Bivector (complex)

{{short description|Vector part of a biquaternion, has three complex dimensions}}

In mathematics, a bivector is the vector part of a biquaternion. For biquaternion {{nowrap|1=q = w + xi + yj + zk}}, w is called the biscalar and {{nowrap|xi + yj + zk}} is its bivector part. The coordinates w, x, y, z are complex numbers with imaginary unit h:

:$x\; =\; x\_1\; +\; \backslash mathrm\{h\}\; x\_2,\backslash \; y\; =\; y\_1\; +\; \backslash mathrm\{h\}\; y\_2,\backslash \; z\; =\; z\_1\; +\; \backslash mathrm\{h\}\; z\_2,\; \backslash quad\; \backslash mathrm\{h\}^2\; =\; -1\; =\; \backslash mathrm\{i\}^2\; =\; \backslash mathrm\{j\}^2\; =\; \backslash mathrm\{k\}^2\; .$

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

:$(x\_1\; \backslash mathrm\{i\}\; +\; y\_1\; \backslash mathrm\{j\}\; +\; z\_1\; \backslash mathrm\{k\})\; +\; \backslash mathrm\{h\}\; (x\_2\; \backslash mathrm\{i\}\; +\; y\_2\; \backslash mathrm\{j\}\; +\; z\_2\; \backslash mathrm\{k\})$

where $r\_1\; =\; x\_1\; \backslash mathrm\{i\}\; +\; y\_1\; \backslash mathrm\{j\}\; +\; z\_1\; \backslash mathrm\{k\}$ and $r\_2\; =\; x\_2\; \backslash mathrm\{i\}\; +\; y\_2\; \backslash mathrm\{j\}\; +\; z\_2\; \backslash mathrm\{k\}$ are vectors.

Thus the bivector $q\; =\; x\; \backslash mathrm\{i\}\; +\; y\; \backslash mathrm\{j\}\; +\; z\; \backslash mathrm\{k\}\; =\; r\_1\; +\; \backslash mathrm\{h\}\; r\_2\; .${{cite journal |first1=W.R. |last1=Hamilton |year=1853 |url=http://www.maths.soton.ac.uk/EMIS/classics/Hamilton/GeoBiQu.pdf |title=On the geometrical interpretation of some results obtained by calculation with biquaternions |journal=Proceedings of the Royal Irish Academy |volume=5 |pages=388–390}} Link from David R. Wilkins collection at Trinity College, Dublin

The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r₁ and r₂ are right versors so that $r\_1^2\; =\; -1\; =\; r\_2^2$, then the biquaternion curve {{nowrap|{exp θr₁ : θ ∈ R} }} traces over and over the unit circle in the plane {{nowrap|{x + yr₁ : x, y ∈ R}.}} Such a circle corresponds to the space rotation parameters of the Lorentz group.

Now {{nowrap|1=(hr₂)² = (−1)(−1) = +1}}, and the biquaternion curve {{nowrap|{exp θ(hr₂) : θ ∈ R} }} is a unit hyperbola in the plane {{nowrap|{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."{{cite journal |first1=Ronald |last1=Shaw |first2=Graham |last2=Bowtell |year=1969 |title=The Bivector Logarithm of a Lorentz Transformation |journal=Quarterly Journal of Mathematics |volume=20 |issue=1 |pages=497–503 |doi=10.1093/qmath/20.1.497 |url=https://academic.oup.com/qjmath/article-abstract/20/1/497/1539677|url-access=subscription }}

The commutator product of this Lie algebra is just twice the cross product on R³, for instance, {{nowrap|1=[i,j] = ij − ji = 2k}}, which is twice {{nowrap|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.{{cite journal |first1=Ronald |last1=Shaw |year=1970 |title=The subgroup structure of the homogeneous Lorentz group |journal=Quarterly Journal of Mathematics |volume=21 |issue=1 |pages=101–124 |doi=10.1093/qmath/21.1.101 |url=https://academic.oup.com/qjmath/article-abstract/21/1/101/1517659|url-access=subscription }}

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).{{rp|665}} The popular text Vector Analysis (1901) used the term.Edwin Bidwell Wilson (1901) Vector Analysis{{rp|249}}

Given a bivector {{nowrap|1=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.{{rp|436}}

In the standard linear representation of biquaternions as 2 × 2 complex matrices acting on the complex plane with basis {{nowrap|{1, h},}}

: represents bivector {{nowrap|1=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 {{nowrap|E + hB}}, where there are three components, each a complex number, known as the Riemann–Silberstein vector.{{Cite journal

|last=Silberstein |first=Ludwik |authorlink=Ludwik Silberstein

|year=1907

|title=Elektromagnetische Grundgleichungen in bivectorieller Behandlung

|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/silberstein_-_em_equations_in_bivector_fomr.pdf

|journal=Annalen der Physik

|volume=327 |issue=3 |pages=579–586

|bibcode=1907AnP...327..579S

|doi=10.1002/andp.19073270313

}}

{{Cite journal

|last=Silberstein |first=Ludwik

|year=1907

|title=Nachtrag zur Abhandlung über 'Elektromagnetische Grundgleichungen in bivectorieller Behandlung'

|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/silberstein_-_addendum.pdf

|journal=Annalen der Physik

|volume=329 |pages=783–4

|bibcode=1907AnP...329..783S

|doi=10.1002/andp.19073291409

|issue=14

}}

"Bivectors [...] help describe elliptically polarized homogeneous and inhomogeneous plane waves – one vector for direction of propagation, one for amplitude."{{cite journal |title=Telegraphic reviews §Bivectors and Waves in Mechanics and Optics |journal=American Mathematical Monthly |volume=102 |issue=6 |year=1995 |page=571 |doi=10.1080/00029890.1995.12004621 }}