The Pseudovetor Mirage

P. J. Kuntz

April 19, 2008

When God saw men building the tower of Babel, he onfounded their speeh

to take them down a peg. It is just possible that axial vetors were inspired by

a divine fear of physiists being too su

essful. Indeed, one has to go a long way

to nd a onept in physis that generates more onfusion in a shorter time than

the idea of the pseudovetor, alias axial vetor. The prex pseudo rightly

onjures up the notion of something whih is not a vetor but resembles it in

some ways. In this ase, it is supposed to be an objet resembling a vetor in

that it has a magnitude, diretion, and orientation (a positive or negative sense)

but does not transform as a vetor under some symmetry operations. \Pseu-

dovetor" is a perfetly valid nomenlature for the said objet but onfusion

sets in when the objet, admittedly not a vetor, is atually represented by a

vetor. One this is done, the pseudovetor beomes a will-o'-the-wisp, eluding

understanding eah time one attempts to fous on it.

How an something whih by denition is not a vetor be represented by

a vetor? Who in his right mind would do suh a thing? Unfortunately, al-

most everyone, if 99% of all physis books are to be believed. Many non-salar

quantities that depend in a fundamental way on the produt of two vetors are

invariably said to be vetors and are dened via the Gibbs ross-produt:

B = a b; (1)

where a and b are the two vetors and B is the vetor representation of the

objet. Angular momentum is a typial example, where a and b are the position

and linear momentumvetors, respetively. Obviously, this enterprise is doomed

to failure beause quantities suh as angular momentumsimply do not transform

like a vetor and should not be represented by one.

It is the Gibbs ross-produt itself that is the ulprit, for it implies that

a vetor an result from the produt of two vetors. It annot. There is a

ontradition that is immediately apparent upon performing a transformation

upon the vetors. Consider a symmetry transformation, inversion in the origin.

Vetors hange their orientation under suh a transformation: a!a, b!b

and B !B so the equation above transforms to

B = (a) (b) = a b = B; (2)

whih an only be true if B = 0, a nonsensial result that ought to set alarm

bells jangling.

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The proper way out of this ul-de-sa is to realize that the ross-produt is

in fat not a bona de produt but only a reipe for obtaining a vetor in 3D

that is perpendiular to two given vetors. As suh it is not permissible to use

it to dene some objet that depends on a produt of two vetors. Instead,

it is neessary to use the exterior produt of the two vetors, sine this is

proportional to eah of the vetors and is a proper produt:

B = a ^ b; (3)

where B is now a bivetor, not a vetor. This equation an be used to dene

how B transforms. Again, under inversion

B = a ^ b! (a) ^ (b) = a ^ b = B; (4)

leaving B unaltered. The ross produt is still a useful omputational tool but

it must be dened in terms of the bivetor:

B = ia b; (5)

where i = e

1

^ e

2

^ e

3

is the pseudosalar (in 3D a trivetor), the exterior

produt of the three independent unit vetors spanning the spae. In this way,

angular momentum an be assoiated with a vetor, even though it is not rep-

resented by one! The ross-produt is dual to the bivetor.

Thus, no inonsisteny arises when quantities like angular momentum are

regarded as bivetors, whih are segments of area having magnitude, diretion

and orientation instead of segments of a line with magnitude, diretion and

orientation. In 3D, any bivetor an be expressed as a linear ombination of

three independent unit bivetors just as a vetor an be expressed as a linear

ombination of three independent unit vetors. The meaning of the term pseu-

dovetor is now lear: a pseudovetor is \not a vetor" beause it is an area

segment, not a line segment; it nevertheless shares ertain properties with a

vetor: magnitude, diretion, orientation and (in 3D) has three omponents.

Still, the term bivetor is preferable, sine it is better to desribe something by

what it is rather than by what it is not!

Unfortunately, the historial and popular way of \working around" the ob-

vious nonsense (Eqn. 2) implied by the denition in terms of the ross-produt

(Eqn. 1) is to invent two dierent types of vetors: polar vetors (ordinary

vetors) and axial vetors. The latter are just like the former exept that they

remain unhanged under inversion in the origin. All vetors generated by the

ross-produt are inluded in this ategory. This smoke-and-mirrors tati is the

soure of muh onfusion and is the diret onsequene of insisting on enoding

things like angular momentum in terms of vetors only, even when, as shown

above, this is impossible.

The mental ontortions required to maintain the illusion of the veraity of

mutually ontraditory things are truly amazing. One author has emphasized

that an axial vetor is not really a vetor (orret) and that therefore the word

\axial" is not an adjetive (onfabulation)! Is this a very helpful approah to

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a student meeting all this for the rst time? Is it sound pedagogy to warn the

reader that he is entering a linguisti mine-eld beause in disussing this topi,

the normal use of english is just plain inadequate? Another author laims that

the unit orthogonal basis vetors e

1

, e

2

, and e

3

in terms of whih all vetors

in 3-spae an be expressed are in fat axial vetors beause eah of them an

be expressed as a ross-produt of the other two: e

i

e

j

= e

k

. Logially, this

implies that all vetors are axial vetors! All of this madness stems from the

need to bear in mind whene eah vetor omes: a polar vetor simply enodes

some vetor quantity like veloity, whereas an axial vetor omes from a ross-

produt. But a vetor is a vetor is a vetor and does not ome equipped with

any speial label telling you where it ame from. If presented with a room full

of vetors, there is no way that you an tell whih of these are axial and whih

are polar. Trying to understand axial vetors is akin to playing the old shell

game|now you see it, now you don't.

Confusion about the pseudovetor an su

essfully be banished by 1) hang-

ing its name to \bivetor" and 2) representing it by a bivetor. Step 1 is not just

a palliative. It is a suggestion to all a spade a spade, always healthy pedagogy.

Step 2 is only ommon sense. This means that a physial quantity suh as angu-

lar momentummust be identied with r^p and not with rp if it is to exhibit

the orret transformation properties. This is not just a matter of nomenlature

but is an essential step if inonsistenies and onfusion are to be avoided. One

must at least make an eort to use mathematial objets appropriate to the

task at hand.

The eort needed may, however, be Herulean. In dispelling the fog sur-

rounding the pseudovetor, we have seen that it is the Gibbs ross-produt that

is to blame. That and the unwillingness to use mathematial objets suh as

the bivetor that arise naturally from bona de vetor produts. But the Gibbs

ross-produt is ubiquitous in physis, appearing not only in mehanis but also

in other areas suh as eletromagnetism. In most ases, it should be replaed

by the appropriate bivetor. Thus the ritiism vented here is not as esoteri as

it might appear but rather applies to the whole of elementary physis, even at

shool level. Nothing less than a omplete revamping of the basi formulations

at all levels will do! This is truly a huge undertaking but it is worth it, for

the gain in understanding is immense and the waste of time and energy hasing

illusions ould be vastly redued.

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