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Hyperbolic quaternion

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A hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association for the Advancement of Science. The idea was criticized for its failure to conform to associativity of multiplication, but it has a legacy in Minkowski space and as an extension of split-complex numbers. Like the quaternions, the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4.

A linear combination

[q = a+bi+cj+dk]
is a hyperbolic quaternion when a, b, c, and d are real numbers and the basis set has these products:

[ij=k=(-j)i]
[jk=i=(-k)j]
[ki=j=(-i)k]
[i^2=1=j^2=k^2]
Though these basis products do not obey associativity, the set

[\]
forms a quasigroup. One also notes that any subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split-complex numbers. If

[q^*=a-bi-cj-dk]
is the conjugate of q, then the product

[q(q^*)=a^2-b^2-c^2-d^2]
is the quadratic form used in Minkowski space.

Later, Macfarlane published an article in the Proceedings of the Royal Society at Edinburgh in 1900. In it he establishes a model for hyperbolic space on the hyperboloid

[q(q^*)=1]
This isotropic model serves as a means to relativize velocity calculations within the limits of the speed of light. Writing in 1967, M.J. Crowe summarized the status of hyperbolic quaternions as follows:
The introduction of another system of vector analysis, even a sort of compromise system such as MacFarlane's, could scarcely be well received by the advocates of the already existing systems and moreover probably acted to broaden the question beyond the comprehension of the as-yet uninitiated reader.
: History of Vector Analysis, p. 191 .
Contents

Basis quasigroup

The basis [\] of the vector space of hyperbolic quaternions is not closed under multiplication: for example, [ji=-k]. Nevertheless, the set [\] is closed under multiplication. In the 1890s there was no structural theory of abstract algebras so this mathematical object could not be labeled, except as a latin square. Loss of the associativity property of multiplication in quasigroup theory is not tenable in linear algebra since all linear transformations compose in an associative manner. Yet physical scientists were calling in the 1890s for mutation of the squares of [i,j,] and [k] to be [+1] instead of [-1] : American physicists Willard Gibbs and Alexander MacFarlane made their cases in pamphlets, and Oliver Heaviside in England wrote columns in the Electrician, a trade paper. The Americans had chairs at Yale University and Texas, while Heaviside expounded in print with vector algebra and differential equations. Cargill Gilston Knott was moved to offer the following:

Theorem (Knott, 1893) If a 4-algebra on basis [\] is associative and off-diagonal products are given by Hamilton's rules,

then [i^2=-1=j^2=k^2].

Proof: [i^2j -i = i^2j + kj =i(i+j)j = i(k+j^2)=-j+ij^2]

Therefore [i^2=-1] and [j^2=-1].Use [j(j+k)k] for [k^2=-1].

This theorem needed statement to justify resistance to the call of the physicists and the electrician. The quasigroup stimulated a considerable stir in the 1890s: the journal Nature was especially conducive to an exhibit of what was known by giving two digests of Knott's work as well as those of several other vector theorists. Michael J. Crowe devotes chapter six of his book History of Vector Analysis to the various published views. Crowe has the benefit of hindsight on vector analysis and the nabla operator, but he does not recognize the quasigroup, being content with the comment:

MacFarlane constructed a new system of vector analysis more in harmony with Gibbs-Heaviside system than with the quaternion system. ...he...defined a full product of two vectors which was comparable to the full quaternion product except that the scalar part was positive, not negative as in the older system.
In retrospect, this quasigroup, with its unusual non-associativity, evoked an attitude of interest in axiomatic basics, an attitude that evolved into abstract algebra with its great variety of axiomatic structures.The contributions of Alfred Tarski, B. L. van der Waerden, and Bourbaki preceded the category and functor theory now used to locate mathematical objects.

MacFarlane's hyperbolic quaternion paper of 1900

The Proceedings of the Royal Society at Edinburgh published "Hyperbolic Quaternions" in 1900, a paper in which MacFarlane regains associativity for multiplication by reverting to complexified quaternions. While there he used some expressions later made famous by Wolfgang Pauli: where MacFarlane wrote
[ij=k\sqrt]
[jk=i\sqrt]
[ki=j\sqrt]
The Pauli matrices satisfy
[\sigma_1\sigma_2=\sigma_3\sqrt]
[\sigma_2\sigma_3=\sigma_1\sqrt]
[\sigma_3\sigma_1=\sigma_2\sqrt]
while referring to the same complexified quaternions.

The opening sentence of the paper is "It is well known that quaternions are intimately connected with spherical trigonometry and in fact they reduce the subject to a branch of algebra." This statement may be verified by reference to the contemporary work Vector Analysis by J.W. Gibbs and E.B. Wilson.In MacFarlane's paper there is an effort to produce "trigonometry on the surface of the equilateral hyperboloids" through the algebra of hyperbolic quaternions, now re-identified in an associative ring of eight real dimensions.The effort is re-enforced by a plate of nine figures on page 181.They illustrate the descriptive power of his "space analysis" method.For example, figure 7 is the common Minkowski diagram used today in special relativity to discuss change of velocity of a frame of reference and simultaneous events.

On page 173 MacFarlane expands on his greater theory of quaternion variables.By way of contrast he notes that Felix Klein appears not to look beyond the theory of Quaternions and spatial rotation.

Minkowski space

Recall that the scalar part of quaternion q = a+bi+cj+dk is the variable a . Using quaternion conjugation q* = a - bi - cj - dk one can express the Minkowski inner product with η(p,q) = scalar part of (pq*) , where there is a hyperbolic quaternion product of p with q*.The inner product generates two structures in Minkowski space: simultaneity of events relative to a given velocity and the Minkowski squared interval
[\eta(q,q) = q q* = a^2 - b^2 - c^2 -d^2]
In particular, the hyperboloid presents a kinematic model since (with appropriate units for a,b,c, and d) it represents the locus of temporal potential for a particle passing throught the origin after a moment of local time.

Simultaneity

Select an arbitrary point from the hyperboloid: u = cosh(a) + r sinh(a). Then relative to u, arbitrary hyperbolic quaternions p and q represent simultaneous events in Minkowski space if the scalar part of the product (p - q)u* is zero.Clearly simultaneity is a function of rapidity a and direction r .Geometrically, the hyperbolic quaternions p - q and u are hyperbolic-orthogonal.

References

External link

 

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