Английская Википедия:Biquaternion

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Шаблон:Short description In abstract algebra, the biquaternions are the numbers Шаблон:Math, where Шаблон:Math, and Шаблон:Mvar are complex numbers, or variants thereof, and the elements of Шаблон: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:

This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844.Шаблон: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 Шаблон:Math, where Шаблон:Math is the field of complex numbers and Шаблон: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 Шаблон:Math complex matrices Шаблон:Math. They are also isomorphic to several Clifford algebras including Шаблон:Math,Шаблон:Sfn the Pauli algebra Шаблон:Math,Шаблон:SfnШаблон:Sfn and the even part Шаблон:Math of the spacetime algebra.Шаблон:Sfn

Definition

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

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

is a biquaternion.Шаблон:Sfn To distinguish square roots of minus one in the biquaternions, HamiltonШаблон:SfnШаблон:Sfn and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field Шаблон:Math by Шаблон:Math to avoid confusion with the Шаблон: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 Шаблон: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 Шаблон: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 Шаблон:Section link below.

Place in ring theory

Linear representation

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 Шаблон: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 Шаблон: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 Шаблон:Math. Given any Шаблон:Math complex matrix, there are complex values Шаблон:Math, Шаблон:Math, Шаблон:Math, and Шаблон:Math to put it in this form so that the matrix ring Шаблон:Math is isomorphicШаблон:Sfn to the biquaternion ring.

Subalgebras

Considering the biquaternion algebra over the scalar field of real numbers Шаблон: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 Шаблон:Math, and Шаблон:Math are all positive one, for example, Шаблон: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 Шаблон:Math and Шаблон: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 Шаблон:Math and Шаблон:Math. It is seen that Шаблон:Math, and that the square of this element is Шаблон:Math. These elements generate the dihedral group of the square. The linear subspace with basis Шаблон:Math thus is closed under multiplication, and forms the coquaternion algebra.

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

Algebraic properties

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 Шаблон: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 transformations

Шаблон:Further

Consider now the linear subspaceШаблон: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>

Шаблон: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, Шаблон:Math cannot form an algebra if it is not even a magma.

Proposition: If Шаблон:Mvar is in Шаблон: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 Шаблон:Mvar satisfy <math>g g^* = 1.</math> Then the Lorentz transformation associated with Шаблон:Mvar is given by

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

Proposition: If Шаблон:Mvar is in Шаблон:Mvar, then Шаблон:Math is also in Шаблон: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

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

Associated terminology

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 Шаблон: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 Шаблон:Math <math>\cong G \cap H .</math> But this subgroup of Шаблон: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 Шаблон:Mvar represent an element of the sphere of square roots of minus one in the real quaternion subalgebra Шаблон:Math. Then Шаблон: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

<math>\exp(ahr) = \cosh(a) + hr\ \sinh(a),\quad a \in R. </math>

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 Шаблон:Math and unit hyperbola in Шаблон:Math are examples of one-parameter groups. For every square root Шаблон:Math of minus one in Шаблон: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 Шаблон:Math-space. With respect to this topology, Шаблон: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 Шаблон:Mvar-vectors to <math>G \cap M.</math> When equipped with the commutator, Шаблон:Mvar forms the Lie algebra of Шаблон:Mvar. Thus this study of a six-dimensional space serves to introduce the general concepts of Lie theory. When viewed in the matrix representation, Шаблон:Mvar is called the special linear group Шаблон:Math in Шаблон:Math.

Many of the concepts of special relativity are illustrated through the biquaternion structures laid out. The subspace Шаблон: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 Шаблон:Math corresponds to a velocity in direction Шаблон:Mvar of speed Шаблон:Math where Шаблон: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 Шаблон:Mvar given by Шаблон: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 Шаблон:Mvar provides a group representation for the Lorentz group.Шаблон: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 Шаблон:Math-representation associated with e.g. the electromagnetic field tensor. Furthermore, particle physics makes use of the Шаблон: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.Шаблон:Sfn

As a composition algebra

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 Шаблон:Math has conjugate Шаблон:Math.

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

<math>(a,b)(c,d) = (a c - d^* b, d a + b c^* ).</math>

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

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

<math>(u, v, w, z)^* = (u, -v, -w, -z). </math>

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

<math>N(u,v,w,z) = u^2 + v^2 + w^2 + z^2 .</math>

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

See also

Citations

Шаблон:Reflist

References

Шаблон:Wikibooks Шаблон:Refbegin

Шаблон:Refend

Шаблон:Number systems

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