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:

x = x 1 + h x 2 ,   y = y 1 + h y 2 ,   z = z 1 + h z 2 , h 2 = − 1 = i 2 = j 2 = k 2 .

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

( x 1 i + y 1 j + z 1 k ) + h ( x 2 i + y 2 j + z 2 k )

where r 1 = x 1 i + y 1 j + z 1 k and r 2 = x 2 i + y 2 j + z 2 k are vectors. Thus the bivector q = x i + y j + z k = r 1 + h r 2 .

The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r₁ and r₂ are right versors so that r 1 2 = − 1 = r 2 2 , 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},

( h v w + h x − w + h x − h v ) 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."