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The biquaternions are the numbers w + xi + yj + zk \ \! where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group. In Group theory, the quaternion group is a non-abelian group of order 8 As there are three types of complex number, so there are three types of biquaternion:

The following article is about the ordinary biquaternions named by William Rowan Hamilton in 1853 (see reference). Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Some of the more prominent proponents of these biquaternions include Alexander MacFarlane, Ludwik Silberstein, Wolfgang Pauli, and Cornelius Lanczos. Alexander Macfarlane may be Alexander Macfarlane (politician (1818-1898 Nova Scotian lawyer and senator Alexander Macfarlane (mathematician Ludwik Silberstein (1872 – 1948 was a Polish-American physicist that helped make Special relativity and General relativity staples of university coursework Cornelius Lanczos ( Lánczos Kornél) (approximate pronunciation 'LAHNT-sawsh') born Löwy Kornél ( February 2, 1893 As developed below, the biquaternions form a natural structure for the presentation of the Lorentz group, which is the foundation of special relativity. In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

The algebra of biquaternions can be considered as a tensor product C⊗H (taken over the reals) where C is the field of complex numbers and H is the algebra of real quaternions. In Mathematics, the Tensor product of two ''R''-algebras is also an R -algebra in a natural way In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In other words, the biquaternions are just the complexification of the real quaternions. In Mathematics, the complexification of a Real vector space V is a vector space V C over the Complex number Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2×2 complex matrices M2(C). In terms of Clifford algebra they can be classified as Cℓ2(C) = Cℓ03(C). In Mathematics, Clifford algebras are a type of Associative algebra.

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Definition

Let {1, i, j, k} be the basis for the (real) quaternions, and let u, v, w, x be complex numbers, then

q = u 1 + v i + w j + x k

is a biquaternion. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The complex scalars are assumed to commute with the quaternion basis vectors (e. g. vj = jv), and the root of -1 in the complex numbers is distinct from all three of the quaternion basis vectors. 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. In Group theory, the quaternion group is a non-abelian group of order 8 In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with The algebra of biquaternions is associative, but not commutative. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result

Place in ring theory

Linear representation

Note the matrix product

\begin{pmatrix}i & 0\\0 & -i\end{pmatrix}\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix} = \begin{pmatrix}0 & i\\i & 0\end{pmatrix}

where each of these three arrays has a square equal to the negative of the identity matrix. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main When the matrix product is interpreted as i j = k, then one obtains a subgroup of the matrix group that is isomorphic to the Quaternion group. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Group theory, the quaternion group is a non-abelian group of order 8 Consequently

\begin{pmatrix}u+iv & w+ix\\-w+ix & u-iv\end{pmatrix} represents biquaternion q.

Given any 2x2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring is isomorphic to the biquaternion ring. In Abstract algebra the matrix ring M( n, R) is the set of all n × n matrices over an arbitrary ring In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those

Alternative complex plane

Suppose we take w to be purely imaginary, w = b ι, where ι ι = - 1. (Here one uses iota instead of i for the complex imaginary to be distinct from quaternion i. ) Now when r = w j, then its square is

r r = (w j )(w j ) = (w w)(j j ) = b b (-1)(-1) = b2.

In particular, when b = 1 or –1, then r 2 = + 1. This development shows that the biquaternions are a source of "algebraic motors" like r that square to +1. Then {a + b ι j : a, b ∈ R } is a subring of biquaternions isomorphic to the split-complex number ring. In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real

Application in relativity physics

Lorentz group presentation

The biquaternions ιk = σ1, ιj = σ2, and −ιi = σ3 were used by Alexander MacFarlane and later, in their matrix form by Wolfgang Pauli. Alexander Macfarlane ( April 21 1851 – August 28, 1913) was a Scottish - Canadian Logician Physicist They have come to be known as Pauli matrices. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. They each square to the identity matrix and hence the subplane {a + b σ ; a, b ∈ R} generated by one of them in the biquaternion ring is isomorphic to the ring of split-complex numbers. Hence a Pauli matrix σ generates a one-parameter group {u : u = exp(a σ), a ∈ R} whose actions on the subplane are hyperbolic rotations. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R The Lorentz group is a six-parameter Lie group, three parameters of which (e. In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group g. subgroups generated by Pauli matrices) are associated with hyperbolic rotations, sometimes called boosts. The other three parameters correspond to ordinary rotations in space, a facility of real quaternion action known as quaternions and spatial rotation. Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions The usual quadratic form view of this presentation is that

u2 + v2 + w2 + x2 = q q*

is preserved by the orthogonal group on the biquaternions when viewed as C4. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n When u is real and v, w, and x are pure imaginary, then one has a subspace M=R4 convenient to model spacetime.

Since the algebra (matrix or biquaternion) centers on the Lorentz group symmetry and the leading idea (spacetime) is relegated to a half of the whole ring, there is the appearance of inverted priority, something of a literary conceit. Aside from its common usage signifying "excessive pride" in literary terms a conceit is an Extended metaphor with a complex Logic that governs The willy-nilly kinematic idea behind the Lorentz group does not take into account concomitants of kinematic orientation such as setting a horizon, acceleration-rotation interaction, or suitable model application such as practiced in traditional analytic geometry. An alternative kinematic approach comes by way of coquaternions. In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the The actual exhibition of individual Lorentz transformations involves extensions of inner automorphisms of the group of units of biquaternions to the singular elements through inversive ring geometry. In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be In Mathematics, inversive ring geometry is the extension to the context of Associative rings of the concepts of Projective line, Homogeneous

See also

References

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