Lorentz boosts and Wigner rotations: self-adjoint complexified quaternions
aa r X i v : . [ g r- q c ] J a n Lorentz boosts and Wigner rotations:self-adjoint complexified quaternions
Thomas Berry ID and Matt Visser ID School of Mathematics and Statistics, Victoria University of Wellington,PO Box 600, Wellington 6140, New Zealand
E-mail: [email protected]; [email protected]
Abstract:
Herein we shall consider Lorentz boosts and Wigner rotations from a (complexified)quaternionic point of view. We shall demonstrate that for a suitably defined self-adjointcomplex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of thequaternion square root of the relative 4-velocity connecting the two inertial frames.Straightforward computations then lead to quite explicit and relatively simple algebraicformulae for the composition of 4-velocities and the Wigner angle. We subsequentlyrelate the Wigner rotation to the generic non-associativity of the composition of three4-velocities, and develop a necessary and sufficient condition for associativity to hold.Finally, we relate the composition of 4-velocities to a specific implementation of theBaker–Campbell–Hausdorff theorem. As compared to ordinary 4 × Date:
15 January 2021; L A TEX-ed January 18, 2021
Keywords: special relativity; Lorentz boosts; combination of velocities; Wigner angle; quaternions.
PhySH: general physics; special relativity. ontents – 1 –
Introduction
The use of Hamilton’s quaternions [1–5] as applied to special relativity has a very long,complicated, and rather fraught history — largely due to a significant number of rathersub-optimal notational choices being made in the early literature [6–8], which was thencompounded by the introduction of multiple mutually disjoint ways of representingthe Lorentz transformations [6–9]. Subsequent developments have, if anything, evenfurther confused the situation [10–13]. See also references [14–16].One reason for being particularly interested in these issues is due to various attemptsto simplify the discussion of the interplay between the Thomas rotation [17–22], therelativistic composition of 3-velocities [23–27], and the very closely related Wignerangle [28–31]. In an earlier article [31] we considered ordinary quaternions and foundthat it was useful to work with the the relativistic half velocities w , defined by v = w w so w = v √ − v = v + O ( v ). (As usual, we set the speed of light to be unity, c → × SO (3), at a computational level the use ofquaternions implies significant savings in storage and significant gains in computationalefficiency.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 simpleresults 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 symmetricversion of the Baker–Campbell–Hausdorff theorem [32–36]. Unfortunately, while cer-tainly elegant, most results based on the Baker–Campbell–Hausdorff expansion seemto not always be computationally useful.– 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 (space)+(time) formalism, in the current article we will focus onthe self-adjoint complexified quaternions in order to develop an integrated space-timeformalism. To set the framework, consider the discussion below.
Ordinary classical quaternions are numbers 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 quaternionunits which satisfy the famous relation i = j = k = ijk = − . (2.2)The quaternions form a four–dimensional number system, commonly denoted H inhonour of Hamilton, that is generally treated as an extension of the complex numbers C . We define the quaternion conjugate of a quaternion q = a + b i + c j + d k to be q ⋆ = a − b i − c j − d k , and the norm of q to be qq ⋆ = | q | = a + b + c + d ∈ R . (2.3)Let us temporarily focus our attention on pure quaternions . That is, quaternions ofthe form a i + b j + c k = ( a, b, c ) · ( i , j , k ). In this instance, the product of two purequaternions 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 } = − ~p · ~q. (2.4)A notable consequence of this formalism is that q = − ~q · ~q = −| q | .– 3 – .2 Complexified quaternions In counterpoint, the complexified quaternions are numbers that can be written in theform 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 tonote at this stage that, although it is quite common to embed the complex numbersinto the quaternions by identifying the complex unit i with the quaternion unit i , it isnow essential that we distinguish between i and i when dealing with the complexifiedquaternions C ⊗ H . As well as the previously defined ⋆ operation, there are now two additional conjugates we can perform on the complexified quaternions: In addition tothe quaternion conjugate q ⋆ = a − b i − c j − d k we can define the ordinary complexconjugate 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 | = q ⋆ q = a + b + c + d ∈ 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 “biquaternions” in the literature.Unfortunately, as the word “biquaternion” has at least two other different possiblemeanings, we will simply call these quantities the complexified quaternions. One of the fundamental issues with trying to reformulate special relativity in terms ofquaternions is that, although both space–time and quaternions are intrinsically four–dimensional, the norm of an ordinary quaternion q = a + a i + a j + a k is given by | q | = a + a + a + a , whereas in contrast the Lorentz invariant norm of a spacetime4-vector A µ is given by || A || = ( A ) − ( A ) − ( A ) − ( A ) . In order to address thisfundamental issue, we consider self-adjoint complexified quaternions satisfying q = q † .That is, we consider complexified quaternions with with real scalar part and imaginaryvectorial part: q = a + ia i + ia j + ia k = a + i ( a i + a j + a k ) . (2.8)Here i ∈ C is the usual complex unit and a , a , a , a ∈ R . Self-adjoint quaternionshave norm | q | = q ⋆ q = ( a − i ( a i + a j + a k ))( a + i ( a i + a j + a k )) = a − a − a − a . (2.9)– 4 –hence 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 invariantnorm of a spacetime 4-vector in a quaternionic framework.Indeed, writing q = a + i ( a i + a j + a k ) and q = b + i ( b i + b j + b k ), for theLorentz invariant inner product of two 4-vectors we can write η ( q , q ) = 12 ( q ⋆ q + q ⋆ q ) = a b − a b − a b − a b ∈ R . (2.10) 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 thelinear 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 † } = L ⋆ † ( q ⋆ { L ⋆ L } q + q ⋆ { L ⋆ L } q ) L † (3.2)Now if L ⋆ L = 1, that is L ⋆ = L − , this simplifies to q ⋆ q + q ⋆ q = L ⋆ † ( q ⋆ q + q ⋆ q ) L † (3.3)But then noting that ( q ⋆ q + q ⋆ q ) ∈ R we have q ⋆ q + q ⋆ q = ( L ⋆ † L † ) ( q ⋆ q + q ⋆ q ) = ( LL ⋆ ) † ( q ⋆ q + q ⋆ q ) = q ⋆ q + q ⋆ q . (3.4)So a necessary and sufficient condition for the quaternionic mapping q → L q L † topreserve the quaternionic form of the Lorentz invariant inner product is L ⋆ L = 1; that is L ⋆ = L − . (3.5)Note that this condition implies that the set of quaternionic Lorentz transformationsforms a group under quaternion multiplication.– 5 – .2 Rotations The rotations form a well-known subgroup of the Lorentz group, and in quaternionicform a rotation about the ˆn axis can be represented by R = exp( θ ˆn ) = cos θ + ˆn sin θ. (3.6)This observation goes back to the days of Hamilton, and the fact that 3-dimensionalrotations can be represented in this quite straightforward manner is one of the reasonsso much effort was put into development of the quaternion formalism. From the pointof view of the (ordinary) quaternions (not complexified in this case) one has R − = exp( − θ ˆn ) = cos θ − ˆn sin θ = R † = R ⋆ . (3.7)Indeed the characterization R − = R † = R ⋆ is both necessary and sufficient for aquaternion to represent a rotation. Let us now see how to factorize a general Lorentz transformation into a boost timesa rotation. (This is effectively a quaternionic form of the notion of “polar decomposi-tion” that one usually encounters in matrix algebra; we make the discussion somewhatpedestrian in the interests of pedagogical clarity.) Without any loss of generalizationwe may always write: L = √ LL † (cid:0) ( LL † ) − / L (cid:1) . (3.8)(To do this we just need to know that the product of 2 complexified quaternions is againa complexified quaternion, and that the multiplication of complexified quaternions isassociative.)Now LL † is self adjoint — so √ LL † is self adjoint, and in turn ( LL † ) − / is self adjoint.Consequently (cid:0) ( LL † ) − / L (cid:1) (cid:0) ( LL † ) − / L (cid:1) † = (cid:0) ( LL † ) − / L (cid:1) (cid:0) L † ( LL † ) − / (cid:1) = 1 . (3.9)Thence (cid:0) ( LL † ) − / L (cid:1) − = (cid:0) ( LL † ) − / L (cid:1) † . (3.10)Furthermore( LL † ) ⋆ = ( L † ) ⋆ L ⋆ = ( L ⋆ ) † L ⋆ = ( L − ) † L − = ( L † ) − L − = ( LL † ) − . (3.11)– 6 –hat 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 thenby the group property (cid:0) ( LL † ) − / L (cid:1) must also be a Lorentz transformation, so (cid:0) ( LL † ) − / L (cid:1) − = (cid:0) ( LL † ) − / L (cid:1) ∗ . (3.12)But this now implies that R = (cid:0) ( LL † ) − / L (cid:1) must be a rotation, and we have shownthat 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 = B R . (3.14) 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 inertialframes. In order to proceed, we must first obtain an explicit expression for the squareroot of a four–velocity V . We represent a position 4-vector ~X = ( t, x, y, z ) = ( t, ~x ) by the self-adjoint quaternion X = t + i ( x i + y j + z k ) . (4.1)Differentiating with respect to the proper time gives a quaternionic notion of 4-velocity V = γ (1 + iv ˆn ); with | V | = V ⋆ V = 1 . (4.2)To explicitly find the square root we first present an elementary discussion: Let usintroduce the notion of rapidity in the usual manner by setting ξ = tanh − v . Then4-velocities can be written in the form V = γ (1 + iv ˆn ) = cosh ξ + i sinh ξ ˆn = e iξ ˆn . (4.3)The square root of the 4-velocity is then easily seen to be √ V = e iξ ˆn / .– 7 –xplicitly, using hyperbolic half-angle formulae, from γ = cosh ξ and v = tanh ξ onegets cosh( ξ/
2) = r γ + 12 , and sinh( ξ/
2) = r γ − . (4.4)Thence √ V = r γ + 12 + i ˆn r γ − . (4.5)In terms of the relativistic half velocity, implicitly defined by v = w w , so that one has w = v √ − v , it is easy to check that w = r γ − γ + 1 ; and γ w = r γ + 12 . (4.6)So we can write √ V = γ w (1 + i ˆn w ) . (4.7)There are many other ways of getting to the same result. The current discussion hasbeen designed to be as straightforward and explicit as possible. Now that we have an expression for the square root of a four-velocity V , we can showhow a pure Lorentz transformation is obtained from quaternion conjugation by √ V .Without any loss of generality, we define V = γ (1 + i i v ), which is the four–velocity foran object travelling with speed v in the ˆ x direction, and represent the four–vector ~X =( t, x, y, z ) by the self-adjoint quaternion X = ( t + i i x + i j y + i k z ) = ( t + i ( i x + j y + k z )).Now consider the transformation of X given by X
7→ √
V X √ V . That is, X
7→ √ V ( t + i i x + i j y + i k z ) √ V = √ V ( t + i i x ) √ V + √ V ( i j y + i k z ) √ V (4.8)= ( t + i i x ) V + ( i j y + i k z ) √ V ⋆ √ V , where in the last equality we have used the fact that √ V commutes with i , and theanti-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( − iξ i /
2) exp( iξ i /
2) = 1 . (4.9)– 8 –hat is, √ V is a unit quaternion. Thus √ V X √ V = ( t + i i x ) V + ( i j y + i k z ) . (4.10)Using our expression for V we find ( t + i i x ) V = γ { ( t + vx ) + i i ( x + vt ) } , giving a finalresult of X = ( t + i i x + i j y + i k z ) γ { ( t + vx ) + i i ( x + vt ) } + ( i j y + i k z ) , (4.11)which are the well–known inverse Lorentz transformations (boosts in the ( − ˆ x ) 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 factcompletely general, due to the general definition of the four-vector ~X .That is, a boost (pure Lorentz transformation without rotation) corresponds to X → √ V X √ V = e iξ ˆn / X e iξ ˆn / . (4.12) Starting in the rest frame of some object, where V = 1, we successively apply twoboosts V = γ (1 + i ˆn v ) and V = γ (1 + i ˆn v ) in directions ˆn and ˆn withvelocities v and v , respectively. The result of this is to shift our rest frame to aframe moving with 4-velocity V ⊕ , which is equivalent to relativistically combiningthe two 4-velocities V and V . This method has the added benefit that it obtainsan expression for the gamma–factor of the frame, γ ⊕ , and hence its speed, withouthaving to take the norm of V ⊕ , thereby avoiding lots of tedious algebra.We begin by boosting our rest frame starting with with V = 1 as in subsection 4.2: V ⊕ = p V p V V p V p V = p V V p V . (4.13)Similarly V ⊕ = p V p V V p V p V = p V V p V . (4.14)That is, the relativistic combination of 4-velocities simply amounts to V ⊕ = p V V p V ; V ⊕ = p V V p V . (4.15)This makes it obvious that V ⊕ = V ⊕ unless [ V , V ] = 0, which in turn requiresthe 3-velocities ~v and ~v to be parallel. In the special case where the 4-velocities docommute we have V ⊕ = V V = V V = V ⊕ . (4.16)– 9 –n terms of rapidities the general case is V ⊕ = e iξ ˆn / e iξ ˆn e iξ ˆn / ; V ⊕ = e iξ ˆn / e iξ ˆn e iξ ˆn / . (4.17)Viewed in this way the relativistic combination of 4-velocities can be interpreted asan application of the symmetrized version of the Baker–Campbell–Hausdorff (BCH)expansion [32–36]. Indeed, taking logarithms: ξ ⊕ ˆn ⊕ = − i ln (cid:8) e iξ ˆn / e iξ ˆn e iξ ˆn / (cid:9) ; (4.18) ξ ⊕ ˆn ⊕ = − i ln (cid:8) e iξ ˆn / e iξ ˆn e iξ ˆn / (cid:9) . (4.19)Unfortunately this formal result, while quite elegant, is not really computationallyeffective. One could for instance fully expand the expression in (4.17) above and isolatethe real part to deduce ξ ⊕ = ξ ⊕ = cosh − (cosh ξ cosh ξ + sinh ξ sinh ξ cos θ ) , (4.20)where θ is the angle between ˆn and ˆn . As expected, for collinear 3-velocities thisreduces to ξ ⊕ = ξ ⊕ → | ξ ± ξ | . (4.21)To check this for consistency, note γ ⊕ = η ( V , V ⊕ ) = 12 ( V ⋆ V ⊕ + V ⋆ ⊕ V ) = 12 ( V ⊕ + V ⋆ ⊕ ) . (4.22)Thence γ ⊕ = 12 (cid:16)p V V p V + p V ⋆ V ⋆ p V ⋆ (cid:17) . (4.23)But γ ⊕ is real, and (cid:0) √ V (cid:1) − = (cid:0) √ V (cid:1) ⋆ so γ ⊕ = (cid:16)p V (cid:17) ⋆ γ ⊕ p V = 12 ( V V + V ⋆ V ⋆ ) = 12 ( V V + ( V V ) ⋆ ) . (4.24)It is then easy (indeed almost trivial) to see that γ ⊕ = γ γ (1 + ~v · ~v ) = γ ⊕ . (4.25)While this very easily yields the magnitude of the combined 3-velocities | ~v ⊕ | = | ~v ⊕ | ,isolating the direction of the combined 3-velocities is much more subtle. Note ˆ v ⊕ =ˆ v ⊕ in general, see equations (4.18)–(4.19).The rapidity formalism is also particularly useful for quickly double-checking formalrelationships such as √ V ⋆ = e − iξ ˆn / = (cid:16) √ V (cid:17) ⋆ = √ V − = (cid:16) √ V (cid:17) − . (4.26)– 10 –e can also use this formalism to write a general Lorentz transformation in the form L = B R = e iξ ˆn / e θ ˆm / . (4.27)Here ˆn is the direction of the boost B , while ˆm is the axis of the rotation R . We shall now derive an explicit quaternionic formula for the Wigner rotation. Forrelevant background see references [28–31]. Note that for 4-velocities V = V † ; V − = V ⋆ ; p ( V ) − = (cid:16) √ V (cid:17) − , (5.1)while for rotations R † = R ⋆ ; R − = R † = R ⋆ . (5.2)Now note V ⊕ = p V V p V = p V p V p V p V (5.3)Let us define a quaternion R by taking p V ⊕ R = p V p V , (5.4)and then checking to see that this quaternion does in fact a correspond to a rotation.First R = q V − ⊕ p V p V ; R † = p V p V q V − ⊕ . (5.5)Now RR † = q V − ⊕ p V p V p V p V q V − ⊕ (5.6)= q V − ⊕ p V V p V q V − ⊕ (5.7)= q V − ⊕ V ⊕ q V − ⊕ (5.8)= 1 . (5.9)That is, R − = R † . Now consider R ⋆ = p V ∗ p V ∗ p V ⊕ , (5.10)and compare it to R † = p V p V q V − ⊕ , (5.11)– 11 –ctually R ⋆ = R † though this might not at first be obvious. Calculate R ⋆ p V ⊕ = p V ∗ p V ∗ p V ⊕ p V ⊕ = p V ∗ p V ∗ V ⊕ . (5.12)Thence R ⋆ p V ⊕ = q V − q V − (cid:16)p V V p V (cid:17) = p V p V . (5.13)Thence R ⋆ p V ⊕ = p V p V = R † p V ⊕ . (5.14)So R ⋆ = R † as claimed. Accordingly R is indeed a well-defined rotation.Explicitly we have R = q V − ⊕ p V p V = rq V − V − q V − p V p V . (5.15)Note that from equation (5.14) we have R † p V ⊕ = p V p V , (5.16)and so, along with the defining relation equation (5.4), we deduce V ⊕ = p V V p V = p V p V p V p V = R † p V ⊕ p V ⊕ R . (5.17)That is V ⊕ = R † V ⊕ R . (5.18)So R is indeed the Wigner rotation as claimed. From the above we note that V (1 ⊕ ⊕ = p V p V V p V p V . (6.1)whereas V ⊕ (2 ⊕ = qp V V p V V qp V V p V . (6.2)This explicitly verifies the general non-associativity of composition of 4-velocities, andfurthermore 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 –iscussion in reference [31], where an equivalent discussion was presented in terms ofquaternionic 3-velocities.From the above V (1 ⊕ ⊕ = p V p V rq V − V − q V − qp V V p V V × qp V V p V rq V − V − p V − p V p V . (6.3)That is V (1 ⊕ ⊕ = p V p V rq V − V − q V − V ⊕ (2 ⊕ rq V − V − p V − p V p V . (6.4)But from our formula for the Wigner rotation R ⊕ = q V − ⊕ p V p V = rq V − V − q V − p V p V , (6.5)this now implies V (1 ⊕ ⊕ = R † ⊕ V ⊕ (2 ⊕ R ⊕ . (6.6)So the Wigner rotation is not just relevant for understanding generic non-commutativitywhen composing two boosts, is also relevant to understanding generic non-associativitywhen composing three boosts.Suppose now we consider a specific situation where the composition of 4-velocities isassociative, that is, we assume: V (1 ⊕ ⊕ = V ⊕ (2 ⊕ . (6.7)Under this condition we would now have p V p V V p V p V = qp V V p V V qp V V p V , (6.8)whence we would need rq V − V − q V − p V p V V p V p V rq V − V − q V − = V . (6.9)We can rewrite this condition in terms of the Wigner rotation as R ⊕ V R † ⊕ = V . (6.10)– 13 –hat is R ⊕ V R − ⊕ = V , (6.11)whence [ R ⊕ , V ] = 0 . (6.12)Thence the combination of velocities is associative V (1 ⊕ ⊕ = V ⊕ (2 ⊕ if and only ifthe boost direction in V is parallel to the rotation axis in R ⊕ . But this holds if andonly if [ v , [ v , v ]] = 0 (6.13)or in more prosaic language, if and only if ~v × ( ~v × ~v ) = 0 . (6.14)(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.) Let us now consider yet another way of understanding combination of velocities, thistime in terms of the (symmetrized) BCH theorem. We have already seen that V ⊕ = exp (cid:16) iξ ˆ ξ / (cid:17) exp (cid:16) iξ ˆ ξ (cid:17) exp (cid:16) iξ ˆ ξ / (cid:17) . (7.1)Now differentiate ∂ V ⊕ ∂ξ = i n ˆ ξ , exp (cid:16) iξ ˆ ξ / (cid:17) exp (cid:16) iξ ˆ ξ (cid:17) exp (cid:16) iξ ˆ ξ / (cid:17)o , (7.2)and rewrite this as ∂ V ⊕ ∂ξ = i (cid:16) iξ ˆ ξ / (cid:17) n ˆ ξ , exp (cid:16) iξ ˆ ξ (cid:17)o exp (cid:16) iξ ˆ ξ / (cid:17) , (7.3)But we note that n ˆ ξ , exp (cid:16) iξ ˆ ξ (cid:17)o = n ˆ ξ , (cid:16) cosh( ξ ) + i sinh( ξ ) ˆ ξ (cid:17)o (7.4)= (cid:16) ξ ) ˆ ξ + i sinh( ξ ) { ˆ ξ , ˆ ξ } (cid:17) . (7.5)And, since { ˆ ξ , ˆ ξ } ∈ R , this implies h ˆ ξ , n ˆ ξ , exp (cid:16) iξ ˆ ξ (cid:17)oi = 0 . (7.6)– 14 –onsequently we can pull the factor exp (cid:16) iξ ˆ ξ / (cid:17) through the anti-commutator, andrewrite the derivative as ∂ V ⊕ ∂ξ = i n ˆ ξ , exp (cid:16) iξ ˆ ξ (cid:17)o exp (cid:16) iξ ˆ ξ (cid:17) . (7.7)Now integrate with respect to ξ . We see V ⊕ = V + i Z ξ n ˆ ξ , exp (cid:16) iξ ˆ ξ (cid:17)o exp (cid:16) iξ ˆ ξ (cid:17) dξ . (7.8)Pull the constant (with respect to ξ ) anti-commutator outside the integral V ⊕ = V + i n ˆ ξ , exp (cid:16) iξ ˆ ξ (cid:17)o Z ξ exp (cid:16) iξ ˆ ξ (cid:17) dξ . (7.9)Perform the integral V ⊕ = V + i n ˆ ξ , exp (cid:16) iξ ˆ ξ (cid:17)o exp (cid:16) iξ ˆ ξ (cid:17) − i ˆ ξ . (7.10)Noting that ( i ˆ ξ ) = 1 this simplifies to V ⊕ = V − n ˆ ξ , exp (cid:16) iξ ˆ ξ (cid:17)o ˆ ξ (cid:16) exp (cid:16) iξ ˆ ξ (cid:17) − (cid:17) . (7.11)That is V ⊕ = V − { ˆ ξ , V } ˆ ξ ( V − . (7.12)A more tractable result is this V ⊕ = V − (cid:16) ˆ ξ V ˆ ξ − V (cid:17) ( V − . (7.13)Thence V ⊕ = V + 12 (cid:16) V − ˆ ξ V ˆ ξ (cid:17) ( V − . (7.14)Rearranging V ⊕ = V V − (cid:16) V + ˆ ξ V ˆ ξ (cid:17) ( V − . (7.15)This gives us the composition of 4-velocities V ⊕ algebraically in terms of V and V and at worst some quaternion multiplication (the need to evaluate √ V has beenside-stepped).Note that this has the right limit for parallel 3-velocities. When [ V , ˆ ξ ] = 0 we see V ⊕ → V V , (7.16)as it should. – 15 –or perpendicular 3-velocities V ⊕ = V − { ˆ ξ , V } ˆ ξ ( V − , (7.17)reduces to V ⊕ → V + γ ( V − . (7.18)Thence V ⊕ → γ (1 + v ) + γ γ (1 + v ) − γ , (7.19)implying V ⊕ → γ γ (cid:18) q − v v + v (cid:19) . (7.20)That is v ⊕ = q − v v + v , (7.21)and | v ⊕ | = v + v − v v , (7.22)exactly as expected for perpendicular 3-velocities. 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 = B R = e iξ ˆn / e θ ˆm / . (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 → √ V X √ V . (8.2) • The relativistic combination of 4-velocities has the simple algebraic form: V ⊕ = p V V p V ; V ⊕ = p V V p V . (8.3) • The Wigner rotation is given by: R = q V − ⊕ p V p V = rq V − V − q V − p V p V . (8.4)– 16 – The Wigner rotation satisfies, in terms of the generic non-commutativity of twoboosts, V ⊕ = R † V ⊕ R ; (8.5)and, in terms of the generic non-associativity of three boosts, V (1 ⊕ ⊕ = R † ⊕ V ⊕ (2 ⊕ R ⊕ . (8.6)Overall, some rather complicated linear algebra involving 4 × C ⊗ H . Acknowledgments
TB was supported by a Victoria University of Wellington MSc scholarship, and wasalso indirectly supported by the Marsden Fund, via a grant administered by the RoyalSociety of New Zealand. MV was directly supported by the Marsden Fund, via a grantadministered by the Royal Society of New Zealand.
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