In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group. As there are three types of complex number, so there are three types of biquaternion:
Contents
- Definition
- Linear representation
- Subalgebras
- Algebraic properties
- Relation to Lorentz transformations
- Associated terminology
- As a composition algebra
- References
This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of Royal Irish Academy 1844 & 1850 page 388). 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 presentation of the Lorentz group, which is the foundation of special relativity.
The algebra of biquaternions can be considered as a tensor product C ⊗ H (taken over the reals) where C is the field of complex numbers and H is the algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the (real) quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2×2 complex matrices M2(C).
Definition
Let {1, i, j, k} be the basis for the (real) quaternions ℍ, and let u, v, w, x be complex numbers, then
q = u 1 + v i + w j + x kis a biquaternion. To distinguish square roots of minus one in the biquaternions, Hamilton and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field ℂ by h since there is an i in the quaternion group. Then
h i = i h, h j = j h, and h k = k h since h is a scalar.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.
Linear representation
Note the matrix product
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 the matrix group 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 is isomorphic to the biquaternion ring.
Subalgebras
Considering the biquaternion algebra over the scalar field of real numbers R, the set {1, h, i, hi, j, hj, k, hk } forms a basis so the algebra has eight real dimensions. Note the squares of the elements hi, hj, and hk are all plus one, for example,
Then the subalgebra given by
A third subalgebra called coquaternions is generated by hj and hk. First note that (hj)(hk) = (−1) i, and that the square of this element is −1. These elements generate the dihedral group of the square. The linear subspace with basis {1, i, hj, hk} thus is closed under multiplication, and forms the coquaternion algebra.
In the context of quantum mechanics and spinor algebra, the biquaternions hi, hj, and hk (or their negatives), viewed in the M(2,C) representation, are called Pauli matrices.
Algebraic properties
The biquaternions have two conjugations:
where
Note that
Clearly, if
Relation to Lorentz transformations
Consider now the linear subspace
M is not a subalgebra since it is not closed under products; for example
Proposition: If q is in M, then
proof:
Definition: Let biquaternion g satisfy
Proposition: If q is in M, then T(q) is also in M.
proof:
Proposition:
proof: Note first that g g * = 1 means that the sum of the squares of its four complex components is one. Then the sum of the squares of the complex conjugates of these components is also one. Therefore,
Associated terminology
As the biquaternions have been a fixture of linear algebra since the beginnings of mathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. The transformation group
To view
Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because
The space of biquaternions has a natural topology through the Euclidean metric on 8-space. With respect to this topology, G is a topological group. Moreover, it has analytic structure making it a six-parameter Lie group. Consider the subspace of bivectors
Many of the concepts of special relativity are illustrated through the biquaternion structures laid out. The subspace M corresponds to Minkowski space, with the four coordinates giving the time and space locations of events in a resting frame of reference. Any hyperbolic versor exp(ahr) corresponds to a velocity in direction r of speed c tanh a where c is the velocity of light. The inertial frame of reference of this velocity can be made the resting frame by applying the Lorentz boost T given by g = exp(0.5ahr) since then
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".
As a composition algebra
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
If
When (a,b)* is written as a 4-vector of ordinary complex numbers,
The biquaternions form an example of a quaternion algebra, and it has norm
Two biquaternions p and q satisfy