Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2015, Article ID 196708, 14 pages http://dx.doi.org/10.1155/2015/196708

Research Article Geometrical Applications of Split Octonions

Merab Gogberashvili1,2 and Otari Sakhelashvili1

1 Tbilisi Ivane Javakhishvili State University, 3 Chavchavadze Avenue, 0179 Tbilisi, Georgia 2Andronikashvili Institute of Physics, 6 Tamarashvili Street, 0177 Tbilisi, Georgia

Correspondence should be addressed to Merab Gogberashvili; [email protected]

Received 16 August 2015; Accepted 28 September 2015

Academic Editor: Yao-Zhong Zhang

Copyright © 2015 M. Gogberashvili and O. Sakhelashvili. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3 + 1)-theory (e.g., number of dimensions, existence of maximal velocities, Heisenberg uncertainty, and particle generations). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie group G2. This group generates specific rotations of (3 + 4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitates standard Poincare transformations. In this picture translations are represented by noncompact Lorentz-type rotations towards the extra time-like coordinates. It is shown how the G2 algebra’s chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin), as well as a parity violating effect on light emitted by a moving quantum system. Elementary particles are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions lead to free particle Lagrangians, which allow virtual trajectories also and exhibit the appearance of spatial horizons governing by mass parameters.

1. Introduction the possible impact of octonions on color symmetry [9–13]; GUTs [14–17]; representation of Clifford algebras [18–20]; Many properties of physical systems can be revealed from the quantum mechanics [21–29]; space-time symmetries [30, 31]; analysis of proper mathematical structures used in descrip- field theory [32–35]; quantum Hall effect [36]; Kaluza-Klein tions of these systems. The geometry of space-time, one of the main physical characteristics of nature, can be understood program without extra dimensions [37–42]; Strings and M- as a reflection of symmetries of physical signals we receive theory [43–48]; SUSY [49]; and so forth. andofthealgebrausedinthemeasurementprocess.Sinceall The essential feature of all normed composition algebras observable quantities we extract from single measurements is the existence of a real unit element and a different number are real, in geometrical applications it is possible to restrict of hypercomplex units. The square of the unit element is ourselves to the field of real numbers. To have a transition always positive, while the squares of the hypercomplex basis fromamanifoldoftheresultsofmeasurementstogeometry, units can be negative as well. In physical applications one onemustbeabletointroduceadistancebetweensome mainly uses division algebras with Euclidean norms, whose objects and an etalon (unit element) for their comparison. In hypercomplex basis elements have negative squares, similar algebraic language all these physical requirements mean that to the ordinary complex unit 𝑖.Theintroductionofvector- to describe geometry we need a composition algebra with the like basis elements (with positive squares) leads to the split unit element over the field of real numbers. algebras with pseudo-Euclidean norms. Besides usual real numbers, according to the Hurwitz In this paper we propose parameterizing world-lines theorem, there are three unique normed division algebras— (paths)ofphysicalobjectsbytheelementsofrealsplit complex numbers, quaternions, and octonions [1–3]. Physical octonions [50–53]: applications of the widest normed algebra of octonions 𝑛 𝑛 are relatively rare (see reviews [4–8]). One can point to 𝑠=𝜔+𝜆 𝐽𝑛 +𝑥 𝑗𝑛 +𝑐𝑡𝐼 (𝑛=1,2,3) . (1) 2 Advances in Mathematical Physics

A pair of repeated upper and lower indices, by the standard So the conjugation of (1) gives 𝑛 𝑛 𝑚 convention, implies a summation; that is, 𝑥 𝑗𝑛 =𝛿𝑛𝑚𝑥 𝑗 , 𝑛𝑚 † 𝑛 𝑛 where 𝛿 is Kronecker’s delta. 𝑠 =𝜔−𝜆𝑛𝐽 −𝑥𝑛𝑗 −𝑐𝑡𝐼. (8) In geometric application four of the eight real parameters 𝑛 𝑛 Using (2), (5), and (8) one can find that the norm of (1) is in (1), 𝑡 and 𝑥 , are space-time coordinates; 𝜔 and 𝜆 are given by interpretedasthephase(classicalaction)andthewavelengths 2 † † 2 2 2 2 2 associated with the octonionic signals [50–53]. 𝑁 =𝑠𝑠 =𝑠𝑠=𝜔 −𝜆 +𝑥 −𝑐 𝑡 , (9) The eight basis units in (1) are represented by one 1 𝐽 2 𝑛 2 𝑛 scalar (denoted by ), the three vector-like objects 𝑛,the (where 𝜆 =𝜆𝑛𝜆 and 𝑥 =𝑥𝑛𝑥 ), representing some kind of 𝑗 three pseudovector-like elements 𝑛,andonepseudoscalar- 8-dimensional space-time with (4+4)-signature and reducing 𝐼 2 2 like unit . The squares (inner products) of seven of the to the classical formula of Minkowski intervals if 𝜔 −𝜆 =0. hypercomplexbasiselementsofsplitoctonionsgivetheunit 1 As for the case of ordinary Minkowski space-time, we element, , with the different signs: assume that for physical events the corresponding “intervals” given by (9) are nonnegative. A second condition is that for 𝐽2 =1, 𝑛 physical signals the vector part of split octonions (1) should 2 be time-like: 𝑗𝑛 =−1, (2) 2 2 𝑛 𝑛 𝑐 𝑡 +𝜆 𝜆 >𝑥𝑥 . (10) 𝐼2 =1. 𝑛 𝑛 The classification of split octonions by the values of their Itisknownthattogenerateacompletebasisofsplitoctonions norms is presented in Appendix A. the multiplication and distribution laws of only three vector- 𝐽 like elements 𝑛 areenough[1–3].Thethreepseudovector- 2. Transformations and Automorphisms like basis units, 𝑗𝑛, can be defined as the binary products, To find the geometry associated with the signals (1) we 1 𝑚 𝑘 𝑗𝑛 = 𝜀𝑛𝑚𝑘𝐽 𝐽 (𝑛,𝑚,𝑘=1,2,3) (3) need to represent rotations by split octonions; due to the 2 nonassociativity of the algebra the group of rotations of 𝑁𝐶 octonionic axes 1, 𝐽𝑛, 𝑗𝑛,and𝐼, 𝐺 is not equivalent to (𝜀𝑛𝑚𝑘 is the totally antisymmetric unit tensor of rank three), 2 and thus describes orthogonal planes spanned by two vector- the group of accompanied passive tensorial transformations of coordinates 𝜔, 𝜆𝑛, 𝑥𝑛,and𝑡 (𝑆𝑂(4, 4)). In Appendix B like elements 𝐽𝑛.Theseventhbasisunit𝐼 (the oriented volume) is defined as the triple product of all three vector-like it is shown that by rotations in four orthogonal planes of elements and has three equivalent representation in terms of octonionic space (one of which always includes the axis of 𝑛 𝑛 𝜔 𝐽 and 𝑗 : real numbers ) the octonion (1) always can be represented as the sum of four elements. The left multiplication, 𝑅𝑠,byan 𝐼=𝐽1𝑗1 =𝐽2𝑗2 =𝐽3𝑗3. (4) octonion with unit norm (Appendix A), 𝑅=𝑒𝜖𝜃, So the algebra of all noncommuting hypercomplex basis units (11) of split octonions has the form where 𝜖 is the (3+4)-vector defined by seven basis units 𝐽𝑛, 𝑗𝑛, 𝐼 𝜃 𝑘 and ,representsrotationsbytheangles in these four planes. 𝐽 𝐽 =−𝐽 𝐽 =𝜀 𝑗 , −1 𝑛 𝑚 𝑚 𝑛 𝑛𝑚𝑘 The right product, 𝑠𝑅 , will reverse the direction of rotation 𝜔 𝑗 𝑗 =−𝑗 𝑗 =𝜀 𝑗𝑘, in the plane of . So we can represent a rotation, which only 𝑛 𝑚 𝑚 𝑛 𝑛𝑚𝑘 affects the (3+4)-vector part of 𝑠, applying the unit half-angle 𝑗 𝐽 =−𝐽𝑗 =𝜀 𝐽𝑘, (5) octonion twice and multiplying on both the left and the right 𝑚 𝑛 𝑛 𝑚 𝑛𝑚𝑘 (with its inverse): 𝐽 𝐼=−𝐼𝐽 =𝑗, 󸀠 −1 † 𝜖𝜃/2 −𝜖𝜃/2 𝑛 𝑛 𝑛 𝑠 =𝑅𝑠𝑅 =𝑅𝑠𝑅 =𝑒 𝑠𝑒 . (12) 𝑗 𝐼=−𝐼𝑗 =𝐽. 𝑛 𝑛 𝑛 These maps are well defined, since the associator of the triplet −1 (𝑅, 𝑠, 𝑅 ) vanishes. The set of rotations (12) of the seven Conjugations of octonionic basis units, which can be 𝑛 𝑛 octonionic coordinates 𝑥 , 𝜆 ,and𝑡,inthreeplanes,whichdo understood as the reflection of vector-like elements, not affect the scalar part 𝜔 of (1), contains 7×3 = 21angles of † the group 𝑆𝑂(3, 4) of passive transformations of coordinates. 𝐽 =−𝐽𝑛, (6) 𝑛 To represent the active rotations in the space of 𝑠, reverses the order of 𝐽𝑛 in products; that is, which preserve the norm (9) and multiplicative structure of octonions(5)aswell,wewouldneedthetransformations † 1 𝑚 𝑘 † 1 𝑘† 𝑚† to be automorphisms (Appendix C). An automorphism 𝑈 of 𝑗 = (𝜀𝑛𝑚𝑘𝐽 𝐽 ) = 𝜀𝑛𝑚𝑘𝐽 𝐽 =−𝑗𝑛, 𝑛 2 2 any two octonions, 𝑂1 and 𝑂2,givestheequation (7) † † † † † (𝑈𝑂 𝑈−1)(𝑈𝑂 𝑈−1)=𝑈𝑂𝑂 𝑈−1, 𝐼 =(𝐽1𝐽2𝐽3) =𝐽3 𝐽2 𝐽1 = −𝐼. 1 2 1 2 (13) Advances in Mathematical Physics 3

𝑛 𝑛 which only holds in general if multiplication of 𝑈, 𝑂1,and 𝛼 and 𝛽 , respectively; boosts, mixing of spatial and time- 𝑛 𝑛 𝑂2 is associative. Combinations of the rotations (12) around like coordinates by the two hyperbolic 3-angle 𝜃 and 𝜙 ; different octonionic axes are not unique. This means that and diagonal boosts of the spatial coordinates, 𝑥𝑛,and not all transformations of 𝑆𝑂(3, 4) form a group and can corresponding time-like parameters, 𝜆𝑛, by the hyperbolic 𝑛 be considered as real rotations. Only the transformations angles 𝜑 . 𝑛 that have a realization as associative multiplications should We notice that if we consider rotations by the angles 𝛼 , 𝑛 𝑛 be considered. Also note that an automorphism of split 𝛽 ,and𝜃 , that is, assuming that octonions can be generated only by the octonions with 𝑚 𝑚 positive norms (Appendix A), since the set of split octonions 𝜙 =𝜑 =0, (16) with negative norms do not form a group (it is not closed under multiplication). So to model space-time symmetries 𝑁𝐶 the passive 𝐺 -transformations (14) of only ordinary space- we need to use the group of automorphisms of split octonions 2 time coordinates, 𝑥𝑛 and 𝑡, will imitate the ordinary infinites- with positive norms. imal Poincare´ transformations of (3 + 1)-Minkowski space: It is known that associative transformations of split octo- nions can be done by the specific simultaneously rotations in 𝑥󸀠 =𝑥 −𝜀 𝛼𝑚𝑥𝑘 −𝜃 𝑐𝑡 +𝑎 , two (and not in three as for 𝑆𝑂(3, 4))orthogonaloctonionic 𝑛 𝑛 𝑛𝑚𝑘 𝑛 𝑛 planes (Appendix C). These rotations form a subgroup of (17) 𝑐𝑡󸀠 =𝑐𝑡−𝜃𝑥𝑛 +𝑎. 𝑆𝑂(3, 4) with 2×7=14parameters, known as the automor- 𝑛 0 𝑁𝐶 phism group of split octonions, 𝐺2 . Generators of this real noncompact form of Cartan’s smallest exceptional Lie group Here the space-time translations, 𝐺 2 is presented in Appendix D. 1 𝑎 = 𝜀 𝜃𝑚𝜆𝑘, Infinitesimal transformations of coordinates which 𝑛 2 𝑛𝑚𝑘 accompany the group of active transformations of octonionic (18) 𝑁𝐶 𝑛 basis units, 𝐺2 , and preserve the diagonal quadratic form, 𝑎 =−𝛽𝜆 , 2 2 2 0 𝑛 𝑥 −𝜆 −𝑑𝑡 ≥0,canbewrittenintheform(AppendixC) are generated by the Lorentz-type rotations toward the 𝑥󸀠 =𝑥 −𝜀 𝛼𝑚𝑥𝑘 −𝜃 𝑐𝑡 𝑛 𝑛 𝑛𝑚𝑘 𝑛 time-like directions 𝜆𝑛. So in the language of octonionic geometry any motion in ordinary space-time is generated 1 󵄨 󵄨 𝑚 𝑚 𝑘 𝑛 𝑛 2 + (󵄨𝜀 󵄨 𝜙 +𝜀 𝜃 )𝜆 by 𝜆 ∼𝑝/𝑝 . Time translations 𝑎0 are smooth, since 𝛽𝑛 󵄨 𝑛𝑚𝑘󵄨 𝑛𝑚𝑘 𝑚 2 are compact angles. However, the angles 𝜃 are hyperbolic 𝑎 1 and for any active spatial translation 𝑛 there exists a horizon +(𝜑𝑛 − ∑𝜑𝑚)𝜆𝑛, (analogues to the Rindler horizon), which is equivalent to the 3 𝑚 introduction of some mass scale or inertia (see Section 6). 󸀠 𝑛 𝑛 For completeness note that there exists a second well- 𝑐𝑡 =𝑐𝑡−𝛽𝑛𝜆 −𝜃𝑛𝑥 , (14) 𝑁𝐶 known representation of 𝐺2 as the symmetry group of 𝜆󸀠 =𝜆 −𝜀 (𝛼𝑚 −𝛽𝑚)𝜆𝑘 +𝛽 𝑐𝑡 aballrollingonalargerfixedballwithoutslippingor 𝑛 𝑛 𝑛𝑚𝑘 𝑛 twisting, when the ratio of the balls radii is 1/3 [54–58]. 1 󵄨 󵄨 Understanding the exceptional Lie groups as the symmetry + (󵄨𝜀 󵄨 𝜙𝑚 −𝜀 𝜃𝑚)𝑥𝑘 groups of naturally occurring objects is a long-standing 2 󵄨 𝑛𝑚𝑘󵄨 𝑛𝑚𝑘 program in mathematics. Symmetries of rolling balls are 𝑥 𝜆 1 visualized in (14) by the presence of two 3-vector, 𝑛 and 𝑛, +(𝜑 − ∑𝜑 )𝑥 , one of which is time-like. The fourth time-like coordinate 𝑛 3 𝑚 𝑛 𝑚 (ordinary time 𝑡), which also is affected by (14), breaks the symmetry between “balls” and the factor 1/3 of the ratio of 𝑛 𝑥󸀠 𝜆󸀠 with no summing over in the last terms of 𝑛 and 𝑛.From their radii corresponds to the existence of the three extra 𝛼𝑚 𝛽𝑚 𝜙𝑚 𝜃𝑚 𝜑𝑚 the five 3-angles in (14), , , , ,and ,only14are time-like coordinates 𝜆𝑛. independent because of the condition 1 3. Boosts, Heisenberg Uncertainty, ∑ (𝜑𝑛 − ∑𝜑𝑚)=0. (15) 𝑛 3 𝑚 and Chirality

The Lorentz-type transformations (14) of the 7-dimen- In the algebra of split octonions there exist only three sional space with four time-like coordinates should describe space-like parameters, 𝑥𝑛, and three independent compact space-time symmetries if the split octonions form relevant rotations around corresponding axes, which explains why algebraic structures for microphysics. The transformations physical space described by octonionic signals has three 𝑁𝐶 (14) formally can be divided into three distinct classes spatial dimensions. Let us analyze new features of the 𝐺2 - (Appendix C): Euclidean rotations of the spatial, 𝑥𝑛,and transformations (14) in comparison with standard Lorentz’s time-like (𝑡 and 𝜆𝑛)coordinatesbythecompact3-angle formulas for (3 + 1)-Minkowski space. 4 Advances in Mathematical Physics

Euclidean rotations around one of the space-like axes, 𝑥1, that is, the quantities 𝜆2 and 𝜆3 are inversely proportional the automorphism (C.1), correspond to the following passive to the velocity, 𝑥1/𝑡.Inthespaceofsplitoctonions(1)there infinitesimal transformations of coordinates: are two classes of time-like parameters (𝑡 and 𝜆𝑛)andtwo different light-cones. So, in addition to 𝑐, there must exist 𝑥󸀠 =𝑥, 1 1 thesecondfundamentalconstant,whichcanbeextracted 𝜆 𝑥󸀠 =𝑥 +𝛼 𝑥 , from 𝑛. In quantum mechanics the quantity with the 2 2 1 3 dimension of length, which is proportional to a fundamental 󸀠 physical constant and is inversely proportional to velocity (or 𝑥3 =𝑥3 −𝛼1𝑥2, momentum, 𝑝𝑛), is called the wavelength. So it is natural to 󸀠 assume that 𝑐𝑡 =𝑐𝑡−𝛽1𝜆1, (19) 𝑝𝑛 ℏ 𝜆󸀠 =𝜆 +𝛽 𝑐𝑡, 𝜆𝑛 =ℏ ∼ , 1 1 1 2 (23) 𝑝 𝑝𝑛 𝜆󸀠 =𝜆 +(𝛼 −𝛽 )𝜆 , 2 2 1 1 3 𝑛 where ℏ is the Planck constant and 𝑝 is the 3-momentum 󸀠 𝜆3 =𝜆3 −(𝛼1 −𝛽1)𝜆2. associated with the octonionic signal. So in our approach the two fundamental physical constants, 𝑐 and ℏ,have When 𝛼1 =𝛽1 this reduces to the standard rotation by geometrical origin and correspond to two kinds of light-cone the Euler angle 𝛼1, which also causes translations of time, signals in the space of split octonions (9). 𝑡, due to mixing with the extra time-like parameter 𝜆1. When 𝜙1 =0from (20) we find that the ratios, In general, any active Euclidean 3-rotation of the spatial 𝑥𝑛 Δ𝑥 𝜆 coordinates changes the time parameter by the amount of 2 =− 3 , 𝜆 the corresponding 𝑛,whichcanbeunderstoodaspassingof Δ𝜆2 𝑥3 time in our world. (24) 𝑁𝐶 𝐺 Δ𝑥3 𝜆2 Analysis of boosts of 2 can help us in the physical =− , interpretation of the extra time-like parameters 𝜆𝑛. Consider Δ𝜆3 𝑥2 the automorphisms (C.2) by the hyperbolic angles 𝜃1 and 𝜙1: Δ𝑥 =𝑥󸀠 −𝑥 Δ𝜆 =𝜆󸀠 −𝜆 󸀠 (where 𝑛 𝑛 𝑛 and 𝑛 𝑛 𝑛), are unchanged 𝑥1 =𝑥1 −𝜃1𝑐𝑡, under infinitesimal transformations (20). Similar relations 1 can be obtained for the boosts along two other space-like axes. 𝑥󸀠 =𝑥 + (𝜙 −𝜃 )𝜆 , From the invariance of octonionic norms (B.14) we know that 2 2 2 1 1 3 󵄨 󵄨 󵄨 󵄨 1 󵄨𝑥𝑛󵄨 ≳ 󵄨𝜆𝑛󵄨 . (25) 𝑥󸀠 =𝑥 + (𝜙 +𝜃 )𝜆 , 󵄨 󵄨 󵄨 󵄨 3 3 2 1 1 2 󸀠 Then inserting (23) and (25) into (24) we can conclude that 𝑐𝑡 =𝑐𝑡−𝜃1𝑥1, (20) the uncertainty relations, 𝜆󸀠 =𝜆 , 1 1 Δ𝑥𝑛Δ𝑝𝑛 ≥ℏ, (26) 1 𝜆󸀠 =𝜆 + (𝜙 +𝜃 )𝑥 , 2 2 2 1 1 3 in our model have the same geometrical meaning as the existence of the maximal velocity, 𝑐 [50–53, 59]. 󸀠 1 From (20) we also notice that in the planes orthogonal to 𝜆3 =𝜆3 + (𝜙1 −𝜃1)𝑥2. 󸀠 󸀠 󸀠 󸀠 2 V1 =𝑥1/𝑡 the pair 𝑥2, 𝜆3 increases, while 𝑥3, 𝜆2 decreases, andforthecase(21)wehave The case 𝜙1 =𝜃1 corresponds to ordinary boost in (𝑡, 𝑥)- planes—transitions to the reference frame moving with the 𝑥2 𝑥3 𝜃 𝑥 =− =𝜃1. (27) velocity 1 along the axis 1. Then if we consider the motion 𝜆3 𝜆2 of the origin of the moving system, Using similar relations for the boosts along other two spatial 𝑥󸀠 =𝑦󸀠 =𝑧󸀠 =0, (21) directions we conclude that there exists some 3-vector (spin): from the first line in (20) we find that 1 𝑥𝑚 𝜎 = 𝜀 𝑥𝑚𝑝𝑘 ∼𝜀 , 𝑥 𝑛 ℏ 𝑛𝑚𝑘 𝑛𝑚𝑘 𝜆 (28) 𝜃 = 1 , 𝑘 1 𝑐𝑡 𝑥 which characterizes relative rotations of 𝑥𝑛 and 𝜆𝑛 in the 𝜆 = 2 , 3 𝜃 (22) direction orthogonal to the velocity planes. One can speculate 1 on the connection of the left-right asymmetry in relative 𝑥3 rotations of 𝑥𝑛 and 𝜆𝑛 in (20) with the right-handed neutrino 𝜆2 =− ; 𝜃1 problem for massless case. Advances in Mathematical Physics 5

̇ 4. Parity Violation where 𝜆2 is the rate of the Doppler shift towards the orthogonal to the (𝑥1,𝑥3)-plane direction. This spatial asym- 𝜙 =𝜃 Forthereferenceframe(21),forthefiniteangles 1 1,from metry of aberration, which distinguishes the left and right (20) one can obtain the standard relativistic expressions: coordinate systems, may be detectable by precise quantum V measurements. 𝜃 = , tanh 1 𝑐 1 5. Spin and Hypercharge cosh 𝜃1 = , √1−V2/𝑐2 (29) Now consider the last class of automorphisms (Appendix C): V/𝑐 1 𝜃 = , 𝑥󸀠 =𝑥 +(𝜑 − ∑𝜑 )𝜆 , sinh 1 2 2 𝑛 𝑛 𝑛 𝑚 𝑛 √1−V /𝑐 3 𝑚

󸀠 where V is the velocity of moving system along 𝑥1.Thenfrom 𝑡 =𝑡, (35) (20) there follows the generalized rule of velocity addition: 󸀠 1 V −𝑐 𝜃 𝜆 =𝜆 +(𝜑 − ∑𝜑 )𝑥 󸀠 1 tanh 1 𝑛 𝑛 𝑛 3 𝑚 𝑛 V1 = , 𝑚 1−tanh 𝜃1 (V1/𝑐) 𝑛 ̇ with no summing by . These transformations represent 󸀠 V2 − tanh 𝜃1𝜆3 rotations of the three pairs of space-like and time-like V2 = , (30) 1−tanh 𝜃1 (V1/𝑐) coordinates, (𝑥1,𝜆1), (𝑥2,𝜆2),and(𝑥3,𝜆3), into each other. We have the three planes, (𝑥1 −𝜆1), (𝑥2 −𝜆2),and(𝑥3 −𝜆3), V + 𝜃 𝜆̇ 𝜑 𝜑 V󸀠 = 3 tanh 1 2 . that undergo rotations through hyperbolic angles, 1, 2,and 3 𝜑 1−tanh 𝜃1 (V1/𝑐) 3 (of the three only two are independent), which are the only hyper-planes in the space of split octonions that are affected We see that the standard expressions are altered only in the by one and not two 3-angles of automorphisms. So it is natural V 𝑁𝐶 planes orthogonal to and that what is important are the to define the Abelian subalgebra of 𝐺2 by generators of two terms with different sign. One consequence of this fact is that independent rotations in these planes. It is known that the 𝑁𝐶 the formula for the aberration of light will be modified. rank of the Cartan subalgebra of 𝐺 isthesameasofthe (𝑥 ,𝑥 ) 2 Consider photons moving in the 1 2 -plane: group 𝑆𝑈(3) [9–11, 60]. Indeed, in terms of two parameters, 𝐾1 and 𝐾2,whicharerelatedtotheangles𝜑𝑛 and (C.4) as V1 =𝑐cos 𝛾12, 1 V =𝑐 𝛾 , 𝐾 =𝑘 = (2𝜑 −𝜑 −𝜑 ), 2 sin 12 1 1 3 1 2 3 (31) V󸀠 =𝑐 𝛾󸀠 , √3 1 (36) 1 cos 12 𝐾 = (𝑘 +𝑘 )=− (2𝜑 −𝜑 −𝜑 ), 2 2 1 1 2√3 3 1 2 󸀠 󸀠 V2 =𝑐sin 𝛾12, the transformations (35) can be written more concisely: 𝛾 𝛾󸀠 V V󸀠 where 12 and 12 are angles between 1 and 1 and the axes 󸀠 󸀠 󸀠 𝜆1 +𝐼𝑥1 𝜆1 +𝐼𝑥1 𝑥1 and 𝑥1,respectively.Using(30),forthecaseV/𝑐 ≪,we 1 󸀠 󸀠 (𝐾1Λ 3+𝐾2Λ 8)𝐼 have (𝜆2 +𝐼𝑥2)=𝑒 (𝜆2 +𝐼𝑥2), (37) 𝛾 −(V/𝑐2) 𝜆̇ 𝜆󸀠 +𝐼𝑥󸀠 𝜆 +𝐼𝑥 󸀠 sin 12 3 2 2 2 2 sin 𝛾12 = , 1−(V/𝑐) cos 𝛾12 2 (32) where 𝐼 is the vector-like octonionic basis unit (𝐼 =1)and 𝛾 − V/𝑐 Λ 3 and Λ 8 are the standard 3×3Gell-Mann matrices: 𝛾󸀠 = cos 12 , cos 12 1−(V/𝑐) 𝛾 cos 12 100 ̇ where 𝜆3 istherateofDoppler’sshiftalonganaxis,𝑥3, Λ 3 =(0−10), orthogonal to the photon’s trajectory. Then angle of aberra- 000 (𝑥 ,𝑥 ) tion in the 1 2 -plane give the value (38) 10 0 󸀠 V V ̇ 1 Δ𝛾12 =𝛾12 −𝛾12 = sin 𝛾12 − 𝜆3. (33) 𝑐 𝑐2 Λ 8 = (01 0). √3 00−2 Analogous to (33) the formula for the aberration angle in the (𝑥1,𝑥3)-plane has the form Then, using analogies with 𝑆𝑈(3),onecanclassifyirre- V V duciblerepresentationsofthespace-timegroupofourmodel, ̇ 𝑁𝐶 Δ𝛾13 = sin 𝛾13 + 𝜆2, (34) 𝑐 𝑐2 𝐺2 ,bythetwofundamentalsimpleroots(𝐾1 and 𝐾2) 6 Advances in Mathematical Physics

corresponding to spin and hypercharge. It is known that where the wavelengths, 𝜆𝑛 ∼ℏ/𝑝𝑛, serve as the parameters of all quarks, antiquarks, and mesons can be imbedded in the trajectory. The relation (40) justifies the interpretation of 𝜔 as 𝑁𝐶 adjoint representation of 𝐺2 [9–11]. So the symmetry (35), the phase function (frequency) corresponding to the classical in addition to the uncertainty relations, probably shows the action of particles. existence of three generations of objects which are necessary For massive particles the time coordinate, 𝑡,isgood to extract three spatial coordinates from any octonionic parameter to describe motion and from (39) we find that the signal 𝑠. To clarify this point note that (9) can be viewed scalar part of a split octonionic signal, 𝜔, which is unchanged 𝑁𝐶 as some kind of space-time interval with four time-like under automorphisms of 𝐺2 , is the conserved function: dimensions. The ordinary time parameter, 𝑡, corresponds to 𝐼 𝑑𝜔 the distinguished octonionic basis unit, , while the other =0. (41) three time-like parameters, 𝜆𝑛, have a natural interpreta- 𝑑𝑡 tion as wavelengths; that is, they do not relate to time as conventionally understood. It is known that a unique Then (39) can be written as physical system in multidimensional geometry generates a 𝐴 V2 𝜆̇2 large variety of “shadows” in (3 + 1)-subspace as different 𝜔= =−𝑐∫ 𝑑𝑡√1− + , (42) dynamical systems (in terms of different Hamiltonians) [61– 𝑚𝑐 𝑐2 𝑐2 70]. The information of multidimensional structures, which 𝐴 𝑚 is retained by these images of the initial system, takes the form where is the classical action of the particle with the mass . of hidden symmetries. For the case of fundamental physical So using (23) the one-particle Lagrangian, signals with three extra time-like dimensions, in addition to V2 𝑝̇2 massless particles (which are not affected by extra times), 𝐿=−𝑚𝑐2√1− +ℏ2 , (43) one can observe three generations of particles with different 𝑐2 𝑐2𝑝4 mass (corresponding to rotations of 𝑡 in one, two, or all three extra time-like planes); see (E.7). Note the factor 1/3 in front contains an extra “quantum” term, which may be relevant for of 𝜔 (action) in the first equation (E.7) that appears due the relativistic velocities, |V|∼𝑐,orforthecaseoflargeforces: to the existence of the three extra time-like parameters, 𝜆𝑛. (𝜔, 𝜆 ) 𝑐𝑝2 Then three independent 𝑛 -rotations in 8-dimensional 𝑝̇ ∼ . (44) octonionicspacewillgivetheappearanceofthreegenerations 𝑛 ℏ of particles in ordinary (3 + 1)-dimensions. Such a possibility ∼ℏ/|𝑝| is not ruled out by problems with ghosts and with unitarity, From (43) we see that on small distance scales, ,a since split octonions with positive norms form the division particle can even exceed the speed of light (become virtual), since the new quantum term in (43) is positive. The condition algebra. 𝑁𝐶 (39) and the invariance of the classical action (42) under 𝐺2 hold only for real trajectories. However, there exists the larger 6. Free Particle Lagrangians group of invariances of the interval (9), 𝑆𝑂(4, 4),thegroup Itisknownthatinsplitalgebrastherecanbeconstructed of passive tensorial transformations of all eight octonionic special elements with zero norms, which are called zero coordinates, which in general mixes space-like and time-like divisors [1]. These zero-norm objects are important structures subspaces and thus introduces virtual trajectories of particles. in physical applications [71]. Zero divisors (light-cone opera- Applyingthecondition(10)(thatforphysicalsignalsthe tors) of split octonions could serve as the unit signals and thus vector part of a split octonion should be time-like) to (43) its may describe elementary particles. Then the number of prim- follows that there should exist some maximal force: itive idempotents (eight) and nilpotents (twelve) numerates 󵄨 󵄨 𝑝2 𝑐3 󵄨𝑝̇󵄨 ≤ |V| =𝑚2 , (45) the types of fundamental particles (Appendix E), bosons, and 󵄨 󵄨 ℏ ℏ fermions, respectively. Indeed, the properties that the product of two projection operators reduces to the same idempotent where 𝑚 denotes the maximal possible value for the mass of (E.2), while the product of two Grassmann numbers is zero particle associated with physical signals. Using estimations of 4 (E.4) naturally explain the validity of Bose and Fermi statistics [72, 73] for the maximum force, 𝐹max =𝑐/4𝐺,where𝐺 is for the corresponding elementary particles. 2 the gravitational constant; from (45) we find that the maximal The zero-norm condition, 𝑁 =0, for the paths (9) mass of particles associated with any octonionic signal is the of elementary particles, be they massless or massive, can be Planck mass: written in the form 2 ℏ𝑐 V2 𝜆̇2 𝑚 ∼ . (46) 𝑑𝑠2 =𝑑𝑠𝑑𝑠† =𝑑𝜔2 −𝑐2𝑑𝑡2 (1 − + )=0. 𝐺 𝑐2 𝑐2 (39)

For photons, |V|=𝑐,thecondition(39)isequivalentto 7. Conclusion the Eikonal equation: To conclude, in this paper, it was analyzed consequences 𝑑𝜔 𝑑𝜔 =1, of describing physical signals in terms of split octonions 𝑛 (40) 𝑑𝜆𝑛 𝑑𝜆 over the field of real numbers. Eight real parameters of Advances in Mathematical Physics 7 splitoctonionswererelatedtospace-timecoordinates,the three classes of split octonions having positive, negative, or phase function (classical action), and wavelengths charac- zero norm: (4 + 4) terizing physical signals. The -norms of the real 2 † † 2 2 split octonions are invariant under the group 𝑆𝑂(4, 4) of 𝑁 =𝑠𝑠 =𝑠𝑠=𝜔 −𝑉 , (A.3) tensorial transformations of parameters (passive transforma- tions of coordinates), while the representation of rotations where 2 2 2 2 2 by octonions themselves (i.e., the active transformations of 𝑉 =−𝑥 +𝑐 𝑡 +𝜆 . (A.4) octonionic basis units) corresponds to the real noncompact 𝑁𝐶 form of Cartan’s exceptional Lie group 𝐺2 (the subgroup In addition one can distinguish split octonions with time-like, of 𝑆𝑂(4, 4)), which under certain conditions imitate the space-like, and light-like vector parts (A.2):

Poincare´ transformations in ordinary space-time. Within this 2 picture, in front of time-like coordinates in the expression 𝑉 <0, (space − like) of pseudo-Euclidean octonionic intervals, there naturally 2 appear two fundamental physical parameters, the light speed 𝑉 >0, (time − like) (A.5) and Planck’s constant. From the requirement of positive 2 𝑁𝐶 𝑉 =0. (light − like). definiteness of norms under 𝐺2 -transformations, together 𝑐 with the introduction of the maximal velocity, ,therefollow 𝜔2 =𝑉̸ 2 conditions which are equivalent to uncertainty relations in Split octonions with nonzero norms, ,have 𝑁𝐶 multiplicative inverses, quantum physics. By examining of 𝐺2 -automorphisms it is explicitly shown that there exists a 3-vector (spin) that is 𝑠† 𝑠−1 = , fully chiral, which corresponds to the observed chirality of 2 (A.6) 𝑁𝐶 𝑁 neutrinos in nature. We examined also the 𝐺2 -effects on a quantum system moving at constant speed with respect with the property to the observer, which emits light in various planes relative −1 −1 to the direction of motion, and new parity-violating effect 𝑠𝑠 =𝑠 𝑠=1, (A.7) on aberration and Doppler shift was derived. It was shown 2 2 𝑁𝐶 while octonions with zero norm, 𝜔 =𝑉,havenoinverses. how a particular class of 𝐺2 -automorphisms of space-time coordinates can be expressed in concise form using two Onecanconstructalsothepolarformofasplitoctonion simple roots of the 𝑆𝑈(3) Lie algebra. Due to their conven- with nonzero norm: tional association with spin and hypercharge in physics, we 𝜖𝜃 𝑠=𝑁𝑒 , (A.8) argued that our split-octonion geometry inherently modeled three generations of fundamental particles. In our approach 2 where 𝜖 is a unit 7-vector (𝜖 =±1). For 𝑁 =0̸ the quantity elementary particles are interpreted as light-cone operators; 𝑠 that is, they are connected with the special elements of the 𝑅= algebra which nullify octonionic intervals. Then the zero- 𝑁 (A.9) 𝑁𝐶 norm conditions in 𝐺 -geometry lead to corresponding free 2 is called unit octonion, which is useful to represent rotations particle Lagrangians, which allow virtual particle trajectories (3 + 4) (exceeding the speed of light, in certain cases) and exhibit the of -vector spaces (A.2). existence of spatial horizons governing by mass parameters. Dependingonthevaluesof(A.3)and(A.4)splitocto- nions have three different representations: 2 2 (i) Every split octonion with negative norm, 𝜔 <𝑉,can Appendices be written in the form A. Classification of Split Octonions 𝑠=𝑁(sinh 𝜃+𝜖cosh 𝜃) , (A.10) The algebra of split octonions is a noncommutative, nonasso- where ciative, nondivision ring. Any element (1) of the ring can be 𝜔 𝜃= , represented as sinh 𝑁 𝑠=𝜔+𝑉, |𝑉| cosh 𝜃= , (A.11) (A.1) 𝑁 𝑠† =𝜔−𝑉, 𝜆𝑛𝐽 +𝑥𝑛𝑗 +𝑐𝑡𝐼 𝜖= 𝑛 𝑛 , |𝑉| where the symbols 𝜔 and and 𝜖 is a unit time-like 7-vector. 𝑉=𝜆𝑛𝐽 +𝑥𝑛𝑗 +𝑐𝑡𝐼 (𝑛 = 1, 2, 3) 2 2 𝑛 𝑛 (A.2) (ii) Every split octonion with positive norm, 𝜔 >𝑉,and 2 with time-like vector part, 𝑉 >0,canbewrittenintheform are called the scalar and vector parts of octonion, respectively. Similar to the case of split quaternions [59, 74], there exist 𝑠=𝑁(cosh 𝜃+𝜖sinh 𝜃) , (A.12) 8 Advances in Mathematical Physics

where where 𝛼𝑛 are three real compact angles. To show this explicitly 𝜔 let us consider the rotation operator 𝑅 in (B.1) when 𝜖= 𝜃= , 𝑗 𝑗 cosh 𝑁 1. In the octonionic algebra (5) the basis unit 1 has three different representations by other basis elements that form an |𝑉| 𝜃= , associative triplets with 𝑗1: sinh 𝑁 (A.13) 𝑗1 =𝐽2𝐽3 =𝑗2𝑗3 =𝐽1𝐼 (B.3) 𝜆𝑛𝐽 +𝑥𝑛𝑗 +𝑐𝑡𝐼 𝜖= 𝑛 𝑛 , |𝑉| or, equivalently, 𝑗 𝐽 =𝐽, so 𝜖 again is a unit time-like 7-vector. 1 3 2 2 2 (iii) Every split octonion with positive norm, 𝜔 >𝑉, 𝑗 𝑗 =𝑗, 2 1 2 3 (B.4) and with space-like vector part, 𝑉 <0,canbewritteninthe form 𝑗1𝐼=𝐽1. 𝑠=𝑁(cos 𝜃+𝜖sin 𝜃) , (A.14) Then it is possible to “rotate out” four octonionic axes and rewrite (1) in the equivalent form where 𝑗 𝛼 𝑗 𝛼 𝜔 𝑠=√𝜔2 +𝑥2𝑒 1 𝜔 + √𝜆2 +𝜆2 𝑒 1 𝜆 𝐽 cos 𝜃= , 1 2 3 3 𝑁 (B.5) 𝑗 𝛼 𝑗 𝛼 √ 2 2 1 𝑥 √ 2 2 1 𝑡 |𝑉| + 𝑥2 +𝑥3𝑒 𝑗2 + 𝑡 +𝜆1𝑒 𝐼, 𝜃= , sin 𝑁 (A.15) where four angles are given by 𝜆𝑛𝐽 +𝑥𝑛𝑗 +𝑐𝑡𝐼 𝜖= 𝑛 𝑛 , 𝜔 |𝑉| cos 𝛼𝜔 = , 2 2 √𝜔 +𝑥1 thus 𝜖 now is a unit space-like 7-vector. 𝜆 The norm of a split octonion (A.3) contains two types of 𝛼 = 3 , 𝑁=0 𝑉=0 cos 𝜆 “light-cones,” and , and we need two constants √𝜆2 +𝜆2 that characterize the maximum spread angles of these cones. 2 3 𝑐 (B.6) The first fundamental parameter, the speed of light ,appears 𝑥2 𝑉2 >0 cos 𝛼𝑥 = , in standard Lorentz boosts for the time-like signals, . √𝑥2 +𝑥2 The second fundamental constant, ℏ, corresponds to maximal 2 3 2 rotations in (𝜔, 𝑉)-planes, when physical events have 𝑁 ≥0 𝑐𝑡 andstillcanbedescribedbytheactions∼𝜔. cos 𝛼𝑡 = . 2 2 √𝑡 +𝜆1

B. One-Side Products 𝑗 𝛼 Now it is obvious that when 𝑅=𝑒1 1 and 𝑠 is represented The 8-dimensional octonionic space (1) always can be rep- by (B.5), the product (B.1) leads to simultaneous rotations in (𝜔, 𝑥 ) (𝜆 ,𝜆 ) (𝑥 ,𝑥 ) (𝜆 ,𝑡) resented as the sum of four elements by rotations in four the four planes, 1 , 2 3 , 2 3 ,and 1 ,bythe 𝛼 orthogonal planes (see (B.5), (B.10), and (B.14) below), one of same compact angle 1. Similar results can be obtained for 𝑗 which always includes the axis of real numbers 𝜔 [75]. Then it the rotations by the other two pseudovector-like units, 2 and 𝑗 can be shown that multiplication of one octonion by another 3. 𝐽 octonion of unit norm (A.9) represents some kind of rotation (ii) For the three vector-like basis elements, 𝑛,withthe 𝐽2 =1 in these four planes. In the algebra of split octonions there positive squares, 𝑛 , we need the unit octonions with 2 exist seven basis units, 𝐽𝑛, 𝑗𝑛,and𝐼,whichdefinethe7-vector 𝑁 >0having time-like vector part: 𝜖 in the formulas for one-sided (e.g., the left) products: 𝐽𝑛𝜃𝑛 𝑒 = cosh 𝜃𝑛 +𝐽𝑛 sinh 𝜃𝑛, (B.7) 𝜖𝜃/2 𝑅s =𝑒 𝑠. (B.1) where 𝜃𝑛 are three real hyperbolic angles. Let us consider the So totally we have 4×7 = 28parameters of the group 𝑆𝑂(4, 4), example when 𝜖=𝐽1 in (B.1). The equivalent representations which preserve the norm (9) [75]. Octonionic hypercomplex of 𝐽1 in the algebra (5) are basis units have different algebraic properties and lead to the 𝐽1 =𝐽2𝑗3 =−𝐽3𝑗2 =𝑗1𝐼 (B.8) three distinct classes of the rotation operator 𝑅: (i) The three pseudovector-like basis elements, 𝑗𝑛,behave 2 or as the ordinary complex unit (𝑗𝑛 =−1) and rotations around 𝐽1𝐽2 =−𝑗3, the corresponding spatial axes, 𝑥𝑛,canbedonebytheunit 𝑁2 >0 octonions with having space-like vector part: 𝐽1𝑗2 =𝐽3, (B.9)

𝑗𝑛𝛼𝑛 𝑒 = cos 𝛼𝑛 +𝑗𝑛 sin 𝛼𝑛, (B.2) 𝐽1𝐼=𝑗1. Advances in Mathematical Physics 9

𝐼𝜎 Then(1)canberewrittenas So the left product (B.1), when 𝑅=𝑒 , corresponds to the (𝜔, 𝑡) (𝑥1,𝜆1) (𝑥2,𝜆2) 𝐽 𝜃 −𝐽 𝜃 simultaneous rotations in the planes , , , √ 2 2 1 𝜔 √ 2 2 1 𝜆 𝑠= 𝜔 −𝜆1𝑒 + 𝑥3 −𝜆2𝑒 𝐽2 and (𝑥3,𝜆3) by the hyperbolic angle 𝜎. (B.10) So the one-side products (B.1) of an octonion 𝑠 by 𝐽 𝜃 𝐽 𝜃 √ 2 2 1 𝑥 √ 2 2 1 𝑡 + 𝑥2 −𝜆3𝑒 𝑗2 + 𝑥1 −𝑡 𝑒 𝐼, each of the seven operators (B.2), (B.7), and (B.12) yield simultaneous rotations in four mutually orthogonal planes of where the angles are given by the octonionic (4 + 4)-spacebythesameangles[75].Oneof 𝜔 these four planes is formed by the unit element, 1,andthe cosh 𝜃𝜔 = , hypercomplex basis unit that defines the axis of rotation. The √ 2 2 𝜔 −𝑥1 remaining orthogonal planes are given by three pairs of other basiselementswhichformassociativetripletswiththechosen 𝜆 2 basis unit. cosh 𝜃𝜆 = , 2 2 The decomposition of a split octonion in the form (B.5) is √𝑥3 −𝜆2 (B.11) valid only if its norm (9) is positive. In contrast with uniform 𝑥2 rotations around 𝑗𝑛, we have limited rotations, (B.10) and cosh 𝜃𝑥 = , 𝐽 𝐼 √𝑥2 −𝜆2 (B.14), around 𝑛 and . For these cases we should require also 2 3 positiveness of the 2 norms of each of the four planes of rota- 𝑡 tions. For example, for (B.14), the expressions of 2 norms of cosh 𝛼𝑡 = . √ 2 2 √ 2 2 √ 2 2 2 2 four orthogonal planes are 𝜔 −𝑡 , 𝑥1 −𝜆1, 𝑥2 −𝜆2, √𝑥1 −𝑡 2 2 and √𝑥3 −𝜆3. These hyperbolic properties are the result of 𝐽 𝜃 𝑅=𝑒1 1 So when , the product (B.1) corresponds to the the existence of zero divisors in split algebras (Appendix E). simultaneous rotations in the planes (𝜔, 𝜆1), (𝜆2,𝑥3), (𝑥2,𝜆3), (𝑥 ,𝑡) 𝜃 and 1 by the hyperbolic angle 1.Wehavesimilarresults 𝐺 for the rotations by the other two vector-like units, 𝐽2 and 𝐽3. C. Automorphisms and 2 (iii) The last vector-like basis element, 𝐼,alsohasthe 2 From the expressions of a hypercomplex unit of split octo- positive square, 𝐼 =1, and to represent corresponding trans- 𝑁2 >0 nions by two other basis elements, which form associative formations we need the unit octonion with having triplets with the selected unit, one can find orthogonal to it time-like vector part: planes of rotations. An automorphism (13) for each basis unit 𝐼𝜎 𝑒 = cosh 𝜎+𝐼sinh 𝜎, (B.12) introduces two angles of rotation in these planes. So there are seven independent automorphisms each by two angles that 𝜎 𝑁𝐶 where is the real hyperbolic angle. When in (B.1) we put correspond to 2×7=14generators of the algebra 𝐺 .For 𝜖=𝐼 2 ,wecanusetheexpressions(4)oftheequivalent example, the automorphism by the two real compact angles, 𝐼 representations of or 𝛼1 and 𝛽1, which leaves unchanged the basis unit 𝑗1 =𝐽2𝐽3 = 𝑗2𝑗3 =𝐽1𝐼,hastheform 𝐼𝑗1 =−𝐽1, 𝐼𝑗 =−𝐽, 󸀠 2 2 (B.13) 𝑗1 =𝑗1, 󸀠 𝐼𝑗3 =−𝐽3. 𝑗2 =𝑗2 cos 𝛼1 +𝑗3 sin 𝛼1, Then the split octonion (1) can be rewritten as 󸀠 󸀠 󸀠 𝑗3 =𝑗1𝑗2 =𝑗3 cos 𝛼1 −𝑗2 sin 𝛼1, 𝐼𝜎 −𝐼𝜎 √ 2 2 𝜔 √ 2 2 1 𝑠= 𝜔 −𝑡 𝑒 + 𝑥1 −𝜆1𝑒 𝑗1 󸀠 𝐽1 =𝐽1 cos 𝛽1 +𝐼sin 𝛽1, (C.1) (B.14) −𝐼𝜎 −𝐼𝜎 󸀠 󸀠 󸀠 √ 2 2 2 √ 2 2 3 + 𝑥2 −𝜆2𝑒 𝑗2 + 𝑥3 −𝜆3𝑒 𝑗3, 𝐼 =𝐽1𝑗1 =𝐼cos 𝛽1 −𝐽1 sin 𝛽1, 󸀠 󸀠 󸀠 where four hyperbolic angles are given by 𝐽2 =𝑗2𝐼 =𝐽2 cos (𝛼1 −𝛽1)+𝐽3 sin (𝛼1 −𝛽1), 𝜔 󸀠 󸀠 󸀠 cosh 𝜎𝜔 = , 𝐽 =𝑗𝐼 =𝐽 (𝛼 −𝛽 )−𝐽 (𝛼 −𝛽 ). √𝜔2 −𝑡2 3 3 3 cos 1 1 2 sin 1 1

𝑥1 It is known that automorphisms in the octonionic algebra cosh 𝜎1 = , 2 2 are completely defined by the images of three basis elements √𝑥1 −𝜆1 thatdonotformassociativesubalgebras(𝑗1, 𝑗2,and𝐽1 in 𝑥2 (B.15) the case of (C.1)). By the definition (13) an automorphism cosh 𝜎2 = , 𝑠 √𝑥2 −𝜆2 does not affect the scalar part of ,whiletheimagesofthe 2 2 other hypercomplex elements (𝑗3, 𝐼, 𝐽2,and𝐽3 in (C.1)) are 𝑥 determined using the algebra (5). Then it is obvious that 𝜎 = 3 . 󸀠 󸀠 󸀠 cosh 3 the transformed basis 𝐽𝑛,𝑗𝑛,𝐼 satisfy the same multiplication √𝑥2 −𝜆2 3 3 rules as 𝐽𝑛,𝑗𝑛,𝐼. 10 Advances in Mathematical Physics

Similar to (C.1) there exist two automorphisms with fixed D. Generators of 𝐺2 𝑗2 =𝐽3𝐽1 =𝑗3𝑗1 =𝐽2𝐼 and 𝑗3 =𝐽1𝐽2 =𝑗1𝑗2 =𝐽3𝐼 axes, each 𝐺𝑁𝐶 generating the two compact angles, 𝛼2,𝛽2 and 𝛼3,𝛽3.Three The group 2 first was studied by Cartan in his thesis [76, 77]. He considered (3 + 4)-space with the seven coordinates, different representations of a basis unit indicate the planes of 𝑛 𝑛 corresponding automorphisms and the positive directions of (𝑦 ,𝑡,𝑧 ), and linear operators of their transformations: rotations. 𝜕 𝜕 One can define also hyperbolic rotations (automor- 𝑋 =−𝑧 +𝑦 𝐽 𝑛𝑛 𝑛 𝑛 phisms) around the three vector-like units 𝑛 by the angles 𝜕𝑧𝑛 𝜕𝑦𝑛 𝜃𝑛 and 𝜙𝑛. For example, similar to the transformation (C.1), automorphism around the axis 𝐽1 =𝑗3𝐽2 =𝐽3𝑗2 =𝑗1𝐼 is 1 3 𝜕 𝜕 + ∑ (𝑧 −𝑦 ), 󸀠 3 𝑚 𝜕𝑧 𝑚 𝜕𝑦 𝐽1 =𝐽1, 𝑚=1 𝑚 𝑚 𝐽 (𝜙 +𝜃 ) 𝑗 (𝜙 +𝜃 ) ( 𝑛) 𝐽󸀠 = 2 cosh 1 1 + 3 sinh 1 1 , no summing over 2 2 2 𝜕 𝜕 𝑗 (𝜃 +𝜙 ) 𝐽 (𝜃 +𝜙 ) 𝑋 =−2𝑡 +𝑦 𝑗󸀠 =𝐽󸀠𝐽󸀠 = 3 cosh 1 1 + 2 sinh 1 1 , 𝑛0 𝜕𝑧 𝑛 𝜕𝑡 3 1 2 2 2 𝑛 (D.1) 𝐼󸀠 =𝐼 𝜃 −𝑗 𝜃 , 1 𝑚 𝜕 𝑘 𝜕 cosh 1 1 sinh 1 (C.2) + 𝜀𝑛𝑚𝑘 (𝑧 −𝑧 ), 2 𝜕𝑦𝑘 𝜕𝑦𝑚 󸀠 󸀠 󸀠 𝑗 =𝐽𝐼 =𝑗1 cosh 𝜃1 −𝐼sinh 𝜃1, 1 1 𝜕 𝜕 𝑋 =−2𝑡 +𝑧 𝑗 (𝜙 −𝜃 ) 𝐽 (𝜙 −𝜃 ) 0𝑛 𝜕𝑦 𝑛 𝜕𝑡 𝑗󸀠 =𝐽󸀠𝐼󸀠 = 2 cosh 1 1 + 3 sinh 1 1 , 𝑛 2 2 2 2 1 𝜕 𝜕 + 𝜀 (𝑦𝑚 −𝑦𝑘 ), 󸀠 󸀠 󸀠 𝐽3 cosh (𝜙1 −𝜃1) 𝑗2 sinh (𝜙1 −𝜃1) 𝑛𝑚𝑘 𝐽 =𝑗𝐼 = + , 2 𝜕𝑧𝑘 𝜕𝑧𝑚 3 3 2 2 𝜕 𝜕 where we chose the representation with half-angle rotations 𝑋𝑛𝑚 =−𝑧𝑚 +𝑦𝑛 , (𝑛 =𝑚̸ ) , in order to write the transformations of 𝜆𝑛 and 𝑥𝑛 in 𝜕𝑧𝑛 𝜕𝑦𝑚 symmetric form. The automorphisms with the fixed 𝐽2 = 𝑗 𝐽 =𝐽𝑗 =𝑗𝐼 𝐽 =𝑗𝐽 =𝐽𝑗 =𝑗𝐼 2 𝑛 1 3 1 3 2 and 3 2 1 2 1 3 axes generate which preserves the quadratic form 𝑑𝑡 +𝑧𝑛𝑦 .Totallythere 𝜃 ,𝜙 𝜃 ,𝜙 hyperbolic angles, 2 2 and 3 3,respectively. are 15 generators in (D.1) since 𝑛, 𝑚 = 1,, 2,3 but the operators 𝐼= Finallyfortherotationsaroundthetimedirection, 𝑋𝑛𝑛 are linearly dependent: 𝐽1𝑗1 =𝐽2𝑗2 =𝐽3𝑗3, we have only two hyperbolic angles, 𝑘1 and 𝑘2: 𝑋11 +𝑋22 +𝑋33 =0, (D.2) 𝐼󸀠 =𝐼, 𝑁𝐶 which explains why 𝐺 is 14- and not 15-dimensional. 𝑗󸀠 =𝑗 𝑘 +𝐽 𝑘 , 2 1 1 cosh 1 1 sinh 1 By transition to our coordinates, 𝑥𝑛 and 𝜆𝑛, 󸀠 𝑗2 =𝑗2 cosh 𝑘2 +𝐽2 sinh 𝑘2, 𝑦𝑛 =𝜆𝑛 +𝑥𝑛, 𝐽󸀠 =𝑗󸀠 𝐼󸀠 =𝐽 𝑘 +𝑗 𝑘 , 1 1 1 cosh 1 1 sinh 1 (C.3) 𝜕 1 𝜕 𝜕 = ( + ), 󸀠 󸀠 󸀠 𝜕𝑦 2 𝜕𝜆 𝜕𝑥 𝐽2 =𝑗2𝐼 =𝐽2 cosh 𝑘2 +𝑗2 sinh 𝑘2, 𝑛 𝑛 𝑛 (D.3) 󸀠 󸀠 󸀠 𝑗3 =𝑗1𝑗2 =𝑗3 cosh (𝑘1 +𝑘2)−𝐽3 sinh (𝑘1 +𝑘2), 𝑧𝑛 =𝜆𝑛 −𝑥𝑛, 󸀠 󸀠 󸀠 𝐽 =𝑗𝐼 =𝐽3 cosh (𝑘1 +𝑘2)−𝑗3 sinh (𝑘1 +𝑘2). 𝜕 1 𝜕 𝜕 3 3 = ( − ); 𝜕𝑧 2 𝜕𝜆 𝜕𝑥 In order to present the transformations of 𝜆𝑛 and 𝑥𝑛 in (14) 𝑛 𝑛 𝑛 in the symmetric form, it is convenient to introduce instead of 𝑘1 and 𝑘2 the three hyperbolic angles 𝜑𝑛: from (D.1) we can find the operators of the transformation 𝑉 1 of the vector part of (1) which preserves the diagonal 𝑉2 =𝜆 𝜆𝑛 +𝑑𝑡2 −𝑥 𝑥𝑛 𝑘1 =𝜑1 − ∑𝜑𝑛, quadratic form, 𝑛 𝑛 : 3 𝑛 1 𝑋 𝑘2 =𝜑2 − ∑𝜑𝑛, (C.4) 𝑛𝑛 3 𝑛 1 𝜕 𝜕 1 3 𝜕 𝜕 =𝑥 +𝜆 − ∑ (𝑥 +𝜆 ), −(𝑘1 +𝑘2)=𝜑3 − ∑𝜑𝑛. 𝑛 𝑛 𝑚 𝑚 3 𝑛 𝜕𝑥𝑛 𝜕𝜆𝑛 3 𝑚=1 𝜕𝑥𝑚 𝜕𝜆𝑚 Advances in Mathematical Physics 11

𝑋 𝑛0 whichobeytherelations

𝜕 𝜕 ±(𝐽) ±(𝐽) ±(𝐽) =(𝜆𝑛 −𝑡 ) 𝐷𝑛 𝐷𝑛 =𝐷𝑛 , 𝜕𝑡 𝜕𝜆𝑛 +(𝐽) −(𝐽) 𝐷𝑛 𝐷𝑛 =0, 𝜕 𝜕 1 𝑚 𝜕 𝑘 𝜕 +(𝑥𝑛 +𝑡 ) + 𝜀𝑛𝑚𝑘 (𝜆 −𝜆 ) (E.2) 𝜕𝑡 𝜕𝑥𝑛 4 𝜕𝜆𝑘 𝜕𝜆𝑚 𝐷±(𝐼)𝐷±(𝐼) =𝐷±(𝐼),

1 𝑚 𝜕 𝑘 𝜕 +(𝐼) −(𝐼) − 𝜀𝑛𝑚𝑘 (𝑥 −𝑥 ) 𝐷 𝐷 =0. 4 𝜕𝑥𝑘 𝜕𝑥𝑚

1 𝑚 𝜕 𝑘 𝜕 We have also the four noncommuting classes (totally + 𝜀𝑛𝑚𝑘 (𝜆 +𝑥 ) twelve) of primitive nilpotents: 4 𝜕𝑥𝑘 𝜕𝜆𝑚 1 1 𝜕 𝜕 ±(𝐽) 𝑚 𝑘 𝐺𝑛 = (𝐽𝑛 ±𝑗𝑛), − 𝜀𝑛𝑚𝑘 (𝑥 +𝜆 ), 2 4 𝜕𝜆𝑘 𝜕𝑥𝑚 1 𝐺±(𝐼) = (𝐼±𝑗 ), (E.3) 𝑋0𝑛 𝑛 2 𝑛 𝜕 𝜕 𝜕 𝜕 (𝑛 = 1, 2, 3) =(𝜆𝑛 −𝑡 )−(𝑥𝑛 +𝑡 ) 𝜕𝑡 𝜕𝜆𝑛 𝜕𝑡 𝜕𝑥𝑛 with the properties 1 𝑚 𝜕 𝑘 𝜕 + 𝜀𝑛𝑚𝑘 (𝜆 −𝜆 ) ±(𝐽) ±(𝐽) 4 𝜕𝜆𝑘 𝜕𝜆𝑚 𝐺𝑛 𝐺𝑛 =0,

1 𝑚 𝜕 𝑘 𝜕 𝐺±(𝐽)𝐺∓(𝐽) =𝐷∓(𝐼), − 𝜀𝑛𝑚𝑘 (𝑥 −𝑥 ) 𝑛 𝑛 4 𝜕𝑥𝑘 𝜕𝑥𝑚 (E.4) 𝐺±(𝐼)𝐺±(𝐼) =0, 1 𝜕 𝜕 𝑛 𝑛 − 𝜀 (𝜆𝑚 +𝑥𝑘 ) 4 𝑛𝑚𝑘 𝜕𝑥 𝜕𝜆 ±(𝐼) ∓(𝐼) ±(𝐽) 𝑘 𝑚 𝐺𝑛 𝐺𝑛 =𝐷𝑛 . 1 𝜕 𝜕 + 𝜀 (𝑥𝑚 +𝜆𝑘 ), From these relations we see that separately nilpotents are 4 𝑛𝑚𝑘 𝜕𝜆 𝜕𝑥 𝑘 𝑚 Grassmann numbers, but different 𝐺𝑛-s do not commute with each other, in contrast to the projection operators (E.2). 𝑋𝑛𝑚 ± Instead the quantities 𝐺𝑛 are elements of the so-called algebra 1 𝜕 𝜕 of Fermi operators with the anticommutators: =− (𝜆𝑛 −𝜆𝑚 ) 2 𝜕𝜆𝑚 𝜕𝜆𝑛 𝐺±(𝐽)𝐺∓(𝐽) +𝐺∓(𝐽)𝐺±(𝐽) =1, 1 𝜕 𝜕 (E.5) − (𝑥𝑛 −𝑥𝑚 ) ±(𝐼) ∓(𝐼) ∓(𝐼) ±(𝐼) 2 𝜕𝑥𝑚 𝜕𝑥𝑛 𝐺 𝐺 +𝐺 𝐺 =1.

1 𝜕 𝜕 The algebra of Fermi operators is some syntheses of the + (𝑥𝑛 +𝜆𝑚 ) 2 𝜕𝜆𝑚 𝜕𝑥𝑛 Grassmann and Clifford algebras. For the completeness we note that the idempotents and 1 𝜕 𝜕 + (𝜆𝑛 +𝑥𝑚 ) (𝑛 =𝑚̸ ) . nilpotents obey the following algebra: 2 𝜕𝑥𝑚 𝜕𝜆𝑛 ±(𝐽) ±(𝐼) ±(𝐼) (D.4) 𝐷𝑛 𝐺𝑛 =𝐺𝑛 , ±(𝐽) ∓(𝐼) 𝐷𝑛 𝐺𝑛 =0, E. Zero Divisors (E.6) 𝐷±(𝐼)𝐺±(𝐽) =0, In the algebra of split octonions two types of zero divisors, 𝑛 idempotent elements (projection operators) and nilpotent 𝐷±(𝐼)𝐺∓(𝐽) =𝐺∓(𝐽). elements (Grassmann numbers), can be constructed [1, 71]. 𝑛 𝑛 There exist four noncommuting (totally eight) primitive Using commuting zero divisors any octonion (1) can be idempotents: writteninthetwoequivalentforms: ±(𝐽) 1 𝐷𝑛 = (1 ± 𝐽𝑛), 1 2 𝑠=𝐷+(𝐽) ( 𝜔+𝜆𝑛)+𝐺+(𝐼) (𝑐𝑡 +𝑥𝑛) 1 𝑛 3 𝑛 𝐷±(𝐼) = (1±𝐼) , (E.1) 2 1 +𝐷−(𝐽) ( 𝜔−𝜆𝑛)+𝐺−(𝐼) (𝑐𝑡 −𝑥𝑛), (𝑛 = 1, 2, 3) 𝑛 3 𝑛 12 Advances in Mathematical Physics

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