Английская Википедия:Bivector (complex)

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Шаблон:Short description In mathematics, a bivector is the vector part of a biquaternion. For biquaternion Шаблон:Nowrap, w is called the biscalar and Шаблон:Nowrap is its bivector part. The coordinates w, x, y, z are complex numbers with imaginary unit h:

<math>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>(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>r_1 = x_1 \mathrm{i} + y_1 \mathrm{j} + z_1 \mathrm{k}</math> and <math>r_2 = x_2 \mathrm{i} + y_2 \mathrm{j} + z_2 \mathrm{k}</math> are vectors. Thus the bivector <math> 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>r_1^2 = -1 = r_2^2</math>, then the biquaternion curve Шаблон:Nowrap traces over and over the unit circle in the plane Шаблон:Nowrap Such a circle corresponds to the space rotation parameters of the Lorentz group.

Now Шаблон:Nowrap, and the biquaternion curve Шаблон:Nowrap is a unit hyperbola in the plane Шаблон:Nowrap 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, Шаблон:Nowrap, which is twice Шаблон:Nowrap. 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).Шаблон:Rp The popular text Vector Analysis (1901) used the term.Шаблон:Rp

Given a bivector Шаблон:Nowrap, the ellipse for which r₁ and r₂ are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r.Шаблон:Rp

In the standard linear representation of biquaternions as 2 × 2 complex matrices acting on the complex plane with basis Шаблон:Nowrap

<math>\begin{pmatrix}hv & w+hx\\-w+hx & -hv\end{pmatrix}</math> represents bivector Шаблон:Nowrap.

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 Шаблон:Nowrap, 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

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book Link from Cornell University Historical Mathematics Collection.
• Шаблон:Cite book