## Biquaternion

In the previous post we have seen that a quaternion is defined as $q = \mathbf{1} z_0 + \mathbf{i} z_1 + \mathbf{j} z_2 + \mathbf{k} z_3$

where $\mathbf{i^2} = \mathbf{j^2} = \mathbf{k^2} = \mathbf{ijk} = -1$ $\mathbf{ij}=-\mathbf{ji}$ $\mathbf{jk}=-\mathbf{kj}$ $\mathbf{ki}=-\mathbf{ik}$

and $z_0$, $z_1$, $z_2$, $z_3$ are real numbers.

Biquaternion

The biquaternion is a complexified quaternion $b = \mathbf{1} z_0 + \mathbf{i} z_1 + \mathbf{j} z_2 + \mathbf{k} z_3$

where $z_0$, $z_1$, $z_2$, $z_3$ are complex numbers.

The basis elements $\displaystyle \mathbf{i}$, $\displaystyle \mathbf{j}$, $\displaystyle \mathbf{k}$ are the same as in quaternions.

Norm

The norm of a biquaternion is defined as $N^2 (b)=$ $z_{0}^{2}$ $+ z_{1}^{2}$ $+ z_{2}^{2}$ $+ z_{3}^{2}$

The definition of the norm is the same as in the quaternion with one important difference. The norm of a quaternion can never be a negative number. The norm of a biquaternion, on the other hand, can be negative or even zero because $z_0$, $z_1$, $z_2$, $z_3$ are complex numbers.

Conjugate

The conjugate of a biquaternion is defined as $\overline{b} = \mathbf{1} z_0 - \mathbf{i} z_1 - \mathbf{j} z_2 - \mathbf{k} z_3$

Inverse

If the norm is non-zero then there is an inverse for a biquaternion. $\displaystyle b^{-1} =\frac{\overline{b}}{N^2 (b)}$

Algebra of biquaternions

The algebra of quaternions $\mathbb{H}$ is a normed division algebra. The algebra of complexified quaternions (biquaternions) $\mathbb{P} = \mathbb{H} \otimes \mathbb{C}$ is not a division algebra since the norm can be zero.

Applications of biquaternions

Biquaternionic formulation of relativistic quantum mechanics is well established as demonstrated by the references in  and a recent review by K.Morita .

The biquaternions also form the core of the “algebrodynamics over complex space” paradigm discussed by V.V.Kassandrov  and the references therein. Another major effort pointing out the “algebraic design” of physics was developed by G.M.Dixon .

Golden biquaternions

In the biquaternionic formulation of relativistic quantum mechanics biquaternions represent the space-time points. In 2009, I proposed  that we can have an alternative view. We can use the biquaternion to represent the particle itself.

In reference  I show that biquaternions satisfying the golden condition $g-1/g=1$  exist. A particular solution $Q$  involving Steiner’s Roman surface may be relevant in modeling 3 generations of fermions.  I call these special solutions ‘golden biquaternions.’

The golden biquaternion and its physical interpretation as the fermion predicts exactly 3 generations and 1/2,  3/2,  2 as the only possibilities for particle spin and hints at the composite nature of bosons. The ‘golden biquaternion as fermion’ idea works if there are 3 fundamental variables that are coupled in a pairwise fashion.

References

 Andre Gsponer and Jean-Pierre Hurni, “Quaternions in mathematical physics (2): Analytical bibliography”, arXiv:math-ph/0511092.

 Katsusada Morita, “Quaternions, Lorentz Group and the Dirac Theory”, Progress of Theoretical Physics, Vol.117, No.3 (2007)

 Vladimir V. Kassandrov, “Algebrodynamics over Complex Space and Phase Extension of the Minkowski Geometry”, arXiv:gr-qc/0602088. (2006)

 Geoffrey M.Dixon, “Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics”, Springer (1994), Kluwer Academic Publishers (2002), ISBN: 0792328906 / 9780792328902 