In
abstract algebra, the biquaternions are the numbers w + xi + yj + zk, where w, x, y, and z are
complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the
quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:
The algebra of biquaternions can be considered as a
tensor productC ⊗RH, where C is the
field of complex numbers and H is the
division algebra of (real)
quaternions. In other words, the biquaternions are just the
complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2 × 2 complex matrices M2(C). They are also isomorphic to several
Clifford algebras including C ⊗RH = Cl[0] 3(C) = Cl2(C) = Cl1,2(R),[2] the
Pauli algebraCl3,0(R),[3][4] and the even part Cl[0] 1,3(R) = Cl[0] 3,1(R) of the
spacetime algebra.[5]
A Biquaternion multiplication table[6] is shown below:
Definition
Let {1, i, j, k} be the basis for the (real)
quaternionsH, and let u, v, w, x be complex numbers, then
is a biquaternion.[7] To distinguish square roots of minus one in the biquaternions, Hamilton[8][9] 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 favour of the real quaternions.
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 ij = k, then one obtains a
subgroup of matrices that is
isomorphic to the
quaternion group. Consequently,
represents biquaternion q = u1 + vi + wj + xk.
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 ringM(2, C) is isomorphic[10] to the biquaternion
ring.
Subalgebras
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.
A third subalgebra called
coquaternions is generated by hj and hk. It is seen 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 M2(C) representation, are called
Pauli matrices.
M is not a subalgebra since it is not
closed under products; for example Indeed, M cannot form an algebra if it is not even a
magma.
Proposition: If q is in M, then
Proof: From the definitions,
Definition: Let biquaternion g satisfy Then the
Lorentz transformation associated with g is given by
Proposition: If q is in M, then T(q) is also in M.
Proof:
Proposition:
Proof: Note first that gg* = 1 implies 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, Now
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 has two parts, and The first part is characterized by ; then the Lorentz transformation corresponding to g is given by since Such a transformation is a
rotation by quaternion multiplication, and the collection of them is SO(3) But this subgroup of G is not a
normal subgroup, so no
quotient group can be formed.
To view it is necessary to show some subalgebra structure in the biquaternions. Let r represent an element of the
sphere of square roots of minus one in the real quaternion subalgebra H. Then (hr)2 = +1 and the plane of biquaternions given by is a commutative subalgebra isomorphic to the plane of
split-complex numbers. Just as the ordinary complex plane has a unit circle, has a
unit hyperbola given by
Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because Hence these algebraic operators on the hyperbola are called
hyperbolic versors. The unit circle in C and unit hyperbola in Dr are examples of
one-parameter groups. For every square root r of minus one in H, there is a one-parameter group in the biquaternions given by
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 boostT given by g = exp(0.5ahr) since then so that
Naturally the
hyperboloid which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with the
hyperboloid model of
hyperbolic geometry. In special relativity, the
hyperbolic angle parameter of a hyperbolic versor is called
rapidity. Thus we see the biquaternion group G provides a
group representation for the
Lorentz group.[12]
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.[13]
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 then the biconjugate
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 N(pq) = N(p) N(q), indicating that N is a quadratic form admitting composition, so that the biquaternions form a
composition algebra.
^Lanczos 1949, See equation 94.16, page 305. The following algebra compares to Lanczos, except he uses ~ to signify quaternion conjugation and * for complex conjugation.
^Hermann 1974, chapter 6.4 Complex Quaternions and Maxwell's Equations.
Conway, Arthur W. (1911), "On the application of quaternions to some recent developments in electrical theory", Proceedings of the Royal Irish Academy, 29A: 1–9.
Silberstein, Ludwik (1914), The Theory of Relativity
Synge, J. L. (1972), "Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices", Communications of the Dublin Institute for Advanced Studies, Series A, 21