Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 072, 18 pages
Clifford Fibrations 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 10, 2009, in final form June 19, 2009; Published online July 14, 2009
doi:10.3842/SIGMA.2009.072
Abstract. Following Herranz and Santander [Herranz F.J., Santander M., Mem. Real
Acad. Cienc. Exact. Fis. Natur. Madrid 32 (1998), 59–84, physics/9702030] we will con-
struct homogeneous spaces based on possible kinematical algebras and groups [Bacry H.,
Levy-Leblond J.-M., J. Math. Phys. 9 (1967), 1605–1614] and their contractions for 2-
dimensional spacetimes. Our construction is different in that it is based on a generalized
Clifford fibration: Following Penrose [Penrose R., Alfred A. Knopf, Inc., New York, 2005]
we will call our fibration a Clifford fibration and not a Hopf fibration, as our fibration is
a geometrical construction. The simple algebraic properties of the fibration describe the
geometrical properties of the kinematical algebras and groups as well as the spacetimes
that are derived from them. We develop an algebraic framework that handles all possible
kinematic algebras save one, the static algebra.
Key words: Clifford fibration; Hopf fibration; kinematic
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 nice role played by quaternions in describing rotations of Euclidean 3-dimensional space,
and the beauty of the Hopf fibration of the 3-sphere, can be simply generalized for the study of
(1+1) kinematics. It is the purpose of this paper to show how this can be done. We can let i, j,
and k denote the basis of the imaginary part of a generalized quaternion number system so that
they also describe a basis for any given kinematical algebra save the static algebra. The space
of unit quaternions (under a suitable choice of norm) then describes a “3-sphere”. If q is a point
on this sphere, then the Hopf flows iq, jq, and kq describe fibrations of the sphere where the
base spaces are the space of events, the space of space-like geodesics, or the space of time-like
geodesics. The description given below is of a unified approach to all kinematical algebras (save
the state algebra) as well as to the three classical Riemannian surfaces of constant curvature.
1 Possible kinematics
It is the purpose of this section to briefly review Bacry and Le´vy-Leblond’s work on possible
kinematics. Bacry and Le´vy-Leblond’s investigations into the nature of all possible Lie algebras
for kinematical groups given three basic principles
(i) space is isotropic and spacetime is homogeneous;
(ii) parity and time-reversal are automorphisms of the kinematical group;
(iii) the one-dimensional subgroups generated by the boosts are non-compact
2 A.S. McRae
Table 1. The 11 possible kinematical groups.
Symbol Name
dS de Sitter groups
adS anti-de Sitter groups
M Minkowski groups
M+ expanding Minkowski groups
M ′ para-Minkowski groups
C Carroll groups
N+ expanding Newtonian Universe groups
N− oscillating Newtonian Universe groups
G Galilei group
SdS static de Sitter Universe groups
St static Universe group
Table 2. The characteristic Lie brackets for the kinematical Lie algebras.
dS adS M M+ M ′ C N+ N− G SdS St
[H,P ] −K K 0 −K K 0 −K K 0 −K 0
[K,H] P P P 0 0 0 P P P 0 0
[K,P ] H H H H H H 0 0 0 0 0
gave rise to 11 possible kinematical algebras. Restricting our attention to 2-dimensional space-
times we still obtain the same 11 kinds of algebras (see [8]), where each of the kinematical groups
is generated by its inertial transformations as well as its spacetime translations. These groups
consist of the de Sitter groups and their 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. The names of the 2-dimensional groups are given in Table 1.
In this paper we will restrict our attention to 2-dimensional spacetimes. So let K denote
the generator of the inertial transformations, H the generator of time translations, and P the
generator of space translations. The kinematical algebras are determined by the structure con-
stants p, h, and k that are given by the commutators
[K,H] = pP, [K,P ] = hH, and [H,P ] = kK.
If we normalize the structure constants to lie in the set {−1, 0, 1}, then the characteristic Lie
brackets for the kinematical Lie algebras are as given in Table 2 (see [1]).
We will follow Herranz, Ortega and Santander (see [5]) and reduce the number of structure
constants from three to two as follows. The kinematical algebras dS, adS, M , N+, N−, and G
(after rescaling) are determined by the structure constants κ1 and κ2 that are given by the
commutators
[K,H] = P, [K,P ] = −κ2H, and [H,P ] = κ1K. (1)
The constant κ1 = ± 1τ2 gives the spacetime curvature κ1 as well as the universe (time) radius τ ,
and the constant κ2 = − 1c2 gives the speed of light
1 c. For the de Sitter groups κ1 < 0 and
κ2 < 0, while for the anti-de Sitter groups κ1 > 0 and κ2 < 0. The remaining kinematical
algebras (save for St) can be obtained by group contractions (κ1 → 0 or κ2 → 0) in possible
conjunction with the symmetries SP , SH , and SK :
SP : {K ↔ H} : [K,H] = −P, [K,P ] = κ1H, and [H,P ] = −κ2K,
1See [4]. We will also demonstrate that κ2 = − 1c2 and that κ1 is the spacetime curvature later on in this paper.
Clifford Fibrations and Possible Kinematics 3
Figure 1. The 11 kinematical and 3 non-kinematical groups.
Table 3. The characteristic Lie brackets for the non-kinematical Lie algebras.
El Eu H
[H,P ] K 0 −K
[K,H] P P P
[K,P ] −H −H −H
SH : {K ↔ P} : [K,H] = −κ1P, [K,P ] = κ2H, and [H,P ] = −K
and
SK : {H ↔ P} : [K,H] = −κ2P, [K,P ] = H, and [H,P ] = −κ1K.
See Fig. 1 for an illustration of how the different groups are related via contractions and symme-
tries: El, Eu, and H denote the (non-kinematical) isometry groups of the elliptical, Euclidean,
and hyperbolic planes (of constant curvature κ1) respectively (see Table 3).
For example, if κ2 → 0 then dS and H contract to N+ while adS and El contract to N−,
while if κ1 → 0 then dS and adS contract to M while H and El contract to Eu. Similarly
a space-time contraction sends either M+ or M ′ to C. In this paper we will specifically work
with the nine kinds of groups indicated in bold in Fig. 1, since the other groups (save St) are
then easily obtained from these nine: the lie algebras of our nine groups have the commutators
as given by (1). Henceforward we will not refer to the other algebras.
We can contract with respect to any subgroup, giving us three fundamental types of con-
traction: 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 → K and P → P into the Lie
algebra and then calculate the singular limit of the Lie brackets as → 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
4 A.S. McRae
Table 4. The 3 basic symmetries are given as reflections of Fig. 1.
Symmetry Reflection across face Corresponding group transformations
SH 1378 M ←→M ′, Eu←→M+, G←→ SdS
SP 1268 C ←→ SdS, M ←→ N+, Eu←→ N−
SK 1458 C ←→ G, M+ ←→ N−, M ′ ←→ N+
Table 5. Important classes of kinematical groups and their geometrical configurations in Fig. 1.
Class of groups Face
Relative-time 1247
Absolute-time 3568
Relative-space 1346
Absolute-space 2578
Cosmological 1235
Local 4678
geodesic, as we are contracting to the subgroup that leaves this worldline invariant, and so are
passing from relativistic to absolute time.
Speed-time contractions. We make the substitutions K → K and H → H into the Lie
algebra and then calculate the singular limit of the Lie brackets as  → 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 → P and H → H into the Lie
algebra and then calculate the singular limit of the Lie brackets as  → 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.
2 Generalized complex numbers
The generalized complex numbers are not new to physics or mathematics (see [12] for example).
It is the purpose of this section to introduce these numbers to the reader who is not already
familiar with them.
Definition 1. By the complex number plane Cκ we will mean the set of numbers of the form
{z = x + iy | (x, y) ∈ R2 and i2 = −κ}, where the constant κ is real and i is not. Cκ is a real
commutative algebra and also has zero divisors when κ ≤ 0. The real part of z is given by
R(z) = x and the imaginary part by I(z) = y.
Zero divisors play an important role in determining the conformal structure of spacetime, and
although it does not make good algebraic sense to divide by them, one can form the Riemann
sphere Σκ.
Definition 2. Let Σκ denote the Riemann sphere consisting of the set of all equivalence classes[
A
B
]
of complex ratios AB , where A,B ∈ Cκ and where either A or B is not a zero-divisor:
A
B ∼
C
D ⇐⇒ A = µC and B = µD for some µ ∈ Cκ where µ is not a zero divisor.
Clifford Fibrations and Possible Kinematics 5
Figure 2. The Riemann sphere Σ0.
We can describe Σκ through stereographic projection, giving a circular cylinder when κ = 0
and a hyperboloid of one sheet when κ < 0 (see [12] for details). Fig. 2 shows such a construction
for Σ0: Here we are projecting from the point P onto the complex number plane C0 (where the
zero divisors consists of all purely imaginary numbers), so that numbers of the form 1ai correspond
to the line ζ of “infinities” on Σ0. So P =
[
1
0
]
, for example.
The unit circle zz = x2 + κy2 = 1 in Cκ, where z = x − iy, is determined by the Hermitian
metric dzdz = dx2 + κdy2. The unit circle can be used to define 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 eiφ = Cκ(φ) + iSκ(φ) is a point on the connected component of the unit circle
containing 1, and φ is the signed distance from 1 to eiφ along the circular arc, defined modulo
the length 2pi√κ of the unit circle when κ > 0
2. The power series for these analytic trigonometric
functions are as follows:
Cκ(φ) = 1−
1
2!
κφ2 +
1
4!
κ2φ4 + · · · ,
Sκ(φ) = φ−
1
3!
κφ3 +
1
5!
κ2φ5 + · · · .
Note that Cκ2(φ) + κSκ2(φ) = 1. We also have that
d
dφ
Cκ(φ) = −κSκ(φ),
d
dφ
Sκ(φ) = Cκ(φ).
2When κ = 0 the distance along the unit circle x2 = 1 is defined by ds2 = dy2, as the Hermitian metric
ds2 = dx2 + κdy2 = dx2 vanishes on the unit circle. This is an instance where it is advantageous to rescale
a metric.
6 A.S. McRae
Definition 3. Let Uκ(1) denote the group (under multiplication) of unit complex numbers
in Cκ.
Before we proceed it might be insightful to see how the algebraic structure of Cκ2 is useful
in describing kinematics for the kinematical groups M (flat Minkowski spacetimes) and G (flat
Galilean spacetime). We will demonstrate the well known fact that classical kinematics is
a limiting case of relativistic kinematics. The Lorentz transformation
(
x′
t′
)
=
1
√
1− v2/c2
(
1 −v
− vc2 1
)(
x
t
)
can be simply written in complex notation as z′ = eiθz in Cκ2 , where z
′ = t′ + ix′, z = t + ix,
and κ2 = − 1c2 . This is because rotation about the origin through the angle θ in the complex
plane Cκ2 can be written as the linear transformation (or boost)
z 7→ eiθz  
(
x
t
)
7→
(
Cκ2(θ) Sκ2(θ)
−κ2Sκ2(θ) Cκ2(θ)
)(
x
t
)
in R2, where θ = T−1κ2 (−v), so that
Tκ2(θ) =
Sκ2(θ)
Cκ2(θ)
= −v and
C2κ2(θ) + κ2S
2
κ2(θ) = C
2
κ2(θ)−
1
c2
S2κ2(θ) = 1.
On the other hand the Galilean transformation
(
x′
y′
)
=
(
1 −v
0 1
)(
x
t
)
can be simply written in complex notation as z′ = eiθz in C0, where κ2 = − 1c2 = 0 as the speed
of light is infinite in Galilean spacetime. Note that C0(θ) = 1 and S0(θ) = θ, so that rotation
about the origin through the angle θ can be written as the linear “shift” transformation (or
boost)
z 7→ eiθz  
(
x
t
)
7→
(
C0(θ) S0(θ)
0 C0(θ)
)(
x
t
)
=
(
1 −θ
0 1
)(
x
t
)
in R2, where θ = −v.
The speed-space contraction κ2 → 0 (passing from relative- to absolute-time) takes Minkowski
spacetime to Galilean spacetime, as Cκ2(θ) → 1 and Sκ2(θ) → θ. The unit circle zz = 1 in
Minkowski spacetime, a hyperbola, transforms into the “degenerate” hyperbola given by t = ±1.
The light cone in Minkowski spacetime, given by t = ±1cx, transforms into the “light cone”
t = 0 in Galilean spacetime, where c = ∞. Locally all spacetimes are equivalent to Minkowski
or Galilean spacetime via space-time contractions where κ1 → 0 (passing from a cosmological
to a local group) so we see that κ2 = − 1c2 .
Although we will not need the theorem stated below, the reader might be interested in seeing
it. Undoubtedly this theorem was known to Yaglom.
Theorem 1 (Yaglom). Let f(t, x) = u(t, x)+iv(t, x) and i2 = −κ, where the partial derivatives
of u and v are continuous on an open set. Then f is holomorphic on that open set if and only
if the Cauchy–Riemann equations
ut = vx, ux = −κvt
are satisfied. Furthermore, f is conformal at any point w where f ′(w) is not a zero-divisor3.
The usual proofs for κ = 1 apply.
3On the complex plane Cκ, the argument function is only defined on the set of non-zero divisors, for a non-zero
divisor w can be written w = reiθ where r is the norm sgn(ww)
√
|ww|2 of w and Arg(reiθ) = θ, see [3].
Clifford Fibrations and Possible Kinematics 7
3 A very brief review of some work
by Ballesteros, Herranz, Ortega and Santander
It is the purpose of this section to introduce some material by Ballesteros, Herranz, Ortega
and Santander that we will refer to in subsequent sections. A real matrix representation for
a kinematical Lie algebra, denoted by soκ1,κ2(3), is given by
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 (1)
[K,H] = P, [K,P ] = −κ2H, and [H,P ] = κ1K.
Elements of a corresponding kinematical Lie group, denoted by SOκ1,κ2(3), are given by real-
linear, orientation-preserving isometries of R3 = {(y, t, x))} imbued with the (possibly indefinite
or degenerate) metric ds2 = dy2 + κ1dt2 + κ1κ2dx2. The one-parameter subgroups H, P, and K
generated respectively by H, P , and K consist of matrices 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)

 .
Ballesteros, Herranz, Ortega and Santander have constructed spacetimes as homogeneous
spaces4 by looking at real representations of their motion groups SOκ1,κ2(3). The spaces
SOκ1,κ2(3)/K, SOκ1,κ2(3)/H, and SOκ1,κ2(3)/P are homogeneous spaces for SOκ1,κ2(3). When
SOκ1,κ2(3) is a kinematical group, then SOκ1,κ2(3)/K can be identified with the manifold of
space-time translations, SOκ1,κ2(3)/P the manifold of time-like geodesics, and SOκ1,κ2(3)/H
the manifold of space-like geodesics.
4 Generalized quaternions
The goal of this section is to develop a simple algebraic description of the kinematical algebras,
using what we already know about the generalized complex numbers. To that end, we begin by
putting the Hermitian norm dzdz = dz1dz1+κ1dz2dz2 on C2κ2 ≡ Cκ2×Cκ2 , where z = (z1, z2) is
an element of C2κ2 . The construction below follows a natural course based on the double covering
of SO(3) by SU(2) as part of the geometry of the standard quaternions.
The Hermitian inner product is obtained as follows. Let z = (z1, z2) and w = (w1, w2). Then
〈z, w〉 =
1
2
(
|z + w|2 − |z|2 − |w|2
)
=
1
2
((z1 + w1) (z1 + w1) + κ1 (z2 + w2) (z2 + w2)− z1z1 − κ1z2z2 − w1w1− κ1w2w2)
4See [2, 5, 6], and also [4], where a special case of the group law is investigated, leading to a plethora of
trigonometric identities.
8 A.S. McRae
=
1
2
(z1w1 + z1w1 + κ1z2w2 + κ1z2w2) = x1x2 + κ2y1y2 + κ1u1u2 + κ1κ2v1v2,
where z1 = x1 + iy1, z2 = u1 + iv1, w1 = x2 + iy2, and w2 = u2 + iv2. So in R4 we can write the
inner product as
(
x1 y1 u1 v1
)




1 0 0 0
0 κ2 0 0
0 0 κ1 0
0 0 0 κ1κ2








x2
y2
u2
v2



 = x1x2 + κ2y1y2 + κ1u1u2 + κ1κ2v1v2.
Definition 4. By the set of generalized quaternions Hκ1,κ2 (or simply quaternions for short) we
will mean the set of numbers of the form {(x+ iy + ju+ kv) | i2 = −κ2, j2 = −κ1,k2 = −κ1κ2}
with the following product rules5
ij = k, ji = −k,
jk = κ1i, kj = −κ1i,
ki = κ2j, ik = −κ2j.
We will show below that Hκ1,κ2 is a real associative algebra over the reals and that the pure
quaternions represent the kinematical algebras given by equation (1).
If q = x+ iy+ ju+ kv, then qq = x2 + κ2y2 + κ1u2 + κ1κ2v2, where q = x− iy− ju− kv. So
if we identify points of Hκ1,κ2 with points of C
2
κ2 = {(z1, z2)} by the correspondence
x+ iy + ju+ kv = z1 + z2j (z1, z2) ,
where z1 = x+iy and z2 = u+iv (in terms of quaternions we can think of z1 and z2 as z1 = x+iy
and z2 = u+ iv), then the norm of q corresponds to the norm of (z1, z2).
Definition 5. Let SUκ1,κ2(2) denote the group of all matrices of the form
(
z1 z2
−κ1z2 z1
)
with determinant z1z1 + κ1z2z2 = 1. It was shown in [8] that SUκ1,κ2(2) is a double cover of
SOκ1,κ2(3), and that suκ1,κ2(2) consists of those elements B of M(2,Cκ2) such that B
?A+AB=0
where A is the matrix
A =
(
κ1 0
0 1
)
.
We will see below that suκ1,κ2(2) can be identified with the space of pure quaternions, a real
algebra, and that finally the space of pure quaternions is a kinematical algebra.
Under the correspondence
x+ iy + ju+ kv  
(
z1 z2
−κ1z2 z1
)
the set of unit quaternions is identified with SUκ1,κ2(2). The context should make it clear as to
whether elements of SUκ1,κ2(2) are to be treated as elements of M(2,Cκ2) or as unit quaternions
in Hκ1,κ2 . The inner product on C
2
κ2 corresponds in Hκ1,κ2 to
〈q1, q2〉 =
1
2
(q1q2 + q2q1) =
1
2
((z1 + z2j) (w1 − w2j) + (w1 + w2j) (z1 − z2j))
5See [11] for another description of the generalized quaternions.
Clifford Fibrations and Possible Kinematics 9
=
1
2
(z1w1 + z1w1 + κ1z2w2 + κ1z2w2) =
1
2
(
|z + w|2 − |z|2 − |w|2
)
= 〈z, w〉,
since j (x+ iy) = (x− iy) j and j2 = −κ1.
We can see that Hκ1,κ2 and the subspace of M(2,Cκ2) consisting of all matrices of the form(
z1 z2
−κ1z2 z1
)
are isomorphic as algebras, for if q1 = z1 + z2j and q2 = w1 + w2j are two
quaternions with corresponding matrices
(
z1 z2
−κ1z2 z1
)
and
(
w1 w2
−κ1w2 w1
)
,
then
q1 + q2  
(
z1 z2
−κ1z2 z1
)
+
(
w1 w2
−κ1w2 w1
)
and
q1q2  
(
z1 z2
−κ1z2 z1
)(
w1 w2
−κ1w2 w1
)
=
(
z1w1 − κ1z2w2 z1w2 + z2w1
−κ1z2w1 − κ1z1w2 z1w1 − κ1z2w2
)
,
since
(z1 + z2j) (w1 + w2j) = (z1w1 − κ1z2w2) + (z1w2 + z2w1) j.
Definition 6. We define the unit one-sphere and two-sphere6
S1κ2 = Uκ2(1) = {z ∈ Cκ2 , |z| = 1} {e
iθ} ⊂ Hκ1,κ2 ,
S3κ1,κ2 = {(z, w) ∈ C
2
κ2 , |(z, w)| = 1} {q ∈ Hκ1,κ2 | |q| = 1},
where the set of unit quaternions is given by numbers of the form eiy+ju+kv. So S3κ1,κ2 can be
identified with SUκ1,κ2(2).
The plane spanned by 1 and i can be easily identified with Cκ2 , and the intersection of this
plane with the sphere of unit quaternions then corresponds to the unit circle of Cκ2 . Similar
remarks hold for Cκ1 or Cκ1κ2 for the planes spanned by 1 and j or 1 and k respectively. So e
it,
ejt, and ekt are all unit quaternions.
If a is a unit quaternion, then a−1 = a. Since ab = ba for any two quaternions a and b, it
follows that |aqb−1| = |q| for any quaternion q, provided that both a and b are unit quaternions.
In fact, the generalized quaternions are a composition algebra, so that |q1q2| = |q1| |q2|. Also,
aqa−1 is a pure quaternion since aqa−1 = −aqa−1, where q denotes the pure part of q. So the
linear transformations (in terms of real coordinates)
R4 → R4 defined by the automorphism q → aqb−1
and
R3 → R3 defined by the inner automorphism q→ aqa−1
respectively give rotations of Hκ1,κ2 and the subspace of pure quaternions. It might appear then
that SUκ1,κ2(2) × SUκ1,κ2(2) is a double cover of the group of rotations of C
2
κ2 with Hermitian
metric dz1dz1+κ1dz2dz2, as (a, b) and (−a,−b) represent the same rotation, but not all rotations
6The context should make it clear as to whether these spheres are to thought of in terms of generalized complex
or quaternion numbers.
10 A.S. McRae
can be so represented by such an automorphism. For example, if both κ1 and κ2 vanish, then
rotations of R4 are of the form




1 0 0 0
0 m22 m23 m24
0 m32 m33 m34
0 m42 m43 m44



 ,
and so the rotation group is 9-dimensional. Yet SUκ1,κ2(2)× SUκ1,κ2(2) has dimension 6. Simi-
larly SUκ1,κ2(2) is not a double cover for the rotation group for the subspace of pure quaternions
7.
Let suκ1,κ2(2) denote the Lie algebra of SUκ1,κ2(2). If we identify SUκ1,κ2(2) with S
3
κ1,κ2 , the
space of unit quaternions, then suκ1,κ2(2) can be represented by the space of pure quaternions:
For if q is a pure quaternion, then eq is a unit quaternion, as qq = qq so that eqeq = eqeq =
eq+q = e0 = 1. SUκ1,κ2(2) acts on its Lie algebra suκ1,κ2(2) by the inner automorphism
p 7→ e
θ
2qpe−
θ
2q where both p and q are pure quaternions. Since
d
dθ
∣
∣
∣
∣
θ=0
e
θ
2qpe−
θ
2q =
1
2
(qp− pq) =
1
2
[q,p] ,
then
1
2
[i, j] =
d
dθ
∣
∣
∣
∣
θ=0
e
θ
2 ije−
θ
2 i =
d
dθ
∣
∣
∣
∣
θ=0
e
θ
2 ie
θ
2 ij = ij = k,
1
2
[i,k] =
d
dθ
∣
∣
∣
∣
θ=0
e
θ
2 ike−
θ
2 i =
d
dθ
∣
∣
∣
∣
θ=0
e
θ
2 ie
θ
2 ik = ik = −κ2j,
1
2
[j,k] =
d
dθ
∣
∣
∣
∣
θ=0
e
θ
2 jke−
θ
2 j =
d
dθ
∣
∣
∣
∣
θ=0
e
θ
2 je
θ
2 jk = jk = κ1i
as jz = zj and kz = zk. We can then represent a given kinematical Lie algebra by
K  2i, H  2j, and P  2k.
In terms of the ordered basis {E1, E2, E3} = {2i, 2j, 2k} for suκ1,κ2(2), the structure con-
stants are given by [Ei, Ej ] = CkijEk, and so the Killing form on suκ1,κ2(2) is given by gij =∑
r,s C
r
isC
s
jr or
(gij) = −2


κ2 0 0
0 κ1 0
0 0 κ1κ2

 .
The Killing form is preserved by the inner automorphism.
We may form three natural homogeneous spaces:
SUκ1,κ2(2)/〈i〉 = SUκ1,κ2(2)/K S
3
κ1,κ2/S
1
κ2 ,
SUκ1,κ2(2)/〈j〉 = SUκ1,κ2(2)/H S
3
κ1,κ2/S
1
κ1 ,
SUκ1,κ2(2)/〈k〉 = SUκ1,κ2(2)/P  S
3
κ1,κ2/S
1
κ1κ2 .
The Killing form naturally determines a metric for each of these homogeneous spaces with
respective inner products
(
κ1 0
0 κ1κ2
)
,
(
κ2 0
0 κ1κ2
)
and
(
κ2 0
0 κ1
)
.
7Note that qq = κ2y2 + κ1u2 + κ1κ2w2 so that, when both κ1 and κ2 vanish, the dimension of the rotation
group of the pure quaternions is 9-dimensional, as any orientation preserving linear map of R3 is a rotation.
Clifford Fibrations and Possible Kinematics 11
Figure 3. Stereographic projection of the Clifford fibration for S3 onto R3. Image courtesy of [Pen-
rose R., Rindler W., Spinors and space-time, Vol. 2, Spinor and twistor methods in space-time geometry,
Cambridge University Press, 1986].
Following Ballesteros, Herranz, Ortega, and Santander we will rescale (even if κ1 or κ2 is equal
to zero) so that, in fact, the respective inner products are as follows:
(
1 0
0 κ2
)
,
(
1 0
0 κ1
)
and
(
κ2 0
0 κ1
)
.
The resulting metrics can be indefinite as well as degenerate.
Theorem 2. Let H, P , and K denote the respective generators for time translations, space
translations, and boosts of the kinematical algebra with commutators
[K,H] = P, [K,P ] = −κ2H, and [H,P ] = κ1K.
Then the kinematical algebra can be represented as the space of pure quaternions in Hκ1,κ2 by
K  2i, H  2j, and P  2k.
If SUκ1,κ2(2) denotes the group of unit quaternions with lie algebra suκ1,κ2(2), then suκ1,κ2(2)
is the space of pure quaternions and the homogeneous space SUκ1,κ2/〈i〉 is the space of events,
SUκ1,κ2/〈j〉 is the space of space-like geodesics, and SUκ1,κ2/〈k〉 is the space of time-like geode-
sics.
5 The generalized Clifford fibration
As pointed out by Urbantke (see [10]), Penrose has called the Clifford fibration an “element of the
architecture of our world”. This fibration can be used to describe two-level quantum systems, the
harmonic oscillator, Taub-NUT space, Robinson congruences, helicity representations, magnetic
monopoles, and the Dirac equation. By generalizing the Clifford fibration we will give yet
another physical application by modeling all kinematical algebras save for the static algebra.
S1κ2 acts freely and smoothly on S
3
κ1,κ2 by left multiplication: q = z + wj 7→ e
iθz + eiθwj. If
eiθq = q, then eiθ = 1 since |q| = 1 and so q is not a zero-divisor. So S3κ1,κ2 is the total space of
a principal S1κ2 bundle. So what can we say about the base space of this bundle?
12 A.S. McRae
We define Cκ2P1 as the space of all complex one-dimensional subspaces of the vector spa-
ce C2κ2 . Each subspace is uniquely described as the solution space to the complex linear equation
Az1 + Bz2 = 0 where
[
A
B
]
defines a point on the Riemann sphere Σκ2 . Note that A and B
cannot both be zero-divisors, for then
[
A
B
]
is not defined, and the set of points (z1, z2) satisfying
Az1 +Bz2 = 0 is no longer one-dimensional.
Alternatively, if we think of the complex line through the point (z1, z2) as being given by
points of the form λ(z1, z2), where λ takes all values in Cκ2 , then we get a line precisely when
either z1 or z2 is not a zero-divisor. For such a line we may let A = z2/z1 and B = −1 if z1 is
not a zero-divisor, and A = −1 and B = z1/z2 otherwise. Taking values in the Riemann sphere
(so that “infinities” are allowed) we may always write
[
A
B
]
= −
[
z2
z1
]
. Note that distinct lines
always intersect at the origin, but they may also intersect at points where both coordinates are
zero-divisors, for λ is allowed be a zero-divisor: So two points do not necessarily determine a
unique line.
So we may identify Cκ2P1 with the Riemann sphere. A complex line will intersect S
3
κ1,κ2
exactly when |z1|2 + κ1|z2|2 = 1 for some point (z1, z2) on the line: Let us call a line null if
|z1|2 + κ1|z2|2 = 0 for all points (z1, z2) on that line. So only the null lines do not intersect
the unit sphere. A complex line that does intersect the unit sphere does so at points of the
form {
(
eiθz1, eiθz2
)
}, where (z1, z2) is any point belonging to the intersection. Let us denote by
Σκ1,κ2 the subset of Σκ2 corresponding to non-null lines. We will see below that Σκ1,κ2 is the
homogeneous space S3κ1,κ2/S
1
κ2 .
So given a null line we must have that both Az1 + Bz2 = 0 and |z1|
2 + κ1 |z2|
2 = 0 for all
points (z1, z2) on the line. Recall that w ≡
[
A
B
]
= −
[
z2
z1
]
represents a point on the Riemann
sphere Σk. When κ1 = 0, Σ0,κ2 is Σκ2 with all infinities removed. If κ1 6= 0, then null lines can
exist only when κ1 < 0, and in this case Σκ1,κ2 is Σκ2 with all points of the form |w|
2 = − 1κ1
removed. It might be useful at this point to consider a familiar case: When κ2 = 1 and κ1 = 1,
0, or −1 we have elliptic, euclidian, and hyperbolic geometry respectively. For elliptic geometry
Σ1,1 = Σ1 is the well known Riemann sphere. For the euclidean plane Σ0,1 = Σ1 \ {∞} is
topologically a plane. And for the hyperbolic plane Σ−1,1 = Σ1 \ S1 is topologically a union of
two planes (each plane giving rise to a model of the hyperbolic plane).
We observe that the vectors iq, jq, and kq span the tangent space Tq(S3κ1,κ2) of S
3
κ1,κ2 at q.
For if q ∈ S3κ1,κ2 , then e
itq, ejtq, and ektq are unit quaternions. Keeping q fixed and letting t
vary, the respective tangents to the curves eitq, ejtq, and ektq passing through q are given by
d
dt
∣
∣
t=0 e
itq = iq, ddt
∣
∣
t=0 e
jtq = jq, and ddt
∣
∣
t=0 e
ktq = kq. Now
iq = i (x+ iy + ju+ kv) = ix− κ2y + ku− κ2jv = z1i+ z2k,
jq = j (x+ iy + ju+ kv) = jx− ky − κ1u+ κ1iv = −κ1z2 + z1j,
kq = k (x+ iy + ju+ kv) = kx+ jκ2y − iκ1u− κ1κ2v = −κ1z2i+ z1k,
and so iq, jq, and kq are linearly independent since |q| = 1. Note that iq, jq and kq are mutually
orthogonal since
〈iq, jq〉 = iq (−qj) + jq (−qi) = −ij− ji = 0,
〈iq,kq〉 = iq (−qk) + kq (−qi) = −ik− ki = 0,
〈jq,kq〉 = jq (−qk) + kq (−qj) = −jk− kj = 0.
However, the frame {iq, jq,kq} is not orthonormal since |iq|2 = κ2, |jq|2 = κ1, and |kq|2 = κ1κ2
(compare with the Killing form on suκ1,κ2(2)). The tangent plane spanned by jq and kq has the
complex structure Cκ2 since multiplying jq on the left by i yields kq. We also see that S
3
κ1,κ2 is
parallelizable: This is no surprise however, since S3κ1,κ2 is topologically either S
3, R3, or S1×R2.
Clifford Fibrations and Possible Kinematics 13
Definition 7. The Clifford fibration is given by
S3κ1,κ2
pi


y
Σκ1,κ2
 
S3κ1,κ2
pi


y
S3κ1,κ2/S
1
κ2
 
S3κ1,κ2
pi


y
S3κ1,κ2/〈i〉
 
S3κ1,κ2
pi


y
S3κ1,κ2/K
where pi−1
([
z1
z2
])
= {eiφ(z1 + z2j)}.
This is a principal fiber bundle over Σκ1,κ2 with fiber given by S
1
κ2 (the curve e
itq is the fiber
passing through q). The Clifford flow is given by the vector field χi(q) = iq, and the canonical
connection is determined by the horizontal planes spanned by jq and kq at each unit quaternion
q ∈ S3κ1,κ2 . Each such plane has the complex structure of Cκ2 . Here Σκ1,κ2 is the spacetime for
the kinematical algebra.
Similarly, we may form the fibrations
S3κ1,κ2
pi


y
S3κ1,κ2/〈j〉
and
S3κ1,κ2
pi


y
S3κ1,κ2/〈k〉
with respective Clifford flows given by χj(q) = jq and χk(q) = kq. These fibrations are principle
fiber bundles of S3κ1,κ2 with respective fibers S
1
κ1 and S
1
κ1κ2 and they give the space of space-like
and time-like geodesics of the spacetime Σκ1,κ2 , as H  2j and P  2k. Note however that the
bases S3κ1,κ2/〈j〉 and S
3
κ1,κ2/〈k〉 are not given by Σκ1,κ2 , as the fibers do not lie in the complex
lines Az1 +Bz2 = 0 which have the complex structure of Cκ2 , not of Cκ1 nor Cκ1κ2 .
Theorem 3. Let H  2j, P  2k, and K  2i denote the respective generators for time
translations, space translations, and boosts of the kinematical algebra with commutators
[K,H] = P, [K,P ] = −κ2H, and [H,P ] = κ1K.
We can construct principal fiber bundles
S3κ1,κ2
pi


y
S3κ1,κ2/〈i〉
and
S3κ1,κ2
pi


y
S3κ1,κ2/〈j〉
and
S3κ1,κ2
pi


y
S3κ1,κ2/〈k〉
on the space S3κ1,κ2 of unit quaternions. Here the respective base spaces are the space of events,
the space of space-like geodesics, and the space of time-like geodesics with corresponding Clifford
flows on S3κ1,κ2 given by χi(q) = iq, χj(q) = jq, and χk(q) = kq. The principal connections are
determined by the distribution of horizontal planes spanned by {jq,kq}, {iq,kq}, and {iq, jq}
with corresponding complex structures Cκ2, Cκ1, and Cκ1κ2 for these planes.
5.1 Optional reading on coordinate charts for Σκ1,κ2
Complex lines will intersect S3κ1,κ2 exactly when Az1 + Bz2 = 0 and z1z1 + κ1z2z2 = 1. Let ω
denote 1w . We can cover Σκ1,κ2 with two coordinate charts: Let U1 denote the set of points[
B
A
]
of Σκ1,κ2 where A is not a zero divisor, and U2 the set of points
[
A
B
]
where B is not
a zero divisor. Then the coordinate charts φ1 : U1 → Cκ2 and φ2 : U2 → Cκ2 are given by
14 A.S. McRae
φ1
([
B
A
])
= ω = − z1z2 ∈ Cκ2 and φ2
([
A
B
])
= w = − z2z1 ∈ Cκ2 respectively
8. If S3κ1,κ2
pi
−→ Σκ1,κ2
defines the projection map, then
(φ1 ◦ pi)
−1 (ω) =
{
eiθ
1 + ωj
√
1 + κ1|ω|2
}
and (φ2 ◦ pi)
−1 (w) =
{
eiθ
w + j
√
|w|2 + κ1
}
,
where eiθ is an arbitrary element of S1κ2 . We can then give a product structure to pi
−1 (U1) ⊂
S3κ1,κ2 and to pi
−1 (U2) ⊂ S3κ1,κ2 by
Φ1
(
ω, eiθ
)
= eiθ
1 + ωj
√
1 + κ1|ω|2
and Φ2
(
w, eiθ
)
= eiθ
w + j
√
|w|2 + κ1
respectively. This trivializing cover of S3κ1,κ2 has a gluing map
Φ−12 ◦ Φ1
(
ω, eiθ
)
=
(
1
ω
, eiθ
ω
|ω|
)
.
We can also map Σκ1,κ2 to the unit sphere in R
3 with metric9 dy2 + κ1dt2 + κ1κ2dx2 by
ω = −
z1
z2
7→
(
R
2ω
1 + κ1|ω|2
, I
2ω
1 + κ1|ω|2
,
−1 + κ1|ω|2
1 + κ1|ω|2
)
or
w = −
z2
z1
7→
(
R
2w
1 + κ1|w|2
, I
2w
1 + κ1|w|2
,
−1 + κ1|w|2
1 + κ1|w|2
)
as can be checked directly.
6 The principal connection form
The right invariant one-forms on S3κ1,κ2 are given by
(dU)U−1 =
(
dz1 dz2
−κ1dz2 dz1
)(
z1 −z2
κ1z2 z1
)
=
(
z1dz1 + κ1z2dz2 −z2dz1 + z1dz2
−κ1z1dz2 + κ1z2dz1 κ1z2dz2 + z1dz1
)
and the left invariant one-forms on S3κ1,κ2 are given by
U−1dU =
(
z1 −z2
κ1z2 z1
)(
dz1 dz2
−κ1dz2 dz1
)
=
(
z1dz1 + κ1z2dz2 −z2dz1 + z1dz2
−κ1z1dz2 + κ1z2dz1 κ1z2dz2 + z1dz1
)
.
We will show that the principal connection form is given by the right invariant form λ =
z1dz1 + κ1z2dz2. Let J denote the almost complex structure on R4 that is compatible with
multiplying (z1, z2) by i in C2κ2 . Then
i(z1, z2) = (−κ2y + ix,−κ2v + iu)
 JX = J




x
y
u
v



 =










0 −κ2 0 0
1 0 0 0
0 0 0 −κ2
0 0 1 0














x
y
u
v



 =




−κ2y
x
−κ2v
u



 ,
8Note that the map w 7→ ω = 1w =
w
ww is conformal on the set of non-zero divisors.
9Recall that SOκ1,κ2(3) is the group of isometries of R
3 = {(y, t, x) | y, t, x ∈ R} with metric dy2 + κ1dt2 +
κ1κ2dx2.
Clifford Fibrations and Possible Kinematics 15
where J2 = −κ2I and I is the identity matrix. Recall that the Hermitian inner product on C2κ2
is given by
〈z, w〉 = z1z2 + κ1w1w2
= (x1x2 + κ2y1y2 + κ1u1u2 + κ1κ2v1v2) + i (y1x2 − x1y2 + κ1v1u2 − κ1u1v2) ,
where z = (z1, z2) = (x1+iy1, u1+iv1) and w = (w1, w2) = (x2+iy2, u2+iv2). In real coordinates
we can write this as 〈X1, X2〉 = 〈〈X1, X2〉〉+ iΦ(X1, X2), where 〈〈X1, X2〉〉 and Φ(X1, X2) give
the respective real and imaginary parts of the inner product 〈X1, X2〉, and where
X1 =




x1
y1
u1
v1



 , X2 =




x2
y2
u2
v2



 .
So
〈〈X1, X2〉〉 = X
T
1




1 0 0 0
0 κ2 0 0
0 0 κ1 0
0 0 0 κ1κ2



X2
and
Φ (X1, X2) = −
1
κ2
〈〈JX1, X2〉〉 =
1
κ2
〈〈X1, JX2〉〉
= −
1
κ2
(−κ2y1 x1 − κ2v1 u1)




1 0 0 0
0 κ2 0 0
0 0 κ1 0
0 0 0 κ1κ2








x2
y2
u2
v2




= −
1
κ2
XT1










0 1 0 0
−κ2 0 0 0
0 0 0 1
0 0 −κ2 0














1 0 0 0
0 κ2 0 0
0 0 κ1 0
0 0 0 κ1κ2



X2
= −
1
κ2
XT1

















0 κ2 0 0
−κ2 0 0 0
0 0 0 κ1κ2
0 0 −κ1κ2 0

















X2
= XT1










0 −1 0 0
1 0 0 0
0 0 0 −κ1
0 0 κ1 0










X2.
16 A.S. McRae
If we let this last matrix describe a (possible degenerate) symplectic form $, then 〈〈X1, JX2〉〉 =
κ2$ (X1, X2), so that $ is compatible with J . In the degenerate case when κ2 = 0 and the
above calculations for Φ (X1, X2) make no sense, we can write κ2Φ (X1, X2) = 〈〈JX1, X2〉〉 and
we then rescale10 by canceling out the factor of κ2 on both the left and the right sides of this
equation.
The unit sphere S3κ1,κ2 in R
4 can be described by the equation 〈〈X,X〉〉 = 1. The orbit of
(z1, z2) in S3κ1,κ2 under the S
1
κ2 action that is given by
(z1, z2) 7→ e
iθ (z1, z2) = (Cκ2(θ) + iSκ2(θ)) (x+ iy, u+ iv)
=
[
Cκ2(θ)x− κ2Sκ2(θ)y + i (Cκ2(θ)y + Sκ2(θ)x) ,
Cκ2(θ)u− κ2Sκ2(θ)v + i (Cκ2(θ)v + Sκ2(θ)u)
]
is, in real terms, given by X 7→ XCκ2(θ)+ JXSκ2(θ). Since X and JX are orthogonal
11, a unit
circle is traced out in the plane of R4 that contains both X and JX: Recall that C2(θ) +
κ2S2(θ) = 1, noting that 〈〈JX, JX〉〉 = κ2 as can be calculated directly taking into account the
fact that 〈〈X,X〉〉 = 1. This calculation also shows that all fibers are of the same size.
The vector tangent to the fiber through (z1, z2) is given by i (z1, z2) or, in real terms, by JX.
If X(t) is a differentiable curve lying in S3κ1,κ2 , then |X(t)|
2 = 1 implies that 〈〈X(t), X˙(t)〉〉 = 0.
If this curve is also orthogonal to the fiber passing through X(t), then we must also have that
〈〈JX(t), X˙(t)〉〉 = 0. So λ = zdz = z1dz1 + κ1z2dz2 is the principal connection form as λ is
clearly equivariant.
The curvature form is dλ = dz1∧dz1+κ1dz2∧dz2. Now if X and Y are curves on S3κ1,κ2 so that
X˙ = jq and Y˙ = kq at a point q ∈ S3κ1,κ2 , then dλ
(
X˙, Y˙
)
= 〈X˙, Y˙ 〉 = jqkq = −jqqk = −κ1i.
Also, recall that [j,k] = 2κ1i. And so the curvature is given by κ1 and λ is a contact form
exactly when κ1 6= 0. The area of the infinitesimal rectangle D defined by jq and δkq has
area δ and the holonomy θ around the rectangle is given by δκ1: So the curvature is defined
by θ =
∫
D K dA or δκ1 = δK so that K = κ1. Our definition of area is in lieu of rescaling
metrics: See [6] or [7] for different calculations of the the curvature. In effect we are treating
{iq, jq,kq} as an orthonormal frame.
Finally we will use the principal connection form to derive the metric of the spacetime Σκ1,κ2
(see also [8]): Recall that the metric is to be rescaled by dividing by κ1. We define a horizontal
curve X(t) (so X(t)X˙(t) = 0) passing through q = z1 + z2j in S3κ1,κ2 as follows
X(t) = (Cκ1(t) + Sκ1(t)j) (z1 + z2j) = (Cκ1(t)z1 − κ1Sκ1(t)z2) + (Cκ1(t)z2 + Sκ1(t)z1) j.
Then
X˙(t) = (−κ1Sκ1(t)z1 − κ1Cκ1(t)z2) + (−κ1Sκ1(t)z2 + Cκ1(t)z1) j,
X˙(0) = −κ1z2 + z1j,
∣
∣X(0)
∣
∣2 = κ1
2 |z2|
2 + κ1 |z1|
2 = 1.
Now X(t) is the horizontal lift of the curve w(t) in Σκ1,κ2 where
w(t) =
Cκ1(t)z2 + Sκ1(t)z1
Cκ1(t)z1 − κ1Sκ1(t)z2
,
w(0) =
z2
z1
= w,
10Similarly we rescaled by dividing by κ1 or κ2, even if they were equal to zero, in order to obtain the metrics
on the homogeneous spaces S3κ1,κ2/〈i〉, S
3
κ1,κ2/〈j〉, and S
3
κ1,κ2/〈k〉.
11Neither X nor JX is the zero vector, and we also have that 〈〈X, JX〉〉 = R (〈X, JX〉) = R(−i) = 0.
Clifford Fibrations and Possible Kinematics 17
w˙(t) = (−κ1Sκ1(t)z2 + Cκ1(t)z1) (Cκ1(t)z1 − κ1Sκ1(t)z2)− (Cκ1(t)z2 + Sκ1(t)z1)
× (−κ1Sκ1(t)z1 − κ1Cκ1(t)z2) (Cκ1(t)z1 − κ1Sκ1(t)z2)
−2 ,
w˙(0) =
z1z1 + κ1z2z2
z21
=
1
z21
.
Since
|z1|
2
(
1 + κ1 |w|
2
)
=
(
1 +
κ1 |z2|
2
|z1|
2
)
|z1|
2 = |z1|
2 + κ1 |z2|
2 = 1,
then |z1|
2 = 1
(1+κ1|w|2)
. As |jq|2 = κ1 and |kq|
2 = κ1κ2, the metric 1κ1ds
2 induced on the base
space is given by
z21z1
2dwdw =
dw dw
(1 + κ1|w|2)
2 .
Theorem 4. Let H  2j, P  2k, and K  2i denote the respective generators for time
translations, space translations, and boosts of the kinematical algebra with commutators
[K,H] = P, [K,P ] = −κ2H, and [H,P ] = κ1K.
Then the principal fiber bundle
S3κ1,κ2
pi


y
S3κ1,κ2/〈i〉
has λ = z1dz1 + κ1z2dz2 as its principal connection form. The base space, which is the space of
events, has induced metric
ds2 =
dw dw
(1 + κ1|w|2)
2
and constant curvature κ1.
In conclusion, it is hoped that the aims of this paper were met, that the nice structure of
the Hopf fibration S3 −→ S2 was generalized in an appealing way, not only for the classical
Riemannian surfaces of constant curvature, but especially for the study of (1 + 1) kinematics.
It is also hoped that the reader will find that these fibrations give a new perspective on these
simple kinematical structures. Finally, I wish to thank the reviewers for their many helpful
suggestions on how this paper could be improved.
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