Using Rotations to Control Observable Relativistic Effects

Abstract

This paper examines the possibility of controlling the outcome of measured (flat space-time) relativistic effects, such as time dilation or length contractions, using pure rotations and their nontrivial interactions with Lorentz boosts in the isometry group ${\mathrm{SO}}^{+}(3,1)$. In particular, boost contributions may annihilate leaving only a geometric phase (Wigner rotation), which we see in the complex solutions of the generalized Euler decomposition problem in ${\mathbb{R}}^3$. We consider numerical examples involving specific matrix factorizations, along with possible applications in special relativity, electrodynamics and quantum scattering. For clearer interpretation and simplified calculations we use a convenient projective biquaternion parametrization which emphasizes the geometric phases and for a large class of problems allows for closed-form solutions in terms of only rational functions.

1. Introduction

The peculiar role of rotations in special relativity (SR) has been puzzling theoretical physicists for more than a century, starting with the famous Erenfest’s paradox: a seeming contradiction between classical rigid body mechanics (‘prohibiting’ shape deformations) and Lorentz length contraction of a spinning disk (cf. [1]). On top of that, we have the infamous Sagnac effect of light circulating around a rotating closed path with different time periods, depending on the direction. This for a while led people to believe there must be something wrong with the postulates of relativity, which was Sagnac’s original intention (see [2]). Nowadays the phenomenon is not only well integrated into SR but also plays a major role in laser interferometers and precise Doppler gyroscopes [3,4,5]. Another interesting discovery from the early days of relativity was the Thomas precession [6] explaining how electron spins are affected by acceleration in non-inertial frames. This mechanism of boost-rotation entanglement, known more generally as Wigner rotation, is quite similar to the Coriolis effect and may be attributed to a classical geometric phase [7,8]. The same pattern is observed in many areas of theoretical physics and engineering: from classical mechanics to geometric optics and even quantum computation [9,10,11,12,13]. A particularly curious context with plenty of practical implementations lies within the realm of electrodynamics [14,15,16], where the magnetic force itself may be regarded as a consequence of such inertial effect and thus linked to a geometric phase following a mechanical analogy [16,17,18]. We refer to [19,20,21,22] for a deeper investigation of the problem of rotating frames in relativity, both from a purely mechanical perspective and with an account of electromagnetic interactions. Instead of diving into the physics and various applications of this topic, the present study discusses a unified geometric approach based on the structure of proper Lorentz transformations ${\mathrm{SO}}^{+}(3,1)\cong \mathrm{SO}(3,\mathbb{C})$ and the corresponding Wigner little groups, starting with the compact subgroup of euclidean rotations in ${\mathbb{R}}^3$. A convenient representation proposed by Rodrigues back in 1840, based on Clifford’s geometric algebras but preceeding their invention (cf. [23,24]) was rediscovered by Fedorov [25] almost a century and a half later in the context of SR and high energy physics. This paper exploits it for the description of interactions between rotations and Lorentz boosts. We begin by briefly revising this useful tool and then explore its applications to the matter of interest: from matrix decompositions to relativistic kinematics, electrodynamics and quantum scattering, providing also examples.

2. Preliminaries

Rotations in 3D Euclidean space provide a first glimpse of the weirdness of motion due to the Coriolis effect which we may attribute to a classical geometric phase, associated with parallel transport (see Figure 1). The complex projective representation of the proper Lorentz group ${\mathrm{SO}}^{+}(3,1)\cong \mathrm{SO}(3,\mathbb{C})$ allows us to extend this weirdness quite smoothly to the realm of special relativity, where the interactions of pure rotations (the compact subgroup) and boosts (the quotient rapidity space ${\mathbb{B}}^3$) create a whole spectrum of effects due to Wigner, Sagnac, Hall, etc. [26,27].

The projective biquaternion (complex Rodrigues’ vector) $\mathbf{c}$$\in {\mathbb{CP}}^3$ is a natural extension of the rational parametrization of $\mathrm{SO}(3)$ (see [25,26] for details) and inherits the group composition law compatible with matrix multiplication

$$\langle {\mathbf{c}}_{\mathbf{2}},{\mathbf{c}}_1\rangle =\frac{{\mathbf{c}}_2+{\mathbf{c}}_1+{\mathbf{c}}_2\times {\mathbf{c}}_1}{1-{\mathbf{c}}_2·{\mathbf{c}}_1},\mathcal{R}(\langle {\mathbf{c}}_{\mathbf{2}},{\mathbf{c}}_1\rangle )=\mathcal{R}({\mathbf{c}}_{\mathbf{2}})\mathcal{R}({\mathbf{c}}_{\mathbf{1}})$$

(1)

where the $\mathrm{SO}(3)$ or $\mathrm{SO}(3,\mathbb{C})$ representation is given, respectively, via the Cayley transform

$$\mathcal{R}(\mathbf{c})=\frac{1+{\mathbf{c}}^{\times }}{1-{\mathbf{c}}^{\times }}=\frac{1-{\mathbf{c}}^2+2\mathbf{c}{\mathbf{c}}^t+2{\mathbf{c}}^{\times }}{1+{\mathbf{c}}^2}·$$

(2)

In the compact (real) setting this coincides with the well-known Rodrigues’ rotation formula

$$\mathcal{R}(\mathbf{n},\phi )=cos\phi +(1-cos\phi )\mathbf{n}{\mathbf{n}}^t+sin\phi {\mathbf{n}}^{\times }$$

(3)

with the Euler substitution $\mathbf{c}=tan\frac{\phi }{2}\mathbf{n}$, i.e., the angle of rotation and the unit normal to the rotation plane is expressed as $\phi =2arctan\left|\mathbf{c}\right|$ and $\mathbf{n}={\left|\mathbf{c}\right|}^{-1}\mathbf{c}$, while $\mathbf{c}=q_0^{-1}$$\mathbf{q}$ is the projection of the unit quaternion $q\in {\mathbb{S}}^3$ describing the lift of our rotation up to the spin covering group $\mathrm{SU}(2)\cong \mathrm{Spin}(3)$. Hence, q does not need to be a unit in the first place: we can start with any element of ${\mathbb{H}}^{\times }$. This simplifies our construction significantly, apart from having to deal with only rational expressions, and it remains true also in the complex setting, as the pseudoscalar $\mathcal{I}$ in the even part of the space-time algebra ${\mathrm{Cliff}}_{3,1}^{\circ }\cong {\mathrm{Cliff}}_{3,1}$ belongs to the center, and therefore, we identify it with the imaginary unit $i=\sqrt{-1}$. Unlike in the compact real case, however, the group does not cover the whole projective space: one has to remove the quadric ${\mathbf{c}}^2+1=0$, which corresponds to the boundary of the unit ball ${\mathbb{B}}^3$. This representation is quite convenient both for practical purposes (due to the reduced number of computations) and theoretical studies, e.g., for the classification of transformations as elliptic (${\mathbf{c}}^2>0$), hyperbolic (${\mathbf{c}}^2<0$), parabolic (${\mathbf{c}}^2=0$) or loxodromic (${\mathbf{c}}^2\notin \mathbb{R}$) in ${\mathrm{SO}}^{+}(3,1)\cong \mathrm{PSL}(2,\mathbb{C})$ seen here as the Möbius group of the 2-sphere ${\mathbb{S}}^2\cong {\mathbb{CP}}^2$. The realization in Minkowski space-time ${\mathbb{R}}^{3,1}$ is also straightforward (see [25] for details):

$$\Lambda (\mathbf{c})=\frac{1}{|1+{\mathbf{c}}^2|}\left(\begin{array}{cc}1-{\left|\mathbf{c}\right|}^2+\mathbf{c}{\overline{\mathbf{c}}}^t+\overline{\mathbf{c}}{\mathbf{c}}^t+{(\mathbf{c}+\overline{\mathbf{c}})}^{\times }& i(\overline{\mathbf{c}}-\mathbf{c}+\overline{\mathbf{c}}\times \mathbf{c})\\ & \\ i{(\overline{\mathbf{c}}-\mathbf{c}-\overline{\mathbf{c}}\times \mathbf{c})}^t& 1+{\left|\mathbf{c}\right|}^2\end{array}\right)$$

(4)

and denoting $\tilde{\Lambda }=\Lambda -\eta {\Lambda }^{\mathrm{t}}\eta$ with $\eta =\mathrm{diag}(1,1,1,-1)$ we easily obtain the inverse map

$$\mathbf{c}=\frac{1}{\mathrm{tr}\Lambda }{\left({\tilde{\Lambda }}_{32}+i{\tilde{\Lambda }}_{14},{\tilde{\Lambda }}_{13}+i{\tilde{\Lambda }}_{24},{\tilde{\Lambda }}_{21}+i{\tilde{\Lambda }}_{34}\right)}^t.$$

(5)

However, one needs to keep in mind that there is a non-trivial Plücker condition (namely $Im{\mathbf{c}}^2=0$) to ensure the existence of an invariant plane $\Sigma$, be it space-like, time-like or null (isotropic). It is determined by the real and imaginary parts of the projective blade $\mathbf{c}=\mathit{\alpha }+i\mathit{\beta }$ (in this case $\mathit{\alpha }\perp \mathit{\beta }$) as $\Sigma =\{{(\mathit{\alpha },0)}^t,{(\mathit{\alpha }\times \mathit{\beta },{\mathit{\alpha }}^2)}^t\}$ if $\mathit{\alpha }\ne 0$ and $\Sigma =\{{({\mathit{\beta }}^{\perp },0)}^t\}$ otherwise (in [28] we use this to realize the embedding into higher-dimensional groups via the twistor correspondence). Note also that Formula (4) is ill-defined for isotropic directions of infinite scale.

3. Generalized Euler Decompositions

The classical factorization result obtained by Euler for $\mathrm{SO}(3)$ is shown to generalize for any set of rotation axes as long as the one in the middle is perpendicular to the other two (Davenport’s condition). Otherwise, such decomposition is also possible in many cases but does not cover the entire group. In [29] we use the projective bivector representation described above to derive a closed form solution with the necessary and sufficient condition

$$\mathcal{R}={\mathcal{R}}_3{\mathcal{R}}_2{\mathcal{R}}_1,{\mathcal{R}}_i=\mathcal{R}({\tau }_i{\mathbf{a}}_i)$$

(6)

where $\{{\mathbf{a}}_i\}$ represent the invariant unit vectors and the angles are ${\phi }_i=2arctan{\tau }_i$. Denoting

$${\omega }_1={\mathbf{a}}_1\times {\mathbf{a}}_2·{\mathcal{R}}^t{\mathbf{a}}_3,{\omega }_2={\mathbf{a}}_1\times {\mathbf{a}}_2·{\mathbf{a}}_3,{\omega }_3=\mathcal{R}{\mathbf{a}}_1\times {\mathbf{a}}_2·{\mathbf{a}}_3$$

as well as $r_{ij}={\mathbf{a}}_i·\mathcal{R}{\mathbf{a}}_j$ and $g_{ij}={\mathbf{a}}_i·{\mathbf{a}}_j$, we use the eigenvector property to express the ${\tau }_i$’s (unless in the so-called ‘gimbal lock’ setting ${\mathbf{a}}_3=\pm \mathcal{R}{\mathbf{a}}_1$, which has a singular solution, also studied in [29])

$${\tau }_i^{\pm }=\frac{{\sigma }_i}{{\omega }_i\pm \sqrt{\Delta }},{\sigma }_i={\epsilon }_{ijk}(g_{jk}-r_{jk}),j>k$$

(7)

where ${\epsilon }_{ijk}$ stands for the Levi–Civita symbol. Finally, ${\tau }_i\in \mathbb{R}\cup \infty$ if and only if (see Figure 1)

$$\Delta =\left|\begin{array}{ccc}1& g_{12}& r_{31}\\ g_{21}& 1& g_{23}\\ r_{31}& g_{32}& 1\end{array}\right|\ge 0$$

(8)

while if the above is not satisfied, the solutions (7) have imaginary counterparts. Before discussing this in detail, let us point out that half-turns correspond to infinite elements in ${\mathbb{RP}}^3$, so (1) and (2) should be applied with the aid of l’Hôpital’s rule. Consider, for example,

$$\mathcal{R}(\infty \mathbf{n})=\mathcal{R}(\mathbf{n},\pi /2)=\frac{1}{3}\left(\begin{array}{ccc}\hfill -1& \hfill 2& \hfill 2\\ \hfill 2& \hfill -1& \hfill 2\\ \hfill 2& \hfill 2& \hfill -1\end{array}\right)\Rightarrow \mathbf{n}=\frac{1}{\sqrt{3}}{(1,1,1)}^t$$

and the set of invariant axes, given by the unit vectors

$${\mathbf{a}}_1=\frac{1}{\sqrt{3}}{(1,-1,-1)}^t,{\mathbf{a}}_2=\frac{1}{\sqrt{3}}{(-1,1,-1)}^t,{\mathbf{a}}_3=\frac{1}{\sqrt{3}}{(-1,-1,1)}^t.$$

Using our algorithm, we obtain for $\mathcal{R}={\mathcal{R}}_3{\mathcal{R}}_2{\mathcal{R}}_1$ only one solution (since we have $\Delta =0$)

$${\tau }_i=\left\{-\sqrt{3},\frac{1}{\sqrt{3}},-\sqrt{3}\right\}\Rightarrow {\phi }_i=\left\{-\frac{2\pi }{3},\frac{\pi }{3},-\frac{2\pi }{3}\right\}$$

and the corresponding rotation matrices in the decomposition (13) take the following form:

$${\displaystyle {\mathcal{R}}_1=\left(\begin{array}{ccc}\hfill 0& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\\ \hfill -1& \hfill 0& \hfill 0\end{array}\right),{\mathcal{R}}_2=\frac{1}{3}\left(\begin{array}{ccc}\hfill 2& \hfill 1& \hfill 2\\ \hfill -2& \hfill 2& \hfill 1\\ \hfill -1& \hfill -2& \hfill 2\end{array}\right),{\mathcal{R}}_3=\left(\begin{array}{ccc}\hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1\\ \hfill -1& \hfill 0& \hfill 0\end{array}\right).}$$

As mentioned above, there is a singular ‘gimbal lock’ setting, determined by the condition ${\mathbf{a}}_3=\pm \mathcal{R}{\mathbf{a}}_1$, to which (7) does not apply, but instead we have a degenerate solution, namely

$${\phi }_2=2arctan\frac{{\mathbf{a}}_1\times {\mathbf{a}}_2·\mathbf{n}}{g_{2[1}{\mathbf{a}}_{2]}·\mathbf{n}},{\phi }_1\pm {\phi }_3=2arctan\frac{{\mathbf{a}}_1\times {\mathbf{a}}_2·\mathbf{n}}{g_{1[2}{\mathbf{a}}_{1]}·\mathbf{n}}$$

(9)

where $a_{[i}{b}_{j]}=a_i{b}_j-a_j{b}_i$ is the alternator and (8) is reduced to $g_{12}=\pm g_{23}$. This expression is valid also in the case of two axes $\mathcal{R}={\mathcal{R}}_2{\mathcal{R}}_1$ (with ${\phi }_3=0$) as long as the necessary and sufficient condition for that $r_{21}=g_{21}$ is satisfied. For closed rotation sequence ${\mathcal{R}}_3{\mathcal{R}}_2{\mathcal{R}}_1=1$ we have the trivial solution ${\phi }_i=0$ for any axis configuration, a singular one with ${\phi }_1\pm {\phi }_3={\phi }_2=0$ if $g_{13}=\pm 1$ (${\mathbf{a}}_3=\pm {\mathbf{a}}_1$) and for ${\omega }_2\ne 0$ we also have a regular solution in the form

$${\phi }_1=2arctan\frac{{\omega }_2}{g_{1[2}{g}_{1]3}},{\phi }_2=2arctan\frac{{\omega }_2}{g_{2[3}{g}_{2]1}},{\phi }_3=2arctan\frac{{\omega }_2}{g_{3[1}{g}_{3]2}}·$$

(10)

Most of the above construction is easily transferred to the hyperbolic case of ${\mathrm{SO}}^{+}(2,1)$ where we have an isotropic singularity, apart from the gimbal lock, which is still present.

4. Complex Solutions

Whenever the condition (8) is not satisfied, one ends up with complex solutions and according to Formula (7), they come in conjugate pairs

$${\tau }_k^{\pm }={\sigma }_k\frac{{\omega }_k\mp i\sqrt{|\Delta |}}{{\omega }_k^2-{\Delta }^2}={\lambda }_k\pm i{\mu }_k\in {\mathbb{CP}}^1$$

which clearly correspond to Lorentz transformations of the loxodromic type. Since there is no invariant plane for these factors, their physical interpretations are somewhat ambiguous (compared to the so-called Wigner little groups). However, we may still factorize using (1):

$$\mathbf{c}=\tau \mathbf{n}=\langle \gamma \mathbf{n},i\delta \mathbf{n}\rangle =\frac{\gamma +i\delta }{1-i\gamma \delta }\mathbf{n},\tau =\lambda +i\mu \in {\mathbb{CP}}^1$$

(11)

and thus obtain the parameter values allowing for such pairs of transformations in the form

$${\gamma }_{\pm }=\frac{{\left|\tau \right|}^2-1\pm \sqrt{{(|\tau |}^2{-1)}^2+4{\lambda }^2}}{2\lambda },{\delta }_{\pm }=\frac{{\left|\tau \right|}^2+1\pm \sqrt{{(|\tau |}^2{+1)}^2-4{\mu }^2}}{2\mu }·$$

(12)

Note, that on the unit circle, ${\mathbb{S}}^1:\tau =e^{i\vartheta }$ one has $\gamma =1$ (quarter-turn) and $\delta =tan\frac{\vartheta }{2}$, so that (1) yields the Cayley transform. In that case $\tau$ cannot be purely imaginary, as it leads to an ill-defined Lorentz boost with ${\tau }^2=-1$. Apart from these two singular cases, the above decomposition yields a pair of well-defined mutually commuting rotations and boosts ${\mathcal{R}}_i={\mathcal{R}}_i^{\circ }{\Lambda }_i={\Lambda }_i{\mathcal{R}}_i^{\circ }$ for each factor in $\mathcal{R}={\mathcal{R}}_3{\mathcal{R}}_2{\mathcal{R}}_1$. This means on the one hand, that a sequence of finite Lorentz transformations may result in a pure rotation, while on the other, it seems possible to eliminate the resultant relativistic effect even for a sequence of pure boosts if we compose them with appropriate rotations. Certainly, not all such sequences would allow it, e.g., if the Davenport condition $g_{12}=g_{23}=0$ in the $\mathrm{SO}(3,\mathbb{C})$ representation of the boosts holds, then the procedure described above cannot result in a pure rotation.

Next, we consider an example of such decomposition, using the same system of axes $\{{\mathbf{a}}_i\}$, but this time the compound rotation will be a (positively oriented) quarter-turn about the x-axis. Since $r_{31}=g_{31}=-1$, both the gimbal lock condition and the one ensuring the decomposition hold, but (8) does not, so we end up with a pair of complex solutions:

$${\tau }_i^{\pm }=\left\{\pm i,0,\frac{2\sqrt{3}\pm 3i}{7}\right\}\Rightarrow \mathbf{c}=\langle \pm i{\mathbf{a}}_3,i{3}^{\pm 1}{\mathbf{a}}_1,\pm 3^{\pm \frac{1}{2}}{\mathbf{a}}_1\rangle$$

where we have used (12) to factorize the loxodromic transformation with invariant axis ${\mathbf{a}}_1$ into a rotation and a pure Lorentz boost. Note, however, that the boosts generated by $\pm i{\mathbf{a}}_3$ lead to ill-defined transformations with ${\mathbf{c}}^2+1=0$, so one may attempt to avoid this by composing $\mathbf{c}=\langle {\tilde{\mathbf{c}}}_{\pm },\pm 3^{\pm \frac{1}{2}}{\mathbf{a}}_1\rangle$ where ${\tilde{\mathbf{c}}}_{\pm }=\langle \pm i{\mathbf{a}}_3,i{3}^{\pm 1}{\mathbf{a}}_1\rangle$, but ${\tilde{\mathbf{c}}}_{+}$ hits infinity in an isotropic direction while ${\tilde{\mathbf{c}}}_{-}^2=-1$, i.e., they are both unfit in this case, yet we may factorize:

$$\mathbf{c}=\mathit{\alpha }+i\mathit{\beta }=\langle i{({\mathit{\alpha }}^2+1)}^{-1}(1+{\mathit{\alpha }}^{\times })\mathit{\beta },\mathit{\alpha }\rangle ,\mathit{\alpha }·\mathit{\beta }=0$$

(13)

using the fact that ${\tilde{\mathbf{c}}}_{\pm }$ in our example satisfy the Plücker condition. However, both solutions yield a ‘forbidden’ boost in the form of $-i{\mathbf{a}}_3$ so in this case the decomposition is impossible even in $\mathrm{SO}(3,\mathbb{C})$ due to the isotropic singularity. That is a somewhat exotic configuration and thus worth taking into account. We shall provide more numerical examples at the end.

5. Cancellation of Boosts

So far we discussed only a quite specific mechanism for boost cancellation associated with the complex solutions in the generalized Euler decomposition problem. Our logic was somewhat backward: constructing the sequence of commuting boost-rotation pairs that produce the desired result as a combination of rotational counterpart and a geometric phase. More generally, one may pose the question ‘What conditions should a set of Lorentz transformations satisfy so that their composition is a pure rotation?’, but even for just two factors the algebraic system is quite cumbersome. However, if we impose once more the additional restriction ${\mathbf{c}}_k={\tau }_k\mathbf{n}$ ($\mathbf{n}\in {\mathbb{S}}^2$), since ${\mathbf{c}}_2\times {\mathbf{c}}_1=0$, the condition is greatly simplified

$$\frac{Im({\tau }_1)}{1+|{\tau }_1{|}^2}+\frac{Im({\tau }_2)}{1+|{\tau }_2{|}^2}=0,Im({\tau }_1{\tau }_2)\ne 0.$$

(14)

On the other hand, one may also demand from the individual factors to be pure boosts, i.e., ${\mathbf{c}}_k=i{\mu }_k{\mathbf{a}}_k$ with ${\mathbf{a}}_k\in {\mathbb{S}}^2$ and ${\mu }_k\in \mathbb{R}/\{\pm 1\}$. In that setting decompositions into pairs are no longer an option: for collinear axes, one has ${\mu }_1+{\mu }_2=0$, which yields a trivial result, while for ${\mathbf{a}}_1\times {\mathbf{a}}_2\ne 0$ the system is inconsistent. In the case of three factors we use (1) to express

$$\langle {\mathbf{c}}_3,{\mathbf{c}}_2,{\mathbf{c}}_1\rangle =\frac{{\mathbf{c}}_3+{\mathbf{c}}_2+{\mathbf{c}}_1+{\mathbf{c}}_3\times {\mathbf{c}}_2+{\mathbf{c}}_3\times {\mathbf{c}}_1+{\mathbf{c}}_2\times {\mathbf{c}}_1+({\mathbf{c}}_3\times {\mathbf{c}}_2)\times {\mathbf{c}}_1-({\mathbf{c}}_3·{\mathbf{c}}_2){\mathbf{c}}_1}{1-{\mathbf{c}}_3·{\mathbf{c}}_2-{\mathbf{c}}_3·{\mathbf{c}}_1-{\mathbf{c}}_2·{\mathbf{c}}_1-{\mathbf{c}}_3\times {\mathbf{c}}_2·{\mathbf{c}}_1}$$

(15)

and consider only odd power terms as they contribute to the imaginary part. It is simplified greatly if we demand the vectors ${\mathbf{a}}_k$ to be coplanar, as in that case we have ${\mathbf{c}}_3\wedge {\mathbf{c}}_2\wedge {\mathbf{c}}_1=0$, which yields the system of quadratic equations for the unknown parameters ${\mu }_i$ in the form

$$g_{12}{\mu }_1{\mu }_2+1=g_{23}{\mu }_2{\mu }_3+1=g_{13}{\mu }_1{\mu }_3-1=0.$$

Note, that the latter is unsolvable in the Davenport setting, but other symmetric configurations allow it, e.g., for $g_{12}=g_{23}=\frac{1}{2}$ and $g_{13}=1$ we easily obtain a dependent solution with ${\mu }_2=-2{\mu }_1$ and ${\mu }_3={\mu }_2^{-1}$. On the other hand, if the condition $g_{12}=g_{23}=0$ that guarantees (8) holds, e.g., in the classical Euler setting ${\mathbf{a}}_1={\mathbf{a}}_3\perp {\mathbf{a}}_2$, we typically end up with either the trivial solution, some contradiction, or the ‘forbidden quadric’ ${\mathbf{c}}^2+1=0$.

One may come to the conclusion that the Davenport condition prevents us from having a complex solution in the decomposition problem, but it would be wrong, for example, if the axes $\{{\mathbf{a}}_i\}$ form an orthonormal basis in ${\mathbb{R}}^3$ (the Bryan setting), there is an interesting solution ${\tau }_1{\tau }_3=-{\tau }_2=1$, such that all three parameters lie on the unit circle, so we can write ${\tau }_1={\overline{\tau }}_3=e^{i\vartheta }$ and align $\{{\mathbf{a}}_k\}$ with the standard basis, which yields the factorization:

$${\displaystyle \left(\begin{array}{ccc}itan\vartheta & -sec\vartheta & 0\\ sec\vartheta & itan\vartheta & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}\hfill 0& \hfill 0& \hfill -1\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 0\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& -itan\vartheta & -sec\vartheta \\ 0& sec\vartheta & -itan\vartheta \end{array}\right)=\left(\begin{array}{ccc}\hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill -1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 0\end{array}\right).}$$

But how is this even possible when real solutions are guaranteed for any set of for orthogonal axes? In short, the result is a half-turn about ${\mathbf{a}}_1+{\mathbf{a}}_3$, so we are once more in a ‘gimbal lock’ mode, which allows for complex continuation as long as ${\phi }_1+{\phi }_3\in \mathbb{R}$ in Formula (9).

Now, let us further factorize the above loxodromic transformations into pairs of mutually commuting boosts and rotations using Formulas (11) and (12). As pointed out earlier, for $\tau =e^{i\vartheta }$ we have $\gamma =1$, $\delta =tan\frac{\vartheta }{2}$, which amounts in this example to a multiplication of the quarter-turn product $\tilde{\mathcal{R}}=\mathcal{R}({\mathbf{a}}_3){\mathcal{R}}_2\mathcal{R}({\mathbf{a}}_1)=\mathcal{R}$ with a pair of boosts ${\Lambda }_{1,3}$ generated, respectively, by $i\delta {\mathbf{a}}_1$ and $-i\delta {\mathbf{a}}_3$. To be explicit, we also provide the ${\mathbb{R}}^{3,1}$ matrix representation

$${\displaystyle {\Lambda }_1=\left(\begin{array}{cccc}sec\vartheta & 0& 0& tan\vartheta \\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ tan\vartheta & 0& 0& sec\vartheta \end{array}\right),{\Lambda }_3=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& sec\vartheta & -tan\vartheta \\ 0& 0& -tan\vartheta & sec\vartheta \end{array}\right)·}$$

Alternatively, one may apply the boosts first, since they commute with the corresponding quarter-turns and the transformation $\mathcal{P}\to {\Lambda }_3\mathcal{P}{\Lambda }_1$ preserves both $\mathcal{R}$ and ${\mathcal{R}}_2$, as it should.

6. Clifford’s Perspective

Let us point out that the matrix representations of our solutions are provided here only because it is a common practice: all calculations are made in the Clifford’s geometric algebra ${\mathrm{Cliff}}_{3,1}^{\circ }\cong {\mathrm{Cliff}}_3\cong {\mathbb{H}}^{\mathbb{C}}$ or rather, its projective version (projective quaternions were discovered by Rodrigues years before Hamilton gave his version, but he had no clue of Rodrigues’ invention which solves the ‘half-angle problem’ and so would help in the scientific debate) introduced in [26]. For example, the matrix transformations in the above decomposition may be expressed in ${\mathrm{Cliff}}_3$ as follows:

$$(1+cos\vartheta {\mathbf{e}}_{12}-sin\vartheta {\mathbf{e}}_3)(1-{\mathbf{e}}_{13})(1+cos\vartheta {\mathbf{e}}_{23}+sin\vartheta {\mathbf{e}}_1)=2cos\vartheta ({\mathbf{e}}_{12}+{\mathbf{e}}_{23})$$

and as the scalar part of the product is zero, projection onto the hyperplane ${\xi }_0=1$ in ${\mathbb{H}}^{\mathbb{C}}\cong {\mathbb{C}}^4$ yields a half-turn (geometrically, an infinite point in ${\mathbb{RP}}^3$) in the plane determined by ${\mathbf{e}}_{12}+{\mathbf{e}}_{23}$. Here, we identify $\{{\mathbf{a}}_i\}$ with the standard orthonormal basis $\{{\mathbf{e}}_i\}$ and denote:

$${\mathbf{e}}_{ij}={\mathbf{e}}_i{\mathbf{e}}_j=-{\mathbf{e}}_j{\mathbf{e}}_i,\mathcal{I}={\mathbf{e}}_1{\mathbf{e}}_2{\mathbf{e}}_3.$$

(16)

A pure rotation in this framework is represented by an even, invertible (i.e., non-vanishing) element $\xi$ of ${\mathrm{Cliff}}_3$ expressed in homogeneous coordinates and after projection onto ${\mathbb{RP}}^3$ as

$${\xi }_0+{\xi }_1{\mathbf{e}}_{23}+{\xi }_2{\mathbf{e}}_{31}+{\xi }_3{\mathbf{e}}_{12}={\langle \xi \rangle }_0+{\langle \xi \rangle }_2\stackrel{p}{\to }\mathbf{c}={\langle \xi \rangle }_0^{-1}{\langle \xi \rangle }_2$$

(17)

where ${\langle ·\rangle }_k$ denotes the grade projection, e.g., pure bivectors ${\langle \xi \rangle }_0=0$ are mapped to the plane at infinity and correspond to half-turns in $\mathrm{SO}(3)$, ${\mathbf{c}}^2=1$ yields a quarter-turn and $\mathbf{c}=0$ represents the trivial element, while inversion is just multiplication with $-1$ since $\langle \mathbf{c},-\mathbf{c}\rangle =0$. If we allow for odd grades as well (vectors and pseudo-scalars), this results in

$${\xi }_0+\mathit{\eta }+\xi -{\eta }_0\mathcal{I},\mathit{\eta }={\eta }_1{\mathbf{e}}_1+{\eta }_2{\mathbf{e}}_2+{\eta }_3{\mathbf{e}}_3$$

where $\mathit{\xi }$ stands for the bivector part, as before. Projections in this setting involve the center of the Lipschitz group ${\mathrm{Cliff}}_3^{\times }$, which includes the pseudo-scalar $\mathcal{I}$ and is isomorphic to ${\mathbb{C}}^{\times }$:

$${\xi }_0+\mathit{\eta }+\mathit{\xi }-{\eta }_0\mathcal{I}\stackrel{p}{\to }\mathbf{c}={\langle \mathbf{c}\rangle }_1+{\langle \mathbf{c}\rangle }_2\stackrel{\iota }{\to }\frac{{\xi }_0\mathit{\xi }+{\mathit{\eta }}_0\eta }{{\xi }_0^2+{\eta }_0^2}+i\frac{{\xi }_0\mathit{\eta }-{\eta }_0\mathit{\xi }}{{\xi }_0^2+{\eta }_0^2}·$$

(18)

Here, $\iota$ denotes the embedding into ${\mathbb{CP}}^3$ and by abuse of notation we let $\mathit{\xi },\mathit{\eta }$ represent both pure quaternions and real vectors. The sign for the pseudo-scalar part is chosen for convenience and $\mathcal{I}$ itself is mapped to the imaginary unit i. Note, that in the case of pure rotation (${\xi }_0\mathit{\eta }={\eta }_0\mathit{\xi }$) we end up with the familiar expression ${\xi }_0^{-1}\mathit{\xi }$, while in order to have a pure Lorentz boost, the condition ${\xi }_0\mathit{\xi }+{\eta }_0\mathit{\eta }=0$ needs to be met, which leads similarly to a parametrization of the unit ball ${\mathbb{B}}^3$ with $i{\xi }_0^{-1}\mathit{\eta }$. The Plücker relation is expressed in the form

$$({\xi }_0^2-{\eta }_0^2)\mathit{\xi }·\mathit{\eta }={\xi }_0{\eta }_0({\mathit{\xi }}^2-{\mathit{\eta }}^2)$$

(19)

and in the case ${\eta }_0=0$ it is reduced to $\mathit{\xi }\perp \mathit{\eta }$. Note, that the above construction yields more than the proper Lorentz group. For example, if (19) holds and $\mathit{\xi }·\mathit{\eta }=-1$, we hit the singularity ${\mathbf{c}}^2+1=0$ which can be written in this case also as $\mathit{\xi }\mathit{\eta }+\mathcal{I}=0$. Setting ${\xi }_0={\eta }_0=0$ leads to a proper half-turn if $\mathit{\eta }=0$ and a ‘forbidden’ boost in the case ${\mathit{\xi }}^2={\mathit{\eta }}^2$.

7. Kinematics and Relativity

Here, we discuss the above ideas in the continuous setting, so a convenient representation of the spatial kinematics is needed: either as a linear quaternion equation or a Ricatti ODE in the projective setting derived as follows: consider the Maurer–Cartan form associated with a smooth spatial trajectory $\mathbb{R}\ni t\to \mathcal{R}(t)\in \mathrm{SO}(3)$, which yields the angular velocity

$${\mathit{\omega }}_{+}^{\times }=\dot{\mathcal{R}}{\mathcal{R}}^t,{\mathit{\omega }}_{-}^{\times }={\mathcal{R}}^t\dot{\mathcal{R}}$$

(20)

respectively, in the stationary (lab) and co-moving (body) frame. Straightforward differentiation of the Cayley transform (2) allows for expressing the above formulas equivalently as

$${\mathit{\omega }}_{\pm }=\frac{2}{1+{\mathbf{c}}^2}(1\pm {\mathbf{c}}^{\times })\dot{\mathbf{c}}$$

(21)

and solving for $\dot{\mathbf{c}}$ via inversion, we end up with the (vector) Riccati equation

$$\begin{array}{c}\hfill \dot{\mathbf{c}}=\frac{1}{2}\left(1+\mathbf{c}{\mathbf{c}}^t\mp {\mathbf{c}}^{\times }\right){\mathit{\omega }}_{\pm }.\end{array}$$

(22)

Note, that the non-commutative term $\mathbf{c}\times \dot{\mathbf{c}}$ in (21) is associated with the geometric phase, e.g., Coriolis effect or Thomas precession [12,27]. In particular, when it is real, for a real $\mathit{\omega }$ it suffices to have $Im(\dot{\mathbf{c}})=0$ even if $\mathbf{c}$ is itself complex, as long as the Plücker condition $Im({\mathbf{c}}^2)=0$ is satisfied, providing a pure rotation on top of a Lorentz boost. One has $\dot{\mathbf{c}}\Vert {\mathit{\omega }}_{\pm }\in {\mathbb{R}}^3$ also from (22) in that setting. Now, let us try to find other complex paths $\mathbf{c}(t)$ in the Lorentz group that lead to real angular velocities (21) as we did in the discrete case using Euler decomposition. Denoting $\mathbf{c}=\mathit{\alpha }+i\mathit{\beta }$ with $\mathit{\alpha },\mathit{\beta }\in {\mathbb{RP}}^3$ and $\mathit{\alpha }\perp \mathit{\beta }$, we arrive at

$${\omega }_{\pm }=\frac{2}{1+{\mathit{\alpha }}^2-{\mathit{\beta }}^2}\left[(1\pm {\mathit{\alpha }}^{\times })\dot{\mathit{\alpha }}-\mathit{\beta }\times \dot{\mathit{\beta }}+i(1\pm {\mathit{\alpha }}^{\times })\dot{\mathit{\beta }}\pm i\mathit{\beta }\times \dot{\mathit{\alpha }}\right]$$

(23)

where the non-commutative terms in the real part account for the Coriolis and Thomas effects, while those in the imaginary part - for the Sagnac and Hall effects (see [26,27] for more details). Making the latter disappear means finding a subset of trajectories, for which

$$(1\pm {\mathit{\alpha }}^{\times })\dot{\mathit{\beta }}=\pm \dot{\mathit{\alpha }}\times \mathit{\beta }$$

(24)

i.e., the two contributions cancel each other. This effectively links the two paths $\mathit{\alpha }(t)$ and $\mathit{\beta }(t)$ on ${\mathbb{RP}}^3$. In particular, for a given $\mathit{\alpha }(t)$, this results in a linear ODE for $\mathit{\beta }(t)$ in the form

$$\dot{\mathit{\beta }}=\left[\frac{\pm 1}{1+{\mathit{\alpha }}^2}\left(\mathcal{I}+\mathit{\alpha }{\mathit{\alpha }}^t\mp {\mathit{\alpha }}^{\times }\right){\dot{\mathit{\alpha }}}^{\times }\right]\mathit{\beta }$$

(25)

which is resolved with a matrix exponent (possibly time-ordered or via Magnus expansion). If for example $\mathit{\alpha }(t)=\tau \mathbf{n}$ and $\tau =\infty$, then one has $\dot{\mathit{\beta }}=\pm \mathit{\alpha }{(\mathbf{n}\times \dot{\mathbf{n}})}^t\mathit{\beta }$. Also, for a given $\mathit{\beta }(t)$

$$({\mathit{\beta }}^2-\mathit{\beta }{\mathit{\beta }}^t)\dot{\mathit{\alpha }}=(\mathit{\beta }·\dot{\mathit{\beta }})\mathit{\alpha }\pm \mathit{\beta }\times \dot{\mathit{\beta }},({\mathit{\beta }}^2{\dot{\mathit{\beta }}}^t-\mathit{\beta }·\dot{\mathit{\beta }}{\mathit{\beta }}^t)\dot{\mathit{\alpha }}=\mathit{\beta }·\dot{\mathit{\beta }}{\dot{\mathit{\beta }}}^t\mathit{\alpha }$$

(26)

which for a spherical orbit $\mathit{\beta }(t)=\left|\mathit{\beta }\right|\mathbf{m}(t)$ in particular simplifies as ${\displaystyle \mathit{\alpha }(t)=\pm \int \mathbf{m}\times \mathrm{d}\mathbf{m}}$. If the unit vector $\mathbf{m}$ processes uniformly in a circular orbit, then $\mathit{\alpha }$ has fixed direction and linear growth, like a resonance amplitude. This corresponds to a plane rotation evolving with the inverse tangent of the time parameter, and the angular velocity remains real along the complex trajectories. Thus, certain kinematic effects of $\mathit{\beta }$ should be ‘canceled out’ by $\mathit{\alpha }$. The above solutions have quite straightforward interpretations in a relativistic context, e.g., a processing pure boost may be complemented by a moment-like rotation vector in order to eliminate the imaginary part of the generalized angular velocity vector (23). However, simply adding the two components in the complex vector parameter $\mathbf{c}=\mathit{\alpha }+i\mathit{\beta }$ is by no means equivalent to a composition, so we use (1) to express the corresponding factorization

$$\mathit{\alpha }+i\mathit{\beta }=\langle \tilde{\mathit{\alpha }},i\mathit{\beta }\rangle ,\mathit{\alpha }\perp \mathit{\beta }$$

(27)

which leads to a self-contradicting system of equations (implying $\tilde{\mathit{\alpha }}·\mathit{\alpha }={\mathit{\alpha }}^2=0$), namely:

$$\tilde{\mathit{\alpha }}={(1-\mathit{\beta }{\mathit{\beta }}^t)}^{-1}\mathit{\alpha },({\tilde{\mathit{\alpha }}}^{\times }+\mathit{\alpha }{\tilde{\mathit{\alpha }}}^t)\mathit{\beta }=0.$$

Hence, one cannot preserve the boost part in such decompositions, at least in the Plücker setting, while on the other hand, it is possible to write instead, as already pointed out in (13)

$$\mathit{\alpha }+i\mathit{\beta }=\langle i{\tilde{\mathit{\beta }}}_{+},\mathit{\alpha }\rangle =\langle \mathit{\alpha },i{\tilde{\mathit{\beta }}}_{-}\rangle ,{\tilde{\mathit{\beta }}}_{\pm }={({\mathit{\alpha }}^2+1)}^{-1}(1\pm {\mathit{\alpha }}^{\times })\mathit{\beta }.$$

(28)

Now, combining the above with (25) we can easily express the decomposition into a pair of mutually orthogonal rotations and boost corresponding to a Lorentz transformation satisfying the Plücker relation, such that the angular velocity vector given by Formula (21) is real.

But how to interpret this real angular velocity corresponding to a complex projective bivector? It helps to go back to the linear quaternion equations from which (22) are derived. They clearly show it depends on the initial condition whether one has a pure rotation or more general Lorentz kinematics even for ${\omega }_{\pm }\in {\mathbb{R}}^3$, so this merely yields invariance of the compact subgroup $\mathrm{SO}(3)$. However, one can use the above solutions to manipulate the observable relativistic effects via boost-rotation interactions. In physical terms we express

$${\displaystyle \mathit{\alpha }=tan\left(\frac{\phi }{2}\right)\mathbf{n},\mathit{\beta }=tanh\left(\frac{\psi }{2}\right)\frac{\mathbf{v}}{v},\psi ={tanh}^{-1}\left(\frac{v}{c}\right)}$$

(29)

where $\mathbf{v}$ is the velocity with $v=\left|\right|\mathbf{v}\left|\right|$ and $\psi$ the corresponding rapidity, c the speed of light in vacuum and $\mathbf{n}$, $\phi$ the axis-angle coordinates of the rotational motion. Thus, we easily evaluate the effect on the Lorentz factor of time dilation $\Delta t\to \gamma \Delta t$ with $\gamma =cosh\psi$ or the Thomas precession phase adjusting the electron spin in non-inertial relativistic frames as:

$${\mathit{\omega }}^{\circ }=\left(1-\gamma \right)\frac{\mathbf{v}\times \dot{\mathbf{v}}}{v^2}·$$

(30)

Note, that the Plücker condition in this context means that the translation motion takes place in the rotation plane. We also discussed the setting of screw motion (11) in which $\mathit{\alpha }\Vert \mathit{\beta }$. Both cases are interesting and have relevant applications but only the former is associated with an invariant plane in the ${\mathbb{R}}^{3,1}$ representation. In particular, for massless particles, e.g., gluons, the Lorentz transformation (2) is parabolic, as ${\mathbf{c}}^2=0$, and reduces to

$$\mathcal{R}(\mathbf{c})=1+2\mathbf{c}{\mathbf{c}}^t+2{\mathbf{c}}^{\times }.$$

(31)

Note, also that Formula (28) yields a straightforward linear relation between the boosts $i{\tilde{\mathit{\beta }}}_{\pm }$:

$$\langle -\mathit{\alpha },i{\tilde{\mathit{\beta }}}_{+},\mathit{\alpha }\rangle =i{\tilde{\mathit{\beta }}}_{+}\Rightarrow {\tilde{\mathit{\beta }}}_{-}={\mathcal{R}}^t(\mathit{\alpha }){\tilde{\mathit{\beta }}}_{+}$$

(32)

confirmed by the Cayley representation (2). As illustrated by the examples below, in some cases this technique provides the usual polar decomposition. However, it only changes the orientation of the invariant plane, keeping the rapidity (hence, the Lorentz factor) constant.

8. Geometric Phases in Electrodynamics

One may consider possible applications in electrodynamics based on the so-called complex representation of the electromagnetic field [14,30] $\mathbf{F}=\mathbf{B}+i\mathbf{E}$ where $\mathbf{E}$, $\mathbf{B}$ are pure quaternions corresponding to the electric and magnetic component (some authors put them in reverse order as they work in a split-quaternion basis spanned by the Pauli matrices). We think of the magnetic field as a relativistic effect (Thomas precession) associated with non-inertial reference frames (see [16]) as a consequence of Lorentz covariance. Note, that the Maxwell field (treated also as a tensor, principal bundle curvature, or an element of Clifford’s algebra of differential forms) is not a representative of $\mathrm{SO}(3,\mathbb{C})$, but rather, of its Lie algebra ${\mathfrak{sl}}_2$ so, in general, it cannot be directly associated to a projective biquaternion (vector-parameter) $\mathbf{c}=\mathit{\alpha }+i\mathit{\beta }$. However, in the isotropic case (31) one has equivalence between Cayley and exponential maps, namely:

$$\mathrm{Cay}({\mathbf{c}}^{\times })=exp(2{\mathbf{c}}^{\times }),{\mathbf{c}}^2=0$$

(33)

so at least for plane waves our representation remains valid and we can write

$$\mathbf{c}=\mathit{\alpha }+i\mathit{\beta }⟷\Theta =\left(\begin{array}{cc}{\mathit{\alpha }}^{\times }& \mathit{\beta }\\ {\mathit{\beta }}^t& 0\end{array}\right)⟶\mathrm{Cay}(\Theta )=exp(2\Theta )$$

(34)

where $\Theta$ is proportional to the Maxwell tensor. Even though (1) may not be relevant in this context, the above kinematic considerations are based solely on the Cayley map and thus still apply. But here (23), (25) and (26) have different interpretations—they describe the aforementioned relativistic concept of magnetism, hence various interactions with the particles’ spins. Manipulating the imaginary part of $\mathit{\omega }$ in this context is equivalent to controlling the Lorentz force and the measured Hall effect. There are other interesting applications of geometric phases in electrodynamics—some related to electronics and engineering, others to the theory of light propagation (we refer to [11] for an extensive review on the matter). One particular example is the interaction of electric fields with particle spins which allows for more convenient hardware in quantum computers (see [31]). In [32] one may find the simple mathematical treatment of the Thomas factor in the spin-orbit interaction, which in our approach emerges naturally from the complex extension of the algebra (see also [26]).

9. The 2 + 1 Setting

From a practical perspective, it would be interesting to see also how the cancellation and phase effects take place in the non-compact real form (Wigner little group [6]) of $\mathrm{SO}(3,\mathbb{C})$:

$${\mathrm{SO}}^{+}(2,1)\cong \mathrm{PSL}(2,\mathbb{R})\cong \mathrm{SU}(1,1)/\{\pm 1\}$$

related to classical hyperbolic geometry and quantum mechanical scattering. It is a toy model for special relativity with real implementations, for instance in graphenes (see [33]). In [27] we consider the Wigner rotation in this setting and its various physical applications: from quantum scattering to the falling cat problem [9,10]. The latter may be regarded as yet another application of classical geometric phases. The cat’s rotation during free fall seemingly violates the conservation principle for angular momenta while it is a result of a Coriolis-type effect emerging in a sequence of shear transformations. In the plane projection (e.g., Arnold’s cat) we may think of composing non-parallel $\mathrm{SL}(2,\mathbb{R})$ boosts generating a $\mathrm{U}(1)$ phase (see [9,27] for details). The monodromy matrix approach in quantum scattering leads to a quite similar mathematical description. Namely, one defines

$${\displaystyle \mathcal{M}=\left(\begin{array}{cc}a& b\\ \overline{b}& \overline{a}\end{array}\right)\in \mathsf{SU}(1,1),a=\frac{1}{\overline{t}},b=-\frac{\overline{r}}{\overline{t}}}$$

to be the monodromy matrix corresponding to a given scattering potential. Its entries are given by the complex transition and reflection coefficients (denoted, respectively, as t and r) relating the left and right free particle asymptotic solutions in the linear scattering problem:

$$\begin{array}{ccc}\hfill \mathrm{\Psi }(k,x)& \sim & {\mathrm{e}}^{ikx}+r(k){\mathrm{e}}^{-ikx},x\to -\infty \hfill \\ & \\ \hfill \mathrm{\Psi }(k,x)& \sim & t(k){\mathrm{e}}^{ikx},x\to \infty .\hfill \end{array}$$

(35)

Thus, a superposition of two scattering potentials $V_1$ and $V_2$ (in this order) yields a compound monodromy matrix $\mathcal{M}={\mathcal{M}}_2{\mathcal{M}}_1$. In particular, if we choose ${\mathcal{M}}_{1,2}$ to be purely hyperbolic, i.e., $t_{1,2}\in \mathbb{R}$, the Wigner rotation angle is given by the elliptic factor ${\mathcal{M}}_{\circ }$ in the composition $\mathcal{M}=\tilde{\mathcal{M}}{\mathcal{M}}_{\circ }$, e.g.,

$$\mathcal{M}=\frac{1}{\left|t\right|}\left(\begin{array}{cc}1& -\overline{r}{\mathrm{e}}^{2i\theta }\\ -r{\mathrm{e}}^{-2i\theta }& 1\end{array}\right)\left(\begin{array}{cc}{\mathrm{e}}^{i\theta }& 0\\ 0& {\mathrm{e}}^{-i\theta }\end{array}\right),\theta =\arg (t)·$$

(36)

Note, that as a real form of ${\mathrm{SO}}^{+}(3,1)$ the three-dimensional Lorentz group inherits the composition law (1) and the Cayley map construction (2) along with the generalized Euler decomposition (both real solutions for time-like axes and imaginary for space-like ones yield rotations) and the kinematic analogy (22). However, the non-euclidean metric $\eta =\mathrm{diag}(1,1,-1)$ takes part in calculating projectors, dot and cross products, which leads to another type of singularity, apart from the gimbal lock, referred to as ‘isotropic’ in [29]. One particular advantage of the 3D setting is that we can use a dual extension and the transfer principle to describe the non-homogeneous Poincaré group, like in the Euclidean case [34].

10. Back to Factorizations

Now, let us come back to matrix factorizations with broadened horizons and more examples. If we consider the generalized Euler decomposition problem for the proper Lorentz group in the $\mathrm{SO}(3,\mathbb{C})$ representation, the results given in (7) and (9) still apply. However, even if we choose normalized vectors ${\mathbf{a}}_k\in {\mathbb{C}}^3$ satisfying the Plücker condition ${\mathbf{a}}_k^2\in \mathbb{R}$, the complex parameters ${\tau }_k$ in the solution may violate it, leaving us with no invariant plane. Thus, one may start with ${\mathbf{a}}_k^2\in {\mathbb{S}}^2$ and then use (11) in order to expand into pairs of mutually commuting pure boosts and rotations (this approach has been demonstrated also in [28]).

Another possibility is to work with fixed ${\mathbf{a}}_1$ and define ${\mathbf{a}}_2={\mathbf{a}}_1\times \mathcal{R}{\mathbf{a}}_1$, thus satisfying the condition $r_{21}=g_{21}$ as long as ${\mathbf{a}}_1$ is not an eigenvector of $\mathcal{R}$ (or one has a singular solution):

$${\displaystyle {\tau }_1=\frac{{\rho }_1}{g_{11}},{\tau }_2=\frac{1}{g_{11}+r_{11}},{\rho }_k=\mathbf{c}·{\mathbf{a}}_k.}$$

(37)

Take for example ${\displaystyle \mathbf{c}={(1+2i,2i-1,-1)}^t}$ and ${\mathbf{a}}_1={(1,-1,0)}^t$. Then ${\tau }_1=1$ and ${\tau }_2{\mathbf{a}}_2=\frac{1}{3}{(2i+1,2i+1,4i-1)}^t$ so the $\mathrm{SO}(3,1)$ factorization (37) yields for $\Lambda (\mathbf{c})$ the result

$${\displaystyle \Lambda =\frac{1}{9}\left(\begin{array}{cccc}\hfill -2& \hfill 4& \hfill 5& \hfill 6\\ \hfill 1& \hfill -2& \hfill 2& \hfill 0\\ \hfill 2& \hfill 5& \hfill 4& \hfill 6\\ \hfill 0& \hfill 6& \hfill 6& \hfill 9\end{array}\right)\left(\begin{array}{cccc}\hfill 1& \hfill -2& \hfill -2& 0\\ \hfill -2& \hfill 1& \hfill -2& 0\\ \hfill 2& \hfill 2& \hfill -1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\end{array}\right)=\left(\begin{array}{cccc}\hfill 1& \hfill 0& \hfill 2& \hfill 2\\ \hfill 0& \hfill -1& \hfill 0& \hfill 0\\ \hfill 2& \hfill 0& \hfill 1& \hfill 2\\ \hfill 2& \hfill 0& \hfill 2& \hfill 3\end{array}\right)\left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill -1& 0\\ \hfill -1& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\end{array}\right)}$$

where we use (28) for the second equality to show the boost-rotation interactions. Our next example is borrowed from [35], where we propose and investigate a generalization of the classical Wigner decomposition. Consider a factorization of proper Lorentz transformations

$$\Lambda (\mathbf{c})=\Lambda (-\tilde{\mathbf{c}})\Lambda ({\mathbf{c}}_{\circ })\Lambda (\tilde{\mathbf{c}}),Re(\tilde{\mathbf{c}})=Im({\mathbf{c}}_{\circ })=0$$

(38)

into a boost and rotation, for which a convenient vector-parametric solution yields (cf. [35])

$${\mathbf{c}}_{=}\mu \mathit{\alpha },\tilde{\mathbf{c}}=\frac{i}{\mu +1}\frac{\mathit{\beta }\times \mathit{\alpha }}{{\mathit{\alpha }}^2},\mu =\sqrt{1-\frac{{\mathit{\beta }}^2}{{\mathit{\alpha }}^2}}·$$

(39)

Let us now take a specific example given by the matrix

$${\displaystyle \Lambda =\frac{1}{5}\left(\begin{array}{cccc}\hfill -4& \hfill 12& \hfill 3& \hfill 12\\ \hfill 12& \hfill -11& \hfill -4& \hfill -16\\ \hfill -3& \hfill 4& \hfill -4& \hfill 4\\ \hfill -12& \hfill 16& \hfill 4& \hfill 21\end{array}\right)}$$

for which we use (5) to obtain $\mathbf{c}={\left(4,3,4i\right)}^t$ and according to (39) arrive at

$${\displaystyle {\Lambda }_{\circ }=\frac{1}{125}\left(\begin{array}{cccc}\hfill 44& \hfill 108& \hfill 45& \hfill 0\\ \hfill 108& \hfill -19& \hfill -60& \hfill 0\\ \hfill -45& \hfill 60& \hfill -100& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1\end{array}\right),\tilde{\Lambda }=\frac{1}{75}\left(\begin{array}{cccc}\hfill 93& \hfill -24& \hfill 0& \hfill -60\\ \hfill -24& \hfill 107& \hfill 0& \hfill 80\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill -60& \hfill 80& \hfill 0& \hfill 125\end{array}\right)·}$$

Similarly, the above allows us to express the pure rotation $\Lambda ({\mathbf{c}}_{\circ })$ in the form

$$\Lambda ({\mathbf{c}}_{\circ })=\Lambda (\tilde{\mathbf{c}})\Lambda (i{\tilde{\mathit{\beta }}}_{+})\Lambda (\mathit{\alpha })\Lambda (-\tilde{\mathbf{c}})=\Lambda (\tilde{\mathbf{c}})\Lambda (\mathit{\alpha })\Lambda (i{\tilde{\mathit{\beta }}}_{-})\Lambda (-\tilde{\mathbf{c}})=\Lambda \left[\Lambda (\tilde{\mathbf{c}})\mathbf{c}\right].$$

where we use (28), so in the end $\Lambda ({\mathbf{c}}_{\circ })$ and $\Lambda (\mathit{\alpha })$ are conjugated with a (symmetric) boost.

Our last example is numerical and concerns a Euler-type decomposition into four factors

$$\mathcal{R}(\mathbf{n},\phi )=\mathcal{R}({\mathbf{a}}_3,{\phi }_3)\mathcal{R}({\mathbf{a}}_1,\psi )\mathcal{R}({\mathbf{a}}_2,{\phi }_2)\mathcal{R}({\mathbf{a}}_1,{\phi }_1)$$

for which the additional parameter $\psi$ allows for optimization of the path length in $\mathrm{SO}(3)$ or avoiding the ‘gimbal lock’ singularity. If we choose the unit vectors ${\mathbf{a}}_i,\mathbf{n}\in {\mathbb{S}}^2$ expressed as

$${\mathbf{a}}_1=\{{28}^{\circ },{32}^{\circ }\},{\mathbf{a}}_2=\{{12}^{\circ },{13}^{\circ }\},{\mathbf{a}}_3=\{0^{\circ },7^{\circ }\},\mathbf{n}=\{{53}^{\circ },{36}^{\circ }\}$$

in spherical coordinates and the angle of compound rotation to be $\phi ={111}^{\circ }$, a generalized version of Formula (7) yields our solution in the form (rounding here is just for convenience)

$${\phi }_1\approx {97}^{\circ }-73i,{\phi }_2\approx 128i-{67}^{\circ },{\phi }_3\approx {40}^{\circ }-80i,\psi \approx {40}^{\circ }.$$

Obviously complex angles correspond to loxodromic Lorentz transformations with parallel rotation and boost counterpart, so one may easily factorize further using (11) and (12), but the interesting feature here is that $\psi$ is real, so we effectively control the relativistic effect (represented by the imaginary part of $\mathcal{R}$) using a pure rotation in the sequence changing the value of $\psi$ results in a generic Lorentz transformation instead of the special orthogonal matrix $\mathcal{R}(\mathbf{n},\phi )$. In other words, inserting an appropriately chosen rotation in the sequence of ${\mathrm{SO}}^{+}(3,1)$ mappings eliminates the net relativistic effect from the observer’s point of view.

11. Discussion

The present study suggests a consistent geometric approach with applications in various branches of physics and engineering. Specific examples are given but without too many technical details that would distract the main focus. One particular benefit is in the context of matrix factorizations, such as the Iwasawa decomposition for $\mathrm{PSL}(2,\mathbb{R})$ or the polar decomposition for the monodromy matrix considered above. These results generalize easily to $\mathrm{SO}(4,\mathbb{C})$ and its other real forms like $\mathrm{SO}(2,2)$ or ${\mathrm{SO}}^{\ast }(4)$, and in the Plücker setting allow for higher-dimensional embeddings as well (once again, we shall refer to [28] for details).

The method used here has many advantages we attempt to emphasize with examples. Firstly, it emerges directly from the Clifford algebra and is thus easily interpretable in geometric terms. Secondly, it provides a quite natural parametrization of the rotation and Lorentz groups, preserving the projective space topology, unlike Euler angles for instance (which use the torus for a model). Hence, no coordinate singularities appear in this representation. Thirdly, it is quite efficient both computation-wise and from a theoretical perspective: no redundant parameters are required and numerical algorithms run much faster (compared for instance with the usual matrix multiplication), while at the same time, the simplicity of the analytic expressions (typically in terms of rational functions) allows for obtaining solutions to non-trivial problems in a casual manner. Lie group factorizations, geometric phases in kinematics and the relativistic extension via complexification are just a few examples to support this. In the present paper we focus mainly on the latter, as it yields deep geometric insight for both theoretical speculations based on analogies and practical problem-solving techniques in engineering, but at the same time, has been neglected in other research papers that aim only at obtaining the end result for a given specific task. Our goal here is to fill this gap (at least partially), discussing mostly complex extensions of already existing solutions in the real setting and their interpretation in the more general Lorentzian context. On the engineering front, this provides plenty of useful applications, such as manipulating the outcome of measurable relativistic effects using pure rotations and vice versa, as well as analogous results in optics and micro-electronics, quantum computation and scattering of elementary particles. Since these solutions have been obtained in other studies, we rely extensively on them without providing detailed derivations and proofs, but that is the price one has to pay for keeping the volume within reasonable limits.