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

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

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About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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