Wednesday, March 19, 2014

The great vectors-versus-quaternions debate

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 vp' = 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 pp' = 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 θSuv = −(TuTv) cos θ  and  Vuv = (TuTv) (sin θn  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 = iddx + jddy + kddz.  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)12, interpreted as a space-time vector q should have length  (w2 − x2 − y2 − z2)12.41

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 +  +  +   (imaginaries α, β, γ) and a  2 × 2  complex matrix

 w+ix y+iz −y+iz w−ix

=  w

 1 0 0 1

+  x

 i 0 0 −i

+  y

 0 1 −1 0

+  z

 0 i i 0

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  iddx + jddy + kddz  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.
• Edmonds, James D.  "Quaternion Quantum Theory: New Physics or Number Mysticism?"  American Journal of Physics 42 (1974): 220–223.
• Finkelstein, David; Jauch, Josef M.; Schiminovich, Samuel; and Speiser, David.  "Foundations of Quaternion Quantum Mechanics."  Journal of Mathematical Physics 3 (1962): 207–220.
• —.  "Principle of General Q Covariance."  Journal of Mathematical Physics 4 (1963): 788–796.
• Fischer, Otto F.  "Hamilton's Quaternions and Minkowski's Potentials."  Philosophical Magazine (7) 27 (Jan.–June 1939): 375–385.
• —.  Universal Mechanics and Hamilton's Quaternions. Stockholm: Axion Institute, 1951.
• Hamilton, William Rowan.  "Quaternions."  Proceedings of the Royal Irish Academy 50 (1945) sect. A: 89–92.
• Hankins, Thomas L.  "Triplets and Triads: Sir William Rowan Hamilton on the Metaphysics of Science."  Isis 68 (1977): 175–193.
• Kennedy, Hubert.  "James Mills Peirce and the Cult of Quaternions." Historia Mathematica 6 (1979): 423–429.
• Kimura, Shunkichi.  "The Nabla of Quaternions."  Annals of Mathematics 10 (1896): 127–155.
• Klein, Felix.  Elementary Mathematics from an Advanced Standpoint. Translated by E. R. Hedrick and C. A. Noble.  New York:  Dover Publications, 1945.
• Kline, Morris.  Mathematical Thought from Ancient through Modern Times.  New York: Oxford University Press, 1972.
• McAulay, Alexander.  "Quaternions."  Encyclopedia Britannica. 11th ed.  1911.
• May, Kenneth O.  "The Impossibility of a Division Algebra of Vectors in Three Dimensional Space."  American Mathematical Monthly 73 (1966): 289–291.
• Murnaghan, Francis D.  "An Elementary Presentation of the Theory of Quaternions."  Scripta Mathematica 10 (1944): 37–49.
• Rastall, Peter.  "Quaternions in Relativity."  Reviews of Modern Physics 36 (1964): 820–832.
• Silberstein, Ludwig.  "Quaternionic Form of Relativity." Philosophical Magazine (6) 23 (Jan.–June 1912): 790–809.
• Whittaker, Edmund T.  "The Hamiltonian Revival."  Mathematical Gazette 24 (1940): 153–158.
• —.  "The Sequence of Ideas in the Discovery of Quaternions."  Proceedings of the Royal Irish Academy 50 (1945) sect. A: 93–98.

Saturday, March 1, 2014

Continuations and term-rewriting calculi

Thinking is most mysterious, and by far the greatest light upon it that we have is thrown by the study of language.
Benjamin Lee Whorf.

In this post, I hope to defuse the pervasive myth that continuations are intrinsically a whole-computation device.  They aren't.  I'd originally meant to write about the relationship between continuations and delimited continuations, but find that defusing the myth is prerequisite to the larger discussion, and will fill up a post by itself.

To defuse the myth, I'll look at how continuations are handled in the vau-control-calculus.  Explaining that calculus involves explaining the unconventional way vau-calculi handle variables.  So, tracing back through the tangle of ideas to find a starting point, I'll begin with some remarks on the use of variables in term-rewriting calculi.

While I'm extracting this high-level insight from the lower-level math of the situation, I'll also defuse a second common misapprehension about continuations, that they are essentially function-like.  This is a subtle point:  continuations are invoked as if they were functions, and traditionally appear in the form of first-class functions, but their control-flow behavior is orthogonal to function application.  This point is (as I've just demonstrated) difficult even to articulate without appealing to lower-level math; but it arises from the same lower-level math as the point about whole-computation, so I'll extract it here with essentially no additional effort.

Contents
Partial-evaluation variables
Continuations by global rewrite
Control variables
Insights
Partial-evaluation variables

From the lasting evidence, Alonzo Church had a thing about variables.  Not as much of a thing as Haskell Curry, who developed a combinatorial calculus with no variables at all; but Church did feel, apparently, that a meaningful logical proposition should not have unbound variables in it.  He had an elegant insight into how this could be accomplished:  have a single binding construct —which for some reason he called λ— for the variable parameter in a function definition, and then —I quite enjoyed this— you don't need additional binding constructs for the existential and universal quantifiers, because you can simply make them higher-order functions and leave the binding to their arguments.  For his quantifiers Π and Σ, Π(F,G) meant for all values v such that F(v) is true, G(v) is true; and Σ(F) meant there exists some value v such that F(v) is true.  The full elegance of this was lost because only the computational subset of his logic survived, under the name λ-calculus, so the quantifiers fell by the wayside; but the habit of a single binding construct has remained.

In computation, though, I suggest that the useful purpose of λ-bound variables is partial evaluation.  This notion dawned on me when working out the details of elementary vau-calculus.  Although I've blogged about elementary vau-calculus in an earlier post, there I was looking at a different issue (explicit evaluation), and took variables for granted.  Suppose, though, that one were centrally concerned only with capturing the operational semantics of Lisp (with fexprs) in a term-rewriting calculus at all, rather than capturing it in a calculus that looks as similar as possible to λ-calculus.  One might end up with something like this:

T   ::=   S | s | (. T) | [wrap T] | A
A   ::=   [eval T T] | [combine T T T]
S   ::=   d | () | e | [operative T T T T]
e   ::=   ⟪ B* ⟫
B   ::=   s ← T
Most of this is the same as in my earlier post (explicit evaluation), but there are three differences:  the internal structure of environments (e) is described; operatives have a different structure, which is fully described; and there are no variables.

Wait.  No variables?

Here, a term (T) is either a self-evaluating term (S), a symbol (s), a pair, an applicative ([wrap T], where T is the underlying combiner), or an active term (A).  An active term is the only kind of term that can be the left-hand side of a rewrite rule: it is either a plan to evaluate something in an environment, or a plan to invoke a combiner with some operands in an environment.  A self-evaluating term is either an atomic datum such as a number (d), or nil, or an environment (e), or an operative — where an operative consists of a parameter tree, an environment parameter, a body, and a static environment.  An environment is a delimited list of bindings (B), and a binding associates a symbol (s) with an assigned value (T).

The rewrite rules with eval on the left-hand side are essentially just the top-level logic of a Lisp evaluator:

[eval S e]   →   S
[eval s e]   →   lookup(s,e)     if lookup(s,e) is defined
[eval (T1 . T2) e]   →   [combine [eval T1 e] T2 e]
[eval [wrap T] e]   →   [wrap [eval T e]]
That leaves two rules with combine on the left-hand side:  one for combining an applicative, and one for combining an operative.  Combining applicatives is easy:
[combine [wrap T0(T1 ... Tn) e]   →   [combine T0 ([eval T1 e] ... [eval Tn e]) e]
Combining operatives is a bit more complicated.  (It will help if you're familiar with how Kernel's `\$vau`  works; see here.)  The combination is rewritten as an evaluation of the body of the operative (its third element) in a local environment.  The local environment starts as the static environment of the operative (its fourth element); then the ordinary parameters of the operative (its first element) are locally bound to the operands of the combination; and the environment parameter of the operative (its second element) is locally bound to the dynamic environment of the combination.
[combine [operative T1 T2 T3 T4] V e]   →   [eval  T3  match(T1,V) · match(T2,e) · T4]     if the latter is defined
where match(X,Y) constructs an environment binding the symbols in definiend X to the corresponding subterms in Y;  echild · eparent  concatenates two environments, producing an environment that tries to look up symbols in echild, and failing that looks for them in eparent;  and a value (V) is a term such that every active subterm is inside a self-evaluating subterm.

Sure enough, there are no variables here.  This calculus behaves correctly.  However, it has a weak equational theory.  Consider evaluating the following two expressions in a standard environment e0.

[eval  `(\$lambda (x) (+ 0 x))`  e0]
[eval  `(\$lambda (x) (* 1 x))`  e0]
Clearly, these two expressions are equivalent; we can see that they are interchangeable.  They both construct an applicative that takes one numerical argument and returns it unchanged.  However, the rewriting rules of the calculus can't tell us this.  These terms reduce to
[wrap [operative  `(x)`  `#ignore`  `(+ 0 x)`  e0]]
[wrap [operative  `(x)`  `#ignore`  `(* 1 x)`  e0]]
and both of these terms are irreducible!  Whenever we call either of these combiners, its body is evaluated in a local environment that's almost like e0; but within the calculus, we can't even talk about what will happen when the body is evaluated.  To do so we would have to construct an active evaluation term for the body; to build the active term we'd need to build a term for the local environment of the call; and to build a term for that local environment, we'd need to bind `x` to some sort of placeholder, meaning "some term, but we don't know what it is yet".

A variable is just the sort of placeholder we're looking for.  So let's add some syntax.  First, a primitive domain of variables.  We call this domain xp, where the "p" stands for "partial evaluation", since that's what we want these variables for (and because, it turns out, we're going to want other variables that are for other purposes).  We can't put this primitive domain under nonterminal S because, when we find out later what a variable stands for, what it stands for might not be self-evaluating; nor under nonterminal A because what it stands for might not be active.  So xp has to go directly under nonterminal T.

T   ::=   xp
We also need a binding construct for these variables.  It's best to use elementary devices in the calculus, to give lots of opportunities for provable equivalences, rather than big monolithic devices that we'd then be hard-put to analyze.  So we'll use a traditional one-variable construct, and expect to introduce other devices to parse the compound definiends that were handled, in the variable-less calculus, by function match.
S   ::=   ⟨λ xp.T⟩
governed by, essentially, the usual β-rule of λ-calculus:
[combine ⟨λ xp.T⟩ V e]   →   T[xp ← V]
That is, combine a λ-expression by substituting its operand (V) for its parameter (xp) in its body (T).  Having decided to bind our variables xp one at a time, we use three additional operative structures to deliver the various parts of the combination one at a time (a somewhat souped-up version of currying):  one structure for processing a null list of operands, one for splitting a dotted-pair operand into its two halves, and one for capturing the dynamic environment of the combination.
S   ::=   ⟨λ0.T⟩ | ⟨λ2.T⟩ | ⟨ε.T⟩
The corresponding rewrite rules are
[combine ⟨λ0.T⟩ () e]   →   T
[combine ⟨λ2.T0⟩ (T1 . T2) e]   →   [combine [combine T0 T1 e] T2 e]
[combine ⟨ε.T0⟩ T1 e]   →   [combine [combine T0 e ⟪⟫] T1 e]

Unlike the variable-less calculus, where the combine rewrite rule initiated evaluation of the body of an operative, here evaluation of the body must be built into the body when the operative is constructed.  This would be handled by the δ-rules (specialized operative-call rewrite rules) for evaluating function definitions.  For example, for variables x,y and standard environment e0,

[eval `(\$lambda (x) (+ 0 x))` e0]   →+   [wrap ⟨ε.⟨λy.⟨λ2.⟨λx.⟨λ0.[eval  `(+ 0 x)`  ⟪`x` ← x⟫ · e0]⟩⟩⟩⟩⟩]
Variable y is a dummy-variable used to discard the dynamic environment of the call, which is not used by ordinary functions.  Variable x is our placeholder, in the constructed term to evaluate the body, for the unknown operand to be provided later.

The innermost redex (reducible expression) here, [eval  `(+ 0 x)`  ⟪`x` ← x⟫ · e0], can be rewritten through a series of steps,

[eval  `(+ 0 x)`  ⟪`x` ← x⟫ · e0]
→   [combine  [eval  `+`  ⟪`x` ← x⟫ · e0]  `(0 x)`  ⟪`x` ← x⟫ · e0]
→   [combine  [wrap +]  `(0 x)`  ⟪`x` ← x⟫ · e0]
→+ [combine  +  `(`0 x`)`  ⟪`x` ← x⟫ · e0]
Where we can go from here depends on additional information of one or another kind.  We may have a rule that tells us the addition operative + doesn't use its dynamic environment, so that we can garbage-collect the environment,
→   [combine  +  `(`0 x`)`  ⟪⟫]
If we have some contextual information that the value of x will be numeric, and a rule that zero plus any number is that number back again, we'd have
→   x
At any rate, we only have the opportunity to even start the partial evaluation of the body, and contemplate these possible further steps, because the introduction of variables allowed us to write a term for the partial evaluation in the first place.

[edit:  I'd goofed, in this post, on the combination rule for λ0; it does not of course induce evaluation of T.  Fixed now.]
Continuations by global rewrite

The idea of using λ-calculus to model programming language semantics goes back at least to Peter Landin in the early 1960s, but there are a variety of programming language features that don't fit well with λ-calculus.  In 1975, Gordon Plotkin proposed a remedy for one of these features — eager argument evaluation, whereas ordinary λ-calculus allows lazy argument evaluation and thereby has different termination properties.  Plotkin designed a variant calculus, the λv-calculus, and proved that on one hand λv-calculus correctly models the semantics of a programming language with eager argument evaluation, while on the other hand it is comparably well-behaved to traditional λ-calculus.  Particularly, the calculus rewriting relation is compatible and Church-Rosser, and satisfies soundness and completeness theorems relative to the intended operational semantics.  (I covered those properties and theorems a bit more in an earlier post.)

In the late 1980s, Matthias Felleisen showed that a technique similar to Plotkin's could be applied to other, more unruly kinds of programming-language behavior traditionally described as "side-effects":  sequential control (continuations), and sequential state (mutable variables).  This bold plan didn't quite work, in that he had to slightly weaken the well-behavedness properties of the calculus.  In both cases (control and state), the problem is to distribute the consequences of a side-effect to everywhere it needs to be known; and Felleisen did this by having special constructs that would "bubble up" through the term, carrying the side-effect with them, until they encompassed the whole term, at which point there would be a whole-term rewriting rule to distribute the side-effect to everywhere it needed to go.  The whole-term rewriting rules were the measure by which the well-behavedness of the calculus would fail, as whole-term rewriting isn't compatible.

For sequential control (our central interest here), Felleisen added two operators, C and A, to λv-calculus.  The syntax of λv-calculus, before the addition, is just that of λ-calculus:

T   ::=   x | (λx.T) | (TT)
In place of the classic β-rule of λ-calculus, λv-calculus has βv, which differs in that the operand in the rule is a value (redexes have to be inside λ-terms):
((λx.T)V)   →   T[x ← V]
The operational semantics, which acts only on whole terms, uses (per Felleisen) an evaluation context E to uniquely determine which subterm is reduced:
E   ::=   ⎕ | (ET) | ((λx.T)E)
E[((λx.T)V)]   ↦   E[T[x ← V]]
For the control calculus, the term syntax adds A and C,
T   ::=   x | (λx.T) | (TT) | (AT) | (CT)
Neither of these operators has the semantics of call-with-current-continuation.  Instead, (AT) means "abort the surrounding computation and just do T", while (CT) means "abort the surrounding computation and apply T to the (aborted) continuation".  Although it's possible to build conventional call-with-current-continuation out of these primitive operators, the primitives themselves are obviously intrinsically whole-term operators.  Operationally, evaluation contexts don't change at all, and the operational semantics has additional rules
E[(AT)]   ↦   T
E[(CT)]   ↦   T(λx.(AE[x]))    for unused variable x
The compatible rewrite relation, →, has rules to move the new operators upward through a term until they reach its top level.  The compatible rules for A are dead easy:
(AT1)T2   →   AT1
V(AT)   →   AT
Evidently, though, once the A operator reaches the top of the term, the only way to get rid of it, so that computation can proceed, is a whole-term rewrite rule,
AT   ᐅ   T
The whole-term rule for C is easy too,
CT   ᐅ   T(λx.(Ax))
but the compatible rewrite rules for C are, truthfully, just a bit frightening:
(CT1)T2   →   C(λx1.(T1(λx2.(A(x1(x2T2))))))    for unused xk
V(CT)   →   C(λx1.(T(λx2.(A(x1(Vx2))))))    for unused xk
This does produce the right behavior (unless I've written it out wrong!), but it powerfully perturbs the term structure; Felleisen's description of this as "bubbling up" is apt.  Imho, it's quite a marvelous achievement, especially given the prior expectation that nothing of the kind would be possible — an achievement in no way lessened by what can now be done with a great deal of hindsight.

The perturbation effect appears to me, in retrospect, to be a consequence of describing the control flow of continuations using function-application structure.  My own approach makes no attempt to imitate function-application and, seemingly as a result, its constructs move upward without the dramatic perturbation of Felleisen's C.

Various constraints can be tampered with to produce more well-behaved results.  Felleisen later proposed to adjust the target behavior — the operational semantics — to facilitate well-behavedness, in work considered formative for the later notion of delimited continuations.  The constraint I've tampered with isn't a formal condition, but rather a self-imposed limitation on what sort of answers can be considered:  I introduce a new binding construct whose form doesn't resemble λ, and whose rewriting rules use a different substitution function than the β-rule.

Control variables

Consider the following Scheme expression:

(call/cc (lambda (c) (c 3)))
Presuming this is evaluated in an environment with the expected binding for call/cc, we can easily see it is operationally equivalent to 3.  Moreover, our reasoning to deduce this is evidently local to the expression; so why should our formalism have to rewrite the whole surrounding term (perturbing it in the process) in order to deduce this?

Suppose, instead of Felleisen's strategy of bubbling-up a side-effect to the top level of a term and then distributing it from there, we were to bubble-up (or, at least, migrate up) a side-effect to some sort of variable-binding construct, and then distribute it from there by some sort of substitution function to all free occurrences of the variable within the binding scope.  The only problem, then, would be what happens if the side-effect has to be distributed more widely than the given scope — such as if a first-class continuation gets carried out of the subterm in which it was originally bound — and that can be solved by allowing the binding construct itself to migrate upward in the term, expanding its scope as much as necessary to encompass all instances of the continuation.

I did this originally in vau-calculus, of course, but for comparison with Felleisen's A/C, let's use λv-calculus instead.  Introduce a second domain of control variables, xc, disjoint from xp, and "catch" and "throw" term structures (κx.T) and (τx.T).

T   ::=   xp | (TT) | (λxp.T) | (κxc.T) | (τxc.T)
Partial-evaluation variables are bound by λ, control variables are bound by κ (catch).  Control variables aren't terms; they can only occur free in τ-expressions, where they identify the destination continuation for the throw.  κ and τ are evaluation contexts; that is,
E   ::=   ... | (κx.E) | (τx.E)

The rewrite rules for τ are pretty much the same as for Felleisen's A, except that there is now a compatible rewrite rule for what to do once the throw reaches its matching catch, rather than a whole-term rewrite for eliminating the A once it reaches the top of the term.

(τx.T1)T2   →   τx.T1
V(τx.T)   →   τx.T
κx.(τx.T)   →   κx.T
What about rewrite rules for κ?  One simple rule we need, in order to relate expressions with κ to expression without, is "garbage collection":
κx.T   →   T    if x does not occur free in T
We also want rules for κ to migrate upward —non-destructively— when it occurs in an evaluation context; but κ may be the target of matching τ expressions, and if we move the κ without informing a matching τ, that τ will no longer do what it was meant to.  Consider a κ, poised to move upward, with a matching τ somewhere in its body (embedded in some context C that doesn't capture the control variable).
V(κx.C[τx.T])
If C happens to be an evaluation context, then it is possible for the τ to move upward to meet the κ and disappear; and, supposing x doesn't occur free in T, we'd have (VT).  Even if C isn't an evaluation context, (τx.T) thus represents the potential to form (VT).  If we move the κ over the V, then, in order for the τ to still represent the same potential it did before, we'd have to change it to (τx.(VT)).  And this has to happen for every matching τ.  So let's fashion a substitution function for control variables, T[x ← C] where C doesn't capture any variables:
y[x ← C]   →   y
(T1T2)[x ← C]   →   ((T1[x ← C])(T2[x ← C]))
(λy.T)[x ← C]   →   (λy.(T[x ← C]))    where y isn't free in C
(κy.T)[x ← C]   →   (κy.(T[x ← C]))    where y isn't x or free in C
(τy.T)[x ← C]   →   (τy.(T[x ← C]))    if y isn't x
(τx.T)[x ← C]   →   (τx.C[T[x ← C]])
The "where" conditions are met by α-renaming as needed.  Now we're ready to write our rewrite rules for moving κ upward:
(κx.T1)T2   →   κx.(T1[x ← ⎕T2] T2)    where x isn't free in T2
V(κx.T)   →   κx.(V(T[x ← V⎕]))    where x isn't free in V
κy.(κx.T)   →   κy.(T[x ← y])
As advertised, κ moves upward without perturbing the term structure (contrast with the bubbling-up rules for C).  If we need a first-class continuation, we can simply wrap τ in λ:  (λy.(τx.y)).  The δ-rule for call/cc would be
(call/cc T)   →   κx.(T(λy.(τx.y)))    for unused x
If this occurs in some larger context, and the first-class continuation escapes into that larger context, then the matching κ will have had to move outward before it over some evaluation context E, and substitutions will have transformed the continuation to (λy.(τx.E[y])).

Insights

The orthogonality of continuation control-flow to function application is, in the lower-level math, rather explicitly demonstrated by the way κ moves smoothly upward through the term, in contrast to the perturbations of the bubbling-up rules for C as it forcibly interacts with function-application structure.  The encapsulation of a τ within a λ to form a first-class continuation seals the deal.

The notion that continuations are a whole-term phenomenon —or, indeed, that any side-effect is a whole-term phenomenon— breaks down under the use of floating binding-constructs such as κ, which doesn't require the side-effect to remain encapsulated within a particular subterm, but does allow it to do so and thus allows local reasoning about it to whatever extent its actual behavior remains local.  Whether or not that makes traditional continuations "undelimited" is a question of word usage:  the κ binding-construct is a delimiter, but a movable one.

As a matter of tangential interest here, the vau-calculus handling of sequential state involves two new kinds of variables and four new syntactic constructs (two of which are binders, one for each of the new kinds of variables).  Here's a sketch:  Mutable data is contained in symbol-value assignments, which in turn are attached to environments; the identity of an environment is a variable, and its binding construct defines the region of the term over which that environment may be accessible.  Assignments are a separate, non-binding syntactic construct, which floats upward toward its matching environment-binding.  When a symbol is evaluated in an environment, a pair of syntax elements are created at the point of evaluation:  a query construct to seek an assignment for the given symbol in the given environment, which binds a query-variable, and within it a matching result construct with a free occurrence of the query-variable.  The result construct is an indivisible term.  The query is a compound, which migrates upward through the term looking for an assignment for the symbol in the environment.  When the query encounters a matching assignment, the query is annihilated (rather as a τ meeting its matching κ), while performing a substitution that replaces all matching result-constructs with the assigned value (there may by this time be any number of matching result-constructs, since the result of the original lookup may have been passed about in anticipation of eventually finding out what its value is).

As a final, bemusing note, there's a curious analogy (which I footnoted in my dissertation) between variables in the full side-effectful vau-calculus, and fundamental forces in physics.  The four forces of nature traditionally are gravity, electromagnetism, strong nuclear force, and weak nuclear force; one of these —gravity— is quite different from the others, with a peculiar sort of uniformity that the others lack (gravity is only attractive).  Whilst in vau-calculus we have four kinds of variables (partial-evaluation, control, environment, and query), of which one —partial-evaluation— is quite different from the others, with a peculiar sort of uniformity that the others lack (each of the other kinds of variable has an α-renaming substitution to maintain hygiene, and one or more separate kinds of substitution to aid its purpose in rewriting; but partial-evaluation variables have only one kind of substitution, of which α-renaming is a special case).

[Note: I later explored the physics analogy in a separate post, here.]