Axial vectors

by Alain Bossavit

Abstract: "Axial" is not a physical property. Axial vectors are geometric objects, defined below, which have specific usefulness in electromagnetism. Enough usefulness to justify using them? That is the question.

  The so-called "axial vectors" seem to be an exclusivity of electromagnetism, which is already suspicious, and discussions about them often tend to degenerate, which is even more suspicious. Some consider axial vectors as confusing entities. Mathematical objects can be more or less difficult to comprehend, but they are born innocent, and should not be blamed for the confusion that may exist in discussions about them. So whence this confusion?

  Vector fields, axial vector fields, and differential forms, make three formalisms which have equal descriptive power with respect to electromagnetism. They differ, however, as regards simplicity, convenience, and conceptual economy. It's a time-honored precept in science that the fewer entities you introduce when building a theory, the better. As a corollary, perhaps less familiar, mathematical structures necessary to describe a theory should be introduced only when called for. Think of numbers: you learn to count before learning to add, multiplication comes after addition, and only then can division and the arcanes of primality be revealed to you.

  The classical introduction to electromagnetism, inherited from Heaviside, does not follow such a layer-over-layer pattern. One introduces a lot of structure from the very beginning: a three-dimensional space, a metric (i.e., a way to measure lengths, areas, etc.), an orientation (right-hand or left-hand rule), and even -- though, thanks heavens, this tends to fall out of fashion -- a system of orthogonal axes. The theory then unfolds. But laying down all cards open at the beginning of the game this way is not mandatory. Let's take 3D space for granted (although even that could be challenged). One may then fully discuss Ampère's theorem and Faraday's law without any metric, by using differential forms [AB]. This is what differential forms are good for. Now when it comes to constitutive laws, more structure is needed, and metric must be introduced. But only then. As for orientation, I claim that it's a useless structure in electromagnetics: Maxwell's theory can entirely be developed without any commitment to an orientation convention.

  The somewhat undertaught concept of "twisted form" must be mobilized for that, and it's not an easy one to understand, granted. A kind of compromise was brokered long ago [W. Voigt: Lehrbuch der Kristallphysik, Teubner (Leipzig), 1910], with the concept of axial vector. These geometric objects allow one to develop Maxwell's theory without orienting space, but they still require a metric from the onset. To this extent, they don't satisfy Occam's principle. But in compensation, they help capitalize on the training in vector algebra that most students have had nowadays, so their use is a viable pedagogical proposition, worth being given a fair hearing.

  What are they? The first thing to understand is what "orientation" means. In real life, losing your orientation means becoming unable to tell East from West. In geometry, it's a different idea. It has to do with being able to tell clockwise from counter-clockwise in the plane, to tell left-handed from right-handed staircases in 3D space. Le'ts have that a little more formal. Take three independent vectors, in 3D space. They form what is called a frame, in which any vector can be represented via its components. Take another frame, form the matrix of its vectors' components with respect to the first frame, and look at the sign of its determinant. (It's well defined: 0 is excluded, by linear independence.) We say that the two frames are "in the same orientation class" if this sign is +. There are two, and only two, such orientation classes. Name one LH (never mind which one -- it's arbitrary), for left-handed, and the other one RH. "Orienting space" consists in choosing one of them for practical purposes, such as deciding whether a given reference frame (three vectors, in a definite order) is politically correct. We then call it a "direct frame" (those of the other class are "skew" frames). An oriented vector space is then, by definition, a pair {X, Or}, where X is a vector space and Or one of its two classes of frames. (Thus Or = either LH or RH. Orientation is a variable with two possible values. For convenience, we shall denote by - Or the opposite orientation, i.e., - Or = RH if Or = LH, and the other way round. This makes for easier algebra.)

  The Euclidean space of school geometry, and of elementary physics, now, is a triple {X, Or, ; }, where " . " is the dot product. This dot product confers metric on X. As for Or, it's usually taken to be RH. Most of us have a fair idea about what RH is, and we agree on this convention. We agree on this CONVENTION -- for this is what is is: a social convention.

  A lot of things depend on this convention, for instance the definition of the cross product: Given two vectors u and v, we know what "orthogonal to both of them" means, thanks to the dot product. We also know which length to attribute to u x v. But telling which way this cross product will point requires an orientation, since the rule is, "u, v, and u x v should make a direct frame". It goes the same with the curl operator, whose definition also requires an orientation convention.

  To the extent that electromagnetic laws require a cross product, and the curl, for their expression in vector formalism, there is no way to state them in this framework without a commitment to one or the other orientation of space. The simplest evidence of that is provided by the magnetic part of Lorentz force, v x B. This force, of course, does not depend on our left-hand or right-hand preference, and neither does the velocity v. But since x depends on orientation, B must depend on it, too. If the orientation is changed, B must be changed into -B in order to represent the same physics. (Note, incidentally, that H also must be changed to -H, since the sign of the permeability mu, obviously, won't change by decree.) So we need an orientation in background if we wish to represent the magnetic induction via a vector field. But does this invalidate the above thesis? Does Maxwell theory require an orientation convention? No, for other geometric objects than vector fields can do the job.

  Let's now consider pairs {v, Or}, where v is a vector and Or, again, one of the two orientation classes. No special name is necessary for such pairs, that will only play a transient role. Say that {v, Or} is "a vector with an orientation piggy-back", and call the pair, therefore, an "orientation-laden vector". Let's agree that {v, Or} and the other pair {-v, -Or} are equivalent. (No reason need be given: we are using our discretionary power to define things as we want. The end will justify the means.) Next, let's group the two orientation-laden vectors {v, Or} and {-v, -Or} into a class, and denote the class by ~v. Now, classes of this kind are, by definition, axial vectors. Note that "axial", here, is not an adjective: Axial vectors are not vectors endowed with this or that special property, but objects of a new kind: An axial vector is a crew of two, a duetto of orientation-laden vectors.

  Before seeing what they are good for, let's dispose of polar vectors: this is not a new kind of objects, just a redundant name that one gives, for emphasis and clarity [?] to ordinary vectors. (You may think this is confusing, and for sure it is, but there is some method in this madness, all things considered: One may repeat the trick, take pairs of orientation-laden axial vectors, and define polar vectors as classes of two. Such classes, then, are easily seen to be in one-to-one correspondence with the vectors one started from. A case of duality, obviously, which may justify the symmetrical terminology.)

  Note that all these manipulations, grouping objects into pairs, defining equivalence relations, hence classes, and using the classes as new "simple" objects, are not something made up for the circumstance, or special to electromagnetism: All of mathematics proceed this way. Think for example of rational numbers, defined as classes of equivalent pairs of integers, and of scores of other "complex" objects which only practice makes us view as "simple" ones, oblivious as we are of their genesis.

  Now, we may represent magnetic induction, without any orientation in background, via a field ~B of axial vectors. Let's argue the case by showing how to obtain the magnetic part of the Lorentz force at point x, where a unit charge happens to pass with velocity v. (The electric part, of course, is E, irrespective of orientation.) Rip open the axial vector, which is a pack containing two orientation-laden vectors {B, Or} and {-B, -Or}, and select the one that carries the standard orientation, RH, on its back. Suppose it happens to be the first one, i.e., Or = RH. Cross-multiply v by this vector, hence the force v x B. No "preference" for RH there, for we could also select {-B, -Or} as representative of the class, and use the other cross product, the one defined by the rule "the frame u, v, and u x v should belong to the class LH", for the same final result, since the two sign inversions cancel out. Axial vectors, we see, live up to expectations: Whichever the orientation convention, ~B predicts the same, well-defined force.

  We could argue about the curl operator in a similar way, but this text is already long enough. What comes out is a coherent theory in which the physical entities electric field, electric induction and current are represented by "polar" vectors E, D, J, and the roles of magnetic induction and magnetic strength are played by axial vectors ~B and ~H, while space is not oriented, and need not be. (Quiz for the reader: What about the vector potential ???)

  By insisting that the device is free of logical contradictions, I am not suggesting that it should be adopted. It is anything but simple, obviously. And for the price of all this complexity, what you get is still metric-dependent. So if one embarks on a crusade to get rid of unnecessary structure, better plan to go all the way, and use differential forms (both ordinary and twisted), which also discard orientation, and put metric at its place, namely, at the level of constitutive laws. (The relation between differential forms, straight and twisted, and vector fields, polar and axial, is not straightforward. See [AB].)

  There are, however, more difficult theories, so it's not the complexity of that one which explains the ongoing confusion. The culprit is the use of careless descriptive sentences in lieu of proper definitions. One often reads, for instance, that "axial vectors are those vectors that change sign by mirror reflection". As a definition, this is pure nonsense. A vector is a vector is a vector, and a mirror reflection will not change its "sign" (a vector has no "sign", anyway), just map it to another vector. How come such a confusing expression got credence?

  Probably because of this: In the first descriptive system (oriented Euclidean space), the electromagnetic field is depicted by four vector fields, E, D, H, B. But there are two ways of orienting space, and if one had chosen the other orientation, the vector fields accounting for the same physics would be E, D, -H, -B. The corresponding physical entities, which I will denote e, d, h, b for contrast, may therefore be said to be of different kinds with respect to how vectors can stand for them. One might say, using the above vocabulary, and with due warning about the informal character of such statements, that e and d have a "polar" nature while h and b have "axial" nature. This is reflected in which kind of geometrical objects, polar or axial vector fields, stand for them in the second system (non-oriented Euclidean space), where one has E, D, ~H, ~B. (This is not all: one might also remark that e and h have "one-dimensional" nature, while d and b have "two-dimensional" nature, for obvious reasons.) But when, losing all elementary caution, one attributes such "nature" not to the physical entities but to the vector fields that serve as proxies for them, confusion sets in for good. It's like saying Chaliapin had tyrannic nature because he sang the part of a tyrant.

  Confusion, therefore, is in the discourse about the theory, not in the theory. One should dismiss it in order to judge the theory on its own merits.

A previous version of this text appeared in ICS Newsletter, 6, 3 (1999), pp. 12-14.


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