What? You've never heard of it? A big knock-down, drag-out fight between great minds of its day over, more-or-less, the philosophy of how to go about mathematical physics. None of this "let's do an experiment to distinguish between these two theories" stuff; that's for wimps. This was the deep stuff: nuts-and-bolts versus mathematical elegance; generic versus well-behaved; even, so we're told, particles versus waves (I kid you not).
Old paradigms get crushed; it's part of how new paradigms establish and maintain themselves. History gets buried. But that doesn't mean we have to like it, or stand for it. As I write this, here's the sum total of what Wikipedia's article History of quaternions has to say about this colorful event in the history of mathematical physics:
From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs and Oliver Heaviside. Both were inspired by the quaternions as used in Maxwell's A Treatise on Electricity and Magnetism, but — according to Gibbs — found that "... the idea of the quaternion was quite foreign to the subject." Vector analysis described the same phenomena as quaternions, so it borrowed ideas and terms liberally from the classical quaternion literature. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics.
Yawn. (Also, so much for Wikipedian neutrality; but that's a different can of worms.)
In this post, I resurrect a paper about the vectors-versus-quaternions debate, written in the Long Ago when we used things called typewriters, and wrote-in special symbols by hand. It's been languishing for years in a file folder, one of those physical things that are the models for the icons on your phone.
Here's how the paper came about. I learned about the vectors-versus-quaternions debate from my father. In fact, I learned about quaternions from my father. I inherited his enthusiasm for them. And then, in my third year at WPI, I seized an opportunity to study the debate in depth.
One of the requirements for the BS degree at WPI was the Humanities Sufficiency project. The idea was that tech students should be well-rounded, so they should take (and pass) a bunch of humanities classes and then, building on those classes, write a term paper on a subject that bridges the gap between humanities and sciences. WPI had an undergraduate grade called "NR", short (I think) for "Not Recorded": if you didn't pass a course, it didn't go on your transcript (though you didn't get your tuition back, naturally). This resulted in students taking some classes because they were interested, and being sometimes more concerned with learning than with getting high grades. So you'll understand when I say, it took me till my third year to accumulate the class credits I needed for the Sufficiency because I only passed about 50% of the humanities classes I took. Though a bunch of people, including me, were rather bemused when I not only passed, but got high marks in, Philosophical Theories of Knowledge and Reality.
I chose vectors-versus-quaternions as my topic, with the no-nonsense title "Quaternions: A Case Study in the Selection of Tools for Mathematical Physics". The Sufficiency was ordinarily a half-semester project, but wasn't required to be, and with such a juicy topic and personal interest, of course I took a bit over that. Professor Parkinson actually apologized for not giving me a top grade on it, explaining that he had a strict rule never to give a top grade to a Sufficiency that took more than the basic half-semester. At the time, what I thought (but at least had enough tact not to say to him) was that I was doing the work for its own sake, not for a mere grade. Later, when that grade caused me to graduate With Distinction instead of With High Distinction, I understood why he'd apologized. And was belatedly a bit put out, after all.
The things I learned from this paper have ever after informed my understanding of how scientific paradigms are chosen — a really major theme in my life since, after all, my master's thesis (pdf) brushed against a rejected paradigm (extensible languages), while my dissertation outright resurrected one (fexprs). The influence from this is also deep in the foundations of my thinking on memetic organisms, which I blogged on some time back.
The writing style is a bit stiffer, here and there, than I strive for now. (I was even worse in highschool.) All in all, though, I'm still fairly pleased with the piece.
The original had both footnotes meant to be read inline, and endnotes with bibliographical details. Here, I've put both in sections at the end, using letters for the erstwhile footnotes, numbers for the endnotes. While I'm being pedantic, this version of the paper has three changes from the original version submitted in Spring 1986. (And, just to prove what a nerd I am, the changes were made in 2002.) Footnotes [m] and [y] have been added; and where Hamilton's nabla is first defined, I've corrected it to use full derivatives, having originally miswritten it with partial derivatives.
Associative memory is a strange thing. Certain details stick with us. I remember worrying about one partial sentence from Crowe that I just couldn't think how to put differently, and in the end deciding to let that passage stand; though at this late date I've no idea which passage that would be.
Contents: Title Body Footnotes Endnotes Bibliography
Quaternions: A Case Study in the Selection of Tools for Mathematical Physics
John N. Shutt
Presented to: Professor Parkinson
Department of Humanities
Term D, 1986
Submitted in Partial Fulfillment
of the Requirements of
the Humanities Sufficiency Program
Worcester Polytechnic Institute
Worcester, Massachusetts
Corrections and additions: 11 March 2002
Quaternions are a form of hypercomplex number with four
components. Mathematically, they are the next most well-behaved
algebra after the complex field. The extent of their usefulness
for mathematical physics has been in doubt since their discovery.
This paper examines historically the principal issues in the
use of quaternions for mathematical physics. The historical and
mathematical background of quaternions is examined, followed by
their application first to classical physics, and then to modern
physics. The paper concludes with an analysis of some of the
major issues.
Vectorial analysis, in its general sense, is the mathematical
treatment of directed magnitudes. It arose in the first
half of the nineteenth century as a synthesis of two major trends
of thought, one in physics and the other in mathematics.1
It has been pointed out2 that the geometry of physics after
Newton differed intrinsically from that of the ancient Greeks.
The difference is that, while Newtonian physics is set on a
Euclidean stage, many of the principal players are (implicitly)
vectors, which are not present in Euclid.
This left physics by the early nineteenth century working
under a handicap. Vectors underlay most of physics, but could
not be handled gracefully. The primary mathematical tool for
handling geometry was the Cartesian coordinate system; Cartesian
coordinates are flexible in principle, but in practice they are
apt to be unwieldy and generally opaque. A natural language for
vectors was needed.a
Mathematics by this time had become overextended.3 The
problem was with the concept of number. The formulae of algebra,
originally developed using only positive numbers, were now being
successfully applied to negative and complex numbers. Since
mathematicians had traditionally grounded their work on intuition,
the more extensive number systems left many mathematicians
uneasy.
The problem led George Peacock, in 1833, to postulate the
principle of permanence of form. This principle said that
"Whatever algebraical forms are equivalent when the symbols are
general in form but specific in value [positive integers], will
be equivalent likewise when the symbols are general in value as
well as in form."4 By 'general in form' is meant that properties
of a particular number cannot be generalized; for example,
14 mod 7 = 0 doesn't imply that x mod 7 = 0.
'General in value' is meant to refer to fractional, irrational, negative,
or complex numbers, but leaves a question almost as big as the one
to which the principle is addressed. Despite its shortcomings, the
principle was important because it did recognize that algebra is
based on rules.
At least six peopleb
had independently devised the geometrical
representation of complex numbers before Gauss finally
published the idea in 1831.5 Several of these people used this
representation as a justification for the complex number field.
(Of the first two of the six to make the discovery, Wessel
embraced this justification but Gauss did not.) It was Gauss's
publication that finally drew general attention to the idea.
However, William Rowan Hamilton (1805–1865) was not aware of
Gauss's 1831 paper until 1852. He was influenced instead by John
Warren, in whose work he would have been exposed to the concepts
of the associative, commutative and distributive laws. Hamilton
did not consider the geometrical approach a sufficient justification.
In 1837 he presented a fresh approach, interpreting
complex numbers as algebraic couples of real numbers.7 He
defined addition, subtraction, multiplication and division of
couples and then derived from them the primitive properties of
complex numbers.
These mathematical developments suggested to many of the
mathematicians involved that a further extension of number might
be analogous to (3-dimensional) space.8 It was generally
expectedc
that the sought-after extension would have three terms,
and obey all the laws of complex algebra (associative, commutative,
and distributive), as well as having close ties to spatial
geometry.
The ties to spatial geometry take many forms. The two that
Hamilton eventually settled on are (1) the Law of the Norms,d and
(2) unique division. In retrospect, only the real and complex
numbers satisfy both all the usual laws and (2), and (1) is
simply impossible in three dimensions.e 9
The idea that triplets (Hamilton's term) might not satisfy
all the usual laws had occurred to Hamilton as early as 1830. He
was acquainted in particular with non-commutative multiplication
from some speculations in set theory.10 On 16 October 1843, as he
walked to a Council of the Royal Irish Academy, several of the
above ideas converged in his mind to produce quaternions. He had
been working recently with triples of the form a+bi+cj, with
i2 = j2 = −1; he now realized that he could satisfy (1) above by
making the assumptions that ij = −ji, and that this product
yielded a third imaginary component k = ij, with k2 = −1.11
The resulting quaternion has the form a+bi+cj+dk, with
i2 = j2 = k2 = ijk = −1. Quaternion multiplication is
distributive over addition, and associative, but not commutative.9
The norm is a modulus of multiplication, and right-division and
left-division are unique. Real and complex numbers and quaternions
are the only three possible division algebras — that is, algebras
with associative and commutative addition, distributive and
associative multiplication, and unique division.f
Hamilton created a plethora of new terms for use in his new
algebra.14 A quaternion q is made up of a real part, called the
scalar of q and denoted Sq, and an imaginary part, called the
vector of q and denoted Vq. Alternatively, it can be expressed
as the product of a positive real number (the 'length,' or square
root of the norm of q), called the tensor of q and denoted Tq,
and a quaternion with tensor equal to one, called the versor of q
and denoted Uq.g Thus q = Sq + Vq = TqUq.
A versor u has a unique decomposition u = cos θ + v sin θ
with angle 0 ≤ θ ≤ π and unit vector v = UVu.h 15 If p is a
vector perpendicular to v, p' = vp is the rotation of p by
angle θ about v. This allows great-circular arcs in space to be
represented by quaternions, leading to elegant proofs in
spherical trigonometry.
Any non-zero quaternion q has a unique inverse q−1 such that
qq−1 = q−1q = 1. Left- and right- division by q are defined
respectively as pre- and post- multiplication by q−1. If q is a
versor q = cos θ + v sin θ, then for an arbitrary vector p,
p' = qp(q−1) is the conical rotation of p by angle 2θ about v.
Another useful decomposition is that of the quaternion
product of vectors into its scalar and vector parts. If u,v are
vectors separated by angle θ, Suv = −(TuTv) cos θ and
Vuv = (TuTv) (sin θ) n where n = UVuv is a unit vector
perpendicular to u and v.i The scalar part is commutative
(Suv = Svu), and the vector part anticommutative (Vuv = −Vvu).
Suv and Vuv were later to form the basis for modern vector
analysis.
Quaternions were the subject of a debate at the British
Association meeting of 1847.16 George Peacock, who favored
quaternions, did not come forward, but Sir John Herschel did, and
called quaternions "a cornucopia of scientific abundance."
Against quaternions it was objected that owing to their
complexity, quaternion calculations are overly prone to mistakes.
There was also at the meeting at least one representative of the
status quo; in Hamilton's words,
Mr. Airy, seeing that the subject could not be
cushioned, rose to speak of his own acquaintance with it
[quaternions], which he avowed to be none at all; but
gave us to understand that what he did not know could
not be worth knowing.
The background to this paper would be incomplete without
some mention of Hermann Günther Grassman (1807–1877).17
In 1844 Grassman published a work, monumental in both size and scope,
entitled Die lineale Ausdehnungslehre (calculus of extension).
The ideas behind the Ausdehnungslehre began in 1832 with
the interpretation of a parallelogram as the geometrical product
of two lines. Grassman generalized this insight to other shapes
and an arbitrary number of dimensions, and placed it on an
abstract mathematical basis. The system of the Ausdehnungslehre
was a very broad mathematical generalization originating from
these geometrical concepts. Several types of multiplication were
defined, the only requirement for a multiplication being
distributivity over addition.
The Ausdehnungslehre of 1844 was written in a strongly
metaphysical style, and was also highly abstract at a time when
mathematics was based on concrete intuition. Grassman was
unknown. Consequently, the Ausdehnungslehre went unnoticed by
the world at large.
Despite Grassman's efforts, including a revised Ausdehnungslehre
in 1862, his work remained obscure throughout his
life, only beginning to attract interest about the time of his
death. One by one, Grassman's discoveries were remade by others,
with Grassman's anticipation unveiled in a subsequent question of
priority.
Although Grassman's inner and outer products are similar
respectively to the scalar and vector parts of Hamilton's
quaternion product of vectors, Grassman was conceptually distant
from quaternions. The significance of the Ausdehnungslehre here
is that it encompasses an n-dimensional system of vectorial
analysis.j
An account will now be given of three significant figures in
the application of quaternions who worked within the quaternion
tradition in the nineteenth century. These three figures are
Hamilton, Tait, and Maxwell. Following this, the circumstances
will be described by which quaternions were abandoned in favor
of vector analysis.
The first major publication on quaternions was Hamilton's
Lectures on Quaternions of 1853.19 The text ran to over 700
pages. It was difficult to read; by 1859, Herschel — a great
enthusiast of quaternions and an able mathematician — had only
managed to read through 129 of its pages.
In 1859 Hamilton began work on the Elements of Quaternions.
It was originally to be an elementary treatise, but became a
reference work longer than the Lectures — though without the
metaphysical emphasis of the earlier work. The Elements
developed quaternions mathematically in great detail, but did not add
to their physical application. By his own admission, Hamilton
was by this time out of touch with contemporary physics.
Hamilton was convinced of the value of quaternions to
physics, and had published scattered such applications. In 1846
he had defined a (nameless) operator
ᐊ = i d⁄dx + j d⁄dy + k d⁄dz.
However, he never did concentrate his own efforts on applications
to physics, choosing instead to develop quaternionic theory. He
had planned for a major section of his Elements on the ᐊ
operator, but the section was never written because of his death
in 1865. The Elements was published posthumously in 1866.
Peter Guthrie Tait purchased and read Hamilton's Lectures in
1853, out of general curiosity.k 21 In 1857 he encountered an
application that reminded him of Hamilton's ᐊ operator;
pursuing this, he shortly became a devoted quaternionist,
ultimately succeeding Hamilton as their chief advocate after the
latter's death.
Tait's interest in quaternions was for their physical
applications. His Elementary Treatise on Quaternions, published
in 1867, was the first accessible introduction to quaternions.
This work went into some detail on the operator ∇,l using it to
express several important theorems (e.g. Green's and Stokes').
Tait did much to further quaternionic applications to
physics. Oddly, he seems to have scrupulously avoided quaternions
in his other work, including all of his lectures at the
University of Edinburgh. Quaternions are also omitted from
Tait's collaboration in mechanics with Lord Kelvin, the Treatise
on Natural Philosophy; speaking of this later, Kelvin said,
We [Kelvin and Tait] have had a thirty-eight years' war
over quaternions.... Times without number I offered
to let quaternions into Thomson and Tait [the
Treatise], if he could only show that in any case our
work would be helped by their use. You will observe
that from beginning to end they were never introduced.
It should be understood that Kelvin was throughout his life
resolutely opposed to all vectorial methods.
James Clerk Maxwell originally derived his equations in the
1860's using component notation.22 He began to study quaternions
in 1870. In his Treatise on Electricity and Magnetism of 1873,
he presented both component and quaternionic notation.
Maxwell was a firm believer in physical analogy. He favored
quaternions as an aid to thinking, because the notation corresponds
more closely than does that of components to physical
reality. For calculation, however, he considered component
notation superior. He made this distinction in the preliminary
chapter of the Treatise, where he advocated "the introduction of
the ideas, as distinguished from the operations and methods of
Quaternions."
Maxwell's use of quaternions in the Treatise was accordingly
limited primarily to the restatement of important results in
quaternionic form. Nonetheless it led some physicists who had
never done so before to study quaternions.
In particular, this was the case with Josiah Willard Gibbs
of America and Oliver Heaviside of England.23 These two men
proceeded independently along very similar lines.
From Maxwell's Treatise, both went to Tait's Elementary
Treatise on Quaternions. Both observed that, as actually used by
Maxwell and for the most part even by Tait, the vector/scalar
partition of quaternions was more important than the quaternions
themselves. Both then developed systems that treated vectors and
scalars as entirely separate entities, V ∇ and S ∇ as separate
operators,m etc.
Heaviside went no further than this. His notation was not
entirely compatible with Tait's, but he never introduced concepts
outside the quaternion tradition. Thus his system was essentially
a subset of quaternion analysis.
Gibbs however, broke all ties with Hamilton, even to citing
Grassman as the main precedent for his system.n His notation is
substantially different from Tait's; in particular he replaced
the prefix operators of Hamilton with infix operators.o (∇ was
an exception to this.) Most significantly, he introduced a
concept totally alien to quaternion analysis — that of the dyad.
(A dyad is neither a vector nor a quaternion, but a tensor.)
Neither Gibbs nor Heaviside shared Tait's scruples about
using their systems in their other work. Gibbs applied his
vector analysis in periodic courses at Yale starting in 1879, and
in some of his physics papers. It was Heaviside who did most to
disseminate vector analysis; he made heavy use of his system in
several important electrical publications, such as his
Electromagnetic Theory, permanently linking vectors to that
rapidly growing field.
A debate took place in the early 1890's, on the proper
vectorial system for mathematical physics.p 25 This debate
involved more than thirty letters and articles over five years
(1890–1894) in eight leading scientific journals, as well as a
scattering of other published writings. It was primarily between
Gibbs and Heaviside on one side and the English quaternionists on
the other.
Superficially, a prominent characteristic of the debate was
its colorful verbiage. Heaviside was the contributor to this for
the vectorists, while considered vectorist ideology was supplied
primarily by the more dignified Professor Gibbs. Metaphors and
the odd pot shot are scattered through the quaternionists'
writings, which are often pervaded by a tone of bitterness.
Particularly fiery are the literary antics of Alexander McAulay,
an unknown youngster who joined the ranks of the quaternionists
in 1893 26,26j with what Tait called "the perfervid outburst of
an enthusiast."
The figures of Grassman and Hamilton became weapons in the
debate. The quaternionists played heavily on Hamilton's fame.
The vectorists dissociated themselves from Hamilton entirely, and
placed themselves firmly behind Grassman,q for whom they built a
reputation. As a result of this and the ultimate triumph of
vector analysis, Hamilton's fame was tarnished. (It was later
resurrected through Hamilton's characteristic function in quantum
mechanics.)
Gibbs was conversant with the systems of Hamilton and
Grassman as well as his own. This served him well on several
occasions in the debate, for Tait was ill acquainted not only
with Grassman's Ausdehnungslehre but with both Gibbs' and
Heaviside's vector analyses.r
The question recurs throughout the debate, of why quaternions
had not been more widely accepted.s
This was generally not itself
an issue (an exception is Gibbs' letter to Nature of 16 March
1893 26g), but served as a focal point for other issues.
The opening shots of the debate were fired by Tait,26a and
were aimed principally not at vector analysis but at component
notation. His arguments centered on expressiveness; no detail
need be given, as the principal interest of the current discussion
is with issues between vectorial systems.t However, it is
significant that Tait apparently failed to appreciate the coming
threat of vector analysis. He seems to have repeatedly
underestimated his opponents for several years into the debate.
What touched off the controversy was the following passage
from the preface to the 1890 (third) edition of Tait's Treatise.
Even Prof. Willard Gibbs must be ranked as one of the
retarders of Quaternion progress, in virtue of his
pamphlet on Vector Analysis; a sort of hermaphrodite
monster, compounded of the notations of Hamilton and
Grassman.
The issue of notations, which Gibbs early subordinated to
that of notions,u 26b
nevertheless was addressed repeatedly.
Gibbs disliked the prefix operators of quaternions, because infix
operators were the existing norm.26b Tait countered that the
prefix operators allowed the use of fewer parentheses, thus
enhancing brevity of expression.26c
C. G. Knott objected to the large number of operators in
Gibbs' notation26e, illustrating his point with Gibbs'
abbreviations 'Pot,' 'New,' 'Lap,' and 'Max,' all of which represent
various combinations of the Nabla operator. Gibbs argued
convincingly26i that the quaternionic equivalents of these
operators were too complicated to be intelligible.
Alexander Macfarlane, like Knott a former student of Tait,
brought out another issue in 1891:26d the sign of the scalar
product. In quaternion analysis, Suv for (positive) vectors
u,v is negative because i2 = j2 = k2 = −1 but the vectorists
had no stake in √−1, so for convenience they defined the scalar
product to be positive.
Macfarlane had his own solution to this. He distinguished
versors from vectors. Since two right turns produce a reversal
(−1), he set i2 = j2 = k2 = −1 for versors; but for convenience
he set i2 = j2 = k2 = 1 for vectors. Thereafter in the debate
he represented a third faction. The vectorists never specifically
addressed his system, so that his net influence was simply
to undermine the quaternionists.
The vectorists never addressed the issue of the sign of the
scalar product; in any case, there was no need for them to do so,
since Macfarlane did it for them. The response to Macfarlane's
innovation came from Knott. In December of 1892, Knott objected26e
that without i2 = j2 = k2 = −1, multiplication ceases to be
associative. Macfarlane answered in May of 1893 26h that, just
as with commutativity, associativity is only a convention and can
be given up with impunity if it is convenient to do so.
The crux of the controversy was the issue of notions.
Everyone in the debate (except Cayley, as noted earlier) agreed
that vectors and scalars are important for physics. The vectorists
maintained that the quaternion has no notable physical
interpretation (other than rotation, Gibbs acknowledged, but
dyadics serve this purpose satisfactorily26b). The burden of
proof throughout the debate thus lay on the quaternionists,
although Tait did once make the same accusation of artificiality
against dyadics.26c
The quaternionists did little during the debate to prove
their case. Knott did address the question twice. In 1892 he
argued simply that the ratio of two vectors is a fundamental
concept, so that since this ratio is a quaternion, quaternions
are fundamental.26e In 1893 he added to this an analogy:
Although sin θ and cos θ occur more frequently than
θ itself, we should not conclude that θ plays no
fundamental role. Similarly we should not infer that αβ
[the full quaternion product] is not fundamental simply
because Vαβ and Sαβ occur more frequently.31
There is another issue which recurs periodically throughout
the debate. It is generalizability to higher numbers of dimensions.
From his first article in 1890, Tait praised quaternions
for being "uniquely adapted to Euclidean [i.e. three-dimensional]
space."26a Gibbs in his
first letter of 1891 praised vectors for
being generalizable "to space of four or more dimensions."26b
These views are representative of the positions taken on this
issue by the respective sides in the debate. There is one
exception: in February of 1893, Dr. William Peddie, Tait's
assistant at the University of Edinburgh, attempted to show that
"quaternions are applicable to space of four or any number of
dimensions."26f
If there was a winning side to the debate, it was the
vectorists. They consistently outmaneuvered (Gibbs) and outspoke
(Heaviside) their opponents. More fundamentally, the
quaternionists did little to justify their position on the crucial
issue of notions. In any event, the decisive factor in the
ultimate acceptance of vector analysis was Heaviside's active use
of it in his published work.v
Passing mention may be made of the International Society for
Promoting the Study of Quaternions and Allied Systems of
Mathematics.32 It was organized shortly after the debate by Shunkichi
Kimura, residing at Yale at the time, and Pieter Molenbroek, and
published a bulletin from 1900 to 1913. The Society was plagued
with difficulties from its inception, the first of which was that
Tait declined the presidency due to ill health. (He died in
1901.) In 1913, all the offices were due for election at the
same time, with no one to arrange it, and the Society slipped
quietly into oblivion.
There was a tendency among quaternionists, which surfaced
on several occasions in the debate, to think of the vectorists as
ungrateful children. This is presumably the source of the bitter
overtone in their writings that was mentioned earlier. It may be
interesting in relation to this to consider the following excerpt
from a review by Heaviside written in the early 1900's.
... as time went on, [after the controversy] ... it
was most gratifying to find that Prof. Tait softened
his harsh judgments, and came to recognize the
existence of rich fields of pure vector analysis, and
to tolerate the workers therein.... I appeased Tait
considerably (during a little correspondence we had) by
disclaiming any idea of discovering a new system.33
Quaternions will now be considered in their application to
twentieth-century physics. It is appropriate in this regard to
recount the early history of the idea of using the real part of a
quaternion to represent time. Priority in this idea belongs to
Hamilton.
In the hierarchy of thought, Hamilton placed mathematics
above physics, and metaphysics above mathematics.34 An important
element of his philosophy was the metaphysical importance of the
number three.35 He attached much significance to the
tridimensionality of space, and this was a major impetus for his search
for algebraic triplets.
After arriving at quaternions by purely algebraic means,
Hamilton struggled to reconcile them with his philosophy. At
least since 1835 he had used time as a metaphysical justification
for the real numbers; this may have contributed to his later
identification of quaternions with the four dimensions of space
and time.
He clearly failed to pass on the idea to the physicist Tait.
Recall that in the 1890's controversy the quaternionists represented
quaternions as "uniquely suited" to three-dimensional
space. Tait did speculate on the possibility of a fourth dimension,
as did Maxwell.37 However, it is not clear that either of
them associated a fourth dimension with time.
In 1896, Kimura pointed out that ∇ is not a full quaternion
operator because it does not include the derivative with respect
to the real component.38 He introduced a full quaternion
differential operator q∇, and used the scalar component to
represent the derivative with respect to time. He had physical
applications in mind.
The possibility of using quaternions occurred to Hermann
Minkowski when he was formulating his space-time in the 1890's
(naturally enough, since this was during the vector-quaternion
controversy). He rejected them completely as "too narrow and
clumsy for the purpose."39
To understand what might have motivated Minkowski to make
this judgment, consider some basic four-dimensional properties
of quaternions. Cayley showed in the 1850's that the general
rotation in Euclidean four-space may be expressed by the quaternion
formula p' = upv, where u,v are arbitrary versors.w 40 Also
in Euclidean four-space, multiplication by an arbitrary versor
u = cos θ + v sin θ may be understood as a relative turning by
angle θ through 4-space.
Unfortunately, Minkowski space-time is non-Euclidean. While
quaternions have 'length' (modulus of multiplication)
Tq = (w2 + x2 + y2 + z2)1⁄2,
interpreted as a space-time vector q
should have length (w2 − x2 − y2 − z2)1⁄2.41
A solution to this problem was found in 1912 by Ludwig
Silberstein.42 He let the scalar component, representing time,
be imaginary by introducing a fourth √−1 independent of the three
of quaternions. He was thus working with what Hamilton called
biquaternions, quaternions whose four coefficients are complex
numbers. Biquaternions do not, of course, have unique division,
although Silberstein did define a biquaternion "inverse."x
By this device, Silberstein was able to express the Lorentz
transformation in the form q' = QqQ, where q is in frame S and
q' is its equivalent in frame S'. Q is a complex versor (i.e.
TQ = 1); further, the coefficient of SQ is real, and the three
coefficients of VQ are imaginary. Thus
Q = cos θ + v sin θ for
imaginary unit vector v and angle θ. The resulting transformation
is a rotation in Minkowski space-time by angle 2θ √−1.
Compare this with the general Euclidean three-space rotation
mentioned earlier.
A different solution to the problem was given in a 1945
paper by P. A. M. Dirac.43 Dirac submitted that the value of
quaternions lies in their special algebraic properties, and that
therefore resorting to biquaternions is not productive.y
Restricting himself to real quaternions, he derived the general
quaternionic linear transformation q' = (aq + b)(cq + d)−1, with
quaternion coefficients a,b,c,d.z
He used this equation to describe a transformation in
five-dimensional projective space, and restricted it to describe the
Lorentz group. He then derived a one-to-one correspondence
between the quaternions q and q' in his transformation and
space-time vectors, through a comparatively involved set of equations.
Finally, he used his quaternionic transformation to derive a
general quaternionic formula for the relativistic addition of
velocities. The elegance of this formula provoked the only
non-mathematical comment in the paper, "The quaternion formulation
appears to be the most suitable one for expressing generally the
law of addition of velocities."
Dirac's treatment is contrary to the traditional usage of
quaternions. Ever since their discovery, much of their perceived
value has been in the physical interpretability of quaternionic
formulae. This perception is evident in the following, written
to Hamilton by John T. Graves.
There is still something in the system which gravels
me. I have not yet any clear views as to the extent to
which we are at liberty arbitrarily to create imaginaries,
and endow them with supernatural properties.
You are certainly justified by the event.... but I
am glad that you have glimpses on physical analogies.44
Physical interpretability was the quaternionists' main argument
against component notation.
Dirac's approach was to set up a correspondence between
quaternions and space-time vectors; but the correspondence was
unintuitive. This abandonment of physical interpretation is
consistent with the general philosophy of much of modern
mathematical physics. However, it is not the way others have applied
quaternions to modern physics.
Papers applying quaternions to modern physics are not as
rare as one might suppose, numbering (as nearly as I can determine)
at least in the dozens.45 These papers deal with a wide
range of topics. It is not within the scope of the current paper
to examine all their areas of application;aa for example, the use
of quaternions to describe elementary particles will be omitted.46
For modern quaternionic ideology, two representative examples
will be used.
The first example is a 1964 paper, "Quaternions in Relativity,"
by Peter Rastall.47 The paper begins with a brief
historical account of quaternionic application to relativity,
along with commentary on why quaternions have been (up to 1964)
repeatedly passed by in favor of other formalisms.bb His own
interest in quaternions is for field equations in curved
(Riemannian) space-time. He argues that for this general case,
neither matrix nor spinor notation yields any clear physical
interpretation because neither can easily be understood in terms
of tetrads of real coordinates (x,y,z,t).
Quaternions, by which he means complex or bi- quaternions,
are to provide this physical interpretability. He uses
Silberstein's form for Lorentz transformations in flat
space-time. (Recall that Silberstein's quaternions have only four
non-zero real coefficients, corresponding directly to coordinates
x,y,z,t.) He describes Riemannian space-time using tetrad
formalism, i.e. with each tetrad of event coordinates he
associates a set of four axes, which need not be orthogonal. By
combining these tools he then derives his general field
equation.
It is important to such quaternionic treatments of
relativity that a quaternionic equivalent is possible for any
matrix formula, and vice versa.cc 48
The most basic quaternion-matrix
equivalence, demonstrated by C. S. Peirce in 1881, is
between a real quaternion w + xα + yβ + zγ
(imaginaries α, β, γ)
and a 2 × 2 complex matrix
The four matrices equivalent to 1,
α,
β,
γ are essentially the
Pauli spin matrices.
The second example consists of two papers written in 1962–63
by a group of physicists on quaternion quantum mechanics.dd These
papers take advantage (respectively) of two basic properties of
quaternions: their close ties to 3-/4- vector spaces, and their
lack of commutativity.
The first paper presents the fundamentals of the theory.49
Their starting point is that
a propositional calculus exists that we can call
general quantum mechanics (as distinguished from
complex quantum mechanics) in as much as no number
system or vector space at all is assumed in its
formulation....
It is always possible to represent the pure states
of a system of "general quantum mechanics" by rays in
a vector space in a one-to-one manner...
and that the
only number system over which this can be done for
all such systems is 𝓠, the quaternions. They suggest that while
real and complex quantum mechanics are very similar, "quaternion
quantum mechanics has many new features that make it much
richer."
The second paper capitalizes on a difficulty that arises
from the non-commutativity of quaternions.ee 50 Multiple mathematical
descriptions arise that should be equivalent. By postulating
invariance of the physical laws over these different
descriptions, they arrive at a new field of which electromagnetism
is a special case.
Perhaps the most extensive example of quaternionic application
in the twentieth century (if not for all time) is the work of
Otto Fischer. Fischer was a Swedish civil engineer who became
interested in quaternions in the years before World War II. In
the 1950's he published two substantial books on the application
of quaternions.51
In Universal Mechanics and Hamilton's Quaternions (I have
not had access to his other book), Fischer set himself a rather
ambitious goal.
This is a book written by a civil engineer on universal
mechanics with an attempt to introduce a certain order
in its mathematical structure by means of Hamilton's
Quaternions. The term "universal mechanics" refers to
the mathematics of ordinary physics of motions,
elasticity, hydrodynamics, aerodynamics, electromagnetism,
together with relativistic and cosmic physics as well
as quantum mechanics.
Fischer's aim is to create a close correspondence between
concepts and explicit mathematical structure. In pursuing this
goal, he correlates several types of mathematical structural
hierarchies to branching specialities within universal mechanics.
One such hierarchy is the "potential pyramid," which expands by
repeated differentiation from an 'apex' potential. Closely
related is an "affinor pyramid",
ff
also formed by repeated
differentiation. The potential pyramid is static, consisting of
functions of simple, quadric or double quadric quaternions,
gg
while an affinor pyramid consists of operators on such functions,
and is therefor dynamic, taking its shape from the function to
which it is applied.
The three different types of quaternions are the basis of
Fischer's other major unifying structure. After introducing and
doing much work with real quaternions in imaginaries i1, i2, i3,
he proceeds to quadric quaternions by introducing "superdirections"
ï, j1, j2, j3 that correspond roughly to different
specialities in physics. Ultimately he applies this technique in
the more general case of double quadric quaternions to his goal
of unifying universal mechanics.hh
Fischer appears to have been widely competent in the
specialities of "universal mechanics." He surely did not,
however, have a talent for presentation. The superficial appearance
of the Universal Mechanics is that of the numerology of the
'crackpot fringe.'
Fischer was part of what has often been called the "Cult of
Quaternions"52 — a tradition of enthusiastic devotees that began
with Hamilton and continues to the present day.ii In 1910, C. S.
Peirce wrote of his brother James Mills Peirce, who had been a
noted quaternionist until his death in 1906, that he "remained to
his dying day a superstitious worshipper of two hostile gods,
Hamilton and the scalar √−1." An earlier reference to the
'cult' is found in the chapter on vector analysis in Heaviside's
Electromagnetic Theory.
"Quaternion" was, I think, defined by an American
schoolgirl to be "an ancient religious ceremony." This
was, however, a complete mistake. The ancients —
unlike Prof. Tait — knew not, and did not worship
Quaternions.53
The members of the 'cult' who have been mentioned in this
paper all exhibited rational reasons for their enthusiasm. Yet,
their reputation as a semi-religious community is not entirely
unsupported in their own writings. A comparatively recent
example is E. T. Whittaker's statement (1940) on quaternionic
research after the turn of the century, that "the good work went
on."jj
In 1871 Maxwell had observed, with no sarcasm intended,
that "The unbelievers are rampant."54
Throughout this paper, the motivating philosophies of
applied quaternionists have been noted. Definite trends are
visible in these philosophies.
From their discovery, the claims of quaternions for physical
application have been interpretability and utility. Their
interpretation is certainly an improvement over component
notation; nevertheless, as F. D. Murnaghan has observed, to the
general public quaternions were the archetype of a baffling
abstract theory until they were supplanted in this role by
Einstein's General Theory of Relativity.56 The quaternionists
simply continued to assert that quaternions are meaningful, and
as a tactic against component notation this was sufficient.
Against vector analysis, however, the tactic failed. Having
extracted the utilitarian part of quaternion analysis, the
vectorists discarded the remainder and branded it meaningless.
The result was that the quaternionists were left clinging
forlornly to their claims of interpretability while practical-minded
physicists flocked to vector analysis.
In the twentieth century, this parting of the ways has led
to a peculiar twist of fate. Following their vectorist and
anti-quaternionist traditions, physicists have adopted the
utilitarian notation of matrices. This has led them away from
interpretability, and ultimately even away from utility into increasing
abstraction. Quaternionists have continued to emphasize
interpretability, finally coming to be its uncontested claimants;
but, being an acknowledged fringe group, have had no general
acceptance to date.
Rastall's paper of 1964 concludes with the following
paragraph. It illustrates clearly the underlying ideology of the
modern physical application of quaternions. It is also a fitting
note on which to end the current paper on quaternions.
The movement towards abstract algebraic and
coordinate-independent formulations of physical theories,
and away from particular matrix representations and
special coordinate systems, is an increasingly popular
one, and our work is in accord with it. Less popular,
and seemingly opposed to this rarefied mathematical
spirit, is our desire to make abstract concepts more
concrete and imaginable. To pure mathematical minds
the aim is unsympathetic. They are happy in their
complex spaces, and would prefer to postulate an affine
connection rather than to align tetrad vectors. It is
a matter of taste. Those, however, who are prepared to
exploit the accident of having been born in space-time
may find this paper useful.
Footnotes
- [a]
This need increased greatly with the development of
electromagnetic theory in the last third of the century.6
- [b]
The six are Wessel (1799) and Gauss (about the same time),
Argand and Buée (1806), and Mourey and Warren (1828).
- [c]
Almost everyone looked for a superset of the complex
numbers. An exception was Servois, who came close to quaternions
in 1815.12
- [d]
In modern terms, the Law of the Norms says that the norm
(the sum of the squares of the real coefficients of the terms)
should be a modulus of multiplication. A modulus of multiplication
is a function M(x) such that M(a) * M(b) = M(a*b). The
Law of the Norms holds for real and complex numbers.
- [e]
In 1844 Augustus de Morgan presented a number of triple
algebras that do not satisfy the Law of the Norms but do have
moduli of multiplication. Given the distributive law, the Law of
the Norms and the uniqueness of division are equivalent, so
naturally de Morgan's triple algebras do not have unique division
either.
- [f]
If the associative law is also dropped, only one further
algebra, Cayley's octonions, is possible.13
- [g]
'Scalar,' 'vector' and 'tensor' seem to be the only three
of Hamilton's original quaternionic terms that have developed
non-quaternionic usages. The modern meaning of 'tensor' is
completely different from Hamilton's.
- [h]
Obviously, v is indeterminate for angles 0 and π. In the
following description of quaternions, special cases such as this
are generally omitted.
- [i]
u,v,n always have the same sense. If a right-handed
coordinate system is used, u,v,n form a dextral set. Hamilton
used a left-handed system, while Tait, Maxwell, Gibbs all used a
right-handed system.18
- [j]
Also implied in Grassman's calculus of extension are matrix
theory and modern tensor analysis.20
- [k]
Remember, this is the same book that so effectively stymied
Herschel.
- [l]
This operator was apparently written as ᐊ by Hamilton but
as ᐁ by Tait. In Tait's form it has been variously
given the names nabla (after a visually similar Assyrian harp), del, and
atled (delta spelled backwards).24
- [m]
In replacing V ∇ and S ∇ by ∇× and ∇·
(Gibbs' notation), it is also necessary to redefine ∇
itself from quaternionic operator i d⁄dx + j d⁄dy + k d⁄dz
to vectorial operator ı ⃗ ∂⁄∂x
+ ȷ ⃗ ∂⁄∂y
+ k⃗ ∂⁄∂z
a subtle and profound (not to say confusing)
change that may serve to suggest the size of the intellectual gulf between
quaternion analysis and vector analysis.
- [n]
There is, nevertheless, substantial evidence that Gibbs'
system was inspired by quaternions, and not by the
Ausdehnungslehre.23
- [o]
'Infix notation' means that the operator symbol appears
between the operands, as in u × v. 'Prefix notation' means that
the operator appears before the operands, as in Vuv. Prefix
notation was later rediscovered by Jan Łukasiewicz,27 and is now
generally called Polish notation.
- [p]
The two sources used for the following discussion of the
1890's' controversy both handle the debate on a roughly
chronological paper-by-paper basis. My discussion is a substantially
different organization of the material.
- [q]
Gibbs provided the initiative that led to the publication
of Grassman's collected works.28
- [r]
In a letter to Nature of January 1893, Tait wrote:
I found that I should not only have to unlearn quaternions (in
whose disfavor much is said) but also to learn a new and most
uncouth parody of notations long familiar to me.... There I
was content to leave the matter.... Dr. Knott [Cargill Gilston
Knott, a former student of Tait and a staunch quaternionist] has
actually had the courage to read the pamphlets of Gibbs and
Heaviside; and, after an arduous journey through trackless
jungles, has emerged a more resolute supporter of Quaternions
than when he entered.29
- [s]
This lack of acceptance was exaggerated. Actually, 168
works were published in the quaternion tradition in the 1890's.30
- [t]
There was only one paper in the debate that argued against
the use of all vectorial systems. This was written by Arthur
Cayley in 1894. It named quaternions in particular, and was
answered by Tait.26k
- [u]
In his response to Tait's preface, Gibbs' wrote,
The criticism relates particularly to notations, but I
believe that there is a deeper issue of notions underlying
that of notations. Indeed, if my offense had been solely in
the matter of notation, it would have been less accurate to
describe my production as a monstrosity, than to characterize
its dress as uncouth.
- [v]
Of course, quaternions didn't vanish overnight. They were
in the final stages of disappearance in about 1910;36 and, as will
be seen, they have never vanished entirely.
- [w]
Cayley considered this result in a purely geometrical
context — i.e. he didn't have time as a fourth dimension in mind.
Note that a special case of this formula is when the rotation is
orthogonal to the real axis; then v = u−1, giving the general
three-dimensional rotation described earlier.
- [x]
For a real quaternion q, the inverse is often defined as q−1 = Kq/Nq,
where Kq is the complement and Nq the norm of q.15
Silberstein simply carried this definition over to biquaternions.
- [y]
Taken in the context of 1945,
Dirac's dismissal of biquaternions is reasonable. However,
in the context of modern theoretical physics,
which favors such rarified creatures as Clifford and Lie algebras,
biquaternions actually do have some interesting properties.
- [z]
The form using left-division is q' = (qa + b)−1(qc + d),
with (of course) different quaternion values for a,b,c,d.
- [aa]
I have not discussed any specific areas of application to
classical physics. It has not been necessary to do so, since the
vector-quaternion controversy was independent of both area and
method of application.
- [bb]
Discussions of the reception (or rather lack of reception)
of quaternions in modern physics appear in most of the modern
works I have examined. Rastall's observations are interesting;
but his historical interpretations do not take into account the
vectorist and anti-quaternionist traditions.
- [cc]
According to A. W. Conway, this equivalence is itself
equivalent to wave-particle duality.
- [dd]
These papers appeared in Journal of Mathematical Physics.
I find no subsequent papers in that journal by this group.
- [ee]
The trouble with the non-commutativity of 𝓠 is that there
is no unique tensor product of the Hilbert space 𝓗𝓠
with itself.
- [ff]
Fischer explains this terminology on page 4 of the
Universal Mechanics:
It is more common in literature to use the term "tensor" for
the general non-commutative "affinor" and speak of symmetric
and antisymmetric tensors instead of tensors and axiators.
But Spielreins terms apparently are more elucidative.
- [gg]
Quadric quaternions are quaternions whose four coefficients
are themselves real quaternions; equivalently they are sums and
products of quaternions in two independent sets of imaginaries.
They have 16 real coefficients each. Double quadric quaternions
are quadric quaternions whose 16 coefficients are themselves
quadric quaternions, or equivalently sums and products of
quaternions in four independent sets of imaginaries. Double quadric
quaternions have 256 real coefficients each.
- [hh]
In gathering the above high-level description of Fischer's
method I have made repeated forays into the Universal Mechanics.
I found the book extremely difficult to read. It is highly
concentrated and moves very rapidly, which is natural considering
that it covers a very large quantity of material. This is
compounded by the erroneous omission from the printing of
scattered pages from the first two chapters of the book.
- [ii]
The most recent representative of the 'cult' in my
bibliography is James D. Edmonds (1974).
- [jj]
Whittaker qualifies as a 'cultist' because of a passage in
the same article (which appeared in the Mathematical Gazette):
... those who were in the outer circles of Hamilton's
influence — e.g. Willard Gibbs in America and Heaviside in
England — wasted their energies in devising bastard
derivatives of the quaternion calculus...
This editorial comment resulted in a brief correspondence in the
Mathematical Gazette between Whittaker and E. A. Milne in the
spirit of the controversy of the 1890's.55
Endnotes
- [1]
Cf. Michael J. Crowe, A History of Vector Analysis
(Notre Dame: University of Notre Dame Press, 1967), p. 1.
- [2]
Crowe, pp. 127–128.
- [3]
On the mathematical ancestry of quaternions, see Morris
Kline, Mathematical Thought from Ancient Through Modern Times
(New York: Oxford University Press, 1972), pp. 772–779. On
extensions to the concept of number prior to 1800, see E. T. Bell,
Development of Mathematics, 2nd ed. (New York: McGraw-Hill,
1945), pp. 172–178.
- [4]
Quoted in Kline, p. 773.
- [5]
Crowe, pp. 5–11.
- [6]
This is remarked on by Crowe, p. 220.
- [7]
Crowe, pp. 23–27.
- [8]
On Hamilton's attempts, see Crowe pp. 26–28; on other
attempts, see [5].
- [9]
Edmund T. Whittaker, "The Sequence of Ideas in the
Discovery of Quaternions," Proceedings of the Royal Irish Academy
50 (1945) sect. A: 97–98.
- [10]
Encyclopedia Britannica, 11th ed., s.v. "Quaternions,"
by Alexander McAulay, p. 720. The relevant part of the article
is the historical profile, which is taken from the corresponding
article in the 9th edition, by Peter Guthrie Tait.
- [11]
William Rowan Hamilton, "Quaternions," Proceedings of
the Royal Irish Academy 50 (1945) sect. A: 89–92. This is the
first publication of some notes made by Hamilton on the day of
his discovery of quaternions.
- [12]
See Crowe, p. 10.
- [13]
Kenneth O. May, "The Impossibility of a Division
Algebra of Vectors in Three Dimensional Space," American
Mathematical Monthly 73 (1966): 289–291. On Cayley's octonions,
see Kline p. 792.
- [14]
On 'scalar' and 'vector,' see Crowe pp. 31–32. On
'tensor' and 'versor,' see Felix Klein, Elementary Mathematics
from an Advanced Standpoint (New York: Dover Publications, 1945),
p. 138. For some examples of others of Hamilton's terms, see
Crowe p. 36.
- [15]
The basic properties of quaternions are taken from
Louis Brand, Vector and Tensor Analysis (New York: John Wiley
& Sons, 1947). Brand devotes the last chapter (chapter X, pp.
403–429) of his book to quaternions.
- [16]
Crowe, pp. 34–35.
- [17]
Crowe, pp. 54–96.
- [18]
Crowe, p. 155.
- [19]
On Hamilton's Lectures and Elements, see Crowe pp. 36–41.
- [20]
Bell, pp. 200, 204.
- [21]
On Tait's quaternionic work see Crowe pp. 117–125.
- [22]
On Maxwell's use of quaternions, see Crowe pp. 127–139.
- [23]
On the development of Gibbs' and Heaviside's systems of
vector analysis, see Crowe pp. 150–177.
- [24]
See Crowe pp. 124, 146.
- [25]
The 1890's' controversy is described in some detail in
Crowe pp. 182–224 (chapter 6). Much of the controversy is also
covered in Alfred M. Bork, " 'Vectors versus Quaternions' — The
Letters in Nature," American Journal of Physics 34 (1966):
202–211. Crowe's treatment is more comprehensive; however, Bork
goes into more detail on the contents of the articles he
discusses, and makes frequent use of quotations.
- [26]
General reference notes for specific papers in the
controversy are ordered chronologically, and indexed by letter
under number 26 (hence notes 26a–26k). References particularly
to one or the other secondary source are numbered separately.
- [26a]
Tait, Philosophical Magazine, January 1890, and the
preface to his Elementary Treatise on Quaternions, 1890
edition. See Crowe pp. 183–185 and (less) Bork pp. 202–203.
- [26b]
Gibbs, Nature, 2 April 1891. See Crowe pp. 185–186
and Bork p. 203.
- [26c]
Tait, Nature, 30 April 1891. See Crowe pp. 186–187
and Bork pp. 203–204.
- [26d]
Macfarlane, Proceedings of the American Association
for the Advancement of Science, published in July 1892. See
Crowe pp. 190–191 and Bork p. 205.
- [26e]
Knott, Proceedings of the Royal Society of
Edinburgh, read 19 December 1892. Crowe pp. 201–203 and
Bork p. 207.
- [26f]
Peddie, Proceedings of the Royal Society of
Edinburgh, read 10 February 1893. See Crowe pp. 208–209.
- [26g]
Gibbs, Nature, 16 March 1893. See Crowe
pp. 198–200 and Bork p. 206.
- [26h]
Macfarlane, Nature, 25 May 1893. See Crowe
pp. 203–204 and Bork p. 207.
- [26i]
Gibbs, Nature, 17 August 1893. See Crowe
pp. 204–205 and Bork p. 208.
- [26j]
McAulay, Utility of Quaternions in Physics, 1893.
See Crowe pp. 194–195.
- [26k]
Arthur Cayley, "Coordinates versus Quaternions,"
and Tait, "On the Intrinsic Nature of the Quaternion
Method," both read before the Royal Society of Edinburgh on
2 July 1894. See Crowe pp. 211–215.
- [27]
Bork, p. 204.
- [28]
Crowe, p. 161.
- [29]
Bork, p. 206.
- [30]
Crowe, p. 111. The supposed slowness of acceptance is
discussed in Crowe pp. 219–220.
- [31]
Crowe, p. 208.
- [32]
A brief account of the history of the International
Society is given in Hubert Kennedy, "James Mills Peirce and the
Cult of Quaternions," Historia Mathematica 6 (1979): 425–426.
- [33]
Crowe, p. 123.
- [34]
Kline, p. 778. These priorities are evident in much of
his work.
- [35]
On Hamilton's metaphysics, see Thomas L. Hankins,
"Triplets and Triads: Sir William Rowan Hamilton on the
Metaphysics of Mathematics." Isis 68 (1977): 175–193.
- [36]
Crowe, p. 240.
- [37]
Alfred M. Bork, "The Fourth Dimension in Nineteenth
Century Physics," Isis 55 (1964): 328–330.
- [38]
The article in which he wrote this is mentioned in
ibid., p. 338. The original article is Shunkichi Kimura, "On the
Nabla of Quaternions," Annals of Mathematics, 10 (1896): 127–155.
- [39]
James D. Edmonds, "Quaternion Quantum Theory: New
Physics or Number Mysticism?" American Journal of Physics 42
(1974): 221. Edmonds derives his information from a 1914 book by
Ludwig Silberstein. These same sentiments are attributed to
Minkowski in Otto F. Fischer, "Hamilton's Quaternions and
Minkowski's Potentials," Philosophical Magazine (7) 27 (1939):
375. Fischer does not identify the source of his information.
- [40]
Ludwig Silberstein, "Quaternionic Form of Relativity,"
Philosophical Magazine 23 (1912): 790.
- [41]
This observation is made in P. A. M. Dirac, "Application
of Quaternions to Lorentz Transformations," Proceedings of the
Royal Irish Academy 50 (1945) sect. A: 261.
- [42]
Silberstein, pp. 790–809.
- [43]
Dirac, pp. 261–270.
- [44]
Crowe, p. 34.
- [45]
For a list of such papers, see Edmonds, p. 220.
- [46]
For references on this topic, see David Finkelstein et
al., "Foundations of Quaternion Quantum Theory," Journal of
Mathematical Physics 3 (1962): 217, and Peter Rastall,
"Quaternions in Relativity," Reviews of Modern Physics 36 (1964):
820.
- [47]
Rastall, pp. 820–832.
- [48]
A. W. Conway, "Quaternions and Matrices," Proceedings of
the Royal Irish Academy 50 (1945) sect. A: 98–103. On the
generality of quaternions, see also William Kingdon Clifford,
"Applications of Grassman's Extensive Algebra," American Journal
of Mathematics 1 (1878): 350–358.
- [49]
Finkelstein et al., "Foundations," pp. 207–220.
- [50]
David Finkelstein et al., "Principle of General Q
Covariance," Journal of Mathematical Physics 4 (1963): 788–796.
- [51]
Crowe, pp. 254–255. The two books are Universal
Mechanics and Hamilton's Quaternions, 1951, and Five Mathematical
Structural Models in Natural Philosophy with Technical Physical
Quaternions, 1957. I have not seen the latter book. It may be
relevant that Silberstein used the name "physical quaternions"
for his specialized biquaternions corresponding to space-time
vectors.
- [52]
For example, the title of Kennedy's paper cited in [32].
- [53]
Crowe, p. 171.
- [54]
Crowe, p. 133.
- [55]
Edmund T. Whittaker, "The Hamiltonian Revival,"
Mathematical Gazette 24 (1940): 153–158. The associated
correspondence appears in Mathematical Gazette 25 (1941): 106–108
and 25 (1941): 298–300.
- [56]
F. D. Murnaghan, "An Elementary Presentation of the
Theory of Quaternions," Scripta Mathematica 10 (1944): 37.
Bibliography
-
Bell, E. T. The Development of Mathematics. 2nd ed. New York:
McGraw-Hill Book Company, Inc., 1945.
-
Bork, Alfred M. "The Fourth Dimension in Nineteenth-Century
Physics." Isis 55 (1964): 326–338.
-
—. " 'Vectors versus Quaternions' — the Letters in
Nature." American Journal of Physics 34 (1966): 202–211.
-
Brand, Louis. Vector and Tensor Analysis. New York: John Wiley
& Sons, Inc., 1947.
-
Clifford, William Kingdon. "Applications of Grassman's Extensive
Algebra." American Journal of Mathematics 1 (1878): 350–358.
-
Conway, A. W. "Quaternions and Matrices." Proceedings of the Royal
Irish Academy 50 (1945) sect. A: 98–103.
-
Crowe, Michael J. A History of Vector Analysis: the Evolution of
the Idea of a Vectorial System. Notre Dame: University of
Notre Dame Press, 1967.
-
Dirac, P. A. M. "Application of Quaternions to Lorentz
Transformations."
Proceedings of the Royal Irish Academy 50 (1945)
sect. A: 261–270.
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