Rotations in the Space of Split Octonions

Abstract

The geometrical application of split octonions is considered. The new representation of products of the basis units of split octonionic having David's star shape (instead of the Fano triangle) is presented. It is shown that active and passive transformations of coordinates in octonionic “eight-space” are not equivalent. The group of passive transformations that leave invariant the pseudonorm of split octonions is 𝑆𝑂(4,4), while active rotations are done by the direct product of 𝑂(3,4)-boosts and real noncompact form of the exceptional group 𝐺₂. In classical limit, these transformations reduce to the standard Lorentz group.

1. Introduction

Nonassociative algebras may surely be called beautiful mathematical entities. However, they have never been systematically utilized in physics, only some attempts have been made toward this goal. Nevertheless, there are some intriguing hints that nonassociative algebras may play essential role in the ultimate theory, yet to be discovered.

Octonions are one example of a nonassociative algebra. It is known that they form the largest normed algebra after the algebras of real numbers, complex numbers, and quaternions [1–3]. Since their discovery in 1844/1845 by Graves and Cayley there have been various attempts to find appropriate uses for octonions in physics (see reviews [4–7]). One can point to the possible impact of octonions on: Color symmetry [8–11]; GUTs [12–15]; Representation of Clifford algebras [16–19]; Quantum mechanics [20–24]; Space-time symmetries [25, 26]; Field theory [27–29]; Formulations of wave equations [30–32]; Quantum Hall effect [33]; Kaluza-Klein program without extra dimensions [34–36]; Strings and 𝑀-theory [37–40]; and so forth.

In this paper we study rotations in the model, where geometry is described by the split octonions [41–43].

2. Octonionic Geometry

Let us review the main ideas behind the geometrical application of split octonions presented in our previous papers [41–43]. In our model some characteristics of physical world (such as dimension, causality, maximal velocities, and quantum behavior) can be naturally described by the properties of split octonions. Interesting feature of the geometrical interpretation of the split octonions is that their pseudonorms, in addition to some other terms, already contain the ordinary Minkowski metric. This property is equivalent to the existence of local Lorentz invariance in classical physics.

To any physical signal we correspond eight-dimensional number, the element of split octonions, 𝑠=𝑐𝑡+𝑥^𝑛𝐽_𝑛+ℏ𝜆^𝑛𝑗_𝑛+𝑐ℏ𝜔𝐼(𝑛=1,2,3). (2.1) Here we have one scalar basis unit (denoted as 1), the three vector-like objects 𝐽_𝑛, the three pseudovector-like elements 𝑗_𝑛, and one pseudoscalar-like unit 𝐼. The eight real parameters that multiply basis elements we treat as the time 𝑡, the special coordinates 𝑥^𝑛, some quantities 𝜆^𝑛 with the dimensions momentum⁻¹, and the quantity 𝜔 having the dimension energy⁻¹. We suppose also that (2.1) contains two fundamental constants of physics: the velocity of light 𝑐 and the Planck constant ℏ.

The squares of basis units of split octonions are inner product resulting unit element, but with the opposite signs, 𝐽²_𝑛=1,𝑗²_𝑛=−1,𝐼²=1. (2.2) Multiplications of different hypercomplex basis units are defined as skew products (𝑛,𝑚,𝑘=1,2,3), 𝐽_𝑛𝐽_𝑚=−𝐽_𝑚𝐽_𝑛=𝜖_𝑛𝑚𝑘𝑗^𝑘,𝑗_𝑛𝑗_𝑚=−𝑗_𝑚𝑗_𝑛=𝜖_𝑛𝑚𝑘𝑗^𝑘,𝐽_𝑛𝑗_𝑚=−𝑗_𝑚𝐽_𝑛=−𝜖_𝑛𝑚𝑘𝐽^𝑘,𝐽_𝑛𝐼=−𝐼𝐽_𝑛=𝑗_𝑛,𝑗_𝑛𝐼=−𝐼𝑗_𝑛=𝐽_𝑛, (2.3) where 𝜖_𝑛𝑚𝑘 is the fully antisymmetric tensor.

From (2.3) we notice that to generate complete basis of split octonions the multiplication and distribution laws of only three vector-like elements 𝐽_𝑛 are needed. In geometrical application this can explain why classical space has three dimensions. The three pseudovector-like basis units 𝑗_𝑛 can be defined as the binary product 𝑗_𝑛=12𝜖_𝑛𝑚𝑘𝐽^𝑚𝐽^𝑘, (2.4) and thus can describe oriented orthogonal planes spanned by two vector-like elements 𝐽_𝑛. The seventh basic unit 𝐼 (the oriented volume) is formed by the products of all three fundamental basis elements 𝐽_𝑛 and has three equivalent representation: 𝐼=𝐽₁𝑗₁=𝐽₂𝑗₂=𝐽₃𝑗₃. (2.5)

The multiplication table of octonionic units is most transparent in graphical form. To visualize the products of ordinary octonions the Fano triangle is used [1–3], where the seventh basic unit 𝐼 is place at the center of the graph. In the algebra of split octonions we have less symmetry, and for a proper description of the products (2.3) the Fano graph should be modified by shifting 𝐼 from the center of the Fano triangle. Also we will use three equivalent representations of 𝐼, (2.5), and, instead of the Fano triangle, we arrive at David's star shaped duality plane for products of the split octonionic basis elements.

On this graph 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 contributes a minus sign to the result. Dashed lines just show that the corners of the triangle with 𝐼 nodes are identified.

Conjugation, which can be understand as a reflection of the vector-like basis units 𝐽_𝑛, reverses the order of octonionic basis elements in any given expression, thus 𝐽⁺_𝑛=−𝐽_𝑛,𝑗⁺_𝑛=12𝜖_𝑛𝑚𝑘(𝐽^𝑚𝐽^𝑘)⁺=12𝜖_𝑛𝑚𝑘𝐽^𝑘+𝐽^𝑚+=−𝑗_𝑛,𝐼⁺=(𝐽_𝑛𝑗_𝑛)⁺=𝑗⁺_𝑛𝐽⁺_𝑛=−𝐼, (2.6) there is no summing in the last formula. So the conjugation of (2.1) gives 𝑠⁺=𝑐𝑡−𝑥_𝑛𝐽^𝑛−ℏ𝜆_𝑛𝑗^𝑛−𝑐ℏ𝜔𝐼. (2.7) Using (2.2) one can find that the pseudonorm of (2.1), 𝑠²=𝑠𝑠⁺=𝑠⁺𝑠=𝑐²𝑡²−𝑥_𝑛𝑥^𝑛+ℏ²𝜆_𝑛𝜆^𝑛−𝑐²ℏ²𝜔², (2.8) has (4+4) signature. If we consider 𝑠 as the interval between two octonionic signals we see that (2.8) reduces to the classical formula of Minkowski space-time in the limit ℏ→0.

Using the algebra of basis elements (2.3) the octonion (2.1) can be written in the equivalent form 𝑠=𝑐(𝑡+ℏ𝜔𝐼)+𝐽^𝑛(𝑥_𝑛+ℏ𝜆_𝑛𝐼). (2.9) We notice that the pseudoscalar-like element 𝐼 introduces the 'quantum' term corresponding to some kind of uncertainty of space-time coordinates. For the differential form of (2.9) the invariance of the pseudonorm (2.8) gives the relation: 𝑑√𝑠²=𝑐𝑑𝑡𝑣1−²𝑐²(1−ℏ²𝑑𝜆^𝑛𝑑𝑥^𝑚𝑑𝜆_𝑛𝑑𝑥_𝑚)−(ℏ𝑑𝜔)𝑑𝑡², (2.10) where 𝑣_𝑛=𝑑𝑥_𝑛/𝑑𝑡 denotes 3-dimensional velocity measured in the frame (2.1). The generalized Lorentz factor (2.10) contains extra terms that vanish in the limit ℏ→0. So the dispersion relation in our model has a form similar to that of double-special relativity models [44, 45].

From the requirement to have the positive pseudonorm (2.8) from (2.10) we obtain several relations 𝑣²≤𝑐²,𝑑𝑥^𝑛𝑑𝜆^𝑛≥ℏ,𝑑𝑡𝑑𝜔≥ℏ. (2.11) Recalling that 𝜆 and 𝜔 have dimensions of momentum⁻¹ and energy⁻¹, respectively, we conclude that the Heisenberg uncertainty principle in our model has the same geometrical meaning as the existence of the maximal velocity in Minkowski space-time.

3. Rotations

To describe rotations in 8-dimensional octonionic space (2.1) with the interval (2.8) we need to define exponential maps for the basis units of split octonions.

Since the squares of the pseudovector-like elements 𝑗_𝑛 are negative, 𝑗²_𝑛=−1, we can define 𝑒^{𝑗𝑛𝜃𝑛}=cos𝜃_𝑛+𝑗_𝑛sin𝜃_𝑛, (3.1) where 𝜃_𝑛 are some real angles.

At the same time for the other basis elements 𝐽_𝑛,𝐼, which have the positive squares 𝐽²_𝑛=𝐼²=1, we have 𝑒^{𝐽𝑛𝑚𝑛}=cosh𝑚_𝑛+𝑗_𝑛sinh𝑚_𝑛,𝑒^𝐼𝜎=cosh𝜎+𝐼sinh𝜎, (3.2) where 𝑚_𝑛 and 𝜎 are real numbers.

In 8-dimensional octonionic “space-time’’ (2.1) there is no unique plane orthogonal to a given axis. Therefore for the operators (3.1) and (3.2) 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 (3.1), (3.2) (e.g., 𝑒^𝑗1𝜃1) yields four simultaneous rotations in four mutually orthogonal planes. For simplicity we consider only the left products since it is known that one side multiplications generate the whole symmetry group that leaves the octonionic norms invariant [46].

So rotations naturally provide splitting of an octonion in four orthogonal planes. To define these planes note that one of them is formed by the hypercomplex element that we chose to define the rotation (𝑗₁ in our example), together with the scalar unit element of the octonion. The rest orthogonal planes are given by the three pairs of other basis elements that lie with the considered basis unit (𝑗₁ in the example) on the lines emerged it in David's star (see Figure 1). Thus the pairs of basis units that are rotated into each other are the pairs that form associative triplets with the considered basis unit. For example, the basis unit 𝑗₁, according to Figure 1, has three different representations in the octonionic algebra: 𝑗₁=𝐽₂𝐽₃=𝑗₂𝑗₃=𝐽₁𝐼. (3.3) So the planes orthogonal to (1−𝑗₁) are (𝐽₂−𝐽₃), (𝑗₂−𝑗₃), and (𝐽₁−𝐼). Using (3.3) and the representation (3.1) it is possible to “rotate out’’ the four octonionic axes, and (2.1) can be written in the equivalent form 𝑠=𝑁_𝑡𝑒^{𝑗1𝜃𝑡}+𝑁_𝑥𝑒^{𝑗1𝜃𝑥}𝐽₃+𝑁_𝜆𝑒^{𝑗1𝜃𝜆}𝑗₂+𝑁_𝜔𝑒^{𝑗1𝜃𝜔}𝐼, (3.4) where 𝑁_𝑡=𝑐²𝑡²+ℏ²𝜆²₁,𝑁_𝑥=𝑥²₂+𝑥²₃,𝑁_𝜆=ℏ𝜆²₂+𝜆²₃,𝑁_𝜔=𝑥²₁+𝑐²ℏ²𝜔² (3.5) are the norms in four orthogonal octonionic planes. The corresponding angles are given by 𝜃_𝑡=arccos(𝑡/𝑁_𝑡),𝜃_𝑥=arccos(𝑥₃/𝑁_𝑥𝜃),_𝜆=arccos(ℏ𝜆₂/𝑁_𝜆),𝜃_𝜔=arccos(𝑐ℏ𝜔/𝑁_𝜔). (3.6) This decomposition of split octonion is valid only if the full pseudonorm of the octonion (2.8) is positive, that is, 𝑠²=𝑁²_𝑡−𝑁²_𝑥+𝑁²_𝜆−𝑁²_𝜔>0. (3.7) A decomposition similar to (3.4) exists if the another pseudovector-like basis unit, 𝑗₂ or 𝑗₃, is fixed.

Figure 1 

Split octonion multiplication as David's star.

In contrast with uniform rotations giving by the operators 𝑗_𝑛 we have limited rotations in the planes orthogonal to (1−𝐽_𝑛) and (1−𝐼). However, we can still perform a decomposition similar to (3.4) of 𝑠 using expressions of the exponential maps (3.2). But now, unlike on (3.5), the norms of the corresponding planes are not positively defined and, instead of the condition (3.7), we should require positiveness of the norms of each four planes. For example, the pseudoscalar-like basis unit 𝐼 has three different representations (2.5), and it can provide the hyperbolic rotations (3.2) in the orthogonal planes (1−𝐼), (𝐽₁−𝑗₁), (𝐽₂−𝑗₂), and (𝐽₃−𝑗₃). The expressions for the 2 norms (3.5) in this case are: 𝑐𝑡²−ℏ²𝜔², 𝑥²₁−ℏ²𝜆²₁, 𝑥²₂−ℏ²𝜆²₂, and 𝑥²₃−ℏ²𝜆²₃.

Now let us consider active and passive transformations of coordinates in 8-dimensional space of signals (2.1). With a passive transformation we mean a change of the coordinates 𝑡, 𝑥_𝑛, 𝜆_𝑛, and 𝜔, as opposed to an active transformation which changes the basis 1, 𝐽_𝑛, 𝑗_𝑛, and 𝐼.

The passive transformations of the octonionic coordinates 𝑡, 𝑥_𝑛, 𝜆_𝑛, and 𝜔, which leave invariant the norm (2.8) form 𝑆𝑂(4,4). We can represent these transformations of (2.1) by the left products 𝑠^′=𝑅𝑠, (3.8) where 𝑅 is one of (3.1), (3.2). The operator 𝑅 simultaneously transforms four planes of 𝑠. However, in three planes 𝑅 can be rotated out by the proper choice of octonionic basis. Thus 𝑅 can represent rotations separately in four orthogonal planes of 𝑠. Similarly we have some four angles for the other six operators (3.1), (3.2) 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 (3.4) we can introduce four arbitrary angles 𝜙_𝑡, 𝜙_𝑥, 𝜙_𝜆, and 𝜙_𝜔, and 𝑠^′=𝑁_𝑡𝑒^{𝑗1(𝜃𝑡+𝜙𝑡)}+𝑁_𝑥𝑒^{𝑗1(𝜃𝑥+𝜙𝑥)}𝐽₃+𝑁_𝜆𝑒^{𝑗1(𝜃𝜆+𝜙𝜆)}𝑗₂+𝑁_𝜔𝑒^{𝑗1(𝜃𝜔+𝜙𝜔)}𝐼. (3.9) Obviously under these transformations the pseudonorm (2.8) is invariant. By the fine tuning of the angles in (3.9) we can define rotations in any single plane from four.

Now let us consider active coordinate transformations, or transformations of basis units 1, 𝐽_𝑛, 𝑗_𝑛, and 𝐼. For them, because of nonassociativity, the results of two different rotations (3.1) and (3.2) are not unique. This means that not all active octonionic transformations (3.1) and (3.2) form a group and can be considered as a real rotation. Thus in the octonionic space (2.1) not to the all passive 𝑆𝑂(4,4)-transformations we can make corresponding active ones, only the transformations that have a realization as associative multiplications should be considered. It is known that associative transformations can be done by the combined rotations of special form in two octonionic planes that form a subgroup of 𝑆𝑂(4,4), known as the automorphism group of split octonions 𝐺^NC₂ (the real noncompact form of Cartan's exceptional Lie group 𝐺₂). Some general results on 𝐺^NC₂ and its subgroup structure can be found in [47,16b].

Let us recall that the automorphism 𝐴 of a algebra is defined as the transformations of the hypercomplex basis units 𝑥 and 𝑦 under which the multiplication table of the algebra is invariant, that is, 𝐴(𝑥+𝑦)=𝐴𝑥+𝐴𝑦,(𝐴𝑥)(𝐴𝑦)=(𝐴(𝑥𝑦)). (3.10) Associativity of these transformations is obvious from the second relation, and the set of all automorphisms of composition algebras form a group. In the case of quaternions, because 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. One has a different situation for octonions. Each automorphism in the octonionic algebra is completely defined by the images of three elements that do not form quaternionic subalgebras, that is, they all not lie on the same David's line [49]. Consider one such set, say (𝑗₁,𝑗₂,𝐽₁). Then there exists an automorphism 𝑗^′₁=𝑗₁,𝑗^′₂=𝑗₂cos(𝛼₁+𝛽₁)/2+𝑗₃sin(𝛼₁+𝛽₁𝐽)/2,_′1=𝐽₁cos𝛽₁+𝐼sin𝛽₁, (3.11) where 𝛼₁ and 𝛽₁ are some independent real angles. By the definition (3.10) the automorphism does not affect unit scalar 1. The images of the other basis elements under automorphism (3.11) are determined by the conditions

𝑗^′₃=𝑗₃cos(𝛼₁+𝛽₁)/2−𝑗₂sin(𝛼₁+𝛽₁)/2=𝑗^′₁𝑗^′₂,𝐼^′=𝐼cos𝛽₁−𝐽₁sin𝛽₁=𝐽^′₁𝑗^′₁,𝐽^′₂=𝐽₂cos(𝛼₁−𝛽₁)/2+𝐽₃sin(𝛼₁−𝛽₁)/2=𝑗^′₂𝐼^′,𝐽^′₃=𝐽₃cos(𝛼₁−𝛽₁)/2−𝐽₂sin(𝛼₁−𝛽₁)/2=𝑗^′₃𝐼^′. (3.12) It can easily be checked that transformed bases 𝐽^′_𝑛, 𝑗^′_𝑛, 𝐼^′ satisfy the same multiplication rules as 𝐽_𝑛, 𝑗_𝑛, 𝐼.

There exist similar automorphisms with fixed 𝑗₂ and 𝑗₃ axes, which are generated by the angles 𝛼₂, 𝛽₂ and 𝛼₃, 𝛽₃, respectively.

One can define also hyperbolic automorphisms for the vector-like units 𝐽_𝑛 by the angles 𝑢_𝑛,𝑘_𝑛. For example, for fixed 𝐽₁ similar to (3.11) and (3.12) transformations are 𝐽^′₁=𝐽₁,𝐽^′₂=𝐽₂cosh(𝑘₁+𝑢₁)/2+𝑗₃sinh(𝑘₁+𝑢₁𝐼)/2,_′=𝐼cosh𝑢₁−𝑗₁sinh𝑢₁,𝑗^′₃=𝑗₃cosh(𝑘₁+𝑢₁)/2+𝑗₂sinh(𝑘₁+𝑢₁)/2=𝐽^′₁𝐽^′₂,𝑗^′₁=𝑗₁cosh𝑢₁−𝐼sinh𝑢₁=𝐽^′₁𝐼^′,𝑗^′₂=𝑗₂cosh(𝑘₁−𝑢₁)/2+𝐽₃sinh(𝑘₁−𝑢₁)/2=𝐽^′₂𝐼^′,𝐽^′₃=𝐽₃cosh(𝑘₁−𝑢₁)/2+𝐽₂sinh(𝑘₁−𝑢₁)/2=𝑗^′₃𝐼^′. (3.13) Analogously in the case of fixed 𝐼 we find that 𝐼^′𝑗=𝐼,_′1=𝑗₁cosh𝜎₁+𝐽₁sinh𝜎₁,𝑗^′₂=𝑗₂cosh𝜎₂+𝐽₂sinh𝜎₂,𝐽^′₁=𝑗^′₁𝐼^′=𝐽₁cosh𝜎₁+𝑗₁sinh𝜎₁,𝐽^′₂=𝑗^′₂𝐼^′=𝐽₂cosh𝜎₂+𝑗₂sinh𝜎₂,𝑗^′₃=𝑗^′₁𝑗^′₂=𝑗₃cosh(𝜎₁+𝜎₂)−𝐽₃sinh(𝜎₁+𝜎₂𝐽),_′3=𝑗^′₃𝐼^′=𝐽₃cosh(𝜎₁+𝜎₂)−𝑗₃sinh(𝜎₁+𝜎₂). (3.14)

So for each octonionic basis there are seven independent automorphisms each introducing two angles that correspond to 2×7=14 generators of the algebra 𝐺^NC₂. For our choice of basis the infinitesimal passive transformation of the coordinates, corresponding to 𝐺^NC₂, has the form 𝑡^′𝑥=𝑡,_′𝑖=𝑥_𝑖−12𝜖_𝑖𝑗𝑘(𝛼^𝑗−𝛽^𝑗)𝑥^𝑘+𝑐ℏ𝛽^𝑖ℏ𝜔+2(𝑈_𝑖𝑘−𝜖_𝑖𝑗𝑘𝑢^𝑗)𝜆^𝑘,𝜔^′1=𝜔−𝛽𝑐ℏ^𝑖𝑥^𝑖−1𝑐𝑢_𝑖𝜆^𝑖,𝜆^′_𝑖=𝜆_𝑖−12𝜖_𝑖𝑗𝑘(𝛼^𝑗+𝛽^𝑗)𝜆^𝑘−𝑐𝑢^𝑖1𝜔+2ℏ(𝑈_𝑖𝑘+𝜖_𝑖𝑗𝑘𝑢^𝑗)𝑥^𝑘, (3.15) where 𝑈_𝑖𝑘 is the symmetric matrix 𝑈=2𝜎₁𝑘₃𝑘₂𝑘₃2𝜎₂𝑘₁𝑘₂𝑘₁−2(𝜎₁+𝜎₂). (3.16)

In the limit (ℏ𝜆,ℏ𝜔→0) the transformations (3.15) reduce to the standard 𝑂(3) rotations of Euclidean 3-space by the Euler angles 𝜙_𝑛=𝛼_𝑛−𝛽_𝑛.

The formulas (3.15) represent rotations of (3,4)-sphere that is orthogonal to the time coordinate 𝑡. To define the boosts note that active and passive forms of mutual transformations of 𝑡 with 𝑥_𝑛,𝜆_𝑛, and 𝜔 are isomorphic and can be described by the seven operators (3.1) and (3.2) (e.g., the first term in (3.9), which form the group 𝑂(3,4). In the case (ℏ𝜆,ℏ𝜔→0) we recover the standard 𝑂(3) Lorentz boost in the Minkowski space-time governing by the operators 𝑒^{𝐽𝑛𝑚𝑛}, where 𝑚_𝑛=arctan𝑣_𝑛/𝑐.

4. Conclusion

In this paper the David's star duality plane, which describes the multiplication table of the basis units of split octonions (instead of the Fano triangle of ordinary octonions), was introduced. Different kind of rotations in the split octonionic space was 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 pseudonorms of split octonions, is 𝑆𝑂(4,4), while active rotations are done by the direct product of the seven 𝑂(3,4)-boosts and fourteen 𝐺^NC₂-rotations. In classical limit these transformations give the standard 6-parametrical Lorentz group.

Acknowledgment

The author would like to acknowledge the support of a 2008-2009 Fulbright Fellowship.