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: A bivector may be written as the sum of real and imaginary parts: where
[1] The Lie algebra of the Lorentz group is expressed by bivectors.
, 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.
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: William Rowan Hamilton coined both the terms vector and bivector.
[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}, 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.