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:
:
A bivector may be written as the sum of real and imaginary parts:
:
where and are vectors.
Thus the bivector {{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 , 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 }}
References
{{Reflist}}
{{refbegin}}
- {{cite book |first1=Ph. |last1=Boulanger |first2=M.A. |last2=Hayes |title=Bivectors and Waves in Mechanics and Optics |url=https://books.google.com/books?id=QN0Ks3fTPpAC |date=1993 |publisher=CRC Press |isbn=978-0-412-46460-7}}
- {{cite book
|title=Modern theory of anisotropic elasticity and applications
|editor1-first=Julian J. |editor1-last=Wu |editor2-first=Thomas Chi-tsai |editor2-last=Ting |editor3-first=David M. |editor3-last=Barnett |chapter-url=https://books.google.com/books?id=2fwUdSTN_6gC&pg=PA280 |page=280 et seq |isbn=0-89871-289-0
|first1=P.H. |last1=Boulanger |first2=M. |last2=Hayes
|chapter=Bivectors and inhomogeneous plane waves in anisotropic elastic bodies
|publisher=Society for Industrial and Applied Mathematics
|year=1991}}
- {{cite book |first1=William Rowan |last1=Hamilton |author1-link=William Rowan Hamilton |year=1853 |url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 |title=Lectures on Quaternions |publisher=Royal Irish Academy}} Link from Cornell University Historical Mathematics Collection.
- {{cite book |editor1-first=William Edwin |editor1-last=Hamilton |year=1866 |url=https://books.google.com/books?id=fIRAAAAAIAAJ |title=Elements of Quaternions |page=219 |publisher=University of Dublin Press}}
{{refend}}