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 : 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."[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. W.R. Hamilton (1853) On the geometrical interpretation of some results obtained by calculation with biquaternions, Proceedings of the Royal Irish Academy 5: 388–90, link from David R. Wilkins collection at Trinity College, Dublin
2. Ronald Shaw and Graham Bowtell (1969) "The Bivector Logarithm of a Lorentz Transformation", Quarterly Journal of Mathematics 20:497–503
3. Ronald Shaw (1970) "The subgroup structure of the homogeneous Lorentz group", Quarterly Journal of Mathematics 21:101–24
4. Edwin Bidwell Wilson (1901) Vector Analysis
5. Silberstein, Ludwik (1907). "Elektromagnetische Grundgleichungen in bivectorieller Behandlung" (PDF). Annalen der Physik. 327 (3): 579–586. Bibcode:1907AnP...327..579S. doi:10.1002/andp.19073270313.
6. Silberstein, Ludwik (1907). "Nachtrag zur Abhandlung über 'Elektromagnetische Grundgleichungen in bivectorieller Behandlung'" (PDF). Annalen der Physik. 329 (14): 783–784. Bibcode:1907AnP...329..783S. doi:10.1002/andp.19073291409.
7. Telegraphic review of Bivectors and Waves in Mechanics and Optics, American Mathematical Monthly 1995 page 571

• P.H. Boulanger & M. Hayes (1993) Bivectors and Waves in Mechanics and Optics, Chapman and Hall.
• P.H. Boulanger & M Hayes (1991). "Bivectors and inhomogeneous plane waves in anisotropic elastic bodies". In Julian J. Wu; Thomas Chi-tsai Ting & David M. Barnett (eds.). Modern theory of anisotropic elasticity and applications. Society for Industrial and Applied Mathematics. p. 280 et seq. ISBN 0-89871-289-0.
• William Rowan Hamilton, (1853) Lectures on Quaternions, Royal Irish Academy, link from Cornell University Historical Mathematics Collection.
• William Edwin Hamilton (editor) (1866) Elements of Quaternions, page 219, University of Dublin Press, link from Google books.