## 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 [1] and a recent review by K.Morita [2].

The biquaternions also form the core of the “algebrodynamics over complex space” paradigm discussed by V.V.Kassandrov [3] and the references therein. Another major effort pointing out the “algebraic design” of physics was developed by G.M.Dixon [4].

Golden biquaternions

In the biquaternionic formulation of relativistic quantum mechanics biquaternions represent the space-time points. In 2009, I proposed [5] that we can have an alternative view. We can use the biquaternion to represent the particle itself.

In reference [5] 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

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

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

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

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