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:

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

where and are vectors. Thus the bivector [1]

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 {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},

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

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