Clifford Algebras and Possible Kinematics

We review Bacry and Lévy-Leblond's work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley-Klein geometries, illustrating how the Cayley-Klein geometries give homogeneous spacetimes for all but one of the kinematical groups. We then...

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Datum:	2007
1. Verfasser:	McRae, A.S.
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spelling	irk-123456789-1473672019-02-15T01:23:13Z Clifford Algebras and Possible Kinematics McRae, A.S. We review Bacry and Lévy-Leblond's work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley-Klein geometries, illustrating how the Cayley-Klein geometries give homogeneous spacetimes for all but one of the kinematical groups. We then construct a two-parameter family of Clifford algebras that give a unified framework for representing both the Lie algebras as well as the kinematical groups, showing that these groups are true rotation groups. In addition we give conformal models for these spacetimes. 2007 Article Clifford Algebras and Possible Kinematics / A.S. McRae // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 28 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 11E88; 15A66; 53A17 http://dspace.nbuv.gov.ua/handle/123456789/147367 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We review Bacry and Lévy-Leblond's work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley-Klein geometries, illustrating how the Cayley-Klein geometries give homogeneous spacetimes for all but one of the kinematical groups. We then construct a two-parameter family of Clifford algebras that give a unified framework for representing both the Lie algebras as well as the kinematical groups, showing that these groups are true rotation groups. In addition we give conformal models for these spacetimes.
format	Article
author	McRae, A.S.
spellingShingle	McRae, A.S. Clifford Algebras and Possible Kinematics Symmetry, Integrability and Geometry: Methods and Applications
author_facet	McRae, A.S.
author_sort	McRae, A.S.
title	Clifford Algebras and Possible Kinematics
title_short	Clifford Algebras and Possible Kinematics
title_full	Clifford Algebras and Possible Kinematics
title_fullStr	Clifford Algebras and Possible Kinematics
title_full_unstemmed	Clifford Algebras and Possible Kinematics
title_sort	clifford algebras and possible kinematics
publisher	Інститут математики НАН України
publishDate	2007
url	http://dspace.nbuv.gov.ua/handle/123456789/147367
citation_txt	Clifford Algebras and Possible Kinematics / A.S. McRae // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 28 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT mcraeas cliffordalgebrasandpossiblekinematics
first_indexed	2025-07-11T01:55:53Z
last_indexed	2025-07-11T01:55:53Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 079, 29 pages Clifford Algebras and Possible Kinematics Alan S. MCRAE Department of Mathematics, Washington and Lee University, Lexington, VA 24450-0303, USA E-mail: mcraea@wlu.edu Received April 30, 2007, in final form July 03, 2007; Published online July 19, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/079/ Abstract. We review Bacry and Lévy-Leblond’s work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley–Klein geo- metries, illustrating how the Cayley–Klein geometries give homogeneous spacetimes for all but one of the kinematical groups. We then construct a two-parameter family of Clifford algebras that give a unified framework for representing both the Lie algebras as well as the kinematical groups, showing that these groups are true rotation groups. In addition we give conformal models for these spacetimes. Key words: Cayley–Klein geometries; Clifford algebras; kinematics 2000 Mathematics Subject Classification: 11E88; 15A66; 53A17 As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection. Joseph Louis Lagrange (1736–1813) The first part of this paper is a review of Bacry and Lévy-Leblond’s description of possible kinematics and how such kinematical structures relate to the Cayley–Klein formalism. We review some of the work done by Ballesteros, Herranz, Ortega and Santander on homogeneous spaces, as this work gives a unified and detailed description of possible kinematics (save for static kinematics). The second part builds on this work by analyzing the corresponding kinematical models from other unified viewpoints, first through generalized complex matrix realizations and then through a two-parameter family of Clifford algebras. These parameters are the same as those given by Ballesteros et. al., and relate to the speed of light and the universe time radius. Part I. A review of kinematics via Cayley–Klein geometries 1 Possible kinematics As noted by Inonu and Wigner in their work [17] on contractions of groups and their repre- sentations, classical mechanics is a limiting case of relativistic mechanics, for both the Galilei group as well as its Lie algebra are limits of the Poincaré group and its Lie algebra. Bacry and Lévy-Leblond [1] classified and investigated the nature of all possible Lie algebras for kinema- tical groups (these groups are assumed to be Lie groups as 4-dimensional spacetime is assumed to be continuous) given the three basic principles that (i) space is isotropic and spacetime is homogeneous, (ii) parity and time-reversal are automorphisms of the kinematical group, and (iii) the one-dimensional subgroups generated by the boosts are non-compact. mailto:mcraea@wlu.edu http://www.emis.de/journals/SIGMA/2007/079/ 2 A.S. McRae Table 1. The 11 possible kinematical groups. Symbol Name dS1 de Sitter group SO(4, 1) dS2 de Sitter group SO(3, 2) P Poincaré group P ′ 1 Euclidean group SO(4) P ′ 2 Para-Poincaré group C Carroll group N+ Expanding Newtonian Universe group N− Oscillating Newtonian Universe group G Galilei group G′ Para-Galilei group St Static Universe group speed-time contraction dS1 dS2 P’ 2 P’ 1 P G’ G C St space-tim e contra ctio n sp eed -sp ace co n tractio n N+ N- Figure 1. The contractions of the kinematical groups. The resulting possible Lie algebras give 11 possible kinematics, where each of the kinematical groups (see Table 1) is generated by its inertial transformations as well as its spacetime trans- lations and spatial rotations. These groups consist of the de Sitter groups and their rotation- invariant contractions: the physical nature of a contracted group is determined by the nature of the contraction itself, along with the nature of the parent de Sitter group. Below we will illustrate the nature of these contractions when we look more closely at the simpler case of a 2- dimensional spacetime. For Fig. 1, note that a “upper” face of the cube is transformed under one type of contraction into the opposite face. Sanjuan [23] noted that the methods employed by Bacry and Lévy-Leblond could be easily applied to 2-dimensional spacetimes: as it is the purpose of this paper to investigate these kine- matical Lie algebras and groups through Clifford algebras, we will begin by explicitly classifying all such possible Lie algebras. This section then is a detailed and expository account of certain parts of Bacry, Lévy-Leblond, and Sanjuan’s work. Clifford Algebras and Possible Kinematics 3 Table 2. The 21 kinematical Lie algebras, grouped into 11 essentially distinct types of kinematics. P −P P −P P −P 0 0 0 0 0 H −H H −H 0 0 H −H 0 0 0 K −K −K K 0 0 0 0 K −K 0 0 0 0 0 P −P P −P P −P H −H H −H 0 0 0 0 H −H K −K −K K K −K −K K 0 0 Table 3. 6 non-kinematical Lie algebras. P −P P −P −P P −H H −H H H −H K −K −K K 0 0 Let K denote the generator of the inertial transformations, H the generator of time transla- tions, and P the generator of space translations. As space is one-dimensional, space is isotropic. In the following section we will see how to construct, for each possible kinematical structure, a spacetime that is a homogeneous space for its kinematical group, so that basic principle (i) is satisfied. Now let Π and Θ denote the respective operations of parity and time-reversal: K must be odd under both Π and Θ. Our basic principle (ii) requires that the Lie algebra is acted upon by the Z2 ⊗ Z2 group of involutions generated by Π : (K,H,P ) → (−K,H,−P ) and Θ : (K,H,P ) → (−K,−H,P ) . Finally, basic principle (iii) requires that the subgroup generated by K is noncompact, even though we will allow for the universe to be closed, or even for closed time-like worldlines to exist. We do not wish for e0K = eθK for some non-zero θ, for then we would find it possible for a boost to be no boost at all! As each Lie bracket [K,H], [K,P ], and [H,P ] is invariant under the involutions Π and Θ as well as the involution Γ = ΠΘ : (K,H,P ) → (K,−H,−P ) , we must have that [K,H] = pP , [K,P ] = hH, and [H,P ] = kK for some constants k, h, and p. Note that these Lie brackets are also invariant under the symmetries defined by SP : {K ↔ H, p↔ −p, k ↔ h}, SH : {K ↔ P, h↔ −h, k ↔ −p}, and SK : {H ↔ P, k ↔ −k, h↔ p}, and that the Jacobi identity is automatically satisfied for any triple of elements of the Lie algebra. We can normalize the constants k, h, and p by a scale change so that k, h, p ∈ {−1, 0, 1}, taking advantage of the simple form of the Lie brackets for the basis elements K, H, and P . There are then 33 possible Lie algebras, which we tabulate in Tables 2 and 3 with columns that have the following form: [K,H] [K,P ] [H,P ] 4 A.S. McRae Table 4. Some kinematical groups along with their notation and structure constants. Anti-de Sitter Oscillating Newtonian Universe Para-Minkowski Minkowski adS N− M ′ M [K,H] P P 0 P [K,P ] H 0 H H [H,P ] K K K 0 Table 5. Some kinematical groups along with their notation and structure constants. de Sitter Expanding Newtonian Universe Expanding Minkowski Universe dS N+ M+ [K,H] P P 0 [K,P ] H 0 H [H,P ] −K −K −K We also pair each Lie algebra with its image under the isomorphism given by P ↔ −P , H ↔ −H, K ↔ −K, and [?, ??] ↔ [??, ?], for both Lie algebras then give the same kinematics. There are then 11 essentially distinct kinematics, as illustrated in Table 2. Also (as we shall see in the next section) each of the other 6 Lie algebras (that are given in Table 3) violate the third basic principle, generating a compact group of inertial transformations. These non-kinematical Lie algebras are the lie algebras for the motion groups for the elliptic, hyperbolic, and Euclidean planes: let us denote these respective groups as El, H, and Eu. We name the kinematical groups (that are generated by the boosts and translations) in concert with the 4-dimensional case (see Tables 4, 5, and 6). Each of these kinematical groups is either the de Sitter or the anti-de Sitter group, or one of their contractions. We can contract with respect to any subgroup, giving us three fundamental types of contraction: speed-space, speed- time, and space-time contractions, corresponding respectively to contracting to the subgroups generated by H, P , and K. Speed-space contractions. We make the substitutions K → εK and P → εP into the Lie algebra and then calculate the singular limit of the Lie brackets as ε → 0. Physically the velocities are small when compared to the speed of light, and the spacelike intervals are small when compared to the timelike intervals. Geometrically we are describing spacetime near a timelike geodesic, as we are contracting to the subgroup that leaves this worldline invariant, and so are passing from relativistic to absolute time. So adS is contracted to N− while dS is contracted to N+, for example. Speed-time contractions. We make the substitutions K → εK and H → εH into the Lie algebra and then calculate the singular limit of the Lie brackets as ε → 0. Physically the velocities are small when compared to the speed of light, and the timelike intervals are small when compared to the spacelike intervals. Geometrically we are describing spacetime near a spacelike geodesic, as we are contracting to the subgroup that leaves invariant this set of simultaneous events, and so are passing from relativistic to absolute space. Such a spacetime may be of limited physical interest, as we are only considering intervals connecting events that are not causally related. Space-time contractions. We make the substitutions P → εP and H → εH into the Lie algebra and then calculate the singular limit of the Lie brackets as ε → 0. Physically the spacelike and timelike intervals are small, but the boosts are not restricted. Geometrically we are describing spacetime near an event, as we are contracting to the subgroup that leaves invariant only this one event, and so we call the corresponding kinematical group a local group as opposed to a cosmological group. Clifford Algebras and Possible Kinematics 5 Table 6. Some kinematical groups along with their notation and structure constants. Galilei Carroll Static de Sitter Universe Static Universe G C SdS St [K,H] P 0 0 0 [K,P ] 0 H 0 0 [H,P ] 0 0 K 0 speed-time contraction dS adS M+ M’ M SdS G C St space-tim e contra ctio n sp eed -sp ace co n tractio n N+ N- 1 2 3 4 5 6 7 8 Figure 2. The contractions of the kinematical groups for 2-dimensional spacetimes. Fig. 2 illustrates several interesting relationships among the kinematical groups. For example, Table 7 gives important classes of kinematical groups, each class corresponding to a face of the figure, that transform to another class in the table under one of the symmetries SH , SP , or SK , provided that certain exclusions are made as outlined in Table 8. The exclusions are necessary under the given symmetries as some kinematical algebras are taken to algebras that are not kinematical. 2 Cayley–Klein geometries In this section we wish to review work done by Ballesteros, Herranz, Ortega and Santander on homogeneous spaces that are spacetimes for kinematical groups, and we begin with a bit of history concerning the discovery of non-Euclidean geometries. Franz Taurinus was the first to explicitly give mathematical details on how a hypothetical sphere of imaginary radius would have a non-Euclidean geometry, what he called log-spherical geometry, and this was done via hyperbolic trigonometry (see [5] or [18]). Felix Klein1 is usually given credit for being the first to give a complete model of a non-Euclidean geometry2: he built his model by suitably adapting 1Roger Penrose [21] notes that it was Eugenio Beltrami who first discovered both the projective and conformal models of the hyperbolic plane. 2 Spherical geometry was not historically considered to be non-Euclidean in nature, as it can be embedded in a 3-dimensional Euclidean space, unlike Taurinus’ sphere. 6 A.S. McRae Table 7. Important classes of kinematical groups and their geometrical configurations in Fig. 2. Class of groups Face Relative-time 1247 Absolute-time 3568 Relative-space 1346 Absolute-space 2578 Cosmological 1235 Local 4678 Table 8. The 3 basic symmetries are represented by reflections of Fig. 2, with some exclusions. Symmetry Reflection across face SH 1378 (excluding M+) SP 1268 (excluding adS and N−) SK 1458 Arthur Cayley’s metric for the projective plane. Klein [19] (originally published in 1871) went on, in a systematic way, to describe nine types of two-dimensional geometries (what Yaglom [28] calls Cayley–Klein geometries) that were then further investigated by Sommerville [26]. Yaglom gave conformal models for these geometries, extending what had been done for both the projective and hyperbolic planes. Each type of geometry is homogeneous and can be determined by two real constants κ1 and κ2 (see Table 9). The names of the geometries when κ2 ≤ 0 are those as given by Yaglom, and it is these six geometries that can be interpreted as spacetime geometries. Following Taurinus, it is easiest to describe a bit of the geometrical nature of these geometries by applying the appropriate kind of trigonometry: we will see shortly how to actually construct a model for each geometry. Let κ be a real constant. The unit circle a2 + κb2 = 1 in the plane R2 = {(a, b)} with metric ds2 = da2 + κdb2 can be used to defined the cosine Cκ(φ) =  cos ( √ κφ), if κ > 0, 1, if κ = 0, cosh (√ −κφ ) , if κ < 0, and sine Sκ(φ) =  1√ κ sin ( √ κφ), if κ > 0, φ, if κ = 0, 1√ −κ sinh (√ −κφ ) , if κ < 0 functions: here (a, b) = (Cκ(φ), Sκ(φ)) is a point on the connected component of the unit circle containing the point (1, 0), and φ is the signed distance from (1, 0) to (a, b) along the circular arc, defined modulo the length 2π√ κ of the unit circle when κ > 0. We can also write down the power series for these analytic trigonometric functions: Cκ(φ) = 1− 1 2! κφ2 + 1 4! κ2φ4 + · · · , Sκ(φ) = φ− 1 3! κφ3 + 1 5! κ2φ5 + · · · . Note that Cκ2(φ)+κSκ 2(φ) = 1. So if κ > 0 then the unit circle is an ellipse (giving us elliptical trigonometry), while if κ < 0 it is a hyperbola (giving us hyperbolic trigonometry). When κ = 0 Clifford Algebras and Possible Kinematics 7 Table 9. The 9 types of Cayley–Klein geometries. Metric Structure Conformal Elliptic Parabolic Hyperbolic Structure κ1 > 0 κ1 = 0 κ1 < 0 Elliptic elliptic Euclidean hyperbolic κ2 > 0 geometries geometries geometries Parabolic co-Euclidean Galilean co-Minkowski κ2 = 0 geometries geometry geometries Hyperbolic co-hyperbolic Minkowski doubly κ2 < 0 geometries geometries hyperbolic geometries the unit circle consists of two parallel straight lines, and we will say that our trigonometry is parabolic. We can use such a trigonometry to define the angle φ between two lines, and another independently chosen trigonometry to define the distance between two points (as the angle between two lines, where each line passes through one of the points as well as a distinguished point). At this juncture it is not clear that such geometries, as they have just been described, are of either mathematical or physical interest. That mathematicians and physicists at the beginning of the 20th century were having similar thoughts is perhaps not surprising, and Walker [27] gives an interesting account of the mathematical and physical research into non-Euclidean geometries during this period in history. Klein found that there was a fundamental unity to these geometries, and so that alone made them worth studying. Before we return to physics, let us look at these geometries from a perspective that Klein would have appreciated, describing their motion groups in a unified manner. Ballesteros, Herranz, Ortega and Santander have constructed the Cayley–Klein geometries as homogeneous spaces3 by looking at real representations of their motion groups. These motion groups are denoted by SOκ1,κ2(3) (that we will refer to as the generalized SO(3) or simply by SO(3)) with their respective Lie algebras being denoted by soκ1,κ2(3) (that we will refer to as the generalized so(3) or simply by so(3)), and most if not all of these groups are probably familiar to the reader (for example, if both κ1 and κ2 vanish, then SO(3) is the Heisenberg group). Later on in this paper we will use Clifford algebras to show how we can explicitly think of SO(3) as a rotation group, where each element of SO(3) has a well-defined axis of rotation and rotation angle. Now a matrix representation of so(3) is given by the matrices H = 0 −κ1 0 1 0 0 0 0 0  , P = 0 0 −κ1κ2 0 0 0 1 0 0  , and K = 0 0 0 0 0 −κ2 0 1 0  , where the structure constants are given by the commutators [K,H] = P, [K,P ] = −κ2H, and [H,P ] = κ1K. By normalizing the constants we obtain matrix representations of the adS, dS, N−, N+, M , and G Lie algebras, as well as the Lie algebras for the elliptic, Euclidean, and hyperbolic motion groups, denoted El, Eu, and H respectively. We will see at the end of this section how the 3See [2, 13, 15], and also [14], where a special case of the group law is investigated, leading to a plethora of trigonometric identities, some of which will be put to good use in this paper: see Appendix A. 8 A.S. McRae Cayley–Klein spaces can also be used to give homogeneous spaces for M ′, M+, C, and SdS (but not for St). One benefit of not normalizing the parameters κ1 and κ2 is that we can easily obtain contractions by letting κ1 → 0 or κ2 → 0. Elements of SO(3) are real-linear, orientation-preserving isometries of R3 = {(z, t, x))} im- bued with the (possibly indefinite or degenerate) metric ds2 = dz2 + κ1dt 2 + κ1κ2dx 2. The one-parameter subgroups H, P, and K generated respectively by H, P , and K consist of matri- ces of the form eαH = Cκ1(α) −κ1Sκ1(α) 0 Sκ1(α) Cκ1(α) 0 0 0 1  , eβP = Cκ1κ2(β) 0 −κ1κ2Sκ1κ2(β) 0 1 0 Sκ1κ2(β) 0 Cκ1κ2(β)  , and eθK = 1 0 0 0 Cκ2(θ) −κ2Sκ2(θ) 0 Sκ2(θ) Cκ2(θ)  (note that the orientations induced on the coordinate planes may be different than expected). We can now see that in order for K to be non-compact, we must have that κ2 ≤ 0, which explains the content of Table 3. The spaces SO(3)/K, SO(3)/H, and SO(3)/P are homogeneous spaces for SO(3). When SO(3) is a kinematical group, then S ≡ SO(3)/K can be identified with the manifold of space- time translations. Regardless of the values of κ1 and κ2 however, S is the Cayley–Klein geometry with parameters κ1 and κ2, and S can be shown to have constant curvature κ1 (also, see [20]). So the angle between two lines passing through the origin (the point that is invariant under the subgroup K) is given by the parameter θ of the element of K that rotates one line to the other (and so the measure of angles is related to the parameter κ2). Similarly if one point can be taken to another by an element of H or P respectively, then the distance between the two points is given by the parameter α or β, (and so the measure of distance is related to the parameter κ1 or to κ1κ2). Note that the spaces SO(3)/H and SO(3)/P are respectively the spaces of timelike and spacelike geodesics for kinematical groups. For our purposes we will also need to model S as a projective geometry. First, we define the projective quadric Σ̄ as the set of points on the unit sphere Σ ≡ {(z, t, x) ∈ R3 \| z2 + κ1t 2 + κ1κ2x 2 = 1} that have been identified by the equivalence relation (z, t, x) ∼ (−z,−t,−x). The group SO(3) acts on Σ̄, and the subgroup K is then the isotropy subgroup of the equivalence class O = [(1, 0, 0)]. The metric g on R3 induces a metric on Σ̄ that has κ1 as a factor. If we then define the main metric g1 on Σ̄ by setting( ds2 ) 1 = 1 κ1 ds2, then the surface Σ̄, along with its main metric (and subsidiary metric, see below), is a projective model for the Cayley–Klein geometry S. Note that in general g1 can be indefinite as well as nondegenerate. The motion exp(θK) gives a rotation (or boost for a spacetime) of S, whereas the mo- tions exp(αH) and exp(βP ) give translations of S (time and space translations respectively for a spacetime). The parameters κ1 and κ2 are, for the spacetimes, identified with the universe time radius τ and speed of light c by the formulae κ1 = ± 1 τ2 and κ2 = − 1 c2 . Clifford Algebras and Possible Kinematics 9 dS adS M + M’ M SdS G C St N + N - 1 2 3 4 5 6 7 8 El H Eu Figure 3. The 9 kinematical and 3 non-kinematical groups. Table 10. The 3 basic symmetries are given as reflections of Fig. 3. Symmetry Reflection across face SH 1378 SP 1268 SK 1458 For the absolute-time spacetimes with kinematical groups N−, G, and N+, where κ2 = 0 and c = ∞, we foliate S so that each leaf consists of all points that are simultaneous with one another, and then SO(3) acts transitively on each leaf. We then define the subsidiary metric g2 along each leaf of the foliation by setting( ds2 ) 2 = 1 κ2 ( ds2 ) 1 . Of course when κ2 6= 0, the subsidiary metric can be defined on all of Σ̄. The group SO(3) acts on S by isometries of g1, by isometries of g2 when κ2 6= 0 and, when κ2 = 0, on the leaves of the foliation by isometries of g2. It remains to be seen then how homogeneous spacetimes for the kinematical groupsM+, M ′, C, and SdS may be obtained from the Cayley–Klein geometries. In Fig. 3 the face 1346 contains the motion groups for all nine types of Cayley–Klein geometries, and the symmetries SH , SP , and SK can be represented as symmetries of the cube, as indicated in Table 104. As vertices 1 and 8 are in each of the three planes of reflection, it is impossible to get St from any one of the Cayley–Klein groups through the symmetries SH , SP , and SK . Under the symmetry SK , respective spacetimes for M+, M ′, and C are given by the spacetimes SO(3)/K for N+, N−, and G, where space and time translations are interchanged. Under the symmetry SH , the spacetime for SdS is given by the homogeneous space SO(3)/P for G, as boosts and space translations are interchanged by SH . Note however that there actually 4Santander [24] discusses some geometrical consequences of such symmetries when applied to dS, adS, and H: note that SH , SP , and SK all fix vertex 1. 10 A.S. McRae are no spacelike geodesics for G, as the Cayley–Klein geometry S = SO(3)/K for κ1 = κ2 = 0 can be given simply by the plane R2 = {(t, x)} with ds2 = dt2 as its line element5. Although SO(3)/P is a homogeneous space for SO(3), SO(3) does not act effectively on SO(3)/P: since both [K,P ] = 0 and [H,P ] = 0, space translations do not act on SO(3)/P. Similarly, inertial transformations do not act on spacetime for SdS, or on St for that matter. Note that SdS can be obtained from dS by P → εP , H → εH, and K → ε2K, where ε → 0. So velocities are negligible even when compared to the reduced space and time translations. In conclusion to Part I then, a study of all nine types of Cayley–Klein geometries affords us a beautiful and unified study of all 11 possible kinematics save one, the static kinematical structure. It was this study that motivated the author to investigate another unified approach to possible kinematics, save for that of the Static Universe. Part II. Another unified approach to possible kinematics 3 The generalized Lie algebra so(3) Preceding the work of Ballesteros, Herranz, Ortega, and Santander was the work of Sanjuan [23] on possible kinematics and the nine6 Cayley–Klein geometries. Sanjuan represents each kine- matical Lie algebra as a real matrix subalgebra of M(2,C), where C denotes the generalized complex numbers (a description of the generalized complex numbers is given below). This is accomplished using Yaglom’s analytic representation of each Caley–Klein geometry as a region of C: for the hyperbolic plane this gives the well-known Poincaré disk model. Sanjuan constructs the Lie algebra for the hyperbolic plane using the standard method, stating that this method can be used to obtain the other Lie algebras as well. Also, extensive work has been done by Gromov [6, 7, 8, 9, 10] on the generalized orthogonal groups SO(3) (which we refer to simply as SO(3)), deriving representations of the generalized so(3) (which we refer to simply as so(3)) by utilizing the dual numbers as well as the standard complex numbers, where again it is tacitly assumed that the parameters κ1 and κ2 have been normalized. Also, Pimenov has given an axiomatic description of all Cayley–Klein spaces in arbitrary dimensions in his paper [22] via the dual numbers ik, k = 1, 2, . . . , where ikim = imik 6= 0 and i2k = 0. Unless stated otherwise, we will not assume that the parameters κ1 and κ2 have been nor- malized, as we wish to obtain contractions by simply letting κ1 → 0 or κ2 → 0. Our goal in this section is to derive representations of so(3) as real subalgebras of M(2,C), and in the process give a conformal model of S as a region of the generalized complex plane C along with a hermitian metric, extending what has been done for the projective and hyperbolic planes7. We feel that it is worthwhile to write down precisely how these representations are obtained in order that our later construction of a Clifford algebra is more meaningful. The first step is to represent the generators of SO(3) by Möbius transformations (that is, linear fractional transformations) of an appropriately defined region in the complex number plane C, where the points of S are to be identified with this region. Definition 1. By the complex number plane Cκ we will mean {w = u+ iv \| (u, v) ∈ R2 and i2 = −κ} where κ is a real-valued parameter. Thus Cκ refers to the complex numbers, dual numbers, or double numbers when κ is nor- malized to 1, 0, or −1 respectively (see [28] and [12]). One may check that Cκ is an associative 5Yaglom writes in [28] about this geometry, “. . . which, in spite of its relative simplicity, confronts the unini- tiated reader with many surprising results.” 6Sanjuan and Yaglom both tacitly assume that both parameters κ1 and κ2 are normalized. 7Fjelstad and Gal [11] have investigated two-dimensional geometries and physics generated by complex numbers from a topological perspective. Also, see [4]. Clifford Algebras and Possible Kinematics 11 t z x C κκ1 2 Cκ2 Cκ1 Σ Figure 4. The unit sphere Σ and the three complex planes Cκ2 , Cκ1 , and Cκ1κ2 . algebra with a multiplicative unit, but that there are zero divisors when κ ≤ 0. For example, if κ = 0, then i is a zero-divisor. The reader will note below that 1 i appears in certain equations, but that these equations can always be rewritten without the appearance of any zero-divisors in a denominator. One can extend Cκ so that terms like 1 i are well-defined (see [28]). It is these zero divisors that play a crucial rule in determining the null-cone structure for those Cayley–Klein geometries that are spacetimes. Definition 2. Henceforward C will denote Cκ2 , as it is the parameter κ2 which determines the conformal structure of the Cayley–Klein geometry S with parameters κ1 and κ2. Theorem 1. The matrices i 2σ1, i 2σ2, and 1 2iσ3 are generators for the generalized Lie algebra so(3), where so(3) is represented as a subalgebra of the real matrix algebra M(2,C), where σ1 = ( 1 0 0 −1 ) , σ2 = ( 0 1 κ1 0 ) and σ3 = ( 0 i −κ1i 0 ) . In fact, we will show that K, H, and P (the subgroups generated respectively by boosts, time and space translations) can be respectively represented by elements of SL(2,C) of the form ei θ 2 σ1, ei α 2 σ2, and e β 2i σ3. Note that when κ1 = 1 and κ2 = 1, we recover the Pauli spin matrices, though my indexing is different, and there is a sign change as well: recall that the Pauli spin matrices are typically given as σ1 = ( 0 1 1 0 ) , σ2 = ( 0 −i i 0 ) and σ3 = ( 1 0 0 −1 ) . We will refer to σ1, σ2, and σ3 as given in the statement of Theorem 1 as the generalized Pauli spin matrices. The remainder of this section is devoted to proving the above theorem. The reader may find Fig. 4 helpful. The respective subgroups K, H, and P preserve the z, x, and t axes as well as the Cκ2 , Cκ1 , and Cκ1κ2 number planes, acting on these planes as rotations. Also, as these groups preserve the unit sphere Σ = {(z, t, x) \| z2 + κ1t 2 + κ1κ2x 2 = 1}, they preserve the respective intersections of Σ with the Cκ2 , Cκ1 , and Cκ1κ2 number planes. These intersections 12 A.S. McRae are, respectively, circles of the form κ1ww̄ = 1 (there is no intersection when κ1 = 0 or when κ1 < 0 and κ2 > 0), ww̄ = 1, and ww̄ = 1, where w, w, and w denote elements of Cκ2 , Cκ1 , and Cκ1κ2 respectively. We will see in the next section how a general element of SO(3) behaves in a manner similar to the generators of K, H, and P, utilizing the power of a Clifford algebra. So we will let the plane z = 0 in R3 represent C (recall that C denotes Cκ2). We may then identify the points of S with a region ς of C by centrally projecting Σ from the point (−1, 0, 0) onto the plane z = 0, projecting only those points (z, t, x) ∈ Σ with non-negative z-values. The region ς may be open or closed or neither, bounded or unbounded, depending on the geometry of S. Such a construction is well known for both the projective and hyperbolic planes RP2 and H2 and gives rise to the conformal models of these geometries. We will see later on how the conformal structure on C agrees with that of S, and then how the simple hermitian metric (see Appendix B) ds2 = dwdw( 1 + κ1 \|w\|2 )2 gives the main metric g1 for S. This metric can be used to help indicate the general character of the region ς for each of the nine types of Cayley–Klein geometries, as illustrated in Fig. 5. Note that antipodal points on the boundary of ς (if there is a boundary) are to be identified. For absolute-time spacetimes (when κ2 = 0) the subsidiary metric g2 is given by g2 = dx2( 1 + κ1t20 )2 and is defined on lines w = t0 of simultaneous events. For all spacetimes, with Here-Now at the origin, the set of zero-divisors gives the null cone for that event. Via this identification of points of S with points of ς, transformations of S correspond to transformations of ς. If the real parameters κ1 and κ2 are normalized to the values K1 and K2 so that Ki =  1, if κi > 0, 0, if κi = 0, −1, if κi < 0 then Yaglom [28] has shown that the linear isometries of R3 (with metric ds2 = dz2 +K1dt 2 + K1K2dx 2) acting on Σ̄ project to those Möbius transformations that preserve ς, and so these Möbius transformations preserve cycles8: a cycle is a curve of constant curvature, corresponding to the intersection of a plane in R3 with Σ̄. We would like to show that elements of SO(3) project to Möbius transformations if the parameters are not normalized, and then to find a realization of so(3) as a real subalgebra of M(2,C). Given κ1 and κ2 we may define a linear isomorphism of R3 as indicated below. κ1 6= 0, κ2 6= 0 κ1 6= 0, κ2 = 0 κ1 = κ2 = 0 z 7→ z′ = z z 7→ z′ = z z 7→ z′ = z t 7→ t′ = 1√ \|κ1\| t t 7→ t′ = 1√ \|κ1\| t t 7→ t′ = t x 7→ x′ = 1√ \|κ1κ2\| x x 7→ x′ = x x 7→ x′ = x This transformation preserves the projection point (−1, 0, 0) as well as the complex plane z = 0, and maps the projective quadric Σ̄ for parameters K1 and K2 to that for κ1 and κ2, and so 8Yaglom projects from the point (z, t, x) = (−1, 0, 0) onto the plane z = 1 whereas we project onto the plane z = 0. But this hardly matters as cycles are invariant under dilations of C. Clifford Algebras and Possible Kinematics 13 P P P P P P P P t t t t t t t t t x x x x x x x x x Figure 5. The regions ς. gives a correspondence between elements of SOK1,K2(3) with those of SOκ1,κ2(3) as well as the projections of these elements. As the Möbius transformations of C are those transformations that preserve curves of the form Im (w′ 1 − w′ 3)(w ′ 2 − w′) (w′ 1 − w′)(w′ 2 − w′ 3) = 0 (where w′ 1, w ′ 2, and w′ 3 are three distinct points lying on the cycle), then if this form is invari- ant under the induced action of the linear isomorphism, then elements of SOκ1,κ2(3) project to Möbius transformations of ς. As a point (z, t, x) is projected to the point ( 0, t z+1 , x z+1 ) corre- sponding to the complex number w = 1 z+1(t + Ix) ∈ CK2 , if the linear transformation sends (z, t, x) to (z′, t′, x′), then it sends w = 1 z+1(t + Ix) ∈ CK2 to w′ = 1 z′+1(t′ + ix′) ∈ Cκ2 = C, where I2 = −K2 and i2 = −κ2. We can then write that κ1 6= 0, κ2 6= 0 κ1 6= 0, κ2 = 0 κ1 = κ2 = 0 w = 1 z+1 (t+ Ix) 7→ w = 1 z+1 (t+ Ix) 7→ w = 1 z+1 (t+ Ix) 7→ w′ = 1 z+1 1√ \|κ1\| ( t+ I√ \|κ2\| x ) w′ = 1 z+1 ( 1√ \|κ1\| t+ Ix ) w′ = w 14 A.S. McRae And so Im (w1 − w3)(w2 − w) (w1 − w)(w2 − w3) = 0 ⇐⇒ Im (w′ 1 − w′ 3)(w ′ 2 − w′) (w′ 1 − w′)(w′ 2 − w′ 3) = 0, as can be checked directly, and we then have that elements of SO(3) project to Möbius trans- formations of ς. The rotations eθK preserve the complex number plane z = t + ix = 0 and so correspond simply to the transformations of C given by w 7→ eiθw, as eiθ = Cκ2(θ) + iSκ2(θ), keeping in mind that i2 = −κ2. Now in order to express this rotation as a Möbius transformation, we can write w 7→ e θ 2 iw + 0 0w + e− θ 2 i . Since there is a group homomorphism from the subgroup of Möbius transformations correspon- ding to SO(3) to the group M(2,C) of 2 × 2 matrices with entries in C, this transformation being defined by aw + b cw + d 7→ ( a b c d ) , each Möbius transformation is covered by two elements of SL(2,C). So the rotations eθK correspond to the matrices ± ( e θ 2 i 0 0 e− θ 2 i ) = ±e θ 2 i 1 0 0 −1 . For future reference let us now define σ1 ≡ ( 1 0 0 −1 ) , where i 2σ1 is then an element of the Lie algebra so(3). We now wish to see which elements of SL(2,C) correspond to the motions eαH and eβP . The x-axis, the zt-coordinate plane, and the unit sphere Σ, are all preserved by eαH . So the zt-coordinate plane is given the complex structure Cκ1 = {w = z + it \| i2 = −κ1}, for then the unit circle ww = 1 gives the intersection of Σ with Cκ1 , and the transformation induced on Cκ1 by eαH is simply given by w 7→ eiαw. Similarly the transformation induced by eβP on Cκ1κ2 = {w = z + ix \| i2 = −κ1κ2} is given by w 7→ eiβw. In order to explicitly determine the projection of the rotation w 7→ eiαw of the unit circle in Cκ1 and also that of the rotation w 7→ eiβw of the unit circle in Cκ1κ2 , note that the projection point (z, t, x) = (−1, 0, 0) lies in either unit circle and that projection sends a point on the unit circle (save for the projection point itself) to a point on the imaginary axis as follows: w = eiφ 7→ iTκ1 ( φ 2 ) , w = eiφ 7→ iTκ1κ2 ( φ 2 ) (where Tκ is the tangent function) for Tκ (µ 2 ) = Sκ(µ) Cκ(µ) + 1 , Clifford Algebras and Possible Kinematics 15 noting that a point a+ib on the unit circle ww of the complex plane Cκ can be written as a+ib = eiψ = Cκ(ψ) + iSκ(ψ). So the rotations eαH and eβP induce the respective transformations iTκ1 ( φ 2 ) 7→ iTκ1 ( φ+ α 2 ) , iTκ1κ2 ( φ 2 ) 7→ iTκ1κ2 ( φ+ β 2 ) on the imaginary axes. We know that such transformations of either imaginary or real axes can be extended to Möbius transformations, and in fact uniquely determine such Möbius maps. For example, if w = iTκ1 ( φ 2 ) , then we have that w 7→ w + iTκ1 ( α 2 ) 1− κ1w i Tκ1 ( α 2 ) or w 7→ Cκ1 ( α 2 ) w + iSκ1 ( α 2 ) −κ1 i Sκ1 ( α 2 ) w + Cκ1 ( α 2 ) with corresponding matrix representation ± ( Cκ1 ( α 2 ) iSκ1 ( α 2 ) iSκ1 ( α 2 ) Cκ1 ( α 2 )) in SL(2, Cκ1), where we have applied the trigonometric identity9 Tκ(µ± ψ) = Tκ(µ)± Tκ(ψ) 1∓ κTκ(µ)Tκ(ψ) . However, it is not these Möbius transformations that we are after, but those corresponding transformations of C. Now a transformation of the imaginary axis (the x-axis) of Cκ1κ2 corresponds to a transfor- mation of the imaginary axis of C (also the x-axis) while a transformation of the imaginary axis of Cκ1 (the t-axis) corresponds to a transformation of the real axis of C (also the t-axis). For this reason, values on the x-axis, which are imaginary for both the Cκ1κ2 as well as the C plane, correspond as iTκ1κ2 ( φ 2 ) = i 1 √ κ1 Tκ2 ( √ κ1 φ 2 ) if κ1 > 0, iTκ1κ2 ( φ 2 ) = i 1√ −κ1 T−κ2 (√ −κ1 φ 2 ) if κ1 < 0, and iTκ1κ2 ( φ 2 ) = i ( φ 2 ) if κ1 = 0, as can be seen by examining the power series representation for Tκ. The situation for the rotation eiα is similar. We can then compute the elements of SL(2,C) corresponding to eαH and eβP as given in tables 13 and 14 in Appendix C. In all cases we have the simple result that those elements of SL(2,C) corresponding to eαH can be written as e α 2i σ3 and those for eβP as ei β 2 σ2 , where σ2 ≡ ( 0 1 κ1 0 ) and σ3 ≡ ( 0 i −κ1i 0 ) . Thus i 2σ1, i 2σ2, and 1 2iσ3 are generators for the generalized Lie algebra so(3), a subalgebra of the real matrix algebra M(2,C). 9For Minkowski spacetimes this trigonometric identity is the well-known formula for the addition of rapidities. 16 A.S. McRae Table 11. The basis elements for Cl3. Subspace of with basis scalars R 1 vectors R3 σ1, σ2, σ3 bivectors ∧2 R3 iσ1, iσ2, 1 i σ3 volume elements ∧3 R3 i 4 The Clifford algebra Cl3 Definition 3. Let Cl3 be the 8-dimensional real Clifford algebra that is identified with M(2,C) as indicated by Table 11, where C denotes the generalized complex numbers Cκ2 . Here we identify the scalar 1 with the identity matrix and the volume element i with the 2× 2 identity matrix multiplied by the complex scalar i: in this case 1 i σ3 can be thought of as the 2×2 matrix( 0 1 −κ1 0 ) . We will also identify the generalized Paul spin matrices σ1, σ2, and σ3 with the vectors î = 〈1, 0, 0〉, ĵ = 〈0, 1, 0〉, and k̂ = 〈0, 0, 1〉 respectively of the vector space R3 = {(z, t, x)} given the Cayley–Klein inner product10. Proposition 1. Let Cl3 be the Clifford algebra given by Definition 3. (i) The Clifford product σ2 i gives the square of the length of the vector σi under the Cayley– Klein inner product. (ii) The center Cen(Cl3) of Cl3 is given by R ⊕ ∧3 R3, the subspace of scalars and volume elements. (iii) The generalized Lie algebra so(3) is isomorphic to the space of bivectors ∧2 R3, where H = 1 2i σ3, P = i 2 σ2, and K = i 2 σ1. (iv) If n̂ = 〈n1, n2, n3〉 and ~σ = 〈iσ1, iσ2, σ3/i〉, then we will let n̂ · ~σ denote the bivector n1iσ1 + n2iσ2 + n3 1 i σ3. This bivector is simple, and the parallel vectors in̂ · ~σ and 1 i n̂ · ~σ are perpendicular to any plane element represented by n̂ · ~σ. Let η denote the line through the origin that is determined by in̂ · ~σ or 1 i n̂ · ~σ. (v) The generalized Lie group SO(3) is also represented within Cl3, for if a is the vector a1σ1+a2σ2+a3σ3, then the linear transformation of R3 defined by the inner automorphism a 7→ e− φ 2 n̂·~σa e φ 2 n̂·~σ faithfully represents an element of SO(3) as it preserves vector lengths given by the Cayley– Klein inner product, and is in fact a rotation, rotating the vector 〈a1, a2, a3〉 about the axis η through the angle φ. In this way we see that the spin group is generated by the elements e θ 2 iσ1 , e β 2 iσ2 , and e α 2i σ3 . (vi) Bivectors n̂ · ~σ act as imaginary units as well as generators of rotations in the oriented planes they represent. Let κ be the scalar − (n̂ · ~σ)2. Then if a lies in an oriented plane determined by the bivector n̂ · ~σ, where this plane is given the complex structure of Cκ, then e− φ 2 n̂·~σae φ 2 n̂·~σ is simply the vector 〈a1, a2, a3〉 rotated by the angle φ in the complex plane Cκ, where ι2 = −κ. So this rotation is given by unit complex multiplication. 10We will use the symbol v̂ to denote a vector v of length one under the standard inner product. Clifford Algebras and Possible Kinematics 17 The goal of this section is to prove Proposition 1. We can easily compute the following: σ1 2 = 1, σ2 2 = κ1, σ3 2 = κ1κ2, σ3σ2 = −σ2σ3 = κ1iσ1, σ1σ3 = −σ3σ1 = iσ2, σ1σ2 = −σ2σ1 = 1 i σ3, σ1σ2σ3 = −κ1i. Recalling that R3 is given the Cayley–Klein inner product, we see that σ2 i gives the square of the length of the vector σi. Note that when κ1 = 0, Cl3 is not generated by the vectors. Cen(Cl3) of Cl3 is given by R⊕ ∧3 R3, and we can check directly that if H ≡ 1 2i σ3, P ≡ i 2 σ2, and K ≡ i 2 σ1, then we have the following commutators: [H,P ] = HP − PH = 1 4 (σ3σ2 − σ2σ3) = κ1iσ1 2 = κ1K, [K,H] = KH −HK = 1 4 (σ1σ3 − σ3σ1) = iσ2 2 = P, [K,P ] = KP − PK = i2 4 (σ1σ2 − σ2σ1) = iσ3 2 = −κ2H. So the Lie algebra so(3) is isomorphic to the space of bivectors ∧2 R3. The product of two vectors a = a1σ1 + a2σ2 + a3σ3 and b = b1σ1 + b2σ2 + b3σ3 in Cl3 can be expressed as ab = a · b + a ∧ b = 1 2(ab + ba) + 1 2(ab − ba), where a · b = 1 2(ab + ba) = a1b1 + κ1a 2b2 + κ1κ2a 3b3 is the Cayley–Klein inner product and the wedge product is given by a ∧ b = 1 2 (ab− ba) = ∣∣∣∣∣∣ −κ1iσ1 −iσ2 1 i σ3 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣ , so that ab is the sum of a scalar and a bivector: here \| ? \| denotes the usual 3× 3 determinant. By the properties of the determinant, if e ∧ f = g ∧ h and κ1 6= 0, then the vectors e and f span the same oriented plane as the vectors g and h. When κ1 = 0 the bivector n̂ · ~σ is no longer simple in the usual way. For example, for the Galilean kinematical group (aka the Heisenberg group) where κ1 = 0 and κ2 = 0, we have that both σ1 ∧ σ3 = iσ2 and (σ1 + σ2) ∧ σ3 = iσ2, so that the bivector iσ2 represents plane elements that do no all lie in the same plane11. Recalling that σ1, σ2, and σ3 correspond to the vectors î, ĵ, and k̂ respectively, we observe that the subgroup P of the Galilean group fixes the t-axis and preserves both of these planes, inducing the same kind of rotation upon each of them: for the plane spanned by î and k̂ we have that eβP : ( î k̂ ) 7→ ( î+ βk̂ k̂ ) while for the plane spanned by î+ ĵ and k̂ we have that eβP : ( î+ ĵ k̂ ) 7→ ( î+ ĵ + βk̂ k̂ ) . 11There is some interesting asymmetry for Galilean spacetime, in that the perpendicular to a timelike geodesic through a given point is uniquely defined as the lightlike geodesic that passes through that point, and this lightlike geodesic then has no unique perpendicular, since all timelike geodesics are perpendicular to it. 18 A.S. McRae If we give either plane the complex structure of the dual numbers so that i2 = 0, then the rotation is given by simply multiplying vectors in the plane by the unit complex number eβi. We will see below that this kind of construction holds generally. What we need for our construction below is that any bivector can be meaningfully expressed as e∧ f for some vectors e and f , so that the bivector represents at least one plane element: we will discuss the meaning of the magnitude and orientation of the plane element at the end of the section. If the bivector represents multiple plane elements spanning distinct planes, so much the better. If n̂ = 〈n1, n2, n3〉 and ~σ = 〈iσ1, iσ2, σ3/i〉, then we will let n̂ · ~σ denote the bivector B = n1iσ1 + n2iσ2 + n3 1 i σ3. Now if a = n1σ3 + κ1n 3σ1, b = −n1σ2 + κ1n 2σ1, c = n3σ2 + n2σ3, then a ∧ c = κ1n 3n̂ · ~σ, b ∧ a = κ1n 1n̂ · ~σ, b ∧ c = κ1n 2n̂ · ~σ, where at least one of the bivectors nin̂ · ~σ is non-zero as n̂ · ~σ is non-zero. If κ1 = 0 and n1 = 0, then σ1∧c = n̂ · ~σ. However, if both κ1 = 0 and n1 6= 0, then it is impossible to have e∧f = n̂ · ~σ: in this context we may simply replace the expression n̂ · ~σ with the expression σ3 ∧σ2 whenever κ1 = 0 and n1 6= 0 (as we will see at the end of this section, we could just as well replace n̂ · ~σ with any non-zero multiple of σ3 ∧ σ2). The justification for this is given by letting κ1 → 0, for then e ∧ f = (√ \|κ1\|n2σ1 − n1√ \|κ1\| σ2 ) ∧ (√ \|κ1\| n3 n1 σ1 + 1√ \|κ1\| σ3 ) = n̂ · ~σ shows that the plane spanned by the vectors e and f tends to the xt-coordinate plane. We will see below how each bivector n̂ · ~σ corresponds to an element of SO(3) that preserves any oriented plane corresponding to n̂ · ~σ: in the case where κ1 = 0 and n1 6= 0, we will then have that this element preserves the tx-coordinate plane, which is all that we require. It is interesting to note that the parallel vectors i(a ∧ b) and 1 i (a ∧ b) (when defined) are perpendicular to both a and b with respect to the Cayley–Klein inner product, as can be checked directly. However, due to the possible degeneracy of the Cayley–Klein inner product, there may not be a unique direction that is perpendicular to any given plane. The vector in̂ · ~σ = −κ2n 1σ1 − κ2n 2σ2 + n3σ1 is non-zero and perpendicular to any plane element corresponding to n̂ · ~σ except when both κ2 = 0 and n3 = 0, in which case in̂ · ~σ is the zero vector. In this last case the vector 1 i n̂ · ~σ = n1σ1 +n2σ2 gives a non-zero normal vector. In either case, let η denote the axis through the origin that contains either of these normal vectors. Before we continue, let us reexamine those elements of SO(3) that generate the subgroups K, P, and H. Here the respective axes of rotation (parallel to σ1, σ2, and σ3) for the generators eθK , eβP , and eαH are given by η, where n̂ · ~σ is given by iσ1 (or σ3 ∧ σ2 by convention), iσ2 = σ1 ∧ σ3, and 1 i σ3 = σ1 ∧ σ2. These plane elements are preserved under the respective rotations. In fact, for each of these planes the rotations are given simply by multiplication by a unit complex number, as the zt-coordinate plane is identified with Cκ1 , the zx-coordinate plane with Cκ1κ2 , and the tx-coordinate plane with Cκ2 as indicated in Fig. 4. Note that the basis bivectors act as imaginary units in Cl3 since( 1 i σ3 )2 = −κ1, (iσ2) 2 = −κ1κ2, and (iσ1) 2 = −κ2. The product of a vector a and a bivector B can be written as aB = a a B + a ∧ B = 1 2(aB − Ba) + 1 2(aB + Ba) so that aB is the sum of a vector a a B (the left contraction of a Clifford Algebras and Possible Kinematics 19 by B) and a volume element a ∧B. Let B = b ∧ c for some vectors b and c. Then 2a a (b ∧ c) = a(b ∧ c)− (b ∧ c)a = 1 2 a(bc− cb)− 1 2 (bc− cb)a so that 4a a (b ∧ c) = cba+ abc− acb− bca = c(b · a+ b ∧ a) + (a · b+ a ∧ b)c− (a · c+ a ∧ c)b− b(c · a+ c ∧ a) = 2(b · a)c− 2(c · a)b+ c(b ∧ a) + (a ∧ b)c− (a ∧ c)b− b(c ∧ a) = 2(b · a)c− 2(c · a)b+ c(b ∧ a)− (b ∧ a)c+ b(a ∧ c)− (a ∧ c)b = 2(b · a)c− 2(c · a)b+ 2 [c a (b ∧ a) + b a (a ∧ c)] = 2(b · a)c− 2(c · a)b− 2a a (c ∧ b) = 2(b · a)c− 2(c · a)b+ 2a a (b ∧ c) where we have used the Jacobi identity c a (b ∧ a) + b a (a ∧ c) + a a (c ∧ b) = 0, recalling that M(2,C) is a matrix algebra where the commutator is given by left contraction. Thus 2a a (b ∧ c) = 2(b · a)c− 2(c · a)b and so a a (b ∧ c) = (a · b)c− (a · c)b. So the vector a a B lies in the plane determined by the plane element b ∧ c. Because of the possible degeneracy of the Cayley–Klein metric, it is possible for a non-zero vector b that b a (b ∧ c) = 0. We will show that if a is the vector a1σ1 + a2σ2 + a3σ3, then the linear transformation of R3 defined by a 7→ e− φ 2 n̂·~σa e φ 2 n̂·~σ faithfully represents an element of SO(3) (and all elements are thus represented). In this way we see that the spin group is generated by the elements e θ 2 iσ1 , e β 2 iσ2 , and e α 2i σ3 . First, let us see how, using this construction, the vectors σ1, σ2, and σ3 (and hence the bivectors iσ1, iσ2, and 1 i σ3) correspond to rotations of the coordinate axes (and hence coordinate planes) given by eθK , eβP , and eαH respectively. Since e θ 2 iσ1 = Cκ2 ( θ 2 ) + iSκ2 ( θ 2 ) σ1, e β 2 iσ2 = Cκ1κ2 ( β 2 ) + iSκ1κ2 ( β 2 ) σ2, e α 2i σ3 = Cκ1 (α 2 ) + 1 i Sκ1 (α 2 ) σ3 and 2Cκ ( φ 2 ) Sκ ( φ 2 ) = Sκ(φ), Cκ 2 ( φ 2 ) − κSκ 2 ( φ 2 ) = Cκ(φ), 20 A.S. McRae Cκ 2 ( φ 2 ) + κSκ 2 ( φ 2 ) = 1 (noting that Cκ is an even function while Sκ is odd) it follows that e− θ 2 iσ1σje θ 2 iσ1 =  σ1 if j = 1, Cκ2(θ)σ2 − Sκ2(θ)σ3 if j = 2, Cκ2(θ)σ3 + κ2Sκ2(θ)σ2 if j = 3, e− β 2 iσ2σje β 2 iσ2 =  Cκ1κ2(β)σ1 + Sκ1κ2(β)σ3 if j = 1, σ2 if j = 2, Cκ1κ2(β)σ3 − κ1κ2Sκ1κ2(β)σ1 if j = 3, e− α 2i σ3σje α 2i σ3 =  Cκ1(α)σ1 + Sκ1(α)σ2 if j = 1, Cκ1(α)σ2 − κ1Sκ1(α)σ1 if j = 2, σ3 if j = 3. So for each plane element, the σj transform as the components of a vector under rotation in the clockwise direction, given the orientations of the respective plane elements: iσ1 is represented by σ3 ∧ σ2, iσ2 = σ1 ∧ σ3, and 1 i σ3 = σ1 ∧ σ2. Now we can write e φ 2 n̂·~σ = 1 + φ 2 n̂ · ~σ + 1 2! ( φ 2 )2 (n̂ · ~σ)2 + 1 3! ( φ 2 )3 (n̂ · ~σ)3 + · · · . If κ is the scalar − (n̂ · ~σ)2, then e φ 2 n̂·~σ = ( 1− 1 2! ( φ 2 )2 κ + 1 4! ( φ 2 )4 κ2 − · · · ) + n̂ · ~σ ( φ 2 − 1 3! ( φ 2 )3 κ + 1 5! ( φ 2 )5 κ2 − · · · ) = Cκ ( φ 2 ) + n̂ · ~σSκ ( φ 2 ) . As a = a1σ1 + a2σ2 + a3σ3 is a vector, we can compute its length easily using Clifford multi- plication as aa = (a1)2 +κ1(a2)2 +κ1κ2(a3)2 = \|a\|2. We would like to show that e− φ 2 n̂·~σae φ 2 n̂·~σ is also a vector with the same length as a. If g and h are elements of a matrix Lie algebra, then so is e−φ ad gh = e−φgheφg (see [25] for example). So if B is a bivector B1iσ1 +B2iσ2 +B3 1 i σ3, then e− φ 2 n̂·~σBe φ 2 n̂·~σ is also a bivector. It follows that e− φ 2 n̂·~σae φ 2 n̂·~σ is a vector as the volume element i lies in Cen(Cl3) so that e− φ 2 n̂·~σσ1e φ 2 n̂·~σ, e− φ 2 n̂·~σσ2e φ 2 n̂·~σ, and e− φ 2 n̂·~σσ3e φ 2 n̂·~σ are all vectors. Since( e− φ 2 n̂·~σae φ 2 n̂·~σ )( e− φ 2 n̂·~σae φ 2 n̂·~σ ) = e− φ 2 n̂·~σ\|a\|2e φ 2 n̂·~σ = \|a\|2e− φ 2 n̂·~σe φ 2 n̂·~σ = \|a\|2 it follows that e− φ 2 n̂·~σae φ 2 n̂·~σ has the same length as a. So the inner automorphism of R3 given by a 7→ e− φ 2 n̂·~σae φ 2 n̂·~σ corresponds to an element of SO(3). We will see in the next section that all elements of SO(3) are represented by such inner automorphisms of R3. Clifford Algebras and Possible Kinematics 21 Finally, note that e− φ 2 n̂·~σ (n̂ · ~σ) e φ 2 n̂·~σ = n̂ · ~σ as n̂ · ~σ commutes with e φ 2 n̂·~σ: so any plane element represented by n̂ · ~σ is preserved by the corresponding element of SO(3). In fact, if n̂ · ~σ = a ∧ b for some vectors a and b and κ is the scalar − (a ∧ b)2, then e− φ 2 a∧b(a)e φ 2 a∧b = [ Cκ ( φ 2 ) − (a ∧ b)Sκ ( φ 2 )] (a) [ Cκ ( φ 2 ) + (a ∧ b)Sκ ( φ 2 )] = C2 κ ( φ 2 ) a+ Cκ ( φ 2 ) Sκ ( φ 2 ) (a ∧ b)a(a ∧ b) − Cκ ( φ 2 ) Sκ ( φ 2 ) (a ∧ b)a− S2 κ ( φ 2 ) a(a ∧ b). Since a(a ∧ b) = −(a ∧ b)a, then e− φ 2 a∧b(a)e φ 2 a∧b = [Cκ(φ)− (a ∧ b)Sκ(φ)] a, and so vectors lying in the plane determined by a ∧ b are simply rotated by an angle −φ, and this rotation is given by simple multiplication by a unit complex number e−iφ where i2 = −κ. Thus, the linear combination ua + vb is sent to ue−iφa + ve−iφb, and so the plane spanned by the vectors a and b is preserved. The significance is that if a lies in an oriented plane determined by the bivector n̂ · ~σ where this plane is given the complex structure of Cκ, then e− φ 2 n̂·~σae φ 2 n̂·~σ is simply the vector a rotated by an angle of −φ in the complex plane Cκ, where ι2 = −κ. Furthermore, the axis of rotation is given by η as η is preserved (recall that i lies in the center of Cl3). Since the covariant components σi of a are rotated clockwise, the contravariant components aj are rotated counterclockwise. So 〈a1, a2, a3〉 is rotated by the angle φ in the complex plane Cκ determined by n̂ · ~σ. If we use b ∧ a instead of a ∧ b to represent the plane element, then κ remains unchanged. Note however that, if c is a vector lying in this plane, then e− φ 2 b∧ace− φ 2 b∧a = [Cκ(φ)− (b ∧ a)Sκ(φ)] c = [Cκ(−φ)− (a ∧ b)Sκ(−φ)] c so that rotation by an angle of φ in the plane oriented according to b∧a corresponds to a rotation of angle −φ in the same plane under the opposite orientation as given by a ∧ b. It would be appropriate at this point to note two things: one, the magnitude of n̂ · ~σ appears to be important, since κ = − (n̂ · ~σ)2, and two, the normalization (n1)2 + (n2)2 + (n3)2 = 1 of n̂ is somewhat arbitrary12. These two matters are one and the same. We have chosen this normalization because it is a simple and natural choice. This particular normalization is not essential, however. For suppose that κ = −(a∧ b)2 while κ′ = −(na∧ b)2, where n is a positive constant. Let Cκ = {t + ix \| i2 = −κ} with angle measure φ and Cκ′ = {t + ιx \| ι2 = −κ′ = −n2κ} with angle measure θ: without loss of generality let κ > 0. Then φ = nθ, for eiθ = cos (√ κ′θ ) − ι√ κ′ sin (√ κ′θ ) = cos ( n √ κθ ) − ι n √ κ sin ( n √ κθ ) = cos (√ κφ ) − i√ κ sin (√ κφ ) = eiφ. So we see that SO(3) is truly a rotation group, where each element has a distinct axis of rotation as well as a well-defined rotation angle. 12Due to dimension requirements some kind of normalization is needed as we cannot have φ, n1, n2, and n3 as independent variables, for so(3) is 3-dimensional. 22 A.S. McRae 5 SU(2) Since the generators of the generalized Lie group SO(3) can be represented by inner automor- phisms of the subspace R3 of vectors of Cl3 (see Definition 3), then every element of SO(3) can be represented by an inner automorphism, as the composition of inner automorphisms is an inner automorphism. On the other hand, we’ve seen that any inner automorphism represents an element of SO(3). In fact, each rotation belonging to SO(3) is then represented by two elements ±e φ 2 n̂·~σ of SL(2,C), where as usual C denotes the generalized complex number Cκ2 : we will denote the subgroup of SL(2,C) consisting of elements of the form ±e φ 2 n̂·~σ by SU(2). Definition 4. Let A be the matrix A = ( κ1 0 0 1 ) . We will now use Definition 4 to show that SU(2) is a subgroup of the subgroup G of SL(2,C) consisting of those matrices U where U?AU = A: in fact, both these subgroups of SL(2,C) are one and the same, as we shall see. Now ( e φ 2 n̂·~σ)?Aeφ 2 n̂·~σ = [ Cκ ( φ 2 ) + (n̂ · ~σ)? Sκ ( φ 2 )] A [ Cκ ( φ 2 ) + n̂ · ~σSκ ( φ 2 )] = C2 κ ( φ 2 ) A+ (n̂ · ~σ)?A (n̂ · ~σ)S2 κ ( φ 2 ) +A (n̂ · ~σ)Cκ ( φ 2 ) Sκ ( φ 2 ) + (n̂ · ~σ)?ACκ ( φ 2 ) Sκ ( φ 2 ) = A because A (n̂ · ~σ) = − (n̂ · ~σ)?A implies that A (n̂ · ~σ)Cκ ( φ 2 ) Sκ ( φ 2 ) + (n̂ · ~σ)?ACκ ( φ 2 ) Sκ ( φ 2 ) = 0 and (n̂ · ~σ)?A (n̂ · ~σ) = −A (n̂ · ~σ)2 = κA implies that C2 κ ( φ 2 ) A+ (n̂ · ~σ)?A (n̂ · ~σ)S2 κ ( φ 2 ) = C2 κ ( φ 2 ) A+ κS2 κ ( φ 2 ) A = A. So SU(2) is a subgroup of the subgroup G of SL(2,C) consisting of those matrices U where U?AU = A. We can characterize this subgroup G as{( α β −κ1β α ) \|α, β ∈ C and αα+ κ1ββ = 1 } . Now e φ 2 n̂·~σ =  C2 κ ( φ 2 ) + n1iS2 κ ( φ 2 ) n2iS2 κ ( φ 2 ) + n3S2 κ ( φ 2 ) n2κ1iS 2 κ ( φ 2 ) − n3κ1S 2 κ ( φ 2 ) C2 κ ( φ 2 ) − n1iS2 κ ( φ 2 )  as can be checked directly, recalling that e φ 2 n̂·~σ = Cκ ( φ 2 ) + (n̂ · ~σ)Sκ ( φ 2 ) , Clifford Algebras and Possible Kinematics 23 where κ = − (n̂ · ~σ)2 = ( n1 )2 κ2 + ( n2 )2 κ1κ2 + ( n3 )2 κ1. Thus det ( e φ 2 n̂·~σ ) = 1, and we see that any element of G can be written in the form e φ 2 n̂·~σ. So the group SU(2) can be characterized by SU(2) = {( α β −κ1β α ) \|α, β ∈ C and αα+ κ1ββ = 1 } . Note that if U(λ) is a curve passing through the identity at λ = 0, then d dλ ∣∣∣∣ λ=0 (U?AU = A) =⇒ U̇?A+AU̇ = 0 so that su(2) consists of those elements B of M(2,C) such that B?A + AB = 0. Although SU(2) is a double cover of SO(3), it is not necessarily the universal cover for SO(3), nor even connected, for sometimes SO(3) is itself simply-connected. Thus we have shown that: Theorem 2. The Clifford algebra Cl3 can be used to construct a double cover of the generalized Lie group SO(3), for a vector a can be rotated by the inner automorphism R3 → R3, a 7→ s−1as where s is an element of the group Spin(3) = {( α β −κ1β α ) \|α, β ∈ C and αα+ κ1ββ = 1 } , where C denotes the generalized complex number Cκ2. Lemma 1. We define the generalized special unitary group SU(2) to be Spin(3). Then su(2) consists of those matrices B of M(2,C) such that B?A+AB = 0. 6 The conformal completion of S Yaglom [28] has shown how the complex plane Cκ may be extended to a Riemann sphere Γ or inversive plane13 (and so dividing by zero-divisors is allowed), upon which the entire set of Möbius transformations acts globally and so gives a group of conformal transformations. In this last section we would like to take advantage of the simple structure of this conformal group and give the conformal completion of S, where S is conformally embedded simply by inclusion of the region ς lying in C and therefore lying in Γ. Herranz and Santander [16] found a conformal completion of S by realizing the conformal group as a group of linear transformations acting on R4, and then constructing the conformal completion as a homogeneous phase space of this conformal group. The original Cayley–Klein geometry S was then embedded into its confor- mal completion by one of two methods, one a group-theoretical one involving one-parameter subgroups and the other stereographic projection. The 6-dimensional real Lie algebra for SL(2,C) consist of those matrices in M(2,C) with trace equal to zero. In addition to the three generators H, P , and K H = 1 2i σ3 = ( 0 1 2 −κ1 2 0 ) , P = i 2 σ2 = ( 0 i 2 κ1i 2 0 ) , K = i 2 σ1 = ( i 2 0 0 − i 2 ) 13Yaglom did this when κ ∈ {−1, 0, 1}, but it is a simple matter to generalize his results. 24 A.S. McRae Table 12. Additional basis elements for sl(2,C). ?� ? ? H P K G1 G2 D H 0 κ1K −P D K −H − κ1G1 P −κ1K 0 κ2H K −κ2D −P + κ1G2 K P −κ2H 0 −S2 κ2G2 0 G1 −D −K S2 0 0 G1 G2 −K κ2D −κ2G2 0 0 G2 D H + κ1G1 P − κ1G2 0 −G1 −G2 0 that come from the generalized Lie group SO(3) of isometries of S, we have three other generators for SL(2,C): one, labeled D, for the subgroup of dilations centered at the origin and two others, labeled G1 and G2, for “translations”. It is these transformations D, G1, G2, that necessitate extending ς to the entire Riemann sphere Γ, upon which the set of Möbius transformations acts as a conformal group. Note that the following correspondences for the Möbius transformations w 7→ w + t and w 7→ w + ti (for real parameter t) are valid only if κ1 6= 0, which explains why our “translations” G1 and G2 are not actually translations: exp [ t ( 0 1 0 0 )] = ( 1 t 0 1 ) � w 7→ w + t, exp [ t ( 0 i 0 0 )] = ( 1 ti 0 1 ) � w 7→ w + ti. Please see Tables 15 and 16. The structure constants [?, ??] for this basis of sl(2,C) (which is the same basis as that given in [15] save for a sign change in G2) are given by Table 12. A Appendix: Trigonometric identities The following trigonometric identities are taken from [14] and [15], and are used throughout Sections 3, 4, and 5 d dφ Cκ(φ) = −κSκ(φ), d dφ Sκ(φ) = Cκ(φ), d dφ T−1 κ (φ) = 1 1 + κφ2 , C2 κ(φ) + κS2 κ(φ) = 1, Cκ(2φ) = C2 κ(φ)− κS2 κ(φ), Sκ(2φ) = 2Cκ(φ)Sκ(φ), Tκ ( φ 2 ) = Sκ(φ) Cκ(φ) + 1 , Tκ(φ± ψ) = Tκ(φ)± Tκ(ψ) 1∓ κTκ(φ)Tκ(ψ) . B Appendix: The Hermitian metric The hermitian metric ds2 = dwdw( 1 + κ1 \|w\|2 )2 was used in Section 3 to construct conformal models for the Cayley–Klein geometries. Following Cayley and Klein we can construct a homomorphism from SL(2,C) to the group of Möbius transformations as follows. Let u and v be complex numbers, where the two component vector ( u v ) will be called a spinor. If ( a b c d ) is an element of SL(2,C), then writing( a b c d )( u v ) = ( u′ v′ ) Clifford Algebras and Possible Kinematics 25 we can define w ≡ u v , w′ ≡ u′ v′ so that w′ = au+ bv cu+ dv = aw + b cw + d . The isometry group of ς with metric g1 is that given by those transformations belonging to Spin(3). After some tedious algebra we have that dw′dw′( 1 + κ1 \|w′\|2 )2 = dwdw( 1 + κ1 \|w\|2 )2 when ( a b c d ) ∈ SU(2) so that ds2 = dwdw( 1 + κ1 \|w\|2 )2 gives the main metric g1 on ς. We have then proved the following lemma. Lemma 2. Those Möbius transformations that correspond to Spin(3) form the isometry group of ς with main metric g1 = dwdw( 1 + κ1 \|w\|2 )2 . We would also like to show, following the proof that is given in [3] for the hyperbolic plane, that d(w1, w2) = Tκ1 −1 (∣∣∣∣ w2 − w1 κ1w1w2 + 1 ∣∣∣∣) where d(w1, w2) is the Cayley–Klein distance between two points w1 and w2 lying in ς. Let M(w) = αw + β −κ1βw + α be a Möbius transformation where( α β −κ1β α ) ∈ SU(2). without loss of generality κ1 > 0, so that if α, β, and c are small positive numbers, then the transformation [0, c] −→ [ β α , β + αc α− κ1βc ] 26 A.S. McRae induced by M is bijective, and the intersection of the real axis with ς is a geodesic14. Since M is an isometry of ς and distances are additive along a geodesic, d ( 0, β + αc α− κ1βc ) = d ( 0, β α ) + d ( β α , β + αc α− κ1βc ) = d ( 0, β α ) + d(0, c). Let us define the quantities ε = d ( 0, β α ) and t = d(0, c) so that d ( 0, β + αc α− κ1βc ) = ε+ t. Let g denote the inverse of d : [0, c] → [0, t], where d(w) is shorthand for d(0, w). Then15 g(t+ ε) = β + αc α− κ1βc = β α + c 1− κ1β α c = g(ε) + g(t) 1− κ1g(ε)g(t) and so g(t+ ε)− κ1g(t+ ε)g(t)g(ε) = g(ε) + g(t) and then we can divide by ε g(t+ ε)− g(t) ε = g(ε) ε [1 + κ1g(t)g(t+ ε)] and take the limit lim ε→0+ g(ε) ε = lim ε→0+ β α Tκ1 −1 ( β α ) = lim φ→0+ φ Tκ1 −1(φ) = lim φ→0+ 1 1 1+κ1φ2 = 1. So g′(t) = 1 + κ1g 2(t). By the inverse function rule for differentiation, d′(w) = 1 1 + κ1w2 and so d(w) = Tκ1 −1(w) as d(0) = 0. If M is the Möbius transformation given by M(w) = cw − cw1 cκ1w1w + c and where c = 1√ 1 + κ1\|w\|2 , then M(w2) → w2 − w1 κ1w1w2 + 1 14Geodesics of ς are projections of the intersections of planes through the origin with the unit sphere Σ: in this case the plane is the zt-coordinate plane. 15We can see from the equation below that g(t) = Tκ1(t). Clifford Algebras and Possible Kinematics 27 Table 13. Elements of SL(2,C) corresponding to eαH . Elements of SL(2,C) corresponding to eαH κ1 κ2 is positive ±  Cκ2 (√ κ1 κ2 α 2 ) √ κ2 κ1 Sκ2 (√ κ1 κ2 α 2 ) −κ2 √ κ1 κ2 Sκ2 (√ κ1 κ2 α 2 ) Cκ2 (√ κ1 κ2 α 2 )  κ1 κ2 is negative ±  C−κ2 (√ −κ1 κ2 α 2 ) √ −κ2 κ1 S−κ2 (√ −κ1 κ2 α 2 ) κ2 √ −κ1 κ2 S−κ2 (√ −κ1 κ2 α 2 ) C−κ2 (√ −κ1 κ2 α 2 )  κ1 = 0 ± ( 1 α 2 0 1 ) κ1 6= 0, κ2 = 0 ± ( Cκ1 ( α 2 ) Sκ1 ( α 2 ) −κ1Sκ1 ( α 2 ) Cκ1 ( α 2 )) Derivatives at α = 0 are given by ± ( 0 1 2 −κ1 2 0 ) as w1 → 0. Since d ( 0, w2 − w1 κ1w1w2 + 1 ) = d ( 0, ∣∣∣∣ w2 − w1 κ1w1w2 + 1 ∣∣∣∣) as rotations are isometries, then d(w1, w2) = d ( 0, w2 − w1 κ1w1w2 + 1 ) = Tκ1 −1 ( w2 − w1 κ1w1w2 + 1 ) = Tκ1 −1 (∣∣∣∣ w2 − w1 κ1w1w2 + 1 ∣∣∣∣) . So we have proven the following lemma. Lemma 3. If w1 and w2 are two points of ς given the metric g1, then the distance between them is given by d (w1, w2) = Tκ1 −1 (∣∣∣∣ w2 − w1 κ1w1w2 + 1 ∣∣∣∣) . C Appendix: Tables Tables 13 and 14 are referred to at the end of Section 3, and Tables 15 and 16 are referred to at the end of Section 6. Acknowledgements I wish to thank the referees for their careful reading of this paper and their suggestions for valuable improvements. 28 A.S. McRae Table 14. Elements of SL(2,C) corresponding to eβP . Elements of SL(2,C) corresponding to eβP κ1 > 0 ±  Cκ2 (√ κ1 β 2 ) i√ κ1 Sκ2 (√ κ1 β 2 ) i √ κ1Sκ2 (√ κ1 β 2 ) Cκ2 (√ κ1 β 2 )  κ1 < 0 ±  C−κ2 (√ −κ1 β 2 ) i√ −κ1 S−κ2 (√ −κ1 β 2 ) −i √ −κ1S−κ2 (√ −κ1 β 2 ) C−κ2 (√ −κ1 β 2 )  κ1 = 0 ± ( 1 iβ2 0 1 ) D erivatives at β = 0 are given by ± ( 0 i 2 κ1i 2 0 ) Table 15. The additional basis elements for sl(2,C) and their one-parameter subgroups in SL(2,C). Additional basis Corresponding one-parameter subgroup elements for sl(2,C) in SL(2,C) G1 = ( 0 0 1 0 ) ( 1 0 t 1 ) G2 = ( 0 0 i 0 ) ( 1 0 ti 1 ) D = 1 2 ( 1 0 0 −1 ) ( e t 2 0 0 e− t 2 ) Table 16. 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[26] Sommerville D.M.Y., Classification of geometries with projective metrics, Proc. Edinb. Math. Soc. 28 (1910– 1911), 25–41. [27] Walker S., The non-Euclidean style of Minkowskian relativity, in The Symbolic Universe, Editor J. Gray, Oxford University Press, Oxford, 1999, 91–127. [28] Yaglom I.M., A simple non-Euclidean geometry and its physical basis: an elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, translated from the Russian by A. Shenitzer, with the editorial assistance of B. Gordon, Springer-Verlag, New York – Heidelberg, 1979. http://arxiv.org/abs/physics/9702030 http://arxiv.org/abs/math-ph/9910041 http://arxiv.org/abs/math-ph/0110019 http://arxiv.org/abs/math-ph/0110019 Part I. A review of kinematics via Cayley-Klein geometries 1 Possible kinematics 2 Cayley-Klein geometries Part II. Another unified approach to possible kinematics 3 The generalized Lie algebra so(3) 4 The Clifford algebra Cl3 5 SU(2) 6 The conformal completion of S A Appendix: Trigonometric identities B Appendix: The Hermitian metric C Appendix: Tables References

Clifford Algebras and Possible Kinematics

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