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Home , Biquaternion, Lorentz transformation, Ludwik Silberstein, Rapidity, Special relativity, Thomas precession

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1 Introduction 2

2 Quaternions 3 2.1 Ordinary classical quaternions ...... 3 2.2 Complexifiedquaternions...... 4 2.3 Self-adjoint complexified quaternions ...... 4

3 Lorentz transformations 5 3.1 Generalform ...... 5 3.2 Rotations ...... 6 3.3 Factorization — quaternionic polar decomposition ...... 6

4 Lorentz boosts 7 4.1 4-velocity, and square root of 4-velocity ...... 7 4.2 Lorentz boosts in terms of relative 4-velocity ...... 8 4.3 Combination of 4-velocities ...... 9

5 11

6 Non-associativity of the combination of velocities 12

7 BCH approach to the combination of 4-velocities 14

8 Summary 16

–1– 1 Introduction

The use of Hamilton’s quaternions [1–5] as applied to has a very long, complicated, and rather fraught history — largely due to a significant of rather sub-optimal notational choices being made in the early literature [6–8], which was then compounded by the introduction of multiple mutually disjoint ways of representing the Lorentz transformations [6–9]. Subsequent developments have, if anything, even further confused the situation [10–13]. See also references [14–16]. One reason for being particularly interested in these issues is due to various attempts to simplify the discussion of the interplay between the Thomas rotation [17–22], the relativistic composition of 3-velocities [23–27], and the very closely related Wigner angle [28–31]. In an earlier article [31] we considered ordinary quaternions and found 2w that it was useful to work with the the relativistic half velocities w, defined by v = 1+w2 v v 3 so w = 2 = + (v ). (As usual, we the of to be unity, 1+√1 v 2 O c 1.) In the− current article we will re-phrase things in terms of self-adjoint complex → quaternionic 4-velocities, arguing for a number of simple compact formulae relating Lorentz transformations and the Wigner angle. A second reason for being interested in quaternions is purely a matter of computational efficiency. Quaternions are often used to deal with 3-dimensional spatial rotations, and while mathematically the use of quaternions is completely equivalent to working with the usual 3 3 orthogonal matrices, that is SO(3), at a computational level the use of × quaternions implies significant savings in storage and significant gains in computational efficiency. A third reason for being interested in quaternions is purely as a matter of pedagogy. Quaternions give one a different viewpoint on the usual physics of special relativity, and in particular the Lorentz transformations. Using quaternions leads to novel simple results for boosts (they are represented by the square-root of the relative 4-velocity) and simple novel results for the Wigner angle. At a deeper level, we shall formally connect composition of 4-velocities to a symmetric version of the Baker–Campbell–Hausdorff theorem [32–36]. Unfortunately, while cer- tainly elegant, most results based on the Baker–Campbell–Hausdorff expansion seem to not always be computationally useful.

–2– 2 Quaternions

It is useful to consider 3 distinct classes of quaternions [1–11, 31]: • Ordinary classical quaternions; • Complexified quaternions; • Self-adjoint complexified quaternions. While the discussion in reference [31] focussed on the ordinary classical quaternions, and so was implicitly a ()+() formalism, in the current article we will focus on the self-adjoint complexified quaternions in order to develop an integrated space-time formalism. To set the framework, consider the discussion below.

2.1 Ordinary classical quaternions

Ordinary classical quaternions are that can be written in the form [1–5, 31]:

q = a + b i + c j + d k (2.1) where a, b, c, and d are real numbers a,b,c,d R, and i, j, and k are the quaternion { } ∈ units which satisfy the famous relation

i2 = j2 = k2 = ijk = 1. (2.2) − The quaternions form a four–dimensional number system, commonly denoted H in honour of Hamilton, that is generally treated as an extension of the complex numbers C. We define the quaternion conjugate of a quaternion q = a + bi + cj + dk to be q⋆ = a bi cj dk, and the norm of q to be − − − qq⋆ = q 2 = a2 + b2 + c2 + d2 R. (2.3) | | ∈ Let us temporarily focus our attention on pure quaternions. That is, quaternions of the form a i + b j + c k = (a, b, c) (i, j, k). In this instance, the product of two pure · quaternions p and q is given by pq = ~p ~q +(~p × ~q) (i, j, k), where, in general, − · · v = ~v (i, j, k). This yields the useful relations · [p, q]=2(~p × ~q) (i, j, k), and p, q = 2 ~p ~q. (2.4) · { } − · A notable consequence of this formalism is that q2 = ~q ~q = q 2. − · −| |

–3– 2.2 Complexified quaternions

In counterpoint, the complexified quaternions are numbers that can be written in the form Q = a + b i + c j + d k, (2.5) where a, b, c, and d are now complex numbers a,b,c,d C. It is important to { } ∈ note at this stage that, although it is quite common to embed the complex numbers into the quaternions by identifying the complex i with the quaternion unit i, it is now essential that we distinguish between i and i when dealing with the complexified quaternions C H. As well as the previously defined ⋆ operation, there are now two ⊗ additional conjugates we can perform on the complexified quaternions: In addition to the quaternion conjugate q⋆ = a b i c j d k we can define the ordinary complex − − − conjugate q =a ¯ + ¯b i +c ¯j + d¯k, and a third type of (adjoint) conjugate given by ⋆ q† =(q) . Note that this now leads to potentially three distinct notions of “norm”:

q 2 = q⋆q = a2 + b2 + c2 + d2 C; qq¯ =¯aa ¯bb cc¯ dd¯ R; (2.6) | | ∈ − − − ∈ and + q†q =¯aa + ¯bb +¯cc + dd¯ R . (2.7) ∈ These complexified quaternions are commonly called “” in the literature. Unfortunately, as the word “” has at least two other different possible meanings, we will simply call these quantities the complexified quaternions.

2.3 Self-adjoint complexified quaternions

One of the fundamental issues with trying to reformulate special relativity in terms of quaternions is that, although both space–time and quaternions are intrinsically four– dimensional, the norm of an ordinary quaternion q = a0 + a1 i + a2 j + a3 k is given by q 2 = a2 + a2 + a2 + a2, whereas in contrast the Lorentz norm of a | | 0 1 2 3 4-vector Aµ is given by A 2 =(A0)2 (A1)2 (A2)2 (A3)2. In order to address this || || − − − fundamental issue, we consider self-adjoint complexified quaternions satisfying q = q†. That is, we consider complexified quaternions with with real part and imaginary vectorial part:

q = a0 + ia1 i + ia2 j + ia3 k = a0 + i(a1 i + a2 j + a3 k). (2.8) Here i C is the usual complex unit and a , a , a , a R. Self-adjoint quaternions ∈ 0 1 2 3 ∈ have norm

q 2 = q⋆q =(a i(a i + a j + a k))(a + i(a i + a j + a k)) = a2 a2 a2 a2. (2.9) | | 0 − 1 2 3 0 1 2 3 0 − 1 − 2 − 3

–4– Thence from the quaternionic point of view the most natural signature choice is the (+ ) “mostly negative” convention. This norm is real, but need not be positive — −−− and is physically and mathematically appropriate for describing the Lorentz invariant norm of a spacetime 4-vector in a quaternionic framework.

Indeed, writing q1 = a0 + i(a1i + a2j + a3k) and q2 = b0 + i(b1i + b2j + b3k), for the Lorentz invariant inner product of two 4-vectors we can write 1 η(q , q )= (q⋆q + q⋆q )= a b a b a b a b R. (2.10) 1 2 2 1 2 2 1 0 0 − 1 1 − 2 2 − 3 3 ∈

3 Lorentz transformations

3.1 General form

Quaternionic Lorentz transformations are now characterized by two key features: • They must be linear mappings from self-adjoint 4-vectors to self-adjoint 4-vectors. • They must preserve the Lorentz invariant inner product. The first condition suggests, (taking L to be a complexified quaternion), looking at the linear mapping q L q L†. (3.1) → (Because this transformation will preserve the self-adjointness of q.) The second condition then requires

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ q q + q q = L † q L Lq L† + L † q L L q L† 1 2 2 1 { 1 }{ 2 } { 2 }{ 1 } ⋆ ⋆ ⋆ ⋆ ⋆ = L † (q L L q + q L L q ) L† (3.2) 1 { } 2 2{ } 1 ⋆ ⋆ 1 Now if L L = 1, that is L = L− , this simplifies to

⋆ ⋆ ⋆ ⋆ ⋆ q1q2 + q2q1 = L †(q1 q2 + q2q1) L† (3.3) But then noting that (q⋆q + q⋆q ) R we have 1 2 2 1 ∈ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ q1q2 + q2q1 = (L †L†)(q1 q2 + q2q1)=(LL )† (q1 q2 + q2q1)= q1q2 + q2q1. (3.4)

So a necessary and sufficient condition for the quaternionic mapping q L q L† to → preserve the quaternionic form of the Lorentz invariant inner product is

⋆ ⋆ 1 L L = 1; that is L = L− . (3.5)

Note that this condition implies that the set of quaternionic Lorentz transformations forms a under quaternion multiplication.

–5– 3.2 Rotations

The rotations form a well-known of the , and in quaternionic form a rotation about the ˆn axis can be represented by

R = exp(θˆn) = cos θ + ˆn sin θ. (3.6)

This observation goes back to the days of Hamilton, and the fact that 3-dimensional rotations can be represented in this quite straightforward manner is one of the reasons so much effort was put into development of the quaternion formalism. From the point of view of the (ordinary) quaternions (not complexified in this case) one has

1 ⋆ R− = exp( θˆn) = cos θ ˆn sin θ = R† = R . (3.7) − − 1 ⋆ Indeed the characterization R− = R† = R is both necessary and sufficient for a quaternion to represent a rotation.

3.3 Factorization — quaternionic polar decomposition

Let us now see how to factorize a general into a boost a rotation. (This is effectively a quaternionic form of the notion of “polar decomposi- tion” that one usually encounters in algebra; we make the discussion somewhat pedestrian in the interests of pedagogical clarity.) Without any loss of generalization we may always write: 1/2 L = √LL† (LL†)− L . (3.8)  (To do this we just need to know that the product of 2 complexified quaternions is again a complexified quaternion, and that the multiplication of complexified quaternions is associative.)

1/2 Now LL† is self adjoint — so √LL† is self adjoint, and in turn (LL†)− is self adjoint. Consequently

1/2 1/2 1/2 1/2 (LL†)− L (LL†)− L † = (LL†)− L L†(LL†)− =1. (3.9)     Thence 1/2 1 1/2 (LL†)− L − = (LL†)− L † . (3.10)   Furthermore

⋆ ⋆ ⋆ ⋆ ⋆ 1 1 1 1 1 (LL†) =(L†) L =(L )†L =(L− )†L− =(L†)− L− =(LL†)− . (3.11)

–6– That is, (LL†) is a Lorentz transformation, (in fact a self adjoint Lorentz transforma- tion), and consequently √LL† is also a (self adjoint) Lorentz transformation. But then 1/2 by the group property (LL†)− L must also be a Lorentz transformation, so  1/2 1 1/2 (LL†)− L − = (LL†)− L ∗ . (3.12)   1/2 But this now implies that R = (LL†)− L must be a rotation, and we have shown that in general 

L = √LL† R. (3.13) Indeed in the next section we shall show that the self-adjoint Lorentz transformation

B = √L†L is actually a boost and that in general one has

L = BR. (3.14)

4 Lorentz boosts

We now show how quaternions can be used to obtain a pure Lorentz transformation (a boost) from the square root of the relative 4-velocity connecting the two inertial frames. In order to proceed, we must first obtain an explicit expression for the square root of a four–velocity V .

4.1 4-velocity, and square root of 4-velocity

We represent a 4-vector X~ =(t, x, y, z)=(t, ~x) by the self-adjoint quaternion

X = t + i(xi + yj + zk). (4.1)

Differentiating with respect to the gives a quaternionic notion of 4-velocity

V = γ(1 + iv ˆn); with V 2 = V⋆V =1. (4.2) | | To explicitly find the square root we first an elementary discussion: Let us 1 introduce the notion of in the usual manner by setting ξ = tanh− v. Then 4-velocities can be written in the form

V = γ (1 + ivˆn) = cosh ξ + i sinh ξ ˆn = eiξˆn. (4.3)

The square root of the 4-velocity is then easily seen to be √V = eiξˆn/2.

–7– Explicitly, using hyperbolic half-angle formulae, from γ = cosh ξ and v = tanh ξ one gets γ +1 γ 1 cosh(ξ/2) = , and sinh(ξ/2) = − . (4.4) r 2 r 2 Thence γ +1 γ 1 √V = + iˆn − . (4.5) r 2 r 2 2w In terms of the relativistic half velocity, implicitly defined by v = 1+w2 , so that one has v w = 2 , it is easy to check that 1+√1 v − γ 1 γ +1 w = − ; and γ = . (4.6) rγ +1 w r 2

So we can write

√V = γw (1 + iˆn w) . (4.7) There are many other ways of getting to the same result. The current discussion has been designed to be as straightforward and explicit as possible.

4.2 Lorentz boosts in terms of relative 4-velocity

Now that we have an expression for the square root of a four-velocity V, we can show how a pure Lorentz transformation is obtained from quaternion conjugation by √V. Without any loss of generality, we define V = γ(1+ iiv), which is the four–velocity for an object travelling with speed v in thex ˆ direction, and represent the four–vector X~ = (t, x, y, z) by the self-adjoint quaternion X =(t+iix+ijy +ikz)=(t+i(ix+jy +kz)). Now consider the transformation of X given by X √VX √V. That is, 7→ X √V(t + iix + ijy + ikz)√V 7→ = √V(t + iix)√V + √V(ijy + ikz)√V (4.8) =(t + iix)V +(ijy + ikz)√V⋆√V, where in the last equality we have used the fact that √V commutes with i, and the anti-commutativity of i with j and k to write √V j = j√V⋆, and √V k = k√V⋆. Explicit calculation of √V⋆√V yields

√V⋆√V = exp( iξi/2) exp(iξi/2)=1. (4.9) −

–8– That is, √V is a unit quaternion. Thus

√VX √V =(t + iix)V +(ijy + ikz). (4.10)

Using our expression for V we find (t+iix)V = γ (t + vx)+ ii(x + vt) , giving a final { } result of

X =(t + iix + ijy + ikz) γ (t + vx)+ ii(x + vt) +(ijy + ikz), (4.11) 7→ { } which are the well–known inverse Lorentz transformations (boosts in the ( xˆ) direc- − tion). Although, for the purpose of simplifying the calculations, we have defined our i axis to lie in the direction of the boost, it should be clear that this argument is in fact completely general, due to the general definition of the four-vector X~ . That is, a boost (pure Lorentz transformation without rotation) corresponds to

X √VX √V = eiξˆn/2 X eiξˆn/2. (4.12) →

4.3 Combination of 4-velocities

Starting in the of some object, where V0 = 1, we successively apply two boosts V1 = γ1(1 + iˆn1v1) and V2 = γ2(1 + iˆn2v2) in directions ˆn1 and ˆn2 with velocities v1 and v2, respectively. The result of this is to shift our rest frame to a frame moving with 4-velocity V1 2, which is equivalent to relativistically combining ⊕ the two 4-velocities V1 and V2. This method has the added benefit that it obtains an expression for the –factor of the frame, γ1 2, and hence its speed, without ⊕ having to take the norm of V1 2, thereby avoiding lots of tedious algebra. ⊕

We begin by boosting our rest frame starting with with V0 = 1 as in subsection 4.2:

V1 2 = V2 V1 V0 V1 V2 = V2 V1 V2. (4.13) ⊕ p p p p p p Similarly

V2 1 = V1 V2 V0 V2 V1 = V1 V2 V1. (4.14) ⊕ p p p p p p That is, the relativistic combination of 4-velocities simply amounts to

V1 2 = V2 V1 V2; V2 1 = V1 V2 V1. (4.15) ⊕ p p ⊕ p p This makes it obvious that V1 2 = V2 1 unless [V1, V2] = 0, which in turn requires ⊕ 6 ⊕ the 3-velocities ~v1 and ~v2 to be parallel. In the special case where the 4-velocities do commute we have

V1 2 = V1V2 = V2V1 = V2 1. (4.16) ⊕ ⊕

–9– In terms of the general case is

iξ2 ˆn2/2 iξ1 ˆn1 iξ2 ˆn2/2 iξ1 ˆn1/2 iξ2 ˆn2 iξ1 ˆn1/2 V1 2 = e e e ; V2 1 = e e e . (4.17) ⊕ ⊕ Viewed in this way the relativistic combination of 4-velocities can be interpreted as an application of the symmetrized version of the Baker–Campbell–Hausdorff (BCH) expansion [32–36]. Indeed, taking logarithms:

iξ2 ˆn2/2 iξ1 ˆn1 iξ2 ˆn2/2 ξ1 2 ˆn1 2 = i ln e e e ; (4.18) ⊕ ⊕ −  iξ1 ˆn1/2 iξ2 ˆn2 iξ1 ˆn1/2 ξ2 1 ˆn2 1 = i ln e e e . (4.19) ⊕ ⊕ − Unfortunately this formal result, while quite elegant, is not really computationally effective. One could for instance fully expand the expression in (4.17) above and isolate the real part to deduce

1 ξ1 2 = ξ2 1 = cosh− (cosh ξ1 cosh ξ2 + sinh ξ1 sinh ξ2 cos θ) , (4.20) ⊕ ⊕ where θ is the angle between ˆn1 and ˆn2. As expected, for collinear 3-velocities this reduces to

ξ1 2 = ξ2 1 ξ1 ξ2 . (4.21) ⊕ ⊕ →| ± | To check this for consistency, note

1 ⋆ ⋆ 1 ⋆ γ1 2 = η(V0, V1 2)= (V0V1 2 + V1 2V0)= (V1 2 + V1 2). (4.22) ⊕ ⊕ 2 ⊕ ⊕ 2 ⊕ ⊕ Thence 1 ⋆ ⋆ ⋆ γ1 2 = V2 V1 V2 + V2 V1 V2 . (4.23) ⊕ 2 p p  1 ⋆ p p But γ1 2 is real, and √V2 − = √V2 so ⊕ ⋆   1 ⋆ ⋆ 1 ⋆ γ1 2 = V2 γ1 2 V2 = (V1V2 + V2 V1)= (V1V2 +(V1 V2) ) . (4.24) ⊕ p  ⊕ p 2 2 It is then easy (indeed almost trivial) to see that

γ1 2 = γ1γ2(1 + ~v1 ~v2)= γ2 1. (4.25) ⊕ · ⊕

While this very easily yields the magnitude of the combined 3-velocities ~v1 2 = ~v2 1 , | ⊕ | | ⊕ | isolating the direction of the combined 3-velocities is much more subtle. Notev ˆ1 2 = ⊕ 6 vˆ2 1 in general, see equations (4.18)–(4.19). ⊕ The rapidity formalism is also particularly useful for quickly double-checking formal relationships such as

⋆ 1 ⋆ iξˆn/2 1 − √V = e− = √V = √V− = √V . (4.26)    

– 10 – We can also use this formalism to write a general Lorentz transformation in the form

L = BR = eiξˆn/2 eθ ˆm/2. (4.27)

Here ˆn is the direction of the boost B, while ˆm is the axis of the rotation R.

5 Wigner rotation

We shall now derive an explicit quaternionic formula for the Wigner rotation. For relevant background see references [28–31]. Note that for 4-velocities

1 1 ⋆ 1 − V = V†; V− = V ; (V)− = √V , (5.1) p   while for rotations ⋆ 1 ⋆ R† = R ; R− = R† = R . (5.2)

Now note

V1 2 = V2 V1 V2 = V2 V1 V1 V2 (5.3) ⊕ p p p p p p Let us define a quaternion R by taking

V1 2 R = V2 V1, (5.4) ⊕ p p p and then checking to see that this quaternion does in fact a correspond to a rotation. First 1 1 R = V1− 2 V2 V1; R† = V1 V2 V1− 2. (5.5) q ⊕ p p p p q ⊕ Now

1 1 RR† = V1− 2 V2 V1 V1 V2 V1− 2 (5.6) q ⊕ p p p p q ⊕ 1 1 = V1− 2 V2V1 V2 V1− 2 (5.7) q ⊕ p p q ⊕ 1 1 = V1− 2V1 2 V1− 2 (5.8) q ⊕ ⊕ q ⊕ = 1. (5.9)

1 That is, R− = R†. Now consider

⋆ R = V1∗ V2∗ V1 2, (5.10) ⊕ p p p and compare it to 1 R† = V1 V2 V1− 2, (5.11) p p q ⊕

– 11 – ⋆ Actually R = R† though this might not at first be obvious. Calculate

⋆ R V1 2 = V1∗ V2∗ V1 2 V1 2 = V1∗ V2∗V1 2. (5.12) ⊕ ⊕ ⊕ ⊕ p p p p p p p Thence ⋆ 1 1 R V1 2 = V1− V2− V2 V1 V2 = V1 V2. (5.13) ⊕ p q q p p  p p Thence ⋆ R V1 2 = V1 V2 = R† V1 2. (5.14) ⊕ ⊕ ⋆ p p p p So R = R† as claimed. Accordingly R is indeed a well-defined rotation. Explicitly we have

R V 1 V V V 1 V 1 V 1 V V = 1− 2 2 1 = r 2− 1− 2− 2 1. (5.15) q ⊕ p p q q p p Note that from equation (5.14) we have

R† V1 2 = V1 V2, (5.16) ⊕ p p p and so, along with the defining relation equation (5.4), we deduce

V2 1 = V1 V2 V1 = V1 V2 V2 V1 = R† V1 2 V1 2 R. (5.17) ⊕ ⊕ ⊕ p p p p p p p p That is

V2 1 = R† V1 2 R. (5.18) ⊕ ⊕ So R is indeed the Wigner rotation as claimed.

6 Non-associativity of the combination of velocities

From the above we note that

V(1 2) 3 = V3 V2 V1 V2 V3. (6.1) ⊕ ⊕ p p p p whereas

V1 (2 3) = V3V2 V3 V1 V3V2 V3. (6.2) ⊕ ⊕ qp p qp p This explicitly verifies the general non-associativity of composition of 4-velocities, and furthermore demonstrates why left-composition is much nicer than right-composition. There has in the past been some confusion in this regard [23–25]. See also the recent

– 12 – discussion in reference [31], where an equivalent discussion was presented in terms of quaternionic 3-velocities. From the above

V V V V 1V 1 V 1 V V V V (1 2) 3 = 3 2 r 3− 2− 3− 3 2 3 1 ⊕ ⊕ p p q q qp p V V V V 1V 1 V 1 V V 3 2 3r 3− 2− − 3 2 3. (6.3) ×qp p q p p p That is

1 1 1 1 1 1 V(1 2) 3 = V3 V2 V3− V2− V3− V1 (2 3) V3− V2− V− 3 V2 V3. ⊕ ⊕ rq q ⊕ ⊕ rq p p p p p(6.4) But from our formula for the Wigner rotation

R V 1 V V V 1 V 1 V 1 V V 1 2 = 1− 2 2 1 = r 2− 1− 2− 2 1, (6.5) ⊕ q ⊕ p p q q p p this now implies

V(1 2) 3 = R2† 3V1 (2 3)R2 3. (6.6) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ So the Wigner rotation is not just relevant for understanding generic non-commutativity when composing two boosts, is also relevant to understanding generic non-associativity when composing three boosts. Suppose now we consider a specific situation where the composition of 4-velocities is associative, that is, we assume:

V(1 2) 3 = V1 (2 3). (6.7) ⊕ ⊕ ⊕ ⊕ Under this condition we would now have

V3 V2 V1 V2 V3 = V3V2 V3 V1 V3V2 V3, (6.8) p p p p qp p qp p whence we would need

V 1V 1 V 1 V V V V V V 1V 1 V 1 V r 3− 2− 3− 3 2 1 2 3r 3− 2− 3− = 1. (6.9) q q p p p p q q We can rewrite this condition in terms of the Wigner rotation as

R2 3 V1 R2† 3 = V1. (6.10) ⊕ ⊕

– 13 – That is 1 R2 3 V1 R2− 3 = V1, (6.11) ⊕ ⊕ whence

[R2 3, V1]=0. (6.12) ⊕ Thence the combination of velocities is associative V(1 2) 3 = V1 (2 3) if and only if ⊕ ⊕ ⊕ ⊕ the boost direction in V1 is parallel to the rotation axis in R2 3. But this holds if and ⊕ only if

[v1, [v2, v3]] = 0 (6.13) or in more prosaic language, if and only if

~v (~v ~v )=0. (6.14) 1 × 2 × 3 (We had almost derived this result in reference [31], but only as a sufficient condition, we never quite got to establishing this as a necessary and sufficient condition.)

7 BCH approach to the combination of 4-velocities

Let us now consider yet another way of understanding combination of velocities, this time in terms of the (symmetrized) BCH theorem. We have already seen that ˆ ˆ ˆ V1 2 = exp iξ2ξ2/2 exp iξ1ξ1 exp iξ2ξ2/2 . (7.1) ⊕       Now differentiate ∂V1 2 i ˆ ˆ ˆ ˆ ⊕ = ξ2, exp iξ2ξ2/2 exp iξ1ξ1 exp iξ2ξ2/2 , (7.2) ∂ξ2 2 n      o and rewrite this as

∂V1 2 i ˆ ˆ ˆ ˆ ⊕ = exp iξ2ξ2/2 ξ2, exp iξ1ξ1 exp iξ2ξ2/2 , (7.3) ∂ξ2 2   n  o  

But we note that ˆ ˆ ˆ ˆ ξ2, exp iξ1ξ1 = ξ2, cosh(ξ1)+ i sinh(ξ1)ξ1 (7.4) n  o n  o ˆ ˆ ˆ = 2 cosh(ξ1)ξ2 + i sinh(ξ1) ξ2, ξ1 . (7.5)  { } And, since ξˆ , ξˆ R, this implies { 2 1} ∈ ˆ ˆ ˆ ξ2, ξ2, exp iξ1ξ1 =0. (7.6) h n  oi

– 14 – ˆ Consequently we can pull the factor exp iξ2ξ2/2 through the anti-, and rewrite the derivative as  

∂V1 2 i ˆ ˆ ˆ ⊕ = ξ2, exp iξ1ξ1 exp iξ2ξ2 . (7.7) ∂ξ2 2 n  o  

Now integrate with respect to ξ2. We see

ξ2 i ˆ ˆ ˆ V1 2 = V1 + ξ2, exp iξ1ξ1 exp iξ2ξ2 dξ2. (7.8) ⊕ 2 Z0 n  o  

Pull the constant (with respect to ξ2) anti-commutator outside the integral

ξ2 i ˆ ˆ ˆ V1 2 = V1 + ξ2, exp iξ1ξ1 exp iξ2ξ2 dξ2. (7.9) ⊕ 2 n  o Z0   Perform the integral ˆ exp iξ2ξ2 1 i ˆ ˆ − V1 2 = V1 + ξ2, exp iξ1ξ1   . (7.10) ⊕ ˆ 2 n  o iξ2 Noting that (iξˆ)2 = 1 this simplifies to

1 ˆ ˆ ˆ ˆ V1 2 = V1 ξ2, exp iξ1ξ1 ξ2 exp iξ2ξ2 1 . (7.11) ⊕ − 2 n  o    −  That is 1 ˆ ˆ V1 2 = V1 ξ2, V1 ξ2 (V2 1) . (7.12) ⊕ − 2{ } − A more tractable result is this 1 ˆ ˆ V1 2 = V1 ξ2V1ξ2 V1 (V2 1) . (7.13) ⊕ − 2  −  − Thence 1 ˆ ˆ V1 2 = V1 + V1 ξ2V1ξ2 (V2 1) . (7.14) ⊕ 2  −  − Rearranging 1 ˆ ˆ V1 2 = V1V2 V1 + ξ2V1ξ2 (V2 1) . (7.15) ⊕ − 2   − This gives us the composition of 4-velocities V1 2 algebraically in terms of V1 and ⊕ V2 and at worst some quaternion multiplication (the need to evaluate √V2 has been side-stepped). ˆ Note that this has the right limit for parallel 3-velocities. When [V1, ξ2] = 0 we see

V1 2 V1V2, (7.16) ⊕ → as it should.

– 15 – For perpendicular 3-velocities

1 ˆ ˆ V1 2 = V1 ξ2, V1 ξ2 (V2 1) , (7.17) ⊕ − 2{ } − reduces to

V1 2 V1 + γ1 (V2 1) . (7.18) ⊕ → − Thence

V1 2 γ1(1 + v1)+ γ1γ2 (1 + v2) γ1, (7.19) ⊕ → − implying 2 V1 2 γ1γ2 1+ 1 v2 v1 + v2 . (7.20) ⊕ →  q −  That is 2 v1 2 = 1 v2 v1 + v2, (7.21) ⊕ q − and 2 2 2 2 2 v1 2 = v1 + v2 v1v2, (7.22) | ⊕ | − exactly as expected for perpendicular 3-velocities.

8 Summary

The method of complexified quaternions allows us to prove several nice results:

• General Lorentz transformations can be factorized into a boost times a rotation:

L = BR = eiξˆn/2 eθ ˆm/2. (8.1)

Here ˆn is the direction of the boost B, while ˆm is the axis of the rotation R.

• Conjugation by the square root of a four velocity implements a Lorentz boost:

X √VX √V. (8.2) → • The relativistic combination of 4-velocities has the simple algebraic form:

V1 2 = V2 V1 V2; V2 1 = V1 V2 V1. (8.3) ⊕ p p ⊕ p p • The Wigner rotation is given by:

R V 1 V V V 1 V 1 V 1 V V = 1− 2 2 1 = r 2− 1− 2− 2 1. (8.4) q ⊕ p p q q p p

– 16 – • The Wigner rotation satisfies, in terms of the generic non-commutativity of two boosts,

V2 1 = R† V1 2 R; (8.5) ⊕ ⊕ and, in terms of the generic non-associativity of three boosts,

V(1 2) 3 = R2† 3 V1 (2 3) R2 3. (8.6) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

Overall, some rather complicated involving 4 4 matrices has been × reduced to relatively simple algebra in the complexified quaternions C H. ⊗

Acknowledgments

TB was supported by a Victoria University of Wellington MSc scholarship, and was also indirectly supported by the Marsden Fund, via a grant administered by the Royal Society of New Zealand. MV was directly supported by the Marsden Fund, via a grant administered by the Royal Society of New Zealand.

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