Dual Hyperquaternion Poincaré Groups

Abstract

A new representation of the Poincaré groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincaré generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators. An extended multivector calculus is introduced (allowing a possible sign change of the metric or of the exterior product). The Poincaré groups are formulated as a dual extension of hyperquaternion pseudo-orthogonal groups. The canonical decomposition and the invariants are discussed. As concrete example, the 4D Poincaré group is examined together with a numerical application. Finally, the hyperquaternion representation is compared to the quantum mechanical one. Potential applications include in particular, moving reference frames and computer graphics.

Appendices

Appendix A: Orthogonal Plane Symmetry

For the convenience of the reader, we derive here the formula of Eq. (4.3) [12, 13]. The orthogonal symmetric $x^{\prime }$ of a vector x with respect to a plane orthogonal to a vector u satisfies the equations

$$\begin{aligned} x^{\prime }=x+ku,\quad u\cdot \left( \frac{x^{\prime }+x}{2}\right) =0 \left( k\in {\mathbb {R}}\right) . \end{aligned}$$

(A.1)

Hence,

$$\begin{aligned} u\cdot \left( x+\frac{ku}{2}\right)= & {} 0\Rightarrow k=\frac{-2u\cdot x}{u\cdot u}, \end{aligned}$$

(A.2)

$$\begin{aligned} x^{\prime }= & {} x-\frac{2\left( u\cdot x\right) }{u\cdot u}=x-\frac{\left( ux+xu\right) u}{uu}=\frac{uxu}{uu_{c}}. \end{aligned}$$

(A.3)

Appendix B: Lie Algebra of the nD-Poincaré Group

We first give the Lie algebra of the restricted Poincaré group $P_{+}^{\uparrow }$ and then of the full group P.

1.1 B.1. Restricted Group

Consider an nD space imbedded in an $n+1$ hyperquaternion algebra having the generators $e_{1},e_{2},\ldots , e_{n},e_{n+1}$. The Lie generators of the rotations are

$$\begin{aligned} M_{ij}=\frac{1}{2}e_{i}e_{j} \quad \left( 1\le i,j\le n,\, i < j\right) . \end{aligned}$$

(B.1)

The Lie commutator being defined as $\left[ A,B\right] =AB-BA,$ one obtains for $i\ne j=r\ne s$ and

$$\begin{aligned} \left[ M_{ij},M_{rs}\right]= & {} \frac{1}{4}\left( e_{i}e_{j} e_{r}e_{s}-e_{r}e_{s}e_{i}e_{j}\right) \nonumber \\= & {} \frac{1}{4}\left( e_{i}e_{j} e_{r}e_{s}+e_{i}e_{r}e_{j}e_{s}\right) \nonumber \\= & {} \frac{1}{2}\eta _{jr}e_{i}e_{s}=\eta _{jr}M_{is} \end{aligned}$$

(B.2)

with $\eta _{jr}=\left( e_{j}e_{r}+e_{r}e_{j}\right) /2.$ Similarly, one has

$$\begin{aligned} \left[ M_{ij},M_{rs}\right]= & {} \eta _{is}M_{jr}\quad \left( j\ne i=s\ne r\right) , \end{aligned}$$

(B.3)

$$\begin{aligned} \left[ M_{ij},M_{rs}\right]= & {} -\eta _{js}M_{ir}\quad \left( i\ne j=s\ne r\right) ,\end{aligned}$$

(B.4)

$$\begin{aligned} \left[ M_{ij},M_{rs}\right]= & {} -\eta _{ir}M_{js}\quad \left( j\ne i=r\ne s\right) ; \end{aligned}$$

(B.5)

combining all possible cases for the rotations one gets

$$\begin{aligned} \left[ M_{ij},M_{rs}\right] =\eta _{jr}M_{is}+\eta _{is}M_{jr}-\eta _{js}M_{ir}-\eta _{ir}M_{js}. \end{aligned}$$

(B.6)

For the nD-translations, the generators are

$$\begin{aligned} M_{\left( n+1\right) i}=\frac{1}{2}\varepsilon e_{n+1}e_{i} \quad \left( 1\le i\le n,\,\varepsilon ^{2}=0,\,e_{n+1}^{2}=-1\right) \end{aligned}$$

(B.7)

(for $e_{n+1}^{2}=1,$ the one takes $M_{i\left( n+1\right) }=-M_{\left( n+1\right) i}$). One has the relations

$$\begin{aligned} \left[ M_{\left( n+1\right) i},M_{\left( n+1\right) j}\right] =0 \quad \left( \forall i,j\right) \end{aligned}$$

(B.8)

and for $i\ne j=k$

$$\begin{aligned} \left[ M_{ij},M_{\left( n+1\right) k}\right]= & {} \frac{\varepsilon }{4}\left( e_{i}e_{j}e_{\left( n+1\right) }e_{k}-e_{\left( n+1\right) }e_{k}e_{i}e_{j}\right) \nonumber \\= & {} \frac{\varepsilon e_{n+1}}{4}\left( e_{i}e_{j}e_{k}+e_{k}e_{j}e_{i}\right) \nonumber \\= & {} \frac{\varepsilon e_{n+1}}{2}\eta _{jk}e_{i}=\eta _{jk}M_{\left( n+1\right) i}; \end{aligned}$$

(B.9)

similarly, for $k=i\ne j$, one has      

$$\begin{aligned} \left[ M_{ij},M_{\left( n+1\right) k}\right] =-\eta _{ik}M_{\left( n+1\right) j}. \end{aligned}$$

(B.10)

Combining the two cases above, one obtains for the translations

$$\begin{aligned} \left[ M_{ij},M_{\left( n+1\right) k}\right] =\eta _{jk}M_{\left( n+1\right) i}-\eta _{ik}\text { }M_{\left( n+1\right) j.} \end{aligned}$$

(B.11)

Projecting the plane $M_{\left( n+1\right) i}$ on the space $V_{n}$ one obtains, for $e_{n+1}^{2}=-1,$ the vector

$$\begin{aligned} P_{i}=e_{n+1}M_{\left( n+1\right) i}=\frac{\varepsilon }{2} e_{n+1}e_{n+1}e_{i}=-\frac{\varepsilon }{2}e_{i}. \end{aligned}$$

(B.12)

For $e_{n+1}^{2}=+1,$ one has

$$\begin{aligned} P_{i}=e_{n+1}M_{i\left( n+1\right) }=\frac{\varepsilon }{2} e_{n+1}e_{i}e_{n+1}=-\frac{\varepsilon }{2}e_{i}. \end{aligned}$$

(B.13)

The complete Lie algebra of the restricted Poincaré group can thus be expressed in the standard abstract form

$$\begin{aligned} \left[ M_{ij},M_{rs}\right]= & {} \eta _{jr}M_{is}+\eta _{is}M_{jr}-\eta _{js}M_{ir}-\eta _{ir}M_{js}, \end{aligned}$$

(B.14)

$$\begin{aligned} \left[ P_{i},P_{j}\right]= & {} 0 ,\end{aligned}$$

(B.15)

$$\begin{aligned} \left[ M_{ij},P_{k}\right]= & {} \eta _{jk}P_{i}-\eta _{ik}P_{j}. \end{aligned}$$

(B.16)

1.2 B.2. Full Group

The other components of the full Poincaré group being obtained from the restricted one through multiplication by a vector $e_{k}$, one has besides the above relations the following ones

$$\begin{aligned} \left[ M_{ij},e_{k}\right]= & {} \eta _{jk}e_{i}-\eta _{ik}e_{j}, \end{aligned}$$

(B.17)

$$\begin{aligned} \left[ M_{\left( n+1\right) i},e_{k}\right]= & {} \eta _{ik}\left( \varepsilon e_{n+1}\right) . \end{aligned}$$

(B.18)

Appendix C: Multivector Structure of $\mathbb {H}\otimes \mathbb {H}\otimes {\mathbb {C}}$

The hyperquaternion algebra $\mathbb {H}\otimes \mathbb {H}\otimes {\mathbb {C}}\simeq C_5(2,3)$ can be viewed as an eight-dimensional vector space over the (real) quaternion algebra with the basis $\{1,l,m,n\}$. Thus, any hyperquaternion A can be written as a linear combination of eight linearly independent basis elements (1, I, J, K, i, iI, iJ, iK) with quaternionic coefficients $\left[ q_{s}\right] =a_{s}+b_{s}l+c_{s}m+d_{s}n$ $(1\le s \le 8)$ as

$$\begin{aligned} A =\left[ q_{1}\right] +I\left[ q_{2}\right] +J\left[ q_{3}\right] +K\left[ q_{4}\right] +i\left[ q_{5}\right] +iI\left[ q_{6}\right] +iJ\left[ q_{7}\right] +iK \left[ q_{8}\right] .\qquad \end{aligned}$$

(C.1)

The multivector structure of A can be derived from Eq. (5.1) as follows:

$$\begin{aligned} I=e_{0123},\, J=e_{1234},\, K=e_{04},\, l=e_{23}, \, m=-e_{13}, \, n=e_{12},\, i=-e_{01234}\nonumber \\ \end{aligned}$$

(C.2)

where $e_{0123}=e_{0}e_{1}e_{2}e_{3},$ etc. Then, the product of two elements A, B can be implemented (algebraically or numerically) in Mathematica with its quaternion product, e.g., $\left[ q_{i}\right] **\left[ q_{j}\right] .$ For the convenience of the reader, the complete multivector structure is given in the table below:

$$\begin{aligned} \left[ \begin{array}{llll} 1 &{} l=e_{2}e_{3} &{} m=e_{3}e_{1} &{} n=e_{1}e_{2}\\ I=e_{0}e_{1}e_{2}e_{3} &{} Il=e_{1}e_{0} &{} Im=e_{2}e_{0} &{} In=e_{3}e_{0}\\ J=e_{1}e_{4}e_{2}e_{3} &{} Jl=e_{4}e_{1} &{} Jm=e_{4}e_{2} &{} Jn=e_{4}e_{3}\\ K=e_{0}e_{4} &{} Kl=e_{0}e_{4}e_{2}e_{3} &{} Km=e_{4}e_{0}e_{1}e_{3} &{} Kn=e_{0}e_{4}e_{1}e_{2}\\ i=e_{0}e_{4}e_{1}e_{2}e_{3} &{} il=e_{4}e_{0}e_{1} &{} im=e_{4}e_{0}e_{2} &{} in=e_{4}e_{0}e_{3}\\ iI=e_{4} &{} iIl=e_{4}e_{2}e_{3} &{} iIm=e_{1}e_{4}e_{3} &{} iIn=e_{4}e_{1}e_{2}\\ iJ=e_{0} &{} iJl=e_{0}e_{2}e_{3} &{} iJm=e_{1}e_{0}e_{3} &{} iJn=e_{0}e_{1}e_{2}\\ iK=e_{1}e_{2}e_{3} &{} iKl=e_{1} &{} iKm=e_{2} &{} iKn=e_{3} \end{array}\right] . \end{aligned}$$