Biquaternion

This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of the Royal Irish Academy 1844 & 1850 page 388[1]). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity.

is a biquaternion.[4]: 639  To distinguish square roots of minus one in the biquaternions, Hamilton[4]: 730 [5] and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field C by h to avoid confusion with the i in the quaternion group. Commutativity of the scalar field with the quaternion group is assumed:

Hamilton introduced the terms bivector, biconjugate, bitensor, and biversor to extend notions used with real quaternions H.

Hamilton's primary exposition on biquaternions came in 1853 in his Lectures on Quaternions. The editions of Elements of Quaternions, in 1866 by William Edwin Hamilton (son of Rowan), and in 1899, 1901 by Charles Jasper Joly, reduced the biquaternion coverage in favor of the real quaternions.

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 C. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor. The algebra of biquaternions forms a composition algebra and can be constructed from bicomplex numbers. See § As a composition algebra below.

Because h is the imaginary unit, each of these three arrays has a square equal to the negative of the identity matrix. When this matrix product is interpreted as i j = k, then one obtains a subgroup of matrices that is isomorphic to the quaternion group. Consequently,

represents biquaternion q = u 1 + v i + w j + x k. Given any 2 × 2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring M(2,C) is isomorphic[6] to the biquaternion ring.

Considering the biquaternion algebra over the scalar field of real numbers R, the set

forms a basis so the algebra has eight real dimensions. The squares of the elements hi, hj, and hk are all positive one, for example, (hi)2 = h2i2 = (−1)(−1) = +1.

is ring isomorphic to the plane of split-complex numbers, which has an algebraic structure built upon the unit hyperbola. The elements hj and hk also determine such subalgebras.

In the context of quantum mechanics and spinor algebra, the biquaternions hi, hj, and hk (or their negatives), viewed in the M2(C) representation, are called Pauli matrices.

After the introduction of spinor theory, particularly in the hands of Wolfgang Pauli and Élie Cartan, the biquaternion representation of the Lorentz group was superseded. The new methods were founded on basis vectors in the set

which is called the complex light cone. The above representation of the Lorentz group coincides with what physicists refer to as four-vectors. Beyond four-vectors, the standard model of particle physics also includes other Lorentz representations, known as scalars, and the (1, 0) ⊕ (0, 1)-representation associated with e.g. the electromagnetic field tensor. Furthermore, particle physics makes use of the SL(2, C) representations (or projective representations of the Lorentz group) known as left- and right-handed Weyl spinors, Majorana spinors, and Dirac spinors. It is known that each of these seven representations can be constructed as invariant subspaces within the biquaternions.[8]

Although W.R. Hamilton introduced biquaternions in the 19th century, its delineation of its mathematical structure as a special type of algebra over a field was accomplished in the 20th century: the biquaternions may be generated out of the bicomplex numbers in the same way that Adrian Albert generated the real quaternions out of complex numbers in the so-called Cayley–Dickson construction. In this construction, a bicomplex number (w,z) has conjugate (w,z)* = (w, – z).

The biquaternion is then a pair of bicomplex numbers (a,b), where the product with a second biquaternion (c, d) is

The biquaternions form an example of a quaternion algebra, and it has norm