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 : 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},

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. 1 2 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. 1 2 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. 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.
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