Bivector (complex)

In mathematics, a bivector is the vector part of a biquaternion. For biquaternion , w is called the biscalar and is its bivector part. The coordinates w, x, y, z are complex numbers with imaginary unit h:

A bivector may be written as the sum of real and imaginary parts:

where and are vectors. Thus the bivector

In the standard linear representation of biquaternions as 2 × 2 complex matrices acting on the complex plane with basis

represents bivector .

The conjugate transpose of this matrix corresponds to −q, so the representation of bivector q is a skew-Hermitian matrix.

History

William Rowan Hamilton coined both the terms vector and bivector. The first term was named with quaternions, and the second about a decade later, as in Lectures on Quaternions (1853). Willard Gibbs included a a note on bivectors in his Elements of Vector Analysis (1884). He used bivectors for Edwin Bidwell Wilson's textbook Vector Analysis (1901) based on his lectures. For instance, given a bivector , the ellipse for which r₁ and r₂ are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r.

Ludwik Silberstein studied a complexified electromagnetic field , where there are three components, each a complex number, known as the Riemann–Silberstein vector.

A consideration of biquaternion representation of special relativity, using Lie theory, brings bivectors into prominence: The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r₁ and r₂ are right versors so that , then the biquaternion curve traces over and over the unit circle in the plane Such a circle corresponds to the space rotation parameters of the Lorentz group.

Now , and the biquaternion curve is a unit hyperbola in the plane The spacetime transformations in the Lorentz group that lead to FitzGerald contractions and time dilation depend on a hyperbolic angle parameter. In the words of Ronald Shaw, "Bivectors are logarithms of Lorentz transformations."

The commutator product of this Lie algebra is just twice the cross product on R³, for instance, , which is twice . As Shaw wrote in 1970:

Now it is well known that the Lie algebra of the homogeneous Lorentz group can be considered to be that of bivectors under commutation. [...] The Lie algebra of bivectors is essentially that of complex 3-vectors, with the Lie product being defined to be the familiar cross product in (complex) 3-dimensional space.

"Bivectors [...] help describe elliptically polarized homogeneous and inhomogeneous plane waves – one vector for direction of propagation, one for amplitude."

References

• Link from Cornell University Historical Mathematics Collection.