Vector part of a biquaternion, has three complex dimensions

Bivector (complex)

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In mathematics, a bivector is the vector part of a biquaternion. For biquaternion $q=w+x\mathrm {i} +y\mathrm {j} +z\mathrm {k}$ , $w$ is called the biscalar and $x\mathrm {i} +y\mathrm {j} +z\mathrm {k}$ is its bivector part. The coordinates $w$ , $x$ , $y$ , $z$ are complex numbers with imaginary unit $\mathrm {h}$ :

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:

(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 $r_{1}=x_{1}\mathrm {i} +y_{1}\mathrm {j} +z_{1}\mathrm {k}$ and $r_{2}=x_{2}\mathrm {i} +y_{2}\mathrm {j} +z_{2}\mathrm {k}$ are vectors. Thus the bivector $q=x\mathrm {i} +y\mathrm {j} +z\mathrm {k} =r_{1}+\mathrm {h} r_{2}.$

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

{\begin{pmatrix}hv&w+hx\\-w+hx&-hv\end{pmatrix}}

represents bivector

q=v\mathrm {i} +w\mathrm {j} +x\mathrm {k}

The conjugate transpose of this matrix corresponds to $-q$ , so the representation of bivector $q$ is a skew-Hermitian matrix.

History

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). Willard Gibbs included a a note on bivectors in his Elements of Vector Analysis (1884). He used bivectors for Edwin Bidwell Wilson's textbook Vector Analysis (1901) based on his lectures. For instance, given a bivector $r=r_{1}+\mathrm {h} r_{2}$ , the ellipse for which $r_{1}$ and $r_{2}$ are a pair of conjugate semi-diameters is called the directional ellipse of the bivector $r$ .

Ludwik Silberstein studied a complexified electromagnetic field $E+\mathrm {h} B$ , where there are three components, each a complex number, known as the Riemann–Silberstein vector.

A consideration of biquaternion representation of special relativity, using Lie theory, brings bivectors into prominence: The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if $r_{1}$ and $r_{2}$ are right versors so that $r_{1}^{2}=-1=r_{2}^{2}$ , then the biquaternion curve $\{\exp \theta \,r_{1}:\theta \in \mathbf {R} \}$ traces over and over the unit circle in the plane $\{x+yr_{1}:x,y\in \mathbf {R} \}$ . Such a circle corresponds to the space rotation parameters of the Lorentz group.

Now $(\mathrm {h} r_{2})^{2}=(-1)(-1)=+1$ , and the biquaternion curve $\{\exp \theta \ (\mathrm {h} r_{2}):\theta \in \mathbf {R} \}$ is a unit hyperbola in the plane $\{x+yr_{2}:x,y\in \mathbf {R} \}$ . 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."

The commutator product of this Lie algebra is just twice the cross product on $\mathbf {R} ^{3}$ , for instance, $[\mathrm {i},\mathrm {j}]=\mathrm {i} \mathrm {j} -\mathrm {j} \mathrm {i} =2\mathrm {k}$ , which is twice $\mathrm {i} \times \mathrm {j}$ . 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.

"Bivectors [...] help describe elliptically polarized homogeneous and inhomogeneous plane waves – one vector for direction of propagation, one for amplitude."

Bivector (complex)

History

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