Hyperbolic quaternion

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In mathematics, 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, so the collection M of all hyperbolic quaternions forms a non-associative ring. It has a legacy in Minkowski space and as an extension of split-complex numbers.

This article describes their algebra and traces a development that preceded prevailing modern insight on what is desirable in a structure. The non-associativity of hyperbolic quaternions poses a complication for constructing transformation geometry from its multiplication, but for a while it was considered very promising. At that time, the split-complex arithmetic was suppressed due to concerns about zero-division, and linear representation was not at all common. Thus, hyperbolic quaternions were succeeded by biquaternions as the popular ring of choice for a while, before generalizing concepts like e.g. tensor algebra, Lie algebra, and Clifford algebra took over. Nevertheless, they remain the only quaternionic number system where the square of all bases is +1 .

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

[edit] Algebraic structure

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 {1,i,j,k} has these products:

ij = k = − ji
jk = i = − kj
ki = j = − ik
i2 = 1 = j2 = k2

Unlike Hamilton's quaternions, of which these are a mutant form, the hyperbolic quaternions are not associative. For example, (ij)j = kj = − i, while i(jj) = i. The first three relations show that products of the (non-real) basis elements are anti-commutative. Although this basis set does not form an group, the set

{1,i,j,k, − 1, − i, − j, − k}

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 * ) = a2 − b2 − c2 − d2

is the quadratic form used in spacetime theory. In fact, the bilinear form called the Minkowski inner product arises as the negative of the real part of the hyperbolic quaternion product pq* :

− p0q0 + p1q1 + p2q2 + p3q3.

Note that the set of units U = {q : qq* ≠ 0 } is not closed under multiplication. See the external reference below for details of this singularity of multiplication within U.

[edit] Geometry

Later, Macfarlane published an article in the Proceedings of the Royal Society at Edinburgh in 1900. In it he treats a model for hyperbolic space H3 on the hyperboloid

H^3 = \{ q : q(q^*)=1 \} \!.

This isotropic model, called the hyperboloid model, serves as a means to relativize velocity calculations within the limits of the speed of light.

[edit] Historical review

The basis \{1,\ i,\ j,\ k\} of the vector space of hyperbolic quaternions is not closed under multiplication: for example, j\ i\ =\ -k. Nevertheless, the set \{1,\ i,\ j,\ k,\ -1,\ -i,\ -j,\ -k\} 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 as found 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 {1,i,j,k} is associative and off-diagonal products are given by Hamilton's rules, then i2 = − 1 = j2 = k2.

Proof:

i2j − i = i2j + kj = i(i + j)j = i(k + j2) = − j + ij2
Therefore i2 = − 1 and j2 = − 1. Use j(j + k)k for k2 = − 1. QED.

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 A 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. Furthermore, the unwieldy nature of hyperbolic quaternions is not encountered when the formal method in ring theory of quotient rings is applied.

[edit] 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{-1}
jk=i\sqrt{-1}
ki=j\sqrt{-1},

the Pauli matrices satisfy

\sigma_1\sigma_2=\sigma_3\sqrt{-1}
\sigma_2\sigma_3=\sigma_1\sqrt{-1}
\sigma_3\sigma_1=\sigma_2\sqrt{-1}

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 reinforced 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.

[edit] 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) being the 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

η(q,q) = qq * = a2 − b2 − c2 − d2.

In particular, the hyperboloid {q:a > 0,qq * = 1} 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 through the origin after a moment of local time.

[edit] Simultaneity

Select an arbitrary point from the hyperboloid: u = cosh(a) + rsinh(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.

[edit] References

  • MacFarlane (1891) "Principles of the Algebra of Physics" Proceedings of the American Association for the Advancement of Science 40:65-117.
  • C.G. Knott (1892) "Recent Innovations in Vector Theory" Proceedings of the Royal Society in Edinburgh and Nature 47:590-3.
  • MacFarlane (1900) "Hyperbolic Quaternions" Proceedings of the Royal Society at Edinburgh, 1899-1900 session, pp. 169-181.
  • J.W. Gibbs and E.B. Wilson (1901) Vector Analysis, Yale.
  • M.J. Crowe (1967) A History of Vector Analysis, University of Notre Dame
  • Alexander MacFarlane and Hyperbolic Quaternions
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