Bivector (complex)

Short description: Vector part of a biquaternion, has three complex dimensions

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

[math]\displaystyle{ x = x_1 + \mathrm{h} x_2,\ y = y_1 + \mathrm{h} y_2,\ z = z_1 + \mathrm{h} z_2, \quad \mathrm{h}^2 = -1 = \mathrm{i}^2 = \mathrm{j}^2 = \mathrm{k}^2 . }[/math]

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

[math]\displaystyle{ (x_1 \mathrm{i} + y_1 \mathrm{j} + z_1 \mathrm{k}) + \mathrm{h} (x_2 \mathrm{i} + y_2 \mathrm{j} + z_2 \mathrm{k}) }[/math]

where [math]\displaystyle{ r_1 = x_1 \mathrm{i} + y_1 \mathrm{j} + z_1 \mathrm{k} }[/math] and [math]\displaystyle{ r_2 = x_2 \mathrm{i} + y_2 \mathrm{j} + z_2 \mathrm{k} }[/math] are vectors. Thus the bivector [math]\displaystyle{ q = x \mathrm{i} + y \mathrm{j} + z \mathrm{k} = r_1 + \mathrm{h} r_2 . }[/math]

The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r₁ and r₂ are right versors so that [math]\displaystyle{ r_1^2 = -1 = r_2^2 }[/math], then the biquaternion curve {exp θr₁ : θ ∈ R} traces over and over the unit circle in the plane {x + yr₁ : x, y ∈ R}. Such a circle corresponds to the space rotation parameters of the Lorentz group.

Now (hr₂)² = (−1)(−1) = +1, and the biquaternion curve {exp θ(hr₂) : θ ∈ R} is a unit hyperbola in the plane {x + yr₂ : x, y ∈ R}. 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, [i,j] = ij − ji = 2k, which is twice i × j. 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.

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). The popular text Vector Analysis (1901) used the term.

Given a bivector r = r₁ + hr₂, the ellipse for which r₁ and r₂ are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r.

In the standard linear representation of biquaternions as 2 × 2 complex matrices acting on the complex plane with basis {1, h},

[math]\displaystyle{ \begin{pmatrix}hv & w+hx\\-w+hx & -hv\end{pmatrix} }[/math] represents bivector q = vi + wj + xk.

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

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

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

References

• Boulanger, Ph.; Hayes, M.A. (1993). Bivectors and Waves in Mechanics and Optics. CRC Press. ISBN 978-0-412-46460-7. https://books.google.com/books?id=QN0Ks3fTPpAC. 
• Boulanger, P.H.; Hayes, M. (1991). "Bivectors and inhomogeneous plane waves in anisotropic elastic bodies". in Wu, Julian J.; Ting, Thomas Chi-tsai; Barnett, David M.. Modern theory of anisotropic elasticity and applications. Society for Industrial and Applied Mathematics. p. 280 et seq. ISBN 0-89871-289-0. https://books.google.com/books?id=2fwUdSTN_6gC&pg=PA280. 
• Hamilton, William Rowan (1853). Lectures on Quaternions. Royal Irish Academy. http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9.  Link from Cornell University Historical Mathematics Collection.
• Hamilton, William Edwin, ed (1866). Elements of Quaternions. University of Dublin Press. p. 219. https://books.google.com/books?id=fIRAAAAAIAAJ.