In the previous post we have seen that a quaternion is defined as

where

and , , , are real numbers.

**Biquaternion**

The biquaternion is a complexified quaternion

where , , , are complex numbers.

The basis elements , , are the same as in quaternions.

**Norm**

The norm of a biquaternion is defined as

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 , , , are complex numbers.

**Conjugate
**

The conjugate of a biquaternion is defined as

**Inverse
**

If the norm is non-zero then there is an inverse for a biquaternion.

**Algebra of biquaternions**

The algebra of quaternions is a normed division algebra. The algebra of complexified quaternions (biquaternions) 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 exist. A particular solution 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

[5] https://sureshemre.wordpress.com/2010/12/05/golden-biquaternions-3-generations-and-spin/