Biquaternion

In mathematics, a biquaternion (or complex quaternion) is an element of the quaternion algebra over the complex numbers. The concept of a biquaterion was first mentioned by William Rowan Hamilton in the nineteenth century. William Kingdon Clifford used the same name in reference to a different algebra. See Clifford biquaternion.

Definition

Let {1, i, j, k} be the basis for the (real) quaternions, and let u, v, w, x be complex numbers, then

q = u 1 + v i + w j + x k

is a biquaternion. The complex scalars are assumed to commute with the quaternion basis vectors (e.g. vj = jv). Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers. The algebra of biquaternions is associative, but not commutative.

The algebra of biquaternions can be consider as a tensor product CH where C is the field of complex numbers and H is the algebra of real quaternions.

Place in ring theory

Linear representation

Note the matrix product

\begin{pmatrix}i & 0\\0 & -i\end{pmatrix}\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}

\begin{pmatrix}0 & i\\i & 0\end{pmatrix}

where each of these three arrays has a square equal to the negative of the identity matrix. When the matrix product is interpreted as i j = k, then one obtains a subgroup of the matrix group that is isomorphic to the Quaternion group.Consequently

\begin{pmatrix}u+iv & w+ix\\-w+ix & u-iv\end{pmatrix}

represents biquaternion q.

Given any 2x2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring is isomorphic to the biquaternion ring.

Alternative complex plane

Suppose we take w to be purely imaginary, w = b ι, where ι ι = - 1. (Here one uses iota instead of i for the complex imaginary to be distinct from quaternion i.) Now when r = w j, then its square is r r = (w j )(w j ) = (w w)(j j ) = b b (-1)(-1) = b². In particular, when b = 1 or –1, then r ² = + 1. This development shows that the biquaternions are a source of "algebraic motors" like r that square to +1. Then {a + b ι j : a, b ∈ R } is a subring of biquaternions isomorphic to the split-complex number ring.

Application in relativity physics

Lorentz group presentation

The biquaternions ιk = σ₁, ιj = σ₂, and −ιi = σ₃ were used by Alexander MacFarlane and later, in their matrix form by Wolfgang Pauli. They have come to be known as Pauli matrices.They each square to the identity matrix and hence the subplane {a + b σ ; a, b ∈ R} generated by one of them in the biquaternion ring is isomorphic to the ring of split-complex numbers.Hence a Pauli matrix σ generates a one-parameter group {u : u = exp(a σ), a ∈ R} whose actions on the subplane are hyperbolic rotations.The Lorentz group is a six-parameter Lie group, three parameters of which (e.g. subgroups generated by Pauli matrices) are associated with hyperbolic rotations, sometimes called boosts.The other three parameters correspond to ordinary rotations in space, a facility of real quaternion action known as quaternions and spatial rotation.The usual quadratic form view of this presentation is that u² + v² + w² + x² = q q*is preserved by the orthogonal group on the biquaternions when viewed as C⁴.When u is real and v, w, and x are pure imaginary, then one has a subspace M=R⁴ convenient to model spacetime.

Since the algebra (matrix or biquaternion) centers on the Lorentz group symmetry and the leading idea (spacetime) is relegated to a half of the whole ring, there is the appearance of inverted priority, something of a literary conceit.The willy-nilly kinematic idea behind the Lorentz group does not take into account concomitants of kinematic orientation such as setting a horizon, acceleration-rotation interaction, or suitable model application such as practiced in traditional analytic geometry.An alternative kinematic approach comes by way of coquaternions.

The actual exhibition of individual Lorentz transformations involves extensions of inner automorphisms of the group of units of biquaternions to the singular elements through inversive ring geometry.

References

Lanczos, Cornelius (1949) The Variational Principles of Mechanics, University of Toronto Press, pp. 304-12.
Silberstein, L. (May 1912) "Quaternionic form of relativity", Philosophy Magazine,series 6, '23:790-809.
Silberstein, L. The Theory of Relativity, 1914.

Synge, J.L. (1972) Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices Communications of the Dublin Institute for Advanced Studies, series A, #21, 67 pages.
Kilmister, C.w.(1994) Eddington's search for a fundamental theory, Cambridge University Press 0521371651, pages 121,122,179,180.

Quaternions | Ring theory | Special relativity