Bivector (complex)

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

${\displaystyle x=x_{1}+\mathrm {h} x_{2},\ y=y_{1}+\mathrm {h} y_{2},\ z=z_{1}+\mathrm {h} z_{2},\quad \mathrm {h} ^{2}=-1=\mathrm {i} ^{2}=\mathrm {j} ^{2}=\mathrm {k} ^{2}.}$

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 ${\displaystyle q=x\mathrm {i} +y\mathrm {j} +z\mathrm {k} =r_{1}+\mathrm {h} r_{2}.}$

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

${\displaystyle {\begin{pmatrix}hv&w+hx\\-w+hx&-hv\end{pmatrix}}}$ represents bivector ${\displaystyle q=v\mathrm {i} +w\mathrm {j} +x\mathrm {k} }$.

The conjugate transpose of this matrix corresponds to ${\displaystyle -q}$, so the representation of bivector ${\displaystyle q}$ 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).^: 665 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.^: 249 For instance, given a bivector ${\displaystyle r=r_{1}+\mathrm {h} r_{2}}$, the ellipse for which ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ are a pair of conjugate semi-diameters is called the directional ellipse of the bivector ${\displaystyle r}$.^: 436

Ludwik Silberstein studied a complexified electromagnetic field ${\displaystyle E+\mathrm {h} B}$, 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 ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ are right versors so that ${\displaystyle r_{1}^{2}=-1=r_{2}^{2}}$, then the biquaternion curve ${\displaystyle \{\exp \theta \,r_{1}:\theta \in \mathbf {R} \}}$ traces over and over the unit circle in the plane ${\displaystyle \{x+yr_{1}:x,y\in \mathbf {R} \}}$. Such a circle corresponds to the space rotation parameters of the Lorentz group.

Now ${\displaystyle (\mathrm {h} r_{2})^{2}=(-1)(-1)=+1}$, and the biquaternion curve ${\displaystyle \{\exp \theta \ (\mathrm {h} r_{2}):\theta \in \mathbf {R} \}}$ is a unit hyperbola in the plane ${\displaystyle \{x+yr_{2}:x,y\in \mathbf {R} \}}$. 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 ${\displaystyle \mathbf {R} ^{3}}$, for instance, ${\displaystyle [\mathrm {i} ,\mathrm {j} ]=\mathrm {i} \mathrm {j} -\mathrm {j} \mathrm {i} =2\mathrm {k} }$, which is twice ${\displaystyle \mathrm {i} \times \mathrm {j} }$. 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."

References