Bivector (complex)

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:

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

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

$\displaystyle{ (x_1 \mathrm{i} + y_1 \mathrm{j} + z_1 \mathrm{k}) + \mathrm{h} (x_2 \mathrm{i} + y_2 \mathrm{j} + z_2 \mathrm{k}) }$

where $\displaystyle{ r_1 = x_1 \mathrm{i} + y_1 \mathrm{j} + z_1 \mathrm{k} }$ and $\displaystyle{ r_2 = x_2 \mathrm{i} + y_2 \mathrm{j} + z_2 \mathrm{k} }$ are vectors. Thus the bivector $\displaystyle{ q = x \mathrm{i} + y \mathrm{j} + z \mathrm{k} = r_1 + \mathrm{h} r_2 . }$[1]

The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r1 and r2 are right versors so that $\displaystyle{ r_1^2 = -1 = r_2^2 }$, then the biquaternion curve {exp θr1 : θR} traces over and over the unit circle in the plane {x + yr1 : x, yR}. 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) : θR} is a unit hyperbola in the plane {x + yr2 : x, yR}. 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."[2]

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.[3]

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).[1]:665 The popular text Vector Analysis (1901) used the term.[4]:249

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.[4]:436

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

$\displaystyle{ \begin{pmatrix}hv & w+hx\\-w+hx & -hv\end{pmatrix} }$ 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.[5][6]

"Bivectors [...] help describe elliptically polarized homogeneous and inhomogeneous plane waves – one vector for direction of propagation, one for amplitude."[7]

References

1. Hamilton, W.R. (1853). "On the geometrical interpretation of some results obtained by calculation with biquaternions". Proceedings of the Royal Irish Academy 5: 388–390.  Link from David R. Wilkins collection at Trinity College, Dublin
2. Shaw, Ronald; Bowtell, Graham (1969). "The Bivector Logarithm of a Lorentz Transformation". Quarterly Journal of Mathematics 20 (1): 497–503. doi:10.1093/qmath/20.1.497.
3. Shaw, Ronald (1970). "The subgroup structure of the homogeneous Lorentz group". Quarterly Journal of Mathematics 21 (1): 101–124. doi:10.1093/qmath/21.1.101.
4. "Telegraphic reviews §Bivectors and Waves in Mechanics and Optics". American Mathematical Monthly 102 (6): 571. 1995. doi:10.1080/00029890.1995.12004621.