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In mathematics, a biquaternion (or complex quaternion) is an element of the (unique) quaternion algebra over the complex numbers. The concept of a biquaterion was first mentioned by William Rowan Hamilton in the nineteenth century. William Kingdon Clifford used the same name in reference to a different algebra. See Clifford biquaternion. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, a quaternion algebra over a field L is a particular kind of central simple algebra A over L, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of L, by extending scalars. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
William Rowan Hamilton Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ...
William Kingdon Clifford William Kingdon Clifford, FRS (May 4, 1845 - March 3, 1879) was an English mathematician who also wrote a fair bit on philosophy. ...
A Clifford biquaternion is a concept from geometric algebra. ...
The algebra of biquaternions can be consider 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 other words, the biquaternions are just the complexification of the real quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2×2 complex matrices M2(C). 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 (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ...
Definition
Let {1, i, j, k} be the basis for the (real) quaternions, and let u, v, w, x be complex numbers, then In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
- q = u 1 + v i + w j + x k
is a biquaternion. The complex scalars are assumed to commute with the quaternion basis vectors (e.g. vj = jv). 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. The algebra of biquaternions is associative, but not commutative. Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ...
In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...
In mathematics, associativity is a property that a binary operation can have. ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
Place in ring theory
Linear representation Note the matrix product  =  where each of these three arrays has a square equal to the negative of the identity matrix. 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.Consequently In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ...
 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 set of all n-by-n matrices over an arbitrary ring R. This forms a ring under matrix addition and multiplication. ...
In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...
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 abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ...
In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...
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. They have come to be known as Pauli 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.The Lorentz group is a six-parameter Lie group, three parameters of which (e.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 Alexander MacFarlane (1851 - 1913) was a Scottish-Canadian logician, physicist, and mathematician. ...
This article is about Austrian-Swiss physicist Wolfgang Pauli. ...
The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. ...
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is...
The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ...
In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...
quaternions and spatial rotation.The usual quadratic form view of this presentation is that To meet Wikipedias quality standards, this article or section may require cleanup. ...
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...
u2 + v2 + w2 + x2 = q q* is preserved by the orthogonal group on the In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
biquaternions when viewed as C4.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.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 literary terms, a conceit is an extended metaphor with a complex logic that governs an entire poem or poetic passage. ...
In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ...
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 is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by ax. ...
The word singular may refer to one of several concepts. ...
In mathematics, inversive ring geometry is the extension, to the context of associative rings, of the concepts of Projective line, homogeneous coordinates, projective transformations, and Cross-ratio, concepts usually built upon rings that happen to be fields. ...
See also A Clifford biquaternion is a concept from geometric algebra. ...
Musean Hypernumbers are a concept envisioned by Charles A. Musès (1919–2000) to form a complete, integrated, connected, and natural number system [1][2][3][4][5]. Musès sketched certain fundamental types of hypernumbers and arranged them in ten levels, each with its own associated arithmetic and geometry. ...
References - Cornelius Lanczos(1949) The Variational Principles of Mechanics, University of Toronto Press, pp. 304-12.
- Ludwik Silberstein (May 1912) "Quaternionic form of relativity", Philosophy Magazine,series 6, 23:790-809.
- Silberstein, L. The Theory of Relativity, 1914.
- Synge, J.L. (1972) Quaternions, Lorentz transformations, and the Conway-Dirac-Eddington matrices Communications of the Dublin Institute for Advanced Studies, series A, #21, 67 pages.
- Kilmister, C.w.(1994) Eddington's search for a fundamental theory, Cambridge University Press [ISBN 0-521-37165-1], pages 121,122,179,180.
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