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2 Octonionic Geometry

Let us review the main ideas behind the geometrical application of split octonions pre- sented in our previous papers [13]. In this model some characteristics of physical world (such as , causality, maximal velocities, quantum behavior, etc.) can be natu- rally described by the properties of split octonions. Interesting feature of the geometrical interpretation of the split octonions is that, in addition to some other terms, their norms already contain the ordinary Minkowski metric. This property is equivalent to the exis- tence of local Lorentz invariance in classical physics. To any real physical signal we correspond 8-dimensional , the element of split octonions, n n s = ct + x Jn +¯hλ jn + chωI¯ , (n =1, 2, 3) (1) where we have one scalar basis unit (which we denote as 1), the three vector-like objects Jn, the three pseudovector-like elements jn and one pseudoscalar-like unit I. The eight real parameters that multiply basis elements in (1) we treat as the time t, the special coordinates xn, some quantities λn with the dimensions momentum−1 and the quantity ω having the dimension energy−1. We suppose also that (1) contains two fundamental constants of physics - the velocity of light c and the Planck constanth ¯. The squares of basis units of split octonions are inner product resulting unit element, but with the opposite signs,

2 2 2 Jn =1 , jn = −1 , I =1 . (2) Multiplications of different hyper-complex basis units are defined as skew products and the algebra of basis elements of split octonions can be written in the form: k JnJm = −JmJn = ǫnmkj , k jnjm = −jmjn = ǫnmkj , k Jnjm = −jmJn = −ǫnmkJ , (3)

JnI = −IJn = jn ,

jnI = −Ijn = Jn , where ǫnmk is the fully (n, m, k =1, 2, 3). From (3) we notice that to generate complete basis of split octonions the multiplica- tion and distribution laws of only three vector-like elements Jn, which describe special directions, are needed. In geometrical application this can explain why classical space has three dimensions. The three pseudovector-like basis units jn can be defined as the binary products 1 j = ǫ J mJ k , (4) n 2 nmk 2 and thus can describe oriented orthogonal planes spanned by two vector-like elements Jn. The seventh basic unit I (oriented volume) is formed by the products of all three fundamental basis elements Jn and has three equivalent representation,

I = J1j1 = J2j2 = J3j3 . (5) Multiplication table of octonionic units is most transparent in graphical form. To visualize the products of ordinary octonions usually the Fano triangle is used, where the seventh basic unit I is place at the center of the graphic. In the algebra of split octonions we have less symmetry and for proper description of the products (3) the Fano graphic should be modified by shifting I from the center of the Fano triangle. Also we shall use three equivalent representations of I, (5), and, instead of Fano triangle, we arrive to King David’s shape duality plane for products of split octonionic basis elements:

J3

II

j1 j2

J2 j3 J1

I

Figure 1: Display of split octonion multiplication rules David’s Star Plane

On this graphic the product of two basis units is determined by following the oriented solid line connecting the corresponding nodes. Moving opposite to the orientation of the line contribute a minus in the result. Dash lines on this picture just show that the corners of the triangle with I nodes are identified. Note that products of triple of basis units lying on a single line is associative and not lying on a single line are precisely anti-associative. Conjugation, what can be understand as a reflections of the vector-like basis units Jn, reverses the of octonionic basis elements in any given expression and thus

+ Jn = −Jn , 1 1 j+ = ǫ (J mJ k)+ = ǫ J k+J m+ = −j , (6) n 2 nmk 2 nmk n + + + + I =(Jnjn) = jn Jn = −I,

3 there is no summing in the last formula. So the conjugation of (1) gives

+ n n s = ct − xnJ − hλ¯ nj − chωI¯ . (7) Using the expressions (2) one can find that the norm of (1),

2 + + 2 2 n 2 n 2 2 2 s = ss = s s = c t − xnx +¯h λnλ − c h¯ ω , (8) has (4 + 4) signature. If we consider s as the interval between two octonionic signals we see that (8) reduce to the classical formula of Minkowski space-time in the limith ¯ → 0. Using the algebra of basis elements (3) the octonion (1) can be written in the equivalent form: n s = c(t +¯hωI)+ J (xn +¯hλnI) . (9) We notice that the pseudoscalar-like element I introduces the ’quantum’ term correspond- ing to some kind of uncertainty of space-time coordinates. For the differential form of (9) the invariance of the norm gives the relation:

2 2 n ds 2 dω v 2 dλ dλn = v 1 − h¯ − 1 − h¯ , (10) dt u dt  c2 dxm dx u ! " m # u t  where vn = dxn/dt denotes 3-dimensional velocity measured in the frame (1). So the generalized Lorentz factor (10) contains an extra terms and the dispersion relation in our model has a form similar to that of double- models [14]. Extra terms in (10) go to zero in the limith ¯ → 0 and ordinary Lorentz symmetry is restored. From the requirement to have the positive norm (8) from (10) we obtain several relations dxn dt v2 ≤ c2 , ≥ h¯ , ≥ h¯ . (11) dλn dω Recalling that λ and ω have dimensions of momentum−1 and energy−1 respectively, we conclude that Heisenberg uncertainty principle in our model has the same geometrical meaning as the existence of the maximal velocity in Minkowski space-time.

3 Generalized Lorentz transformations

To describe rotations in 8-dimensional octonionic space (1) with the interval (8) we need to define exponential maps for the basis units of split octonions. Since the squares of the pseudovector-like elements jn, as it is for ordinary complex 2 unit, is negative, jn = −1, we can define

jnθn e = cos θn + jn sin θn , (12) where θn are some real angles. At the same time for the other four basis elements Jn,I, which have the positive 2 2 squares Jn = I = 1, we have

Jnmn e = cosh mn + jn sinh mn , eIσ = cosh σ + I sinh σ , (13)

4 where mn and σ are real numbers. In 8-dimensional octonionic ’space-time’ (1) there is no unique plane orthogonal to a given axis. Therefore for the operators (12) and (13) it is not sufficient to specify a single rotation axis and an angle of rotation. It can be shown that the left multiplication of the octonion s by one of the operators (12), (13) (e.g. exp(j1θ1)) yields four simultaneous rotations in four mutually orthogonal planes. For the simplicity we consider only the left products since it is known that for octonions one side multiplications generate all the symmetry group that leave octonionic norms invariant [15]. So rotations naturally provide splitting of a octonion in four orthogonal planes. To define these planes note that one of them is formed just by the hyper-complex element that we choice to define the rotation (j1 in our example), together with the scalar unit element of the octonion. Other three orthogonal planes are given by the three pairs of other basis elements that lay with the considered basis unit (j1 in the example) on the lines emerged it in the David’s Star (see Figure 1). Thus the pairs of basis units that are rotate into each other are just the pairs products of which give considered basis unit and thus form an associative triplets with it. For example, the basis unit j1, according to Figure 1, have three different representations in the octonionic algebra,

j1 = J2J3 = j2j3 = J1I. (14)

Than orthogonal to (1 − j1) planes are (J2 − J3), (j2 − j3) and (J1 − I). Using (14) and the representation (12) its possible to ’rotated out’ four octonionic axis and (1) can be written in the equivalent form

j1θt j1θx j1θλ j1θω s = Nte + Nxe J3 + Nλe j2 + Nωe I, (15) where

2 2 2 2 2 2 Nt = c t +¯h λ1 , Nx = x2 + x3 , q q 2 2 2 2 2 2 Nλ =h ¯ λ2 + λ3 , Nω = x1 + c h¯ ω , (16) q q are the norms in four orthogonal octonionic planes and the angles are done by:

θt = arccos(t/Nt) , θx = arccos(x3/Nx) ,

θλ = arccos(¯hλ2/Nλ) , θω = arccos(chω/N¯ ω) .

This decomposition of split octonion is valid only if the full norm of the octonion (8) is positive, i.e. 2 2 2 2 2 s = Nt − Nx + Nλ − Nω > 0 . (17)

Similar to (15) decomposition exists for the other two pseudovector-like basis units j2 and j3. In contrast with uniform rotations giving by the operators jn we have limited rota- tions in the planes orthogonal to (1 − Jn) and (1 − I). However, we can still perform similar to (15) decomposition of s using expressions of the exponential maps (13). But now, unlike to (16), the norms of corresponding planes are not positively defined and, instead of the condition (17), we should require positiveness of the norms of each four

5 planes. For example, the pseudoscalar-like basis unit I have three different represen- tations (5) and it can provide the hyperbolic rotations (13) in the orthogonal planes (1 − I), (J1 − j1), (J2 − j2) and (J3 − j3). The expressions for the 2-norms (16) in this case 2 2 2 2 2 2 2 2 2 2 2 2 are: c t − h¯ ω , x1 − h¯ λ1, x2 − h¯ λ2 and x3 − h¯ λ3. Nowq let us considerq active andq passive transformationsq of coordinates in 8-dimensional space of signals (1). With a passive transformation we mean a change of the coordinates t, xn,λn and ω, as opposed to an active transformation which changes the basis 1,Jn, jn and I. The passive transformations of the octonionic coordinates t, xn,λn and ω, which leave invariant the norm (8) form just SO(4, 4), obviously. We can represent these transforma- tions of (1) by the left products s′ = Rs , (18) where R is one of (12), (13). The operator R simultaneously transform four planes of s. However, in three planes R can be rotate out by the proper choice of octonionic basis. Thus R can represent passive independent rotations in four orthogonal planes of s separately. Similarly we have some four angles for other six operators (12), (13) and thus totally 4 × 7 = 28 parameters corresponding to SO(4, 4) group of passive coordinate transformations. For example, in the case of the decomposition (15) we can introduce four arbitrary angles φt,φx,φλ and φω and

′ j1(θt+φt) j1(θx+φx) j1(θλ+φλ) j1(θω+φω) s = Nte + Nxe J3 + Nλe j2 + Nωe I. (19)

Obviously under this transformations the norm (8) is invariant. By the fine tuning of the angles in (19) we can define rotations in any single plane from four. Now let us consider active coordinate transformations, or transformations of eight oc- tonionic basis units 1,Jn, jn and I. For them, because of non-associativity, result of two different rotations (12) and (13) are not unique. This means that not all active octonionic transformation generated by (12) and (13) form a group and can be considered as a real ro- tation. Thus in the octonionic space (1) not to the all passive SO(4, 4)-transformations we can correspond active ones, only the transformations that have realization with associative multiplications should be considered. It was found that these associative transformations can be done by the combine rotations of special form in at least two octonionic planes. This kind of rotations also form a group ( of SO(4, 4)), known as the automor- NC phism group of split octonions (the real non-compact form of Cartan’s exceptional NC Lee group G2). Some general results on G2 and its subgroup structure one can find in [16]. Let us remind that the A of a algebra is defined as the transformations of the hyper-complex basis units x and y under which multiplication table of the algebra is invariant, i.e.

A(x + y)= Ax + Ay , (Ax)(Ay)=(A(xy)) . (20)

Associativity of this transformations is obvious from the second relation and the set of all of composition algebras form a group. In the case of quaternions, because

6 of associativity, active and passive transformations, SU(2) and SO(3) respectively, are isomorphic and quaternions are useful to describe rotations in 3-dimensional space. There is different situation for octonions. Each automorphism in the octonionic algebra is com- pletely defined by the images of three elements that are not form quaternionic , or all are not lay on the same David’s lime [17]. Consider one such set, say (j1, j2,J1). Then there exists an automorphism such that ′ j1 = j1 , ′ j2 = j2 cos(α1 + β1)/2+ j3 sin(α1 + β1)/2 , (21) ′ J1 = J1 cos β1 + I sin β1 , where α1 and β1 are some independent real angles. By definition (20) automorphism does not affect unit scalar 1. The images of the other basis elements under automorphism (21) are determined by the conditions: ′ ′ ′ j3 = j1j2 = j3 cos(α1 + β1)/2 − j2 sin(α1 + β1)/2 , ′ ′ ′ I = J1j1 = I cos β1 − J1 sin β1 , ′ ′ ′ J2 = j2I = J2 cos(α1 − β1)/2+ J3 sin(α1 − β1)/2 , (22) ′ ′ ′ J3 = j3I = J3 cos(α1 − β1)/2 − J2 sin(α1 − β1)/2 . ′ ′ ′ It can easily be checked that transformed basis Jn, jn,I satisfy the same multiplication rules as Jn, jn,I. There exists similar automorphisms with fixed j2 and j3 axis, which are generated by the angles α2, β2 and α3, β3 respectively. One can define also hyperbolic automorphisms for the vector-like units Jn by the angles un,kn. For example, if we fix the axis J1 then corresponding to (21) and (22) transformations are ′ J1 = J1 , ′ J2 = J2 cosh(k1 + u1)/2+ j3 sinh(k1 + u1)/2 , ′ I = I cosh u1 − j1 sinh u1 , ′ ′ ′ j3 = J1J2 = j3 cosh(k1 + u1)/2+ j2 sinh(k1 + u1)/2 , (23) ′ ′ ′ j1 = J1I = j1 cosh u1 − I sinh u1 , ′ ′ ′ j2 = J2I = j2 cosh(k1 − u1)/2+ J3 sinh(k1 − u1)/2 , ′ ′ ′ J3 = j3I = J3 cosh(k1 − u1)/2+ J2 sinh(k1 − u1)/2 . Similarly automorphism with the fixed seventh axis I has the form: I′ = I, ′ j1 = j1 cosh σ1 + J1 sinh σ1 , ′ j2 = j2 cosh σ2 + J2 sinh σ2 , ′ ′ ′ J1 = j1I = J1 cosh σ1 + j1 sinh σ1 , (24) ′ ′ ′ J2 = j2I = J2 cosh σ2 + j2 sinh σ2 , ′ ′ ′ j3 = j1j2 = j3 cosh(σ1 + σ2) − J3 sinh(σ1 + σ2) , ′ ′ ′ J3 = j3I = J3 cosh(σ1 + σ2) − j3 sinh(σ1 + σ2) .

7 So for each octonionic basis there are seven independent automorphism each introduc- NC ing two angles that correspond to 2×7 = 14 generators of the algebra G2 . For our choice of octonionic basis infinitesimal passive transformation of the coordinates, corresponding NC to G2 , can be written as t′ = t , 1 h¯ x′ = x − ǫ αj − βj xk + chβ¯ iω + U − ǫ uj λk , i i 2 ijk 2 ik ijk 1  1    ω′ = ω − β xi − u λi , (25) ch¯ i c i 1 1 λ′ = λ − ǫ αj + βj λk − cuiω + U + ǫ uj xk , i i 2 ijk 2¯h ik ijk     where Uik is the symmetric

2σ1 k3 k2 U =  k3 2σ2 k1  . (26) k k −2(σ + σ )  2 1 1 2    In the limit (¯hλ, hω¯ → 0) the transformations (25) reduce to the standard O(3) rota- tions of Euclidean 3-space by the Euler angles φn = αn − βn. However, for some problems in quantum regime extra symmetries can be retrieved. The formulas (25) represent rotations of (3,4)-sphere that is orthogonal to the time co- ordinate t. To define the busts note that active and passive form of mutual transformations of t with xn,λn and ω are isomorphic and can be described by the seven operators (12) and (13) (e.g. first term in (19)), which form the group O(3, 4). In the case (¯hλ, hω¯ → 0) we stay with the standard O(3) Lorentz boost in the Minkowski space-time governing by Jnmn the operators e , where mn = arctan vn/c.

4 Conclusion

In this paper the David’s Star shape duality plane that describe multiplications table of basis units of split octonions (instead of the Fano triangle of ordinary octonions) was introduced. Different kind of rotations in the split octonionic space were considered. It was shown that in octonionic space active and passive transformations of coordinates are not equivalent. The group of passive coordinate transformations, which leave invariant the norms of split octonions, is SO(4, 4), while active rotations are done by the direct NC product of the seven O(3, 4)-boosts and fourteen G2 -rotations. In classical limit these transformations give the standard 6-parametrical Lorentz group.

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