Biquaternion

Template:Short description In abstract algebra, the biquaternions are the numbers Template:Math, where Template:Math, and Template:Mvar are complex numbers, or variants thereof, and the elements of Template:Math multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

Biquaternions when the coefficients are complex numbers.
Split-biquaternions when the coefficients are split-complex numbers.
Dual quaternions when the coefficients are dual numbers.

This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844.Template:Sfn Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity.

The algebra of biquaternions can be considered as a tensor product Template:Math, where Template:Math is the field of complex numbers and Template:Math is the division algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of Template:Math complex matrices Template:Math. They are also isomorphic to several Clifford algebras including Template:Math,Template:Sfn the Pauli algebra Template:Math,Template:Sfn Template:Sfn and the even part Template:Math of the spacetime algebra.Template:Sfn

DefinitionEdit

Let Template:Math be the basis for the (real) quaternions Template:Math, and let Template:Math be complex numbers, then

<math>q = u \mathbf 1 + v \mathbf i + w \mathbf j + x \mathbf k</math>

is a biquaternion.Template:Sfn To distinguish square roots of minus one in the biquaternions, HamiltonTemplate:Sfn Template:Sfn and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field Template:Math by Template:Math to avoid confusion with the Template:Math in the quaternion group. Commutativity of the scalar field with the quaternion group is assumed:

<math> h \mathbf i = \mathbf i h,\ \ h \mathbf j = \mathbf j h,\ \ h \mathbf k = \mathbf k h .</math>

Hamilton introduced the terms bivector, biconjugate, bitensor, and biversor to extend notions used with real quaternions Template:Math.

Hamilton's primary exposition on biquaternions came in 1853 in his Lectures on Quaternions. The editions of Elements of Quaternions, in 1866 by William Edwin Hamilton (son of Rowan), and in 1899, 1901 by Charles Jasper Joly, reduced the biquaternion coverage in favour of the real quaternions.

Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers Template:Math. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor. The algebra of biquaternions forms a composition algebra and can be constructed from bicomplex numbers. See Template:Section link below.

Place in ring theoryEdit

Linear representationEdit

Note that the matrix product

<math>\begin{pmatrix}h & 0\\0 & -h\end{pmatrix}\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix} = \begin{pmatrix}0 & h\\h & 0\end{pmatrix}</math>.

Because Template:Math is the imaginary unit, each of these three arrays has a square equal to the negative of the identity matrix. When this matrix product is interpreted as Template:Math, then one obtains a subgroup of matrices that is isomorphic to the quaternion group. Consequently,

<math>\begin{pmatrix}u+hv & w+hx\\-w+hx & u-hv\end{pmatrix}</math>

represents biquaternion Template:Math. Given any Template:Math complex matrix, there are complex values Template:Math, Template:Math, Template:Math, and Template:Math to put it in this form so that the matrix ring Template:Math is isomorphicTemplate:Sfn to the biquaternion ring. This representation also shows that the 16-element group

<math>\{\pm\mathbf 1, \pm h, \pm\mathbf i, \pm h\mathbf i, \pm\mathbf j, \pm h\mathbf j, \pm\mathbf k, \pm h\mathbf k \}</math>

is isomorphic to the Pauli group, the central product of a cyclic group of order 4 and the dihedral group of order 8. Concretely, the Pauli matrices

<math>X =

\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix},\quad Y = \begin{pmatrix} 0&-h\\ h&0 \end{pmatrix},\quad Z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}</math> correspond respectively to the elements Template:Math, and Template:Math.

SubalgebrasEdit

Considering the biquaternion algebra over the scalar field of real numbers Template:Math, the set

<math>\{\mathbf 1, h, \mathbf i, h\mathbf i, \mathbf j, h\mathbf j, \mathbf k, h\mathbf k \}</math>

forms a basis so the algebra has eight real dimensions. The squares of the elements Template:Math, and Template:Math are all positive one, for example, Template:Math.

The subalgebra given by

<math>\{ x + y(h\mathbf i) : x, y \in \R \} </math>

is ring isomorphic to the plane of split-complex numbers, which has an algebraic structure built upon the unit hyperbola. The elements Template:Math and Template:Math also determine such subalgebras.

Furthermore,

<math>\{ x + y \mathbf j : x,y \in \Complex \} </math>

is a subalgebra isomorphic to the bicomplex numbers.

A third subalgebra called coquaternions is generated by Template:Math and Template:Math. It is seen that Template:Math, and that the square of this element is Template:Math. These elements generate the dihedral group of the square. The linear subspace with basis Template:Math thus is closed under multiplication, and forms the coquaternion algebra.

In the context of quantum mechanics and spinor algebra, the biquaternions Template:Math, and Template:Math (or their negatives), viewed in the Template:Math representation, are called Pauli matrices.

Algebraic propertiesEdit

The biquaternions have two conjugations:

the biconjugate or biscalar minus bivector is <math>q^* = w - x\mathbf i - y\mathbf j - z\mathbf k \!\ ,</math> and
the complex conjugation of biquaternion coefficients <math>\bar{q} = \bar{w} + \bar{x}\mathbf i + \bar{y} \mathbf j + \bar{z}\mathbf k </math>

where <math>\bar{z} = a - bh</math> when <math>z = a + bh,\quad a,b \in \reals,\quad h^2 = -\mathbf 1.</math>

Note that <math>(pq)^* = q^* p^*, \quad \overline{pq} = \bar{p} \bar{q}, \quad \overline{q^*} = \bar{q}^*.</math>

Clearly, if <math>q q^* = 0 </math> then Template:Math is a zero divisor. Otherwise <math>\lbrace q q^* \rbrace^{-\mathbf 1} </math> is a complex number. Further, <math>q q^* = q^* q </math> is easily verified. This allows the inverse to be defined by

<math>q^{-1} = q^* \lbrace q q^* \rbrace^{-1}</math>, if <math>qq^* \neq 0.</math>

Relation to Lorentz transformationsEdit

Template:Further

Consider now the linear subspaceTemplate:Sfn

<math>M = \lbrace q\colon q^* = \bar{q} \rbrace = \lbrace t + x(h\mathbf i) + y(h \mathbf j) + z(h \mathbf k)\colon t, x, y, z \in \reals \rbrace .</math>

Template:Math is not a subalgebra since it is not closed under products; for example <math>(h\mathbf i)(h\mathbf j) = h^2 \mathbf{ij} = -\mathbf k \notin M.</math> Indeed, Template:Math cannot form an algebra if it is not even a magma.

Proposition: If Template:Mvar is in Template:Mvar, then <math>q q^* = t^2 - x^2 - y^2 - z^2.</math>

Proof: From the definitions,

<math>\begin{align}

q q^* &= (t+xh\mathbf i+yh\mathbf j+zh\mathbf k)(t-xh\mathbf i-yh\mathbf j-zh\mathbf k)\\ &= t^2 - x^2(h\mathbf i)^2 - y^2(h\mathbf j)^2 - z^2(h\mathbf k)^2 \\ &= t^2 - x^2 - y^2 - z^2. \end{align} </math>

Definition: Let biquaternion Template:Mvar satisfy <math>g g^* = 1.</math> Then the Lorentz transformation associated with Template:Mvar is given by

Proposition: If Template:Mvar is in Template:Mvar, then Template:Math is also in Template:Math.

Proof: <math>(g^* q \bar{g})^* = \bar{g}^* q^* g = \overline{g^*} \bar{q} g = \overline{g^* q \bar{g})}.</math>

Proposition: <math>\quad T(q) (T(q))^* = q q^* </math>

Proof: Note first that gg* = 1 implies that the sum of the squares of its four complex components is one. Then the sum of the squares of the complex conjugates of these components is also one. Therefore, <math>\bar{g} (\bar{g})^* = 1.</math> Now

Associated terminologyEdit

As the biquaternions have been a fixture of linear algebra since the beginnings of mathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. The transformation group <math>G = \lbrace g : g g^* = 1 \rbrace </math> has two parts, <math>G \cap H</math> and <math>G \cap M.</math> The first part is characterized by <math>g = \bar{g}</math> ; then the Lorentz transformation corresponding to Template:Mvar is given by <math>T(q) = g^{-1} q g </math> since <math>g^* = g^{-1}. </math> Such a transformation is a rotation by quaternion multiplication, and the collection of them is Template:Math <math>\cong G \cap H .</math> But this subgroup of Template:Mvar is not a normal subgroup, so no quotient group can be formed.

To view <math>G \cap M</math> it is necessary to show some subalgebra structure in the biquaternions. Let Template:Mvar represent an element of the sphere of square roots of minus one in the real quaternion subalgebra Template:Math. Then Template:Math and the plane of biquaternions given by <math>D_r = \lbrace z = x + yhr : x, y \in \mathbb R \rbrace</math> is a commutative subalgebra isomorphic to the plane of split-complex numbers. Just as the ordinary complex plane has a unit circle, <math>D_r </math> has a unit hyperbola given by

Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because <math>\exp(ahr) \exp(bhr) = \exp((a+b)hr). </math> Hence these algebraic operators on the hyperbola are called hyperbolic versors. The unit circle in Template:Math and unit hyperbola in Template:Math are examples of one-parameter groups. For every square root Template:Math of minus one in Template:Math, there is a one-parameter group in the biquaternions given by <math>G \cap D_r.</math>

The space of biquaternions has a natural topology through the Euclidean metric on Template:Math-space. With respect to this topology, Template:Mvar is a topological group. Moreover, it has analytic structure making it a six-parameter Lie group. Consider the subspace of bivectors <math>A = \lbrace q : q^* = -q \rbrace </math>. Then the exponential map <math>\exp:A \to G</math> takes the real vectors to <math>G \cap H</math> and the Template:Mvar-vectors to <math>G \cap M.</math> When equipped with the commutator, Template:Mvar forms the Lie algebra of Template:Mvar. Thus this study of a six-dimensional space serves to introduce the general concepts of Lie theory. When viewed in the matrix representation, Template:Mvar is called the special linear group Template:Math in Template:Math.

Many of the concepts of special relativity are illustrated through the biquaternion structures laid out. The subspace Template:Mvar corresponds to Minkowski space, with the four coordinates giving the time and space locations of events in a resting frame of reference. Any hyperbolic versor Template:Math corresponds to a velocity in direction Template:Mvar of speed Template:Math where Template:Mvar is the velocity of light. The inertial frame of reference of this velocity can be made the resting frame by applying the Lorentz boost Template:Mvar given by Template:Math since then <math>g^{\star} = \exp(-0.5ahr) = g^*</math> so that <math>T(\exp(ahr)) = 1 .</math> Naturally the hyperboloid <math>G \cap M,</math> which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with the hyperboloid model of hyperbolic geometry. In special relativity, the hyperbolic angle parameter of a hyperbolic versor is called rapidity. Thus we see the biquaternion group Template:Mvar provides a group representation for the Lorentz group.Template:Sfn

After the introduction of spinor theory, particularly in the hands of Wolfgang Pauli and Élie Cartan, the biquaternion representation of the Lorentz group was superseded. The new methods were founded on basis vectors in the set

<math>\{ q \ :\ q q^* = 0 \} = \left\{ w + x\mathbf i + y\mathbf j + z\mathbf k \ :\ w^2 + x^2 + y^2 + z^2 = 0 \right\} </math>

which is called the complex light cone. The above representation of the Lorentz group coincides with what physicists refer to as four-vectors. Beyond four-vectors, the standard model of particle physics also includes other Lorentz representations, known as scalars, and the Template:Math-representation associated with e.g. the electromagnetic field tensor. Furthermore, particle physics makes use of the Template:Math representations (or projective representations of the Lorentz group) known as left- and right-handed Weyl spinors, Majorana spinors, and Dirac spinors. It is known that each of these seven representations can be constructed as invariant subspaces within the biquaternions.Template:Sfn

As a composition algebraEdit

Although W. R. Hamilton introduced biquaternions in the 19th century, its delineation of its mathematical structure as a special type of algebra over a field was accomplished in the 20th century: the biquaternions may be generated out of the bicomplex numbers in the same way that Adrian Albert generated the real quaternions out of complex numbers in the so-called Cayley–Dickson construction. In this construction, a bicomplex number Template:Math has conjugate Template:Math.

The biquaternion is then a pair of bicomplex numbers Template:Math, where the product with a second biquaternion Template:Math is

If <math>a = (u, v), b = (w,z), </math> then the biconjugate <math>(a, b)^* = (a^*, -b).</math>

When Template:Math is written as a 4-vector of ordinary complex numbers,

The biquaternions form an example of a quaternion algebra, and it has norm

Two biquaternions Template:Math and Template:Math satisfy Template:Math, indicating that Template:Math is a quadratic form admitting composition, so that the biquaternions form a composition algebra.