Symmetry, Integrability and Geometry: Methods and Applications Clifford Algebras and Possible Kinematics

We review Bacry and Levy-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.

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 representations, 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 kinematical 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.  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 translations and spatial rotations. These groups consist of the de Sitter groups and their rotationinvariant 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 2dimensional 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 kinematical 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.
Let K denote the generator of the inertial transformations, H the generator of time translations, 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 Z 2 ⊗ Z 2 group of involutions generated by 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 e 0K = e θK 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 [ 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 3 3 possible Lie algebras, which we tabulate in Tables 2 and 3 with columns that have the following form: Table 4. Some kinematical groups along with their notation and structure constants. Table 5. Some kinematical groups along with their notation and structure constants.

Anti-de Sitter Oscillating Newtonian Universe Para-Minkowski Minkowski
de Sitter Expanding Newtonian Universe Expanding Minkowski Universe 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, speedtime, 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 → ǫK and P → ǫP into the Lie algebra and then calculate the singular limit of the Lie brackets as ǫ → 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 → ǫK and H → ǫH into the Lie algebra and then calculate the singular limit of the Lie brackets as ǫ → 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 → ǫP and H → ǫH into the Lie algebra and then calculate the singular limit of the Lie brackets as ǫ → 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. Table 6. Some kinematical groups along with their notation and structure constants.   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 S H , S P , or S K , 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.

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 Klein 1 is usually given credit for being the first to give a complete model of a non-Euclidean geometry 2 : he built his model by suitably adapting  [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 κ 1 and κ 2 (see Table 9). The names of the geometries when κ 2 ≤ 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 a 2 + κb 2 = 1 in the plane R 2 = {(a, b)} with metric ds 2 = da 2 + κdb 2 can be used to defined the cosine ) 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: Note that C κ 2 (φ) + κS κ 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 Hyperbolic co-hyperbolic Minkowski doubly κ 2 < 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 spaces 3 by looking at real representations of their motion groups. These motion groups are denoted by SO κ 1 ,κ 2 (3) (that we will refer to as the generalized SO(3) or simply by SO(3)) with their respective Lie algebras being denoted by so κ 1 ,κ 2 (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 κ 1 and κ 2 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 where the structure constants are given by the commutators 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 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 κ 1 and κ 2 is that we can easily obtain contractions by letting κ 1 → 0 or κ 2 → 0. Elements of SO(3) are real-linear, orientation-preserving isometries of R 3 = {(z, t, x))} imbued with the (possibly indefinite or degenerate) metric ds 2 = dz 2 + κ 1 dt 2 + κ 1 κ 2 dx 2 . The one-parameter subgroups H, P, and K generated respectively by H, P , and K consist of matrices of the form (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 κ 2 ≤ 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 spacetime translations. Regardless of the values of κ 1 and κ 2 however, S is the Cayley-Klein geometry with parameters κ 1 and κ 2 , and S can be shown to have constant curvature κ 1 (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 κ 2 ). 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 κ 1 or to κ 1 κ 2 ). 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 quadricΣ as the set of points on the unit sphere Σ ≡ {(z, t, x) ∈ R 3 | z 2 + κ 1 t 2 + κ 1 κ 2 x 2 = 1} that have been identified by the equivalence relation (z, t, x) ∼ (−z, −t, −x). The group SO(3) acts onΣ, and the subgroup K is then the isotropy subgroup of the equivalence class O = [(1, 0, 0)]. The metric g on R 3 induces a metric onΣ that has κ 1 as a factor. If we then define the main metric g 1 onΣ by setting then the surfaceΣ, along with its main metric (and subsidiary metric, see below), is a projective model for the Cayley-Klein geometry S. Note that in general g 1 can be indefinite as well as nondegenerate.
The motion exp(θK) gives a rotation (or boost for a spacetime) of S, whereas the motions exp(αH) and exp(βP ) give translations of S (time and space translations respectively for a spacetime). The parameters κ 1 and κ 2 are, for the spacetimes, identified with the universe time radius τ and speed of light c by the formulae

Symmetry Reflection across face
For the absolute-time spacetimes with kinematical groups N − , G, and N + , where κ 2 = 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 g 2 along each leaf of the foliation by setting Of course when κ 2 = 0, the subsidiary metric can be defined on all ofΣ. The group SO(3) acts on S by isometries of g 1 , by isometries of g 2 when κ 2 = 0 and, when κ 2 = 0, on the leaves of the foliation by isometries of g 2 . It remains to be seen then how homogeneous spacetimes for the kinematical groups M + , 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 S H , S P , and S K can be represented as symmetries of the cube, as indicated in Table 10  Under the symmetry S H , the spacetime for SdS is given by the homogeneous space SO(3)/P for G, as boosts and space translations are interchanged by S H . Note however that there actually are no spacelike geodesics for G, as the Cayley-Klein geometry S = SO(3)/K for κ 1 = κ 2 = 0 can be given simply by the plane R 2 = {(t, x)} with ds 2 = dt 2 as its line element 5 . 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 → ǫP , H → ǫH, and K → ǫ 2 K, where ǫ → 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 nine 6 Cayley-Klein geometries. Sanjuan represents each kinematical 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 κ 1 and κ 2 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 i k , k = 1, 2, . . . , where i k i m = i m i k = 0 and i 2 k = 0. Unless stated otherwise, we will not assume that the parameters κ 1 and κ 2 have been normalized, as we wish to obtain contractions by simply letting κ 1 → 0 or κ 2 → 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 planes 7 . 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 C κ we will mean {w = u + iv | (u, v) ∈ R 2 and i 2 = −κ} where κ is a real-valued parameter.
Thus C κ refers to the complex numbers, dual numbers, or double numbers when κ is normalized to 1, 0, or −1 respectively (see [28] and [12]). One may check that C κ is an associative 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 C κ 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 C κ 2 , as it is the parameter κ 2 which determines the conformal structure of the Cayley-Klein geometry S with parameters κ 1 and κ 2 .
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 e i θ 2 σ 1 , e i α 2 σ 2 , and e β 2i σ 3 .
Note that when κ 1 = 1 and κ 2 = 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 We will refer to σ 1 , σ 2 , and σ 3 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 C κ 2 , C κ 1 , and C κ 1 κ 2 number planes, acting on these planes as rotations. Also, as these groups preserve the unit sphere Σ = {(z, t, x) | z 2 + κ 1 t 2 + κ 1 κ 2 x 2 = 1}, they preserve the respective intersections of Σ with the C κ 2 , C κ 1 , and C κ 1 κ 2 number planes. These intersections are, respectively, circles of the form κ 1 ww = 1 (there is no intersection when κ 1 = 0 or when κ 1 < 0 and κ 2 > 0), ww = 1, and ww = 1, where w, w, and w denote elements of C κ 2 , C κ 1 , and C κ 1 κ 2 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 R 3 represent C (recall that C denotes C κ 2 ). 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 RP 2 and H 2 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) gives the main metric g 1 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 κ 2 = 0) the subsidiary metric g 2 is given by and is defined on lines w = t 0 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 κ 1 and κ 2 are normalized to the values K 1 and K 2 so that then Yaglom [28] has shown that the linear isometries of R 3 (with metric ds 2 = dz 2 + K 1 dt 2 + K 1 K 2 dx 2 ) acting onΣ project to those Möbius transformations that preserve ς, and so these Möbius transformations preserve cycles 8 : a cycle is a curve of constant curvature, corresponding to the intersection of a plane in R 3 withΣ. 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 κ 1 and κ 2 we may define a linear isomorphism of R 3 as indicated below.
This transformation preserves the projection point (−1, 0, 0) as well as the complex plane z = 0, and maps the projective quadricΣ for parameters K 1 and K 2 to that for κ 1 and κ 2 , and so gives a correspondence between elements of SO K 1 ,K 2 (3) with those of SO κ 1 ,κ 2 (3) as well as the projections of these elements. As the Möbius transformations of C are those transformations that preserve curves of the form = 0 (where w ′ 1 , w ′ 2 , and w ′ 3 are three distinct points lying on the cycle), then if this form is invariant under the induced action of the linear isomorphism, then elements of SO κ 1 ,κ 2 (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 corresponding to the complex number w = 1 z+1 (t + Ix) ∈ C K 2 , if the linear transformation sends (z, t, x) to (z ′ , t ′ , x ′ ), then it sends w = 1 z+1 (t + Ix) ∈ C K 2 to w ′ = 1 z ′ +1 (t ′ + ix ′ ) ∈ C κ 2 = C, where I 2 = −K 2 and i 2 = −κ 2 . We can then write that as can be checked directly, and we then have that elements of SO(3) project to Möbius transformations of ς.
The rotations e θK preserve the complex number plane z = t + ix = 0 and so correspond simply to the transformations of C given by w → e iθ w, as e iθ = C κ 2 (θ) + iS κ 2 (θ), keeping in mind that i 2 = −κ 2 . Now in order to express this rotation as a Möbius transformation, we can write Since there is a group homomorphism from the subgroup of Möbius transformations corresponding to SO(3) to the group M (2, C) of 2 × 2 matrices with entries in C, this transformation being defined by For future reference let us now define where i 2 σ 1 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 e αH and e βP . The x-axis, the zt-coordinate plane, and the unit sphere Σ, are all preserved by e αH . So the zt-coordinate plane is given the complex structure C κ 1 = {w = z + it | i 2 = −κ 1 }, for then the unit circle ww = 1 gives the intersection of Σ with C κ 1 , and the transformation induced on C κ 1 by e αH is simply given by w → e iα w. Similarly the transformation induced by e βP on In order to explicitly determine the projection of the rotation w → e iα w of the unit circle in C κ 1 and also that of the rotation w → e iβ w of the unit circle in C κ 1 κ 2 , 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: (where T κ is the tangent function) for noting that a point a+ib on the unit circle ww of the complex plane C κ can be written as a+ib = e iψ = C κ (ψ) + iS κ (ψ). So the rotations e αH and e βP induce the respective transformations 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 = iT κ 1 φ 2 , then we have that with corresponding matrix representation in SL(2, C κ 1 ), where we have applied the trigonometric identity 9 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 C κ 1 κ 2 corresponds to a transformation of the imaginary axis of C (also the x-axis) while a transformation of the imaginary axis of C κ 1 (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 C κ 1 κ 2 as well as the C plane, correspond as if κ 1 = 0, as can be seen by examining the power series representation for T κ . The situation for the rotation e iα is similar. We can then compute the elements of SL(2, C) corresponding to e αH and e βP 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 e αH can be written as e α 2i σ 3 and those for e βP as e i β 2 σ 2 , where Thus i 2 σ 1 , i 2 σ 2 , and 1 2i σ 3 are generators for the generalized Lie algebra so (3), a subalgebra of the real matrix algebra M (2, C).

Subspace of with basis scalars
The Clifford algebra Cl 3 Definition 3. Let Cl 3 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 C κ 2 . 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 σ 3 can be thought of as the 2 × 2 matrix 0 1 −κ 1 0 . We will also identify the generalized Paul spin matrices σ 1 , σ 2 , and σ 3 with the vectorsî = 1, 0, 0 ,ĵ = 0, 1, 0 , andk = 0, 0, 1 respectively of the vector space R 3 = {(z, t, x)} given the Cayley-Klein inner product 10 . (iv) Ifn = n 1 , n 2 , n 3 and σ = iσ 1 , iσ 2 , σ 3 /i , then we will letn · σ denote the bivector n 1 iσ 1 + n 2 iσ 2 + n 3 1 i σ 3 . This bivector is simple, and the parallel vectors in · σ and 1 in · σ are perpendicular to any plane element represented byn · σ. Let η denote the line through the origin that is determined by in · σ or 1 in · σ. (vi) Bivectorsn · σ act as imaginary units as well as generators of rotations in the oriented planes they represent. Let κ be the scalar − (n · σ) 2 . Then if a lies in an oriented plane determined by the bivectorn · σ, where this plane is given the complex structure of C κ , then e − φ 2n · σ ae φ 2n · σ is simply the vector a 1 , a 2 , a 3 rotated by the angle φ in the complex plane C κ , where ι 2 = −κ. So this rotation is given by unit complex multiplication.
The goal of this section is to prove Proposition 1. We can easily compute the following: Recalling that R 3 is given the Cayley-Klein inner product, we see that σ 2 i gives the square of the length of the vector σ i . Note that when κ 1 = 0, Cl 3 is not generated by the vectors. Cen(Cl 3 ) of Cl 3 is given by R ⊕ 3 R 3 , and we can check directly that if then we have the following commutators: So the Lie algebra so (3) is isomorphic to the space of bivectors 2 R 3 . The product of two vectors a = a 1 σ 1 + a 2 σ 2 + a 3 σ 3 and b

is the Cayley-Klein inner product and the wedge product is given by
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 κ 1 = 0, then the vectors e and f span the same oriented plane as the vectors g and h. When κ 1 = 0 the bivectorn · σ is no longer simple in the usual way. For example, for the Galilean kinematical group (aka the Heisenberg group) where κ 1 = 0 and κ 2 = 0, we have that both σ 1 ∧ σ 3 = iσ 2 and (σ 1 + σ 2 ) ∧ σ 3 = iσ 2 , so that the bivector iσ 2 represents plane elements that do no all lie in the same plane 11 . Recalling that σ 1 , σ 2 , and σ 3 correspond to the vectorsî,ĵ, andk 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 byî andk we have that If we give either plane the complex structure of the dual numbers so that i 2 = 0, then the rotation is given by simply multiplying vectors in the plane by the unit complex number e βi .
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. Ifn = n 1 , n 2 , n 3 and σ = iσ 1 , iσ 2 , σ 3 /i , then we will letn · σ denote the bivector B = n 1 iσ 1 + n 2 iσ 2 + n 3 1 where at least one of the bivectors n in · σ is non-zero asn · σ is non-zero. If κ 1 = 0 and n 1 = 0, then σ 1 ∧c =n · σ. However, if both κ 1 = 0 and n 1 = 0, then it is impossible to have e∧f =n · σ: in this context we may simply replace the expressionn · σ with the expression σ 3 ∧ σ 2 whenever κ 1 = 0 and n 1 = 0 (as we will see at the end of this section, we could just as well replacen · σ with any non-zero multiple of σ 3 ∧ σ 2 ). The justification for this is given by letting κ 1 → 0, for then shows that the plane spanned by the vectors e and f tends to the xt-coordinate plane. We will see below how each bivectorn · σ corresponds to an element of SO(3) that preserves any oriented plane corresponding ton · σ: in the case where κ 1 = 0 and n 1 = 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 in · σ = −κ 2 n 1 σ 1 − κ 2 n 2 σ 2 + n 3 σ 1 is non-zero and perpendicular to any plane element corresponding tô n · σ except when both κ 2 = 0 and n 3 = 0, in which case in · σ is the zero vector. In this last case the vector 1 in · σ = n 1 σ 1 + n 2 σ 2 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 σ 1 , σ 2 , and σ 3 ) for the generators e θK , e βP , and e αH are given by η, wheren · σ is given by iσ 1 (or σ 3 ∧ σ 2 by convention), iσ 2 = σ 1 ∧ σ 3 , and 1 i σ 3 = σ 1 ∧ σ 2 . 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 C κ 1 , the zx-coordinate plane with C κ 1 κ 2 , and the tx-coordinate plane with C κ 2 as indicated in Fig. 4. Note that the basis bivectors act as imaginary units in Cl 3 since The product of a vector a and a bivector B can be written as aB = a ⊣ B + a ∧ B = 1 2 (aB − Ba) + 1 2 (aB + Ba) so that aB is the sum of a vector a ⊣ B (the left contraction of a by B) and a volume element a ∧ B. Let B = b ∧ c for some vectors b and c. Then where we have used the Jacobi identity recalling that M (2, C) is a matrix algebra where the commutator is given by left contraction. Thus and so So the vector 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 ⊣ (b ∧ c) = 0. We will show that if a is the vector a 1 σ 1 + a 2 σ 2 + a 3 σ 3 , then the linear transformation of R 3 defined by · 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 First, let us see how, using this construction, the vectors σ 1 , σ 2 , and σ 3 (and hence the bivectors iσ 1 , iσ 2 , and 1 i σ 3 ) correspond to rotations of the coordinate axes (and hence coordinate planes) given by e θK , e βP , and e αH respectively. Since (noting that C κ is an even function while S κ is odd) it follows that So for each plane element, the σ j transform as the components of a vector under rotation in the clockwise direction, given the orientations of the respective plane elements: If κ is the scalar − (n · σ) 2 , then As a = a 1 σ 1 + a 2 σ 2 + a 3 σ 3 is a vector, we can compute its length easily using Clifford multiplication as aa = (a 1 ) 2 + κ 1 (a 2 ) 2 + κ 1 κ 2 (a 3 ) 2 = |a| 2 . We would like to show that e − φ 2n · σ ae φ 2n · 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 g h = e −φg he φg (see [25] for example). So if B is a bivector B 1 iσ 1 + B 2 iσ 2 + B 3 1 i σ 3 , then e − φ 2n · σ Be φ 2n · σ is also a bivector. It follows that e − φ 2n · σ ae φ 2n · σ is a vector as the volume element i lies in Cen(Cl 3 ) so that e − φ 2n · σ σ 1 e φ 2n · σ , e − φ 2n · σ σ 2 e φ 2n · σ , and e − φ 2n · σ σ 3 e φ 2n · σ are all vectors. Since · σ ae φ 2n · σ has the same length as a. So the inner automorphism of R 3 given by a → e − φ 2n · σ ae φ 2n · 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 R 3 .
Finally, note that e − φ 2n · σ (n · σ) e φ 2n · σ =n · σ asn · σ commutes with e φ 2n · σ : so any plane element represented byn · σ is preserved by the corresponding element of SO (3). In fact, if n · σ = a ∧ b for some vectors a and b and κ is the scalar − (a ∧ b) 2 , then 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 i 2 = −κ. Thus, the linear combination ua + vb is sent to ue −iφ a + ve −iφ b, 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 bivectorn · σ where this plane is given the complex structure of C κ , then e − φ 2n · σ ae φ 2n · σ is simply the vector a rotated by an angle of −φ in the complex plane C κ , where ι 2 = −κ. Furthermore, the axis of rotation is given by η as η is preserved (recall that i lies in the center of Cl 3 ). Since the covariant components σ i of a are rotated clockwise, the contravariant components a j are rotated counterclockwise. So a 1 , a 2 , a 3 is rotated by the angle φ in the complex plane C κ determined byn · 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 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 ofn · σ appears to be important, since κ = − (n · σ) 2 , and two, the normalization (n 1 ) 2 + (n 2 ) 2 + (n 3 ) 2 = 1 ofn is somewhat arbitrary 12 . 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 C κ = {t + ix | i 2 = −κ} with angle measure φ and C κ ′ = {t + ιx | ι 2 = −κ ′ = −n 2 κ} with angle measure θ: without loss of generality let κ > 0. Then φ = nθ, for 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.

SU (2)
Since the generators of the generalized Lie group SO(3) can be represented by inner automorphisms of the subspace R 3 of vectors of Cl 3 (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 2n · σ of SL(2, C), where as usual C denotes the generalized complex number C κ 2 : we will denote the subgroup of SL(2, C) consisting of elements of the form ±e φ 2n · σ by SU (2).

Definition 4. Let A be the matrix
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 and (n · σ) ⋆ A (n · σ) = −A (n · σ) 2 = κA implies that So SU (2) is a subgroup of the subgroup G of SL(2, C) consisting of those matrices U where We can characterize this subgroup G as as can be checked directly, recalling that where κ = − (n · σ) 2 = n 1 2 κ 2 + n 2 2 κ 1 κ 2 + n 3 2 κ 1 .
Thus det e φ 2n · σ = 1, and we see that any element of G can be written in the form e φ 2n · σ . So the group SU (2) can be characterized by Note that if U (λ) is a curve passing through the identity at λ = 0, then where s is an element of the group where C denotes the generalized complex number C κ 2 .
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 C κ may be extended to a Riemann sphere Γ or inversive plane 13 (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 R 4 , 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 conformal completion by one of two methods, one a group-theoretical one involving one-parameter subgroups and the other stereographic projection.
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 G 1 and G 2 , for "translations". It is these transformations D, G 1 , G 2 , 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 → w + t and w → w + ti (for real parameter t) are valid only if κ 1 = 0, which explains why our "translations" G 1 and G 2 are not actually translations: 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 G 2 ) 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 .

B Appendix: The Hermitian metric
The hermitian metric The isometry group of ς with metric g 1 is that given by those transformations belonging to Spin(3). After some tedious algebra we have that when a b c d ∈ SU (2) so that gives the main metric g 1 on ς. We have then proved the following lemma. We would also like to show, following the proof that is given in [3] for the hyperbolic plane, that and so g(t + ǫ) − κ 1 g(t + ǫ)g(t)g(ǫ) = g(ǫ) + g(t) and then we can divide by ǫ g(t + ǫ) − g(t) ǫ = g(ǫ) ǫ [1 + κ 1 g(t)g(t + ǫ)] and take the limit So g ′ (t) = 1 + κ 1 g 2 (t). By the inverse function rule for differentiation, then M (w 2 ) → w 2 − w 1 κ 1 w 1 w 2 + 1 14 Geodesics 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. 15 We can see from the equation below that g(t) = Tκ 1 (t). Elements of SL(2, C) corresponding to e αH κ 1 κ 2 is positive ±