# Geometrical Applications of Split Octonions

Authors: Merab Gogberashvili; Otari Sakhelashvili
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

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 groupG2. 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.


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## Body

## 1. Introduction

Many properties of physical systems can be revealed from the analysis of proper mathematical structures used in descriptions of these systems. The geometry of space-time, one of the main physical characteristics of nature, can be understood as a reflection of symmetries of physical signals we receive and of the algebra used in the measurement process. Since all observable quantities we extract from single measurements are real, in geometrical applications it is possible to restrict ourselves to the field of real numbers. To have a transition from a manifold of the results of measurements to geometry, one must be able to introduce a distance between some objects and an etalon (unit element) for their comparison. In algebraic language all these physical requirements mean that to describe geometry we need a composition algebra with the unit element over the field of real numbers.Besides usual real numbers, according to the Hurwitz theorem, there are three unique normed division algebras—complex numbers, quaternions, and octonions [1–3]. Physical applications of the widest normed algebra of octonions are relatively rare (see reviews [4–8]). One can point to the possible impact of octonions on color symmetry [9–13]; GUTs [14–17]; representation of Clifford algebras [18–20]; quantum mechanics [21–29]; space-time symmetries [30, 31]; field theory [32–35]; quantum Hall effect [36]; Kaluza-Klein program without extra dimensions [37–42]; Strings and M-theory [43–48]; SUSY [49]; and so forth.The essential feature of all normed composition algebras is the existence of a real unit element and a different number of hypercomplex units. The square of the unit element is always positive, while the squares of the hypercomplex basis units can be negative as well. In physical applications one mainly uses division algebras with Euclidean norms, whose hypercomplex basis elements have negative squares, similar to the ordinary complex uniti. The introduction of vector-like basis elements (with positive squares) leads to the split algebras with pseudo-Euclidean norms.In this paper we propose parameterizing world-lines (paths) of physical objects by the elements of real split octonions [50–53]:(1)s=ω+λnJn+xnjn+ctIn=1,2,3.A pair of repeated upper and lower indices, by the standard convention, implies a summation; that is, xnjn=δnmxnjm, where δnm is Kronecker’s delta.In geometric application four of the eight real parameters in (1), t and xn, are space-time coordinates; ω and λn are interpreted as the phase (classical action) and the wavelengths associated with the octonionic signals [50–53].The eight basis units in (1) are represented by one scalar (denoted by 1), the three vector-like objects Jn, the three pseudovector-like elements jn, and one pseudoscalar-like unit I. The squares (inner products) of seven of the hypercomplex basis elements of split octonions give the unit element, 1, with the different signs:(2)Jn2=1,jn2=-1,I2=1.It is known that to generate a complete basis of split octonions the multiplication and distribution laws of only three vector-like elements Jn are enough [1–3]. The three pseudovector-like basis units, jn, can be defined as the binary products,(3)jn=12εnmkJmJkn,m,k=1,2,3(εnmk is the totally antisymmetric unit tensor of rank three), and thus describes orthogonal planes spanned by two vector-like elements Jn. The seventh basis unit I (the oriented volume) is defined as the triple product of all three vector-like elements and has three equivalent representation in terms of Jn and jn:(4)I=J1j1=J2j2=J3j3.So the algebra of all noncommuting hypercomplex basis units of split octonions has the form(5)JnJm=-JmJn=εnmkjk,jnjm=-jmjn=εnmkjk,jmJn=-Jnjm=εnmkJk,JnI=-IJn=jn,jnI=-Ijn=Jn.Conjugations of octonionic basis units, which can be understood as the reflection of vector-like elements,(6)Jn†=-Jn,reverses the order of Jn in products; that is,(7)jn†=12εnmkJmJk†=12εnmkJk†Jm†=-jn,I†=J1J2J3†=J3†J2†J1†=-I.So the conjugation of (1) gives(8)s†=ω-λnJn-xnjn-ctI.Using (2), (5), and (8) one can find that the norm of (1) is given by(9)N2=ss†=s†s=ω2-λ2+x2-c2t2,(where λ2=λnλn and x2=xnxn), representing some kind of 8-dimensional space-time with (4+4)-signature and reducing to the classical formula of Minkowski intervals if ω2-λ2=0.As for the case of ordinary Minkowski space-time, we assume that for physical events the corresponding “intervals” given by (9) are nonnegative. A second condition is that for physical signals the vector part of split octonions (1) should be time-like:(10)c2t2+λnλn>xnxn.The classification of split octonions by the values of their norms is presented in Appendix A.


## 2. Transformations and Automorphisms

To find the geometry associated with the signals (1) we need to represent rotations by split octonions; due to the nonassociativity of the algebra the group of rotations of octonionic axes 1, Jn, jn, and I, G2NC is not equivalent to the group of accompanied passive tensorial transformations of coordinates ω, λn, xn, and t (SO(4,4)). In Appendix B it is shown that by rotations in four orthogonal planes of octonionic space (one of which always includes the axis of real numbers ω) the octonion (1) always can be represented as the sum of four elements. The left multiplication, Rs, by an octonion with unit norm (Appendix A),(11)R=eϵθ,where ϵ is the (3+4)-vector defined by seven basis units Jn, jn, and I, represents rotations by the angles θ in these four planes. The right product, sR-1, will reverse the direction of rotation in the plane of ω. So we can represent a rotation, which only affects the (3+4)-vector part of s, applying the unit half-angle octonion twice and multiplying on both the left and the right (with its inverse):(12)s′=RsR-1=RsR†=eϵθ/2se-ϵθ/2.These maps are well defined, since the associator of the triplet (R,s,R-1) vanishes. The set of rotations (12) of the seven octonionic coordinates xn, λn, and t, in three planes, which do not affect the scalar part ω of (1), contains 7×3=21 angles of the group SO(3,4) of passive transformations of coordinates.To represent the active rotations in the space ofs, which preserve the norm (9) and multiplicative structure of octonions (5) as well, we would need the transformations to be automorphisms (Appendix C). An automorphism U of any two octonions, O1 and O2, gives the equation(13)UO1U-1UO2U-1=UO1O2U-1,which only holds in general if multiplication of U, O1, and O2 is associative. Combinations of the rotations (12) around different octonionic axes are not unique. This means that not all transformations of SO(3,4) form a group and can be considered as real rotations. Only the transformations that have a realization as associative multiplications should be considered. Also note that an automorphism of split octonions can be generated only by the octonions with positive norms (Appendix A), since the set of split octonions with negative norms do not form a group (it is not closed under multiplication). So to model space-time symmetries we need to use the group of automorphisms of split octonions with positive norms.It is known that associative transformations of split octonions can be done by the specific simultaneously rotations in two (and not in three as forSO(3,4)) orthogonal octonionic planes (Appendix C). These rotations form a subgroup of SO(3,4) with 2×7=14 parameters, known as the automorphism group of split octonions, G2NC. Generators of this real noncompact form of Cartan’s smallest exceptional Lie group G2 is presented in Appendix D.Infinitesimal transformations of coordinates which accompany the group of active transformations of octonionic basis units,G2NC, and preserve the diagonal quadratic form, x2-λ2-dt2≥0, can be written in the form (Appendix C)(14)xn′=xn-εnmkαmxk-θnct+12εnmkϕm+εnmkθmλk+φn-13∑mφmλn,ct′=ct-βnλn-θnxn,λn′=λn-εnmkαm-βmλk+βnct+12εnmkϕm-εnmkθmxk+φn-13∑mφmxn,with no summing over n in the last terms of xn′ and λn′. From the five 3-angles in (14), αm, βm, ϕm, θm, and φm, only 14 are independent because of the condition(15)∑nφn-13∑mφm=0.The Lorentz-type transformations (14) of the 7-dimensional space with four time-like coordinates should describe space-time symmetries if the split octonions form relevant algebraic structures for microphysics. The transformations (14) formally can be divided into three distinct classes (Appendix C): Euclidean rotations of the spatial, xn, and time-like (t and λn) coordinates by the compact 3-angle αn and βn, respectively; boosts, mixing of spatial and time-like coordinates by the two hyperbolic 3-angle θn and ϕn; and diagonal boosts of the spatial coordinates, xn, and corresponding time-like parameters, λn, by the hyperbolic angles φn.We notice that if we consider rotations by the anglesαn, βn, and θn, that is, assuming that(16)ϕm=φm=0,the passive G2NC-transformations (14) of only ordinary space-time coordinates, xn and t, will imitate the ordinary infinitesimal Poincaré transformations of (3+1)-Minkowski space:(17)xn′=xn-εnmkαmxk-θnct+an,ct′=ct-θnxn+a0.Here the space-time translations,(18)an=12εnmkθmλk,a0=-βnλn,are generated by the Lorentz-type rotations toward the time-like directions λn. So in the language of octonionic geometry any motion in ordinary space-time is generated by λn~pn/p2. Time translations a0 are smooth, since βn are compact angles. However, the angles θm are hyperbolic and for any active spatial translation an there exists a horizon (analogues to the Rindler horizon), which is equivalent to the introduction of some mass scale or inertia (see Section 6).For completeness note that there exists a second well-known representation ofG2NC as the symmetry group of a ball rolling on a larger fixed ball without slipping or twisting, when the ratio of the balls radii is 1/3 [54–58]. Understanding the exceptional Lie groups as the symmetry groups of naturally occurring objects is a long-standing program in mathematics. Symmetries of rolling balls are visualized in (14) by the presence of two 3-vector,  xn and λn, one of which is time-like. The fourth time-like coordinate (ordinary time t), which also is affected by (14), breaks the symmetry between “balls” and the factor 1/3 of the ratio of their radii corresponds to the existence of the three extra time-like coordinates λn.


## 3. Boosts, Heisenberg Uncertainty, and Chirality

In the algebra of split octonions there exist only three space-like parameters,xn, and three independent compact rotations around corresponding axes, which explains why physical space described by octonionic signals has three spatial dimensions. Let us analyze new features of the G2NC-transformations (14) in comparison with standard Lorentz’s formulas for (3+1)-Minkowski space.Euclidean rotations around one of the space-like axes,x1, the automorphism (C.1), correspond to the following passive infinitesimal transformations of coordinates:(19)x1′=x1,x2′=x2+α1x3,x3′=x3-α1x2,ct′=ct-β1λ1,λ1′=λ1+β1ct,λ2′=λ2+α1-β1λ3,λ3′=λ3-α1-β1λ2.When α1=β1 this reduces to the standard rotation by the Euler angle α1, which also causes translations of time, t, due to mixing with the extra time-like parameter λ1. In general, any active Euclidean 3-rotation of the spatial coordinates xn changes the time parameter by the amount of the corresponding λn, which can be understood as passing of time in our world.Analysis of boosts ofG2NC can help us in the physical interpretation of the extra time-like parameters λn. Consider the automorphisms (C.2) by the hyperbolic angles θ1 and ϕ1:(20)x1′=x1-θ1ct,x2′=x2+12ϕ1-θ1λ3,x3′=x3+12ϕ1+θ1λ2,ct′=ct-θ1x1,λ1′=λ1,λ2′=λ2+12ϕ1+θ1x3,λ3′=λ3+12ϕ1-θ1x2.The case ϕ1=θ1 corresponds to ordinary boost in (t,x)-planes—transitions to the reference frame moving with the velocity θ1 along the axis x1. Then if we consider the motion of the origin of the moving system,(21)x′=y′=z′=0,from the first line in (20) we find that(22)θ1=x1ct,λ3=x2θ1,λ2=-x3θ1;that is, the quantities λ2 and λ3 are inversely proportional to the velocity, x1/t. In the space of split octonions (1) there are two classes of time-like parameters (t and λn) and two different light-cones. So, in addition to c, there must exist the second fundamental constant, which can be extracted from λn. In quantum mechanics the quantity with the dimension of length, which is proportional to a fundamental physical constant and is inversely proportional to velocity (or momentum, pn), is called the wavelength. So it is natural to assume that(23)λn=ħpnp2~ħpn,where ħ is the Planck constant and pn is the 3-momentum associated with the octonionic signal. So in our approach the two fundamental physical constants, c and ħ, have geometrical origin and correspond to two kinds of light-cone signals in the space of split octonions (9).Whenϕ1=0 from (20) we find that the ratios,(24)Δx2Δλ2=-λ3x3,Δx3Δλ3=-λ2x2,(where Δxn=xn′-xn and Δλn=λn′-λn), are unchanged under infinitesimal transformations (20). Similar relations can be obtained for the boosts along two other space-like axes. From the invariance of octonionic norms (B.14) we know that(25)xn≳λn.Then inserting (23) and (25) into (24) we can conclude that the uncertainty relations,(26)ΔxnΔpn≥ħ,in our model have the same geometrical meaning as the existence of the maximal velocity, c [50–53, 59].From (20) we also notice that in the planes orthogonal to v1=x1/t the pair x2′, λ3′ increases, while x3′, λ2′ decreases, and for the case (21) we have(27)x2λ3=-x3λ2=θ1.Using similar relations for the boosts along other two spatial directions we conclude that there exists some 3-vector (spin):(28)σn=1ħεnmkxmpk~εnmkxmλk,which characterizes relative rotations of xn and λn in the direction orthogonal to the velocity planes. One can speculate on the connection of the left-right asymmetry in relative rotations of xn and λn in (20) with the right-handed neutrino problem for massless case.


## 4. Parity Violation

For the reference frame (21), for the finite angles ϕ1=θ1, from (20) one can obtain the standard relativistic expressions:(29)tanh⁡θ1=Vc,cosh⁡θ1=11-V2/c2,sinh⁡θ1=V/c1-V2/c2,where V is the velocity of moving system along x1. Then from (20) there follows the generalized rule of velocity addition:(30)v1′=v1-ctanh⁡θ11-tanh⁡θ1v1/c,v2′=v2-tanh⁡θ1λ˙31-tanh⁡θ1v1/c,v3′=v3+tanh⁡θ1λ˙21-tanh⁡θ1v1/c.We see that the standard expressions are altered only in the planes orthogonal to V and that what is important are the terms with different sign. One consequence of this fact is that the formula for the aberration of light will be modified.Consider photons moving in the(x1,x2)-plane:(31)v1=ccos⁡γ12,v2=csin⁡γ12,v1′=ccos⁡γ12′,v2′=csin⁡γ12′,where γ12 and γ12′ are angles between v1 and v1′ and the axes x1 and x1′, respectively. Using (30), for the case V/c≪1, we have(32)sin⁡γ12′=sin⁡γ12-V/c2λ˙31-V/ccos⁡γ12,cos⁡γ12′=cos⁡γ12-V/c1-V/ccos⁡γ12,where λ˙3 is the rate of Doppler’s shift along an axis, x3, orthogonal to the photon’s trajectory. Then angle of aberration in the (x1,x2)-plane give the value(33)Δγ12=γ12′-γ12=Vcsin⁡γ12-Vc2λ˙3.Analogous to (33) the formula for the aberration angle in the (x1,x3)-plane has the form(34)Δγ13=Vcsin⁡γ13+Vc2λ˙2,where λ˙2 is the rate of the Doppler shift towards the orthogonal to the (x1,x3)-plane direction. This spatial asymmetry of aberration, which distinguishes the left and right coordinate systems, may be detectable by precise quantum measurements.


## 5. Spin and Hypercharge

Now consider the last class of automorphisms (AppendixC):(35)xn′=xn+φn-13∑mφmλn,t′=t,λn′=λn+φn-13∑mφmxnwith no summing by n. These transformations represent rotations of the three pairs of space-like and time-like coordinates, (x1,λ1), x2,λ2, and (x3,λ3), into each other. We have the three planes, (x1-λ1), x2-λ2, and (x3-λ3), that undergo rotations through hyperbolic angles, φ1, φ2, and φ3 (of the three only two are independent), which are the only hyper-planes in the space of split octonions that are affected by one and not two 3-angles of automorphisms. So it is natural to define the Abelian subalgebra of G2NC by generators of two independent rotations in these planes. It is known that the rank of the Cartan subalgebra of G2NC is the same as of the group SU(3) [9–11, 60]. Indeed, in terms of two parameters, K1 and K2, which are related to the angles φn and (C.4) as(36)K1=k1=132φ1-φ2-φ3,K2=32k1+k1=-1232φ3-φ1-φ2,the transformations (35) can be written more concisely:(37)λ1′+Ix1′λ2′+Ix2′λ2′+Ix2′=eK1Λ3+K2Λ8Iλ1+Ix1λ2+Ix2λ2+Ix2,where I is the vector-like octonionic basis unit (I2=1) and Λ3 and Λ8 are the standard 3×3 Gell-Mann matrices:(38)Λ3=1000-10000,Λ8=1310001000-2.Then, using analogies with SU(3), one can classify irreducible representations of the space-time group of our model, G2NC, by the two fundamental simple roots (K1 and K2) corresponding to spin and hypercharge. It is known that all quarks, antiquarks, and mesons can be imbedded in the adjoint representation of G2NC [9–11]. So the symmetry (35), in addition to the uncertainty relations, probably shows the existence of three generations of objects which are necessary to extract three spatial coordinates from any octonionic signal s. To clarify this point note that (9) can be viewed as some kind of space-time interval with four time-like dimensions. The ordinary time parameter, t, corresponds to the distinguished octonionic basis unit, I, while the other three time-like parameters, λn, have a natural interpretation as wavelengths; that is, they do not relate to time as conventionally understood. It is known that a unique physical system in multidimensional geometry generates a large variety of “shadows” in (3+1)-subspace as different dynamical systems (in terms of different Hamiltonians) [61–70]. The information of multidimensional structures, which is retained by these images of the initial system, takes the form of hidden symmetries. For the case of fundamental physical signals with three extra time-like dimensions, in addition to massless particles (which are not affected by extra times), one can observe three generations of particles with different 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 of ω (action) in the first equation (E.7) that appears due to the existence of the three extra time-like parameters, λn. Then three independent (ω,λn)-rotations in 8-dimensional octonionic space will give the appearance of three generations of particles in ordinary (3+1)-dimensions. Such a possibility is not ruled out by problems with ghosts and with unitarity, since split octonions with positive norms form the division algebra.


## 6. Free Particle Lagrangians

It is known that in split algebras there can be constructed special elements with zero norms, which are called zero divisors [1]. These zero-norm objects are important structures in physical applications [71]. Zero divisors (light-cone operators) of split octonions could serve as the unit signals and thus may describe elementary particles. Then the number of primitive idempotents (eight) and nilpotents (twelve) numerates the types of fundamental particles (Appendix E), bosons, and fermions, respectively. Indeed, the properties that the product of two projection operators reduces to the same idempotent (E.2), while the product of two Grassmann numbers is zero (E.4) naturally explain the validity of Bose and Fermi statistics for the corresponding elementary particles.The zero-norm condition,N2=0, for the paths (9) of elementary particles, be they massless or massive, can be written in the form(39)ds2=dsds†=dω2-c2dt21-v2c2+λ˙2c2=0.For photons,|v|=c, the condition (39) is equivalent to the Eikonal equation:(40)dωdλndωdλn=1,where the wavelengths, λn~ħ/pn, serve as the parameters of trajectory. The relation (40) justifies the interpretation of ω as the phase function (frequency) corresponding to the classical action of particles.For massive particles the time coordinate,t, is good parameter to describe motion and from (39) we find that the scalar part of a split octonionic signal, ω, which is unchanged under automorphisms of G2NC, is the conserved function:(41)dωdt=0.Then (39) can be written as(42)ω=Amc=-c∫dt1-v2c2+λ˙2c2,where A is the classical action of the particle with the mass m. So using (23) the one-particle Lagrangian,(43)L=-mc21-v2c2+ħ2p˙2c2p4,contains an extra “quantum” term, which may be relevant for the relativistic velocities, |v|~c, or for the case of large forces:(44)p˙n~cp2ħ.From (43) we see that on small distance scales, ~ħ/|p|, a particle can even exceed the speed of light (become virtual), since the new quantum term in (43) is positive. The condition (39) and the invariance of the classical action (42) under G2NC hold only for real trajectories. However, there exists the larger group of invariances of the interval (9), SO(4,4), the group of passive tensorial transformations of all eight octonionic coordinates, which in general mixes space-like and time-like subspaces and thus introduces virtual trajectories of particles.Applying the condition (10) (that for physical signals the vector part of a split octonion should be time-like) to (43) its follows that there should exist some maximal force:(45)p˙≤vp2ħ=m2c3ħ,where m denotes the maximal possible value for the mass of particle associated with physical signals. Using estimations of [72, 73] for the maximum force, Fmax=c4/4G, where G is the gravitational constant; from (45) we find that the maximal mass of particles associated with any octonionic signal is the Planck mass:(46)m2~ħcG.


## 7. Conclusion

To conclude, in this paper, it was analyzed consequences of describing physical signals in terms of split octonions over the field of real numbers. Eight real parameters of split octonions were related to space-time coordinates, the phase function (classical action), and wavelengths characterizing physical signals. The(4+4)-norms of the real split octonions are invariant under the group SO(4,4) of tensorial transformations of parameters (passive transformations of coordinates), while the representation of rotations by octonions themselves (i.e., the active transformations of octonionic basis units) corresponds to the real noncompact form of Cartan’s exceptional Lie group G2NC (the subgroup of SO(4,4)), which under certain conditions imitate the Poincaré transformations in ordinary space-time. Within this picture, in front of time-like coordinates in the expression of pseudo-Euclidean octonionic intervals, there naturally appear two fundamental physical parameters, the light speed and Planck’s constant. From the requirement of positive definiteness of norms under G2NC-transformations, together with the introduction of the maximal velocity, c, there follow conditions which are equivalent to uncertainty relations in quantum physics. By examining of G2NC-automorphisms it is explicitly shown that there exists a 3-vector (spin) that is fully chiral, which corresponds to the observed chirality of neutrinos in nature. We examined also the G2NC-effects on a quantum system moving at constant speed with respect to the observer, which emits light in various planes relative to the direction of motion, and new parity-violating effect on aberration and Doppler shift was derived. It was shown how a particular class of G2NC-automorphisms of space-time coordinates can be expressed in concise form using two simple roots of the SU(3) Lie algebra. Due to their conventional association with spin and hypercharge in physics, we argued that our split-octonion geometry inherently modeled three generations of fundamental particles. In our approach elementary particles are interpreted as light-cone operators; that is, they are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions in G2NC-geometry lead to corresponding free particle Lagrangians, which allow virtual particle trajectories (exceeding the speed of light, in certain cases) and exhibit the existence of spatial horizons governing by mass parameters.



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