentum. Gj1 = (1 j)i , Gj2 = (1 j)l , 2 ± 2 ± If we extract the two fundamental constants c andh ¯ 1 G± = (1 j)(il) , from ω, we find a quantity with the dimensions of energy j3 2 ± 1 1 c¯h G± = (1 l)i , G± = (1 l)j , (5.8) E = . (5.12) l1 2 ± l2 2 ± ω 1 G± = (1 l)(ij) , So the norm of the octonion l3 2 ± 2 2 2 ± 1 ± 1 ¯h ¯h ¯h G = (1 (ij)l)i , G = (1 (ij)l)j , N = OO∗ = c2t2 + + + (ij)l1 2 ± (ij)l2 2 ± 2 2 2 Px Py Pz − ± 1 G = (1 (ij)l)l . c2¯h2 (ij)l3 2 ± x2 y2 z2 , (5.13) − − − − E2 Only the numbers G1, G2 and G3 from each class with the same sign mutually commute. So we have eight in- becomes similar to a interval in some kind of phase space.
8 The appearance of Planck’s constant in definitions of in addition to (4.24) and (4.25), following from the pos- the intervals (5.13) means that the difference of the pro- itive definiteness of the norm (5.13) do not have the posed model from the standard approach mainly will take same meaning of the uncertainty relation as in (4.27). place for large energies and nonlinear fields. For exam- However, (5.16) leads to generalizations of standard un- ple, the non-associativity can be connected with Green’s certainty relation similar as considered in various ap- functions of strongly non-linear Quantum Field Theory proaches [23]. considered in [22]. Non-associativity of octonions follows from the prop- Specific to the representation (5.10) is the existence erty of triple vector products and appears in multiplica- of three zero divisors forming full Grassmann algebras tions of three different basis elements of octonion neces- with three elements for any direction. So for octonions sarily including as the one of the terms the eighth basis we have four type of commuting critical signals. One element ((ij)l). For example, we have corresponds to the rotations in (t x) plane giving the constant c, two correspond to the− different rotations in j (l ((ij)l)) = j ( (ij)) = i , − (x P ) planes, which give uncertainty relations similar (jl) ((ij)l) = (jl)((jl)i)= i . (5.17) to− (4.25) containing ¯h, and the last one corresponds to − the rotations in (P E) plane. Non-associativity of octonions is difficult to under- Note that in spite− of the fact that in the split-octonion stand if we only study the properties of basis elements. algebra there are four real basis vectors i,j,l and ((ij)l) For example, if one introduce so-called open multiplica- we can consider it to present only three ordinary space tion laws for the basis elements [21] (i.e. does not place dimensions, since there could be constructed only the the brackets) and formally use only the anti-commutation three independent rotation operators with the property properties , he find R+R− = 1, ijl = lij . (5.18) 1 R± = [1 (ij)] , ij √2 ± But anti-commuting comes from the properties of vector products and it is impossible to use it separately. If we ± 1 R = [1 (il)] , (5.14) put the brackets back then, from the properties of vector il √2 ± products, we discover the another result 1 R± = [1 (jl)] , jl √2 ± ((ij)l)= (l(ij)) . (5.19) − corresponding to rotations in the (i j), (i l) and (j l) This relation is obvious because expresses the vector − − − planes respectively. Another fact is that in the formulae product of two vectors l and (ij). (5.7) and (5.8) there is no symmetry between the zero Non-associativity of octonions is the result of the divisors along i, j, or l and along ((ij)l). The reason pseudo-vector character of the basis elements (ij), (il) is non-associativity of octonions resulting in non-unique and (jl), which have the properties of an imaginary unit, ± ± answers for the products of D(ij)l and G(ij)l. This means i.e. there square (scalar product) is negative. Even in that the commutating relations of these operators de- three dimensions the ordinary vector multiplication is pends on the order of multiplication of the basis elements non-associative and we could not write a unique decomposition (5.10) of octonion using D± and G± . [[ab]c] = [a[bc]] . (5.20) (ij)l (ij)l 6 The basis element ((ij)l) always appears for the rota- a b tions around i, j and l. For example, the rotation by π/2 For example, if and are two unit orthogonal 3-vectors around i (i.e. in the (j l) plane) by the operators (5.14) we have gives ((jl)i). After a− rotation on π we receive i and − [[aa]b]=0 , [a[ab]] = b . (5.21) only after the full rotation bt 2π do we come back to i, − indeed In 3-space the triple vector product + − + − RjliRjl = i(jl) = (ij)l , Rjl((ij)l)Rjl = i . (5.15) [a[bc]] = (ac)b (ab)c (5.22) − − − A similar situation occurs for the rotations around j and of three orthogonal vectors (describing by the determi- l. nant of three-on-three matrix) is zero. Considering vec- So non-associativity, which results in non-visibility of tor products of orthogonal vectors in more than three the fourth real direction, introduces fundamental uncer- dimensions, it is possible to construct extra vectors, nor- tainty in 3-space. Again because of non-associativity the mal to all three (using matrix with the more than three extra relations columns and rows). Then the triple vector product of ∆E three orthogonal vectors will be not zero and we shall ∆t∆E > ¯h , >c¯h , (5.16) ∆P automatically discover their non-associative properties.
9 In particular, when we have a product of three differ- c). From the positive definiteness of the norms there ent basis elements of octonion, and when one of them is arise uncertainty relations and fundamental uncertainty the eighth basis unit ((ij)l), for some order of multiplica- in measuring of coordinates (because of existence of a tion there necessarily arises the scalar product (square) fourth space-like direction with non-associative proper- of two pseudo-vectors from (ij), (il) and (jl), which give ties). the opposite sign in the result. We do not yet introduce any physical fields or equa- In general, the familiar 3-dimensional identity for triple tions. We have only shown that the proper algebra of vector product (5.22) does not follows from the experi- split-octonions could in principle describe the observed mentally established properties of vectors (1.6). Also, geometry and have introduced many necessary charac- it is not valid in seven dimensions, the only dimension teristics of the future particle physics model. other than three where the vector product can be writ- ten in principle [3,5]. Octonions have a weak form of associativity called al- ACKNOWLEDGEMENTS ternativity. In the language of basis elements this means that in the expressions with only two fundamental basis Author would like to acknowledge the hospitality ex- elements the placing of brackets is arbitrary. For exam- tended during his visits at Theoretical Divisions of CERN ple, and SLAC where this work was done. ((ii)l) = (i(il)) , ((li)i) = (l(ii)) . (5.23)
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