Talk:Pseudovector

More on pseudovectors

The cross product of two polar vectors or two pseudovectors will be a pseudovector while cross product of a pseudovector and a polar vector is again a polar vector. Moreover, dot product of a pseudovector with a polarvector gives a pseudoscalar while dot product of two polar vectors or two pseudovectors is a scalar.

Microprocessors

Shouldn't there at least be a mention of the microprocessor application of pseudo-vectors? (John2kx 21:05, 27 April 2007 (UTC))

too technical

I have no idea what this article is talking about; can it be explained in a manner comprehensible to somebody who majored in a subject other than mathematics? As it stands now, the article is of use only to those who already understand the topic. 69.140.159.215 (talk) 03:22, 10 January 2008 (UTC)

I think you are right. Furthermore, I want a more technical definition of what this is all about. The "transforms like" is very vague. I sounds like it has a life of its own and "transforms" out of free will! —Bromskloss (talk) 16:11, 18 January 2008 (UTC)

I tried to clarify the lead. However, we can expect people to read up on what a vector is, and how vectors transform. I've opted to put links to the relevant articles in the lead in so far as they were not already present. Before you learn to run, first learn to walk. It's useless to memorise the differences between vectors and pseudovectors if you don't have a general idea of how vectors behave in the first place. Shinobu (talk) 00:47, 28 November 2008 (UTC)

the conspiracy of pseudo vectors

This article needs a LOT of work. It absolutely confuses me what the author means by 'inversion':

do you invert all vectors including the given ones a and b and the basis vectors? do you invert only the vectors a and b but not the basis vectors? do you invert the basis vectors but keep the same a and b?

are you talking about active or passive transformation? when you say a becomes -a after inversion do you mean it is the opposite vector geometricaly i. e. the components of a in the old basis flip or you mean it has the opposite components in the flipped vector basis which compensates and its the same geometrical vector?

does the geometrical, right screw, definition of vector product changes when you do 'inversion' to a left screw?

I haven't seen a single place where those questions were addressed systematically and I've been asking myself for years if the right screw for vector product turns into left screw in left coordinate system ... All textbooks just gloss over the problem and never explain what exactly they mean by 'inversion'. How can you talk about operation without clearly defining it? I have the feeling nobody in the world actually knows it, they just repeat like parrots 'vector' and 'pseudo vector'... —Preceding unsigned comment added by 128.135.235.188 (talk) 02:59, 14 January 2009 (UTC)

You're probably reading physics textbooks. It sounds like you may enjoy looking into the mathematical physics literature, where things like active versus passive transformations are treated carefully and rigorously. Anyway, I happen to prefer the active-transformation approach, in which case the following are not changed: (Direction of the +x axis, direction of the +y axis, direction of the +z axis), and the following are inverted: (a, b, any other vector). The right-hand rule is always the right-hand rule. If you simultaneously invert the coordinate system, vectors, and definition of right-hand rule, then you haven't done anything at all, of course.

Did you read the example about the car wheels in the article?

Anyway, I'm not implying that the article doesn't need improvement. :-) --Steve (talk) 05:00, 14 January 2009 (UTC)

It's me again from the pseudo vector conspiracy.

The article talks about 'coordinate inversion'. To me that implies inversion of the coordinate system axes but the article should be more explicit about what is being inverted and not leave that to the reader to guess. Also the article says that x goes to -x and y to -y. These are in bold face implying they are vectors probably the basis vectors. Again that is not specified implicitly, the reader has to guess the meaning of the symbols. So a goes to -a, b goes to -b, and a x b remains the same (according to right screw rule) which is the opposite of what a vector would do so it's a pseudo vector. This is the active transformation view. But wait a minute, active view is when you invert all vectors except the basis vectors (the coordinate axes). So what is the meaning then of 'x goes to -x' .....

Now let's consider the passive transformation i.e. inversion only of the basis vectors (coordinate axes) and all other vectors remaining the same. Then a remains a, b remains b, a x b remains a x b, and any other vector remains the same. So in this case a x b behaves like any other vector, it remains the same geometrically. Where is the 'pseudo-vector' in this case?

The article should be rewritten so that these questions are so clearly addressed that the above confusion does not arise at all. And a clear definition of what is meant by inversion cannot be substituted with the usual confusing and not clarifying 'examples from the real world' about wheels, magnets, mirror worlds and whatever.

I personally have NEVER seen a clear address of the above question in ANY book, in neither mathematics nor physics. If such book exists, a reference should be put pointing to it for the readers that want to understand the stuff better than some car wheel example. If the article is not rewritten it simply propagates the usual confusion found in thousands of textbooks and is simply of no value to Wikipedia. —Preceding unsigned comment added by 75.22.195.143 (talk) 09:22, 14 January 2009 (UTC)

I guess I only understood this stuff well later on in my physics education--after courses in GR and QFT. I can't think of a book off-hand that addresses this topic well at an intro-physics level. I vaguely remember Feynman's Lectures on Physics discussing this, but I don't have it with me and I could be wrong.

Does the new section I wrote help? I put in the "Examples" section to show how you can do things from the ground up. (I'm not implying that the introduction section doesn't also need improvement.)

To answer your specific question, your argument is correct: If you do a passive transformation, you have to switch to the left-hand rule. Sorry that I implied otherwise! More specifically, if you have right-hand coordinate axes you need the right-hand rule, and if you have left-hand coordinate axes you need the left-hand rule. This has to be the case, because it must always be true that (1,0,0) X (0,1,0) = (0,0,1), i.e. x-hat cross y-hat equals z-hat. (When in doubt, forget about left hands and right hands, and just use the algebraic definition, which never changes.) So the a vector doesn't move, but its coordinates change from (a_x,a_y,a_z) to (-a_x,-a_y,-a_z), and ditto for b, and if you algebraically compute the cross product of a and b with their new coordinates, and then plot the result using the new coordinate axes, you'll see that a x b has in fact switched directions. Again, this is why I prefer the active-transformation approach. :-) --Steve (talk) 21:30, 14 January 2009 (UTC)

Feynman discusses the definition of vectors as things that transform like the coordinates (i.e. are contravariant), but he doesn't discuss pseudovectors as far as I can tell. Jackson's Classical Electrodynamics and Arfken & Weber's Mathematical Methods for Physicists both have good discussions of the subject (and are cited as references in our article), especially the latter. Wikipedia's article on covariance and contravariance of vectors is, unfortunately, rather incomprehensible IMO (partly because it starts too generally, with linear functionals and curved manifolds). —Steven G. Johnson (talk) 21:47, 14 January 2009 (UTC)

There seems to be an enormous amount of confusion about this topic. Vectors and vector cross products are defined so that they are independent of coordinate system. Whether you use a right handed or left handed coordinate system has no bearing on the use of the right hand rule to determine the direction of the cross product of two vectors. It is conventional to always use a right hand rule to determine the direction of the cross product of two vectors. In a right handed basis i x j =k. In a left handed basis i x j=-k. Since the directions of both vectors and pseudovectors (cross products) are fixed independently of coordinate system, they must transform in the same way under a passive rotation regardless of whether it is proper or improper. The distinction between vectors and pseudovectors arise only for active improper rotations.Jcpaks3 (talk) 23:17, 27 June 2009 (UTC)

Jcpaks3, if you use the definition

$(a\times b)_{i}=\epsilon _{ijk}a_{j}b_{k}$

then (1,0,0) x (0,1,0) = (0,0,1) always. So in a left-handed coordinate system, you need to use the left-hand rule.

I'm sure other definitions of the cross product are possible, but this one is pretty standard as far as I can tell. Also, with this definition, everything is the same whether you take the point of view of active or passive transformations. With the definition you like, they're quite different, which I view as a sign that your definition is not a great definition. After all, many (probably most) physicists only use passive transformations, and yet have no problem defining and discussing pseudovectors. --Steve (talk) 02:24, 28 June 2009 (UTC)

The definition you quote is not a fundamental definition of cross product and only applies to a right handed coordinate system. The Levi Civita symbol is a tensor density, not a tensor. If you transform to a left-handed basis set, the relation you use acquires an additional factor of det(R) where R is the orthogonal transformation matrix. The cross product of two vectors (pseudo vector) is independent of coordinate system. If A and B are two known vectors, you don't need a coordinate system to determine A x B. By convention, we have made the choice of using a right hand rule to determine the direction of A x B. The determinant that we use to find the cross product in elementary physics (and where your "definition" comes from) implicitly assumes a right hand coordinate system. For example, in classical physics the electric field (E) is a vector and the magnetic field (B) is a pseudo vector (whose direction is determined by a small bar magnet at the point in question, B is parallel to the magnet and directed from the south to the north pole). It is easy to generate static electric and magnetic fields at the same point that are parallel. Simply changing to a different coordinate system does nothing to E and B. They must remain parallel. This is obviously true under a proper or improper rotation. Therefore E and B must transform in the same way under any passive orthogonal transformation. It is only under an active (physical system changes, coordinate system remains the same) orthogonal transformation that E and B transform differently.Jcpaks3 (talk) 17:36, 28 June 2009 (UTC)

Hmm, I checked all the textbooks I have on hand, and none discusses pseudovectors (or cross-products) from the passive-transformation approach. Do you have a textbook that does this? Or says that it's impossible? If so, that would be a nice addition to the article.

You're using the definition "cross product always follows the right-hand-rule, even in a left-handed coordinate system". In which case, as long as Maxwell's equations don't change, E and B obviously "transform" the same way. I'm using the definition "cross product follows the right-hand rule in a right-handed coordinate system and left-hand rule in a left-handed coordinate system". In which case a passive parity inversion would obviously change the direction of B but not E. It would also switch all the "right-hand rules" with "left-hand rules" in magnetism, and given a bar magnet it would switch which pole was labeled "north" and which was labeled "south". The physics all works out, despite the changing direction of B, and that includes your bar magnet example.

I would call my transformation "passive coordinate inversion". You would call it "passive coordinate inversion plus changing the meaning of the cross product". I don't know which is the more common thing to call it. Both are probably acceptable. But can we agree that, whatever you call it, this is a way to define and discuss pseudovectors in a (basically) passive-transformation point-of-view? --Steve (talk) 19:02, 28 June 2009 (UTC)

I am a retired physics professor who taught an intermediate E&M class last year. The text I used had a discussion of passive transformations of vectors and then introduced the notion of pseudo vectors in a problem. I emailed the author and tried to convince him, with limited success, that vectors and pseudo vectors had to transform the same way under passive coordinate rotations, proper or improper. You and I seem to disagree about how the cross product is defined. I think that you want to choose a coordinate system first and then define the cross product according to whether the coordinate system is right or left handed. My interpretation of cross product is as follows. Two vectors determine a plane. The cross product of the two vectors is defined to be perpendicular to that plane. The only ambiguity is which direction to give the cross product. This ambiguity is resolved by using the right hand rule. This makes the definition independent of any coordinate system. The magnetic field is a pseudo vector. If we define its direction by the bar magnet that I mentioned previously, then the magnetic force on a moving charge is q(v x B) if you use a right hand rule for the cross product. This vector equation should be the same in any coordinate system, and it will be if the right rule is always used to determine the cross product. The right hand rule has absolutely nothing to do with the coordinate system one chooses to use. Consistency requires that the direction of A x B is always determined by using a right hand rule. I think that the confusion arises when we attempt to find the components of the cross product in a particular coordinate system. The fact that an improper rotation (or left handed coordinate system) introduces an extra negative sign seems to be the source of confusion. This sign change is necessary for a pseudo vector to remain the same under a passive coordinate change. It would be a strange world if inverting a coordinate system changed the direction of a magnetic field (and hence the poles of a bar magnetic).

I have not been able to find a source that discusses this issue in detail. In fact the sources that I have found seem to say that cross products transform differently from vectors under passive improper rotations. I don't believe that this is the case for the reasons I have stated. That is why I found your article in the first place. I noticed that one of your earlier responders also said that vectors and pseudo vectors transform the same under improper rotations. Since there seems to be a lot of confusion about this, I think it would be great if you were able to shed some light on it or convince me that I am wrong.Jcpaks3 (talk) 20:27, 28 June 2009 (UTC)

I agree with essentially everything you're saying. Insofar as you can define things in a coordinate-independent way, passive transformations are meaningless, obviously. In fact, I had a physics professor who said "active transformations are superior in every way to passive transformations"...not coincidentally, his main research interest was proving physics theorems in a coordinate-independent way. Anyway, how about this:

Pseudovectors are usually discussed using active transformations. An alternate approach, more along the lines of "passive transformations", is to keep the universe fixed, but switch "right-hand rule" with "left-hand rule" and vice-versa everywhere in physics, in particular in the definition of the cross-product. Any polar vector (e.g., a translation vector) would be unchanged, but pseudovectors (e.g., the magnetic field vector at a point) would switch signs. Quantities defined via right-hand rules would likewise switch, such as the "north pole" and "south pole" of a magnet.

(I left out links and references, but have some.) Would you agree with this? I tried to skirt the issue of how the cross-product is fundamentally defined. --Steve (talk) 22:34, 28 June 2009 (UTC)

There are times when active transformations have advantages, but in relativity passive transformations (to my knowledge) are almost exclusively studied.

I don't understand your desire to skirt the fundamental definition of the cross product. I also think that using a left hand rule in a left handed coordinate system is inconsistent. For example, suppose that the velocity of a charged particle is along the x-axis and the magnetic field is along the y-axis in a right handed coordinate system. If the charge is positive, the force is along the z-axis. Now consider a passive inversion: i'=-1, j'=-j, k'=-k. The force doesn't change. It is given by

f=q(vi)x(Bj)=qvbk =q(-vi')x(-Bj').

According to your prescription, the last term would be evaluated using a left hand rule giving i'xj'=k'=-k and f=vBk=-vBk. Using a right hand rule gives i'xj'=-k which is consistent. Unless I have done something wrong, this is clearly inconsistent.Jcpaks3 (talk) 15:33, 29 June 2009 (UTC)

When you switch to left-hand rule, you need to change the sign of the magnetic field (like any other pseudovector).

What I'm trying to get at is Feynman Volume 1 Chapter 52: "...if some demon were to sneak into all the physics laboratories and replace the word "right" for "left" in every book in which "right-hand rules" are given, and instead we were to use all "left-hand rules," uniformly, then it should make no difference whatever in the physical laws." (Except phenomena involving the weak force, of course.) If you read the chapter, you'll see he gives examples with magnets, it all works out.

I don't know how the cross product is "supposed" to be defined in a left-handed coordinate system. Instead I propose saying explicitly whether we're using the left-hand rule or right-hand rule. If we do that, everything is clear and explicit, and the conventional definition, whatever it is, doesn't come up. --Steve (talk) 15:53, 29 June 2009 (UTC)

The right hand rule is just a convention to determine which of two possible normals to a plane to use to define a cross product. It is obvious that a left hand rule would work just as well. If we kept the same form for the Biot-Savart law, but adopted a left hand rule the direction of the magnetic field (and other pseudo vectors) would change. But if you let your choice of whether to use a right or left hand rule depend on whether you use a right or left handed coordinate system, things would be pretty chaotic. I don't think you really want to advocate a formalism where the direction of the earth's magnetic field depends on whether you use a right or left handed coordinate system.Jcpaks3 (talk) 19:53, 29 June 2009 (UTC)

Good, we're in agreement. I'm very happy to forget about the handedness of the coordinate system. We can just talk about right-hand-rule versus left-hand-rule, without thinking about coordinates. Would you agree with the addition of this text (or something like it) to the article?...

Pseudovectors are usually discussed using active transformations. An alternate approach, more along the lines of "passive transformations", is to keep the universe fixed, but switch "right-hand rule" with "left-hand rule" and vice-versa everywhere in physics, in particular in the definition of the cross-product. Any polar vector (e.g., a translation vector) would be unchanged, but pseudovectors (e.g., the magnetic field vector at a point) would switch signs.

--Steve (talk) 23:18, 29 June 2009 (UTC)

The choice of right hand vs left hand rule is simply a matter of convention. I am not aware that this is really a point that anyone is worried about. Feynman's comments are perfectly obvious. We also made a choice in assuming that the velocity vector points in the direction a particle is moving rather than the opposite direction. Although it would be unnatural, we could reformulate mechanics if we change the conventional direction of velocity and get a perfectly consistent formalism. In quantum mechanics we choose the commutator of x and p to be i (in natural units). We could have chosen it to be -i and developed an equivalent theory. We make choices like this all the time. The important thing is to be consistent once a choice is made. That is why you can't switch between right and left hand rules depending on the orientation of an arbitrary coordinate system.

The thing that concerns me is that there seems to be some confusion as to whether cross products transform differently from vectors under both passive and active improper rotations. The sources that I have found indicate that cross products do transform differently from vectors in both cases. I don't believe that because I don't believe the earth's magnetic field changes direction under a coordinate inversion. I think that it is only under active transformations that vectors and pseudo vectors transform differently. This is what I was hoping you could discuss.Jcpaks3 (talk) 14:54, 30 June 2009 (UTC)

(1) Pseudovectors acquire an extra sign-flip under active improper rotations.

(2) Pseudovectors acquire an extra sign-flip under the "convention-change" from right-hand-rule to left-hand-rule.

I'm saying that (1) is the active-transformation definition of pseudovectors and (2) is the closest thing there is to a passive-transformation definition of pseudovectors. It's not literally a passive transformation, but it's reminiscent of passive transformations, insofar as you're not actively transforming the system. Plus, some would argue that (2) is a passive transformation to a left-handed coordinate system. Personally, I don't know about that last sentence, since it depends on how you define cross-product, and I've seen both possible definitions in textbooks. So I would duck that controversy by not saying that (2) is a passive-transformation statement, but instead just saying that (2) is "more along the lines of a 'passive transformation' approach", or something like that. --Steve (talk) 15:48, 30 June 2009 (UTC)

I think that would help. You seem like a reasonable guy. It has been nice talking to you.Jcpaks3 (talk) 20:46, 30 June 2009 (UTC)

I revised the German version of the article last year and would like to add some remarks. As Jcpaks3 wrote it is difficult to find a decent description of pseudovectors in the literature. The only book I found that has a good introduction to axial vectors is Feynman's Lectures on physics. Vol. 1, p. 52-6ff. (also cited by Steve above). I agree with Jcpaks3 (and Feynman) that the topic is discussed clearest when considering active transformations instead of using pseudomathematical gibberish about coordinate transformations. (Vectors don't change under an change of basis!)

To take account of the use of coordinate transformations in many books, I came to the same conclusion as Steve: The passive transformation is really a change from right-hand-rule to left-hand-rule. It's like an observer is attached to the coordinate system. He is inverted with the coordinate system, his right hand becomes a left hand, so for him the pseudovector points to the opposite direction. It's really an active transformation of the observer instead of a coordinate change. Steve wondered if there is a textbook that is explicit about using the passive-transformation approach. Arnold Sommerfeld states in Lectures on Theoretical Physics: Mechanics (§22, translated by me from the german edition):

The rectangular components of the axial vector transform under a pure rotation of the coordinate system like the components of the corresponding arrow, i.e. orthogonally; under an inversion, however, they don't invert their sign. Then the right-screw rule is to be replaced by the left-screw rule, according to the fact that under an inversion a right-handed coordinate system transforms into a left-handed coordinate system.

A more recent account of the same idea is given in Goldstein, Poole, Safko: Classical Mechanics, 3ed., 2000, p. 169:

On the passive interpretation of the transformation, it is easy to see why polar vectors behave as they do under inversion.... What then is different for an axial vector? It appears that an axial vector always carries with it a "handedness" convention, as implied, e.g., by the definition Eq. (4.77), of a cross product. Under inversion a right-handed coordinate system changes to a left-handed system, and the cyclic order requirement of Eq. (4.77) implies a similar change from the right-hand screw convention to a left-hand convention. Hence, even on the passive interpretation, there is an actual change in the direction of the cross product upon inversion.

(This book is not the best source on the topic. On page 167 the author boldly claims: As is well known from elementary vector algebra, there are two kinds of vectors in regard to transformation properties under an inversion. I think this claim is ridiculous. I doubt that mathematicians, which are in charge of teaching vector algebra, care about pseudovectors.)

There is a more mathematical approach, which is described in Alain Bossavit: "On axial and polar vectors", ICS Newsletter, 6, 3 (1999), pp. 12-1, see also Applied Differential Geometry (A compendium) and the discussion Intrinsic definitions of pseudovectors/&c. in sci.math.research.

--Theowoll (talk) 17:53, 20 July 2009 (UTC)

Note that the claim you find "ridiculous" is qualified by a footnote (at least in my 2nd edition of Goldstein): "As is well known*", where "*" references the terminology as used in a particular text. The general difficulty here is that the term "vector" has become rather overloaded in science. Most mathematicians would hardly recognize that there is any definition other than "any element of a vector space", but the usage in classical mechanics and electrodynamics relies on a more restrictive definition (closer to the 19th-century definition) requiring certain transformation properties relative to a spatial coordinate system. In that context, it is certainly well known that there are "two types" of vectors. As far as I can tell, Goldstein never uses the more general algebraic concept of "vector space". I should also point out that Goldstein is not particularly "modern" — my (2nd) edition is from 1980, and the first edition was from 1950; the passage you quoted is certainly in the 2nd edition, although I haven't checked whether it is in the 1st. — Steven G. Johnson (talk) 19:36, 20 July 2009 (UTC)

Goldstein/Poole/Safko apparently don't explain the abstract concept of a vector space. But in Section 4.2 Orthogonal Transformations they mention the difference between active transformation and coordinate transformation: When the transformation is taken as acting only on the coordinate system, we speak of the passive role of the transformation. In the active sense, the transformation is looked on as changing the vector or other physical quantity. If a coordinate transformation doesn't change the vector or physical quantity, why do we learn later in the book that axial vectors flip to the opposite direction when we change from right-handed to left-handed coordinate systems? The answer is: that actually shouldn't happen. What makes the axial vector flip is a change from the right-hand rule to the left-hand rule, which is used to calculate the cross product. This choice of orientation is independent of the choice of coordinates, as User:Jcpaks3 has already pointed out. An orientation doesn't fix the handedness of coordinate systems we may use, instead it tells us, which coordinate systems are positive and which are negative. Goldstein and Sommerfeld (and Steve) don't separate these two concepts - choice of coordinates and choice of orientation. At least these authors explain what they mean (see the quotations I gave above). In texts of many other authors it remains completely unclear, what they mean by "flip the sign under inversion", which causes confusion and complaints like the one that started this discussion.

The footnote in the 2nd edition of Goldstein, which refers to J. B. Marion, Principles of Vector Analysis, pp. 42-49, is missing in the 3rd edition. I don't have access to this text. I suppose it is about the classification of geometric quantities by irreducible representations of the general linear group acting on space (which seems to be the highbrow formulation for "certain transformation properties relative to a spatial coordinate system"). I don't believe this classification is "well known" among students who use Goldberg as an introductory text to theoretical classical mechanics. I provide some references, which might be interesting in this context:

• Schouten, J. A. (1951). Tensor analysis for physicists. Oxford University Press: A comprehensive classification of geometric quantities, which are characterized by a collection of components with certain transformation properties. Introduces pictorial representation for these quantities.
• Weinreich, G. (1998). Geometrical vectors. The University of Chicago Press: Gives a less pretentious introduction into the different kind of vectors using pictorial nicknames.
• Burke W. L. (1985). Applied Differential Geometry. Cambridge University Press: Seems to be one of the main references recommended by Bossavit (who is recommended by me, see below). I don't have access to this text, but the author published also an article Manifestly parity invariant electromagnetic theory and twisted tensors and there is another (unfinished) book Div, Grad, Curl are Dead (PDF file from R. Montgomery's website).
• Hehl, F. W. and Obukhov, Y. N. (2001, draft). Foundations of classical electrodynamics: Give a modern introduction to the business of geometric quantities in physics (tensors and "twisted tensors").
• Bossavit, A. (1998). On the geometry of electromagnetism (1)–(4). J. Japan Soc. Appl. Electromagn., 6 (all parts in one PDF file available from FIRB): I discovered this article only recently. It explains twisted vectors intelligibly on an elementary level.

--Theowoll (talk) 15:33, 4 August 2009 (UTC)

It sounds like we're all in agreement. One shouldn't say "passive coordinate inversion" when one really means "switching right-hand-rule to left-hand-rule". And passive transformations by themselves don't do anything to help define axial vectors, only the switching of right-hand-rule to left-hand-rule does. And we agree that a lot of textbooks don't do a good job explaining this. :-) --Steve (talk) 19:42, 4 August 2009 (UTC)

In my opinion the distinction between “polar” and “axial” vectors, as far as coordinate transformations are concerned, is phony and should be abolished. Consider the following argument. When we set up a coordinate system, we can start by choosing x and y axes with unit vectors. Given the x-y plane, we must choose a direction for the z-axis. This is commonly done by using a right hand rule. This means that the unit vector along the z-axis is a pseudovector. Thus our basis set consists of two “polar” vectors and one “axial” vector which a surprisingly large number of people claim have different transformation properties under passive improper rotations. This is obviously inconsistent unless the cross product transforms in the same way as “polar” vectors in any situation. It is actually easy to see that the formalism is consistent with this claim.

I am also puzzled by the continued insistence on using a left hand rule for left-handed coordinate system. If you consistently use a right hand rule, a right hand basis set of vectors satisfy i X j = k and a left hand basis set satisfy i X j = -k. If you use a left hand rule you get the correct orientation for the inverted z-axis, but you still can’t use a left hand rule to compute the direction of the cross product. You seem to be saying that the angular momentum of a spinning sphere or the magnetic moment of a current carrying coil depends on whether you use a right or left handed coordinate system? Please tell me that I have misunderstood. That would be absurd.

I believe that when passive coordinate transformation are discussed in classical physics, it is assumed that there is only one observer who is simply looking at the relations between vector components that are expressed in different coordinate systems. A Lorentz transformation is different. There the emphasis is on understanding and reconciling the results observed by two different observers in two different reference frames. In this case we know the raw results will be different, but we can still check to see if they are actually consistent.

Consider the following experiment. Let a current carrying coil in the x-y plane be stationary at the origin. There are two observers in stationary inertial frames with this common origin, but one system has been inverted. The z-components of the magnetic field in the systems have the same value (and sign) under a Lorentz transformation, but since the axes have been inverted the two observers will claim that the magnetic fields are in opposite directions. These apparently inconsistent results are actually consistent since the observer in the inverted system has also been inverted and his right and left have been interchanged. He is still using a right hand rule in his system, but the other observer sees him using a left hand rule and that explains the discrepancy. And maybe this explains some of the confusion about right and left handed rules. Jcpaks3 (talk) 16:05, 22 August 2009 (UTC)

As Steve said, the participants in this discussion are in agreement about the unfortunate tradition of coupling the handedness to the coordinate system. Or as A. Bossavit wrote in On the geometry of electromagnetism: Pupils asked to “orient the figure” are, unfortunately, often confused by that, for they tend to believe that this is the same as selecting coordinate axes. Not so. Orienting the paper sheet (n = 2) means deciding on a “direct” sense of rotation (anticlockwise, most often), but one is not committed to definite axes by that. Maybe this common misconception should be pointed out more explicitly in the article. What you wrote in your gedanken experiment about the change of handedness of the inverted observer is the same I wrote in the German article. It is assuring to know that someone else has the same idea.--Theowoll (talk) 21:32, 23 August 2009 (UTC)

If we agree that the right hand rule should be used for any coordinate system, do we also agree that vector cross products must transform as vectors do for proper and improper passive rotations?Jcpaks3 (talk) 14:51, 25 August 2009 (UTC)

Let me remind you that our opinions are irrelevant here; what matters is what convention the references use. — Steven G. Johnson (talk) 15:18, 25 August 2009 (UTC)

I think we've found legitimate sources that used both possible conventions: "Cross product is right-hand rule always" and "Cross product is right-hand rule in right-handed coordinate systems and left-hand rule in left-handed coordinate systems". That's why we all agree that talking about passive coordinate transformations is an unhelpful way to explain about pseudovectors. :-) --Steve (talk) 16:29, 25 August 2009 (UTC)

Yes, vector cross products transform like tensors of rank one. I think that's obviously true. The interesting question is why we often read atrocities like "vector products flip their sign under inversion of coordinate axes". That's because people (physicists) do not always separate the two concepts choice of bases and choice of orientation. If we want to distinguish polar vectors from axial vectors we need to examine what happens when switching between right-hand rule and left-hand rule.--Theowoll (talk) 21:50, 25 August 2009 (UTC)

I don’t agree that talking about passive rotations is an “unhelpful way to explain about pseudovectors.” In Euclidean space, a three component quantity is a vector if it follows the same transformation rules as a displacement vector. Vectors are not defined by their properties under active rotations, so any discussion of the properties of vectors must involve passive transformations. Active rotations are more interesting physically, but I believe they are irrelevant to the question of whether or not a cross product is a vector or a pseudovector. We need new terms to distinguish between vectors with different transformation properties under active rotations.

I have seen comments that a polar vector has an inherent direction whereas an axial vector does not. I don’t agree with that. The velocity vector of a particle is tangent to the trajectory, but we arbitrarily choose it to point in the direction the particle is traveling in. Someone more interested in the past, might choose to define its direction otherwise. The electric field (a polar vector) has a defined, not inherent, direction. I don’t think that a polar vector has any real claim to an inherent direction. In both cases it is a matter of definition so it doesn’t distinguish between two types of vectors.

One problem that no one else has mentioned or commented on is that in 3-space, one of the basis vectors is a cross product. It is clearly inconsistent to use a basis in which one basis vector transforms differently from the others.

Steve is right that Wikipedia is not the appropriate place for this type of discussion. He would need to reference a published paper to include these ideas. A related question is whether Wikipedia is a place to restate incorrect ideas.Jcpaks3 (talk) 21:05, 27 August 2009 (UTC)

Wikipedia is not the place to determine whether ideas published in reputable sources are incorrect. Wikipedia only claims to report what reputable sources say, it is not an arbiter of truth. See WP:V. — Steven G. Johnson (talk) 23:53, 27 August 2009 (UTC)

The problem here is that some "reputable" sources are not even wrong. The object of the discussion should be to find out which sources make sense to use them as a primary source.--Theowoll (talk) 16:44, 29 August 2009 (UTC)

As has been pointed out in the article and the discussion, there are two ways to distinguish polar and axial vectors: Examine a physical system and it's mirror – that's the "active" method – or observe what happens when you change the orientation (the "handednes" to calculate the cross product) – that's the "passive" method. I can't see how coordinate transformations are of any use to distinguish polar and axial vectors. There seems to be another source of confusion: When people say "axial vectors", do they mean the geometric quantity (in the sense of Shouten, Burke, Bossavit) or are they talking about a vector that represents that quantity (and depends on the chosen orientation)? That's often not made clear.--Theowoll (talk) 16:44, 29 August 2009 (UTC)

I have not contributed to this discussion for several years. Many people believe that you can prove that the cross product of two vectors is a pseudovector. In fact, the formula for the components of axb=c, c_i=ε_ijka_jb_k is actually a definition and cannot be derived. The correct expression for the components is found by simply expressing the vectors in terms of their components (using e_i as a set of orthonormal basis vectors) and using the fact that the cross product is distributive is c_i=e_i^.c = e_i^.(e_jxe_k)a_jb_k. It is easily shown that e_i^.(e_jxe_k)=e₁^.(e₂xe₃)ε_ijk for a general Euclidean coordinate system if "x" implies a right hand rule. c_i=ε_ijka_jb_k assumes that e₁^.(e₂xe₃)=1. This assumption is unnecessary and it requires using a left hand rule for the cross product in a left hand coordinate system. We should ask if this assumption makes sense. Here are some reasons not to do this. (1) First of all it requires the magnetic field direction to depend on the orientation of the coordinate system. This alone should be enough to make any physicist reject it. (2) I have tried to use the argument that if e₁ and e₂ are vectors, then e₃ must be a pseudovector if we choose it by using a right hand rule. One person argued that at every point there is a vector and pseudovector with the same magnitude and direction. All we have to do is choose the vector for the third basis vector. He was essentially arguing that it made sense to use e₃ as symbol for both a vector and a pseudovector. That made sense to him. After all he used "x" to stand for both a right hand and a left hand rule. (3) Not only do we have two quantities as every point with the same magnitude and direction along with some mysterious property that requires them to transform differently under a coordinate inversion, but we use the same symbol for both a scalar and a pseudo-scalar. There are an infinite number of ways to choose vectors so that a^.(bxc)=1 and d^.e=1. The "1" is a pseudoscalar in one case and a scalar in the other. This makes no sense to me. It continues to bother me that we present such garbage to students as factual. We should teach students to be skeptical and think critically, not to accept everything in a text or lecture as fact. I have tried to publish a paper in AJP presenting these arguments with no success. I don't know what else to do. I would welcome your suggestions.Jcpaks3 (talk) 17:50, 7 January 2016 (UTC)

Pseudovector different from axial vector?

I read a little about pseudovectors in "Mathematical Methods for Physics and Engineering" by Riley, Hobson and Bence. They went to great pains to distinguish between axial vectors (such as L= r x p in physics), which are "proper" vectors in the sense that they are independent of any coordinate system used to describe them and so can be used to describe physical quantities, from "pseudovectors" which they claim are dependent on the coordinate system and so can never describe physical quantities. I'm afraid that I don't have the book with me and I can't remember the detail of their arguments. I suspect that the way they define axial vectors is the same as you are defining axial/pseudovectors in this article, and that their "pseudovector" is some different object that you (and maybe all mathematicians?) do not consider here, but I thought it was worth pointing this out. Perhaps the article should mention these alternative definitions in case people like me are coming from a textbook that treats the words differently? Thanks! —Preceding unsigned comment added by 86.170.205.35 (talk) 11:53, 19 December 2009 (UTC)

As far as I can tell they use the terms interchangeably. What page is it where they "take great pains to distinguish between axial vectors and pseudovectors"? I'm looking at pages 947-9 on google books. On p948 in the box you're asked to prove the cross product of two vectors is a pseudovector. Angular momentum is the cross product of two vectors, so it's a pseudovector. But angular momentum is also their example of an "axial vector".

I thought they were saying that angular momentum, torque, etc. are not "physical quantities" because the right-hand-rule for cross-products and/or right-handed coordinate systems are an arbitrary conventions. That's a fine point of view, but I wish they had been less pedantic about it. :-) --Steve (talk) 14:22, 19 December 2009 (UTC)

You stumped me there for a bit with that point but I think I get what they're on about now. They get you to prove that a vector $\mathbf {a}$, whose components $a_{i}$ equal $\epsilon _{ijk}b_{j}c_{k}$, is a pseudovector. But they go on to say (in a really convoluted way!) that $\mathbf {a} \neq \mathbf {b} \times \mathbf {c}$: you get the components right in a RH system, but not LH. They're defining the cross product geometrically (a vector of magnitude $bc\sin \theta$ orthogonal to the plane spanned by $b$ and $c$ in a right handed direction). Sure, there's the arbitrary right-handed convention, but it is nothing to do with the handedness of the coordinate system (not that you were implying that it is). The components of a cross product can only be written in terms of the epsilon (pseudo!) tensor in a right handed coordinate system.

They say a few lines down on the same page that "...angular momentum [which is a cross product] can only be described by a vector, not a pseudovector". Also, on the following page, they say: "This suggests that vectors can be divided into two categories as follows: polar vectors [...] which reverse direction under active inversion [...] and axial vectors (such as angular momentum) which remain unchanged. It should be emphasised that at no point in this discussion have we used the concept of a pseudovector to describe a real physical quantity." So they're definitely saying that angular momentum is an axial vector, but not a pseudovector. The more I think about it, the more I think this is actually a nice distinction. It's (very!) pedantic, but in terms of dealing with the kind of active vs. passive confusion that has been mentioned in posts above, I think it makes for a good pedagogical tool. Has anyone else come across this way of defining axial/pseudovectors in any of the literature? If so, maybe it should be included (e.g. "some authors define..."). But if it's just a crusade by RHB then I guess there's no need. —Preceding unsigned comment added by 86.170.205.35 (talk) 21:07, 19 December 2009 (UTC)

Ah, a "pseudovector" is a cross product with "right-hand-rule in right-handed coordinate system, left-hand-rule in left-handed coordinate system", and an "axial vector" is a cross product with "right-hand-rule always". Yikes. This is why people shouldn't even try to use passive transformations to define pseudovectors, it's not worth the effort and confusion. In the words of my old physics professor, "Active transformations are superior in every way"!

The only other source I found (in 1 minute of searching) that distinguishes pseudovectors and axial vectors does it in the exact opposite way! footnote 10. Hmmmm. :-) --Steve (talk) 05:36, 20 December 2009 (UTC)

Compare this with axial vs. polar vectors and this and this. It appears some discussion in the article is necessary. It's another one of those terminology differences that plague physics. Brews ohare (talk) 15:40, 20 December 2009 (UTC)

I added a footnote and two sources about this topic. Brews ohare (talk) 04:03, 22 December 2009 (UTC)

I tried rewriting your footnote. I found it a bit misleading for you to give Baylis the final say on what is a pseudovectors and what is an axial vectors, when after all Riley uses definitions that amount to the exact opposite. I also described the distinction with a simple example (a cross-product in a left-handed vs right-handed coordinate system) rather than getting into wedge products, since a lot more of the article's likely readers will be familiar with cross products than the Grassmann algebra. Anyway, the real distinction is that one is a cross-product defined coordinate-wise, and one is a cross-product defined geometrically (coordinate-free). It hardly matters whether you use the (coordinate-free) wedge product or the (coordinate-free) right-hand rule, as far as this distinction goes. I also tried to briefly make the point that I've made over and over again on this talk-page, that we can use active transformations to avoid the issue altogether. With active transformations you never have left-handed coordinate systems, and even Baylis and Riley would agree that pseudovectors and axial vectors are the exact same thing in this case.

Do you agree with what I wrote? :-) --Steve (talk) 16:02, 30 December 2009 (UTC)

Hi Steve: I agree there is ambiguity in usage. It appears that Riley, Hobson, Bence define the pseudovector as ε_ijkb_jc_k, which is what Baylis calls the axial vector. Baylis distinguishes between pseudovector and axial vector. The definition of an axial vector in RH&B doesn't appear to be accessible on-line, so I can't say if they differentiate between them; if they don't they are in a big group. This Google book search shows that a large percentage of texts (in fact, Baylis may be a rare exception) do not differentiate between axial vectors and pseudovectors. If few distinguish between axial vectors and pseudovectors, maybe those that do all agree with Baylis? Maybe you have some idea of how pseudovector coupling works in particle field theory? Maybe its really axial-field coupling? Brews ohare (talk) 19:25, 7 January 2010 (UTC)

Brews, see the start of this section for my discussion with 86.170.205.35 on how Riley defines "axial vector". I was confused at first but I'm now pretty confident that what I wrote in the article is correct: Baylis's pseudovector is Riley's axial vector; and Riley's pseudovector is Baylis's axial vector. I'm glad you agree with me that the vast majority of authors think that pseudovector = axial vector. In my recent edit to the article's lead, I tried to make that extremely clear. I'm only aware of these two exceptions (Baylis and Riley). --Steve (talk) 22:04, 7 January 2010 (UTC)

Steve: You encouraged me to look back at your discussion with 86.170.205.35, which led me to read what Riley, Hobson, Bence say following their introduction of ε_ijkb_jc_k. All I can say about their discussion is that it is quite possibly incomprehensible, and certainly not worth the time it would take to decide what they are trying to say. Brews ohare (talk) 23:53, 7 January 2010 (UTC)

To put it more bluntly, I wouldn't trust this source as being an exponent of an opinion other their own. The geometric algebra approach uses their own pseudovector terminology and authors in this area have the same opinion about cross product, that of Baylis, and identify the cross product with an axial vector, that is, a dual to the pseudovector (bivector), and therefore, definitely not the same thing. So failing any additional support for the views of Riley, Hobson, Bence, it doesn't seem an adequate basis to support a view of inconsistency in the literature, and I'd cut this footnote. I'm not an expert here, but so far as I can tell, the basic divergence in the literature is just between those that don't distinguish between pseudovectors and the cross product, and the geometric algebraists, who do. Brews ohare (talk) 22:49, 8 January 2010 (UTC)

I'm not sure what you're referring to. Can you please link directly to at least one source besides Baylis or Riley that says pseudovectors and axial vectors are not the same thing? --Steve (talk) 23:30, 8 January 2010 (UTC)

Hi Steve: Yes, I've linked several in the added section. Basically throughout geometric algebra the pseudovector in 3-D is a bivector, while the cross product is dual to the bivector, and so is not the same thing. Please look at the new section "Geometric algebra" for these links. Brews ohare (talk) 23:34, 8 January 2010 (UTC) Please compare also with this. Brews ohare (talk) 23:37, 8 January 2010 (UTC)

The topic that I'm trying to talk about with you is: "Is there anyone besides Baylis and Riley who thinks that the terms 'pseudovector' and 'axial vector' are not synonyms." What I'm looking for is a book that uses both terms and more specifically defines both terms in non-synonymous ways and consistently uses each term when appropriate. For example, the book by Pezzaglia never uses the term "axial vector" at all, and neither does this. Do you know of a book that meets these criteria? --Steve (talk) 23:56, 8 January 2010 (UTC)

This and this appear to be geometric algebra books that say pseudovector and axial vector are synonyms. (This one too, but it's subscription-only. Quote: "Consequently, such vectors are called "pseudovectors" or axial vectors (they are closely associated with rotations).") Perhaps the terminology isn't as unanimous as you thought? Or am I misunderstanding? --Steve (talk) 00:05, 9 January 2010 (UTC)

Are there any sources that that think pseudvector and axial vector are not synonyms? Baylis defines the axial vector as

$c_{i}=\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a_{j}b_{k}\ ,$

which is the standard formula for the cross product, and says it is the dual of the bivector. He says the bivector is the pseudovector in 3D as does this author & others. I'd say that means the axial vector is not the same as the pseudovector because it is not the same as the bivector. These authors seem to say the axial vector and the cross product of two polar vectors are the same thing. Thus, the axial vector is not the same as the pseudovector either. If a = axial vector and c = crossproduct and b = bivector and p=pseudoscalar, we have sources that say: a = c, b = p and c ≠ b. So c = a ≠ b = p Not just what you want, but not WP:OR either, eh? Brews ohare (talk) 01:00, 9 January 2010 (UTC)

Is the literature unanimous about bivectors and axial vectors? This calls the cross-product a pseudovector. That usage is mathematically incompatible with the language of geometric algebra, and appears to be a hang-over from the vector context that the article starts with, where the two are not separated. However, this author does agree that, in geometric algebra, cross-products are replaced by bivectors. He is therefore in the predicament of saying pseudovectors are not bivectors, placing him at variance with the geometric terminology that says the pseudovector is the bivector in 3D. On the other hand this says that axial vectors can be mapped onto bivectors using a^b = i a×b as Jancewicz points out. This mapping shows explicitly that the bivector is a different animal than the cross product, its dual. However, Jancewicz also calls the axial vector (cross product) the pseudovector. Bottom line: there is some variety of usage here. Brews ohare (talk) 01:19, 9 January 2010 (UTC)

I added a final aside on usage differences in the Geometric algebra section. Brews ohare (talk) 15:38, 9 January 2010 (UTC)

The cross product isn't generalizable beyond 3-D (see this too), so the generalization of pseudovector used in geometric algebra as the (n-1)-blade of the n-dimensional case will generalize, but using the identification of the pseudovector with the cross product will not. Brews ohare (talk) 01:52, 9 January 2010 (UTC)

A tabulation

OK, just to simplify I'll stick to 3D for now. What I have is:

Who	"Bivector" means...	"Pseudovector" means...	"Axial vector" means...
Virtually all physicists	Bivector	Hodge-dual of bivector	Hodge-dual of bivector
Riley	Bivector	Coordinate-wise cross product	Hodge-dual of bivector
Baylis	Bivector	Hodge-dual of bivector(?)	Coordinate-wise cross product
Some geometric algebraists(?)	Bivector	Hodge-dual of bivector	(not used)
Other geometric algebraists(?)	Bivector	Bivector	Hodge-dual of bivector
Yet other geometric algebraists	Bivector	Hodge-dual of bivector	Hodge-dual of bivector

(Hodge-dual of bivector would be, for example, a cross-product always using the right-hand rule. Coordinate-wise cross product is $\epsilon _{ijk}a_{j}b_{k}$, i.e. a cross-product using the right-hand rule in a right-handed orthonormal basis and the left-hand-rule in a left-handed orthonormal basis.) I haven't gone through source-by-source, this is just my recollection. Is this roughly consistent with your own impression? Or does anything strike you as way off? --Steve (talk) 22:37, 9 January 2010 (UTC)

I'm not comfortable with the table, especially the attributions to "Who". Basically there are several groups:

• There are those who don't use the term bivector and refer to pseudovector as synonymous with cross product. That group has been well attested in the article.

• There are those who come at things from the viewpoint of tensors.

• There are those within geometric algebra. They do use the term bivector.

Everyone in geometric algebra refers to the cross product as the dual of the bivector and agrees on the formula a^b = i a×b. Not to do so is to contradict simple algebraic manipulation.

Of these that use the term bivector, a few don't use the term pseudovector at all.

A majority of those using bivector refer to the pseudovector as the n-1 blade in n dimensions (as does John Blackburne). That is, the pseudovector in 3D is the 2-blade or bivector, and is not the cross product.

A very few that use bivector continue the common identification of pseudovector with axial vector, but still agree that the bivector is the dual of the cross product (that is, they assert by implication that the bivector is not the pseudovector in 3D, contradicting the authorities like Hestenes). These last I discount as antiquated within the realm of algebraic geometry where "bivector" is commonly used: they are continuing to use old terminology in the geometric algebra context where a different terminology is current.

I think the present subsection Geometric Algebra is pretty clear and adequately sourced. How about suggesting some rewording of that if you don't like it?

It may be that you wish to stray farther afield and present the tensor standpoint that stands outside geometric algebra? If so, yet another section should be written.

Forgive me if I am somewhat unresponsive as I'm out of town for a while. Brews ohare (talk) 03:29, 10 January 2010 (UTC)

As my name's been mentioned I thought I should add my thoughts on this. In GA the pseudoscalar is a special object: the highest grade element of the algebra, used in duals, identified with the unit imaginary, usually with other properties depending on the dimension and signature. E.g. the exterior product of n vectors in n dimensions is the determinant of the matrix with rows those vectors times the unit pseudovector. The pseudoscalar is used analogously but is far less special: it's more a label attached to different things than a name for any one thing, to emphasise a connection rather than define the objects. E.g. in 3D the elements of grade two pseudovectors but they are primarily bivectors. --JohnBlackburne^words_deeds 12:27, 10 January 2010 (UTC)

The tabulation is meant for 3D, but 4D also has to be considered: e.g. Hestenes & Datta. Once you have 3D and 4D, it's natural to ask: "What about n-D?" Brews ohare (talk) 18:17, 10 January 2010 (UTC)

Higher dimensional spaces

It would be a good addition if the more general use of "pseudovector" were addressed. Brews ohare (talk) 22:30, 7 January 2010 (UTC)

Here is a source suggesting that in a Dirac algebra the pseudovector is the same as a trivector, while in 3-D this source says the pseudovector is a bivector. (in 3-D the trivector is a pseudoscalar an oriented volume.) Apparently the assignment of the adjective "pseudovector" to an entity in geometric algebra depends upon the dimensionality of the space. Brews ohare (talk) 19:43, 7 January 2010 (UTC)

Yes, in geometric algebra or Clifford algebra the pseudovector is the dual of the vector, so its dimension is n - 1, where n is the dimension of the space and algebra. So in 3D bivectors are pseudovectors. In 4D (including spacetime that stuff like the Dirac algebra is over) trivectors are pseusovectors, etc. The pseudoscalar is similarly defined to be the dual of the scalar, so is a trivector in 3D and 4-vector in 4D. Those are the only two commonly used.--John Blackburne (words ‡ deeds) 21:02, 7 January 2010 (UTC)

Geometric algebra

I added a footnote on Baylis and another on the connection to Dirac algebra. These notes are relevant to the geometric algebra, and so are not for everyone. However, a heads up on how geometric algebra uses these terms and some WP links are appropriate.

The geometric algebra idea is to remain coordinate free, so it isn't appropriate to characterize its terminology in terms of right and left handed coordinate systems. It should stand within its own axioms. Brews ohare (talk) 17:26, 8 January 2010 (UTC)

Perhaps a subsection on the meaning of pseudovector in geometric algebra would serve better? Brews ohare (talk) 17:35, 8 January 2010 (UTC)

Here is a proposal:

Geometric algebra

In geometric algebra the basic elements are vectors, and these are used to build an hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors.

The basic multiplication in the geometric algebra is the geometric product, denoted by simply juxtaposing two vectors as in ab. This product is expressed as:

${\boldsymbol {ab}}={\boldsymbol {a\cdot b}}+{\boldsymbol {a\wedge b}}\ ,$

where the leading term is the customary vector dot product and the second term is called the wedge product. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a multivector is a summation of k-fold wedge products of various k-values. A k-fold wedge product also is referred to as a k-blade.

In the present context the pseudovector is one of these combinations. This term is attached to a different mulitvector depending upon the dimensions of the space (that is, the number of linearly independent vectors in the space). In three dimensions, the most general 2-blade or bivector can be expressed as a single wedge product and is a pseudovector. In four dimensions, however, the pseudovectors are trivectors. In general, it is a (n - 1)-blade, where n is the dimension of the space and algebra.

The transformation properties of the pseudovector in three dimensions has been compared to that of the vector cross product by Baylis. To paraphrase Baylis: Although the terms axial vector and pseudovector are often treated as synonyms, a distinction often is useful, as follows. Given two polar vectors (that is, true vectors) a and b in three dimensions, the axial vector composed from a and b is the vector normal to their plane given by a × b. On the other hand, the plane of the two vectors is represented by the exterior product or wedge product, denoted by a ʌ b = i a × b, where i= e₁ ʌ e₂ ʌ e₃ is called the unit pseudoscalar, and the basis vectors are the e_ℓ. In this context of geometric algebra, this bivector is called a pseudovector, and is the dual of the cross product. For further detail see the article Comparison of aʌb and a×b. If the vectors a and b are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the axial vector are invariant. On the other hand, if the components are fixed and the basis vectors e_ℓ are inverted, then the pseudovector is invariant, but the axial vector changes sign. This behavior of axial vectors is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.

Notes

Any comments? Brews ohare (talk) 18:40, 8 January 2010 (UTC)

Comments on proposed section

• I think this subsection would be helpful in guiding the interested reader through some of the WP geometric algebra morass to find what a pseudovector is in this context. Roughly, the basic extension geometric algebra provides is the simultaneous consideration of multiple subspaces within the same notation. Those that aren't interested can skip it, of course. Brews ohare (talk) 18:40, 8 January 2010 (UTC)

I added this section and removed a couple of footnotes that now appear in this section. Hope this action doesn't raise furor; I'm leaving town & thought maybe I'd save some trouble, assuming the section to be OK. Brews ohare (talk) 19:51, 8 January 2010 (UTC)

I'd like to add some examples like this & this of the physics use of geometric-algebra pseudovectors, but I am forbidden to do so. Maybe someone can help? Brews ohare (talk) 22:15, 8 January 2010 (UTC)

Lead sentence

The lead sentence of this article is a verbatim quote from the AbsoluteAstronomy.com web page. Brews ohare (talk) 01:53, 19 January 2010 (UTC)

The reason is because it came from Wikipedia in the first place! See the bottom of that page where it says "The source of this article is wikipedia, the free encyclopedia." I think it's funny ... until it happens to me! :) CosineKitty (talk) 03:15, 19 January 2010 (UTC)

It's funny how wikipedia can create its own truth...like in the Argusto Emfazie hoax article, where an article about a made-up person was on wikipedia for four years...at the end of the four years, this "person" had 4000 google hits! :-) --Steve (talk) 04:42, 19 January 2010 (UTC)

Undoing a notation-changing edit

See this edit. I feel bad undoing it, because someone did a lot of work, but I think it has to be undone...

I checked the first ten google books results. Nine of them used the notation a × b for cross product, and the tenth used the notation a × b. None of the ten used the notation ab, which is the notation introduced by this edit.

Obviously we want the article to use the same notation that is almost-universally used in the world, in order to help readers easily understand the article (among many other reasons). So I am undoing the edit to restore the common notation a × b. --Steve (talk) 21:34, 31 May 2012 (UTC)

I agree with your revert: most of the formatting changes were unnecessary while the ones you identify were plain wrong. I noticed that the geometric algebra section had similar formatting issues, which may be why the editor tried to 'fix' the rest of the article, so I've tidied up that section to match the rest of the article and the manual of style.--JohnBlackburne^words_deeds 21:58, 31 May 2012 (UTC)

Constrain the first definition to 3D

The definition at the start of the lead

a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection

applies only in an odd number of dimensions, yet the concept makes sense in any number of dimensions in the sense of geometric algebra. This statement must either be qualified (and I'd propose restricting it to three dimensions), and using a general definition (that does not make reference to sign flips) for any more general number of dimensions. — Quondum 13:53, 5 October 2013 (UTC)

Assessment comment

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Comment(s)

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In my opinion the distinction between “polar” and “axial” vectors, as far as coordinate transformations are concerned, is phony and should be abolished. Consider the following argument. When we set up a coordinate system, we can start by choosing x and y axes with unit vectors. Given the x-y plane, we must choose a direction for the z-axis. This is commonly done by using a right hand rule. This means that the unit vector along the z-axis is a pseudovector. Thus our basis set consists of two “polar” vectors and one “axial” vector which a surprisingly large number of people claim have different transformation properties under passive improper rotations. This is obviously inconsistent unless the cross product transforms in the same way as “polar” vectors in any situation. It is actually easy to see that the formalism is consistent with this claim. I am puzzled by the continued insistence on using a left hand rule for left-handed coordinate system. If you consistently use a right hand rule, a right hand basis set of vectors satisfy i X j = k and a left hand basis set satisfy i X j = -k. If you use a left hand rule you get the correct orientation for the inverted z-axis, but you still can’t use a left hand rule to compute the direction of the cross product. Are you actually saying that the angular momentum of a spinning sphere or the magnetic moment of a current carrying coil depends on whether you use a right or left handed coordinate system? Please tell me that I have misunderstood. That would be absurd. It is my understanding that when passive coordinate transformation are discussed in classical physics, it is assumed that there is only one observer who is simply looking at the relations between vector components that are expressed in different coordinate systems. A Lorentz transformation is different. There the emphasis is on understanding and reconciling the results observed by two different observers in two different reference frames. In this case we know the raw results will be different, but we can still check to see if they are consistent. Consider the following experiment. A current carrying coil in the x-y plane is stationary at the origin. There are two observers in stationary inertial frames with this common origin, but one system has been inverted. The z-components of the magnetic field in the systems have the same value (and sign) under a Lorentz transformation, but since the axes have been inverted the two observers will claim that the magnetic fields are in opposite directions. These apparently inconsistent results are actually consistent since the observer in the inverted system has also been inverted and his right and left have been interchanged. He is still using a right hand rule in his system, but the other observer sees him using a left hand rule and that explains the discrepancy. And maybe this explains some of the confusion about right and left handed rules.Jcpaks3 (talk) 03:31, 22 August 2009 (UTC)

Last edited at 03:31, 22 August 2009 (UTC). Substituted at 03:30, 30 April 2016 (UTC)

Pseudovector and Pseudoscalar and Pseudotensor factual errors

Before we get started, let me just say that vectors and scalars and tensors are the same thing. Technical details available upon request. None of the arguments made in Pseudovector, Pseudoscalar, and Pseudotensor make any sense at all because they assume the vector, scalar, and tensor counterpart is somehow oriented wrong. Therefore, I'm going to boldly propose that these 3 articles be deleted and redirected to vector, scalar, and tensor. Let me know what you think. Brian Everlasting (talk) 07:37, 22 April 2018 (UTC)

I’m sorry, but you clearly do not understand the contents of the articles Pseudovector, Pseudoscalar, and Pseudotensor. They are distinct and well understood topics, with rigorous mathematical definitions and concrete physical applications. Looking at just this article, see its lengthy and authoritative list of references. I suggest you consult some of them to gain a better understanding of the topic. Until you do please do not make the edits you proposed. Such would be effectively deleting the articles, and could be seen as disruptive without a proper deletion discussion.--JohnBlackburne^words_deeds 08:16, 22 April 2018 (UTC)

JohnBlackburne, I have consulted the references and I believe I am knowledgable about this topic. You resort to personal attacks

"you clearly do not understand"

instead of discussing the issue. Please familiarize yourself with WP:NPA before we continue our discussion. Thanks. Brian Everlasting (talk) 17:26, 22 April 2018 (UTC)

OK, I hereby request the technical details that support your claim "that vectors and scalars and tensors are the same thing". I'm also interested in whether you figured this out all by yourself, or if you learned it from a book or other reference, and if the latter, please tell me where to find it. Thanks, --Steve (talk) 17:55, 22 April 2018 (UTC)

Sbyrnes321, thank you for your message and your great contributions to wikipedia. Because this is a very complicated issue, I will require a little more time to give you a good response. Sorry I am so slow. Brian Everlasting (talk) 17:39, 24 April 2018 (UTC)

reverting new addition by PAR

PAR (talk · contribs) here has added some paragraphs and equations to "Physical examples". I reverted for reasons discussed here.

My main complaint is, if I understand correctly, the new text says that it is impossible to define and talk about polar vectors and pseudovectors except in the context of representations in a coordinate system: "polar vectors and pseudovectors are matrix representations of physical vectors". I disagree entirely, and the reason why is already in the article. Take a physical system, and point me to a "physical vector", and I can tell you whether it is a polar vector or a pseudovector, without ever defining a coordinate system. How would I do it? I would create an atom-by-atom mirror copy of that physical system, look at the corresponding "physical vector", and see whether it's sign-flipped (relative to its surroundings) compared to the original. That's the "active transformation" approach to defining pseudovectors, as in the "Details" section of the article.

There's a second alternative choice, which also doesn't require defining a coordinate system: Sneak into all the world's physics textbooks at night, and replace "right-hand rule" by "left-hand rule" and vice-versa everywhere. Now re-calculate your "physical vector". Is it the same as yesterday? Then it's a polar vector. Is it sign-flipped vs yesterday? Then it's a pseudovector. Now, you might say "switching "right-hand rule" with "left-hand rule" is actually defining a coordinate system!" I disagree! If I want to use the left-hand rule to define angular momentum in a right-hand coordinate system, why not?? Even if you insist that the left-hand-rule somehow requires a left-hand coordinate system, that's still not fully specifying a coordinate system and using it, it's just stating the handedness of the coordinate system, i.e. one bit of information.

So again, there are two good ways to define pseudovectors, active and right-hand-vs-left-hand-rule, and these are already given in Sections 2 & 3 respectively. The PAR addition is I think misleading and inaccurate, uses a strange definition of "physical" that seems inconsistent with standard physics terminology, and glosses over tricky but essential issues like how cross product and wedge products are defined in a left-handed coordinate system (different textbooks make different choices, see extensive and exhausting discussions of this earlier on this talk page). I also find the text unnecessarily technical and abstract.

Just my opinion, happy to talk more. --Steve (talk) 02:02, 20 March 2019 (UTC)

I fully agree to the reversion, especially for pseudo-vectors being denied their physical relevance. However, I am not so sure about the argumentation claiming not to refer to coordinate systems (=ordered bases). Imho, the relevant property is this one bit called orientation, deciding whether an ordered basis belongs to one of two equivalence classes or to the other, i.e., two coordinate systems are necessary to decide upon their belonging to different orientations. Calling one class the right-handed one, appeals to a tradition of already existing systems. Orientation is inherently bound to coordinate systems, imho. Purgy (talk) 16:09, 20 March 2019 (UTC)

The bottom line is, any supposedly physical equation which requires one to ask, "wait, what handed rule are you using?" before it can deliver a concrete equivalence or prediction or answer, is NOT physical. Mother nature does need to know what hand we have chosen to use before deciding how to answer a question. Steve wrote:

How would I do it? I would create an atom-by-atom mirror copy of that physical system, look at the corresponding "physical vector", and see whether it's sign-flipped (relative to its surroundings) compared to the original.

But you CREATED the pseudovector using your right hand! The physical system only knows the polar vectors, and if we want to create an intermediate vector or vector field that has any chance at being real, its direction cannot depend on which hand we choose to use to define it.

Let's take a concrete example, an "infinitely" long straight wire with current J flowing in it, J being a physical vector parallel to the wire, pointing in the direction of current flow. The pseudovector B field at a point p is B(r)=Jxr where r is a physical vector, perpendicular to J, of length equal to the perpendicular distance from the wire to point p.

Temporary file used in Pseudovector article

Now consider two such wires each carrying equal current in the same direction. The force per unit length on each wire is F=JxB, directed perpendicularly towards the other wire. The F, J, and r vectors are physical vectors, and we CREATE a device called the B field to express the relation between the three true vectors. Is the B field "real" or is it a figment of our imagination?. The B vector at one wire is perpendicular to the plane of the two wires, and, depending on what coordinate system we use it may point one way or the other, and is therefore not physical. If we asked mother nature "which way is the B field pointing?", mother nature would have to ask you "what handed rule are you using?" and mother nature NEVER needs to know that to answer to a question about a truly physical entity. That means B is not "real", not physical.

The exterior product (∧) does not produce a pseudovector, it produces a polar tensor. If we define B=J∧B then that 2-tensor does not "change sign" or some such thing, depending on the coordinate system. If we look in the mirror, the image of the B tensor is the exterior product of the images of the J and r vectors. It has a chance at being a real physical field. The force on the wire is then F=B.J and mother nature doesn't need to ask any questions. We don't have to have needless asterisks attached to our equations saying "remember - right hand rule!"

Steve wrote:

Sneak into all the world's physics textbooks at night, and replace "right-hand rule" by "left-hand rule" and vice-versa everywhere.

Yes, and change all diagrams illustrating the right-hand rule to use a left hand, change all the "up" B vectors to "down" B vectors, etc., etc., I agree, changing the rule does not change the coordinate system, but it does change the cross products of the unit vectors defining that coordinate system, and it changes the direction of the B pseudovector.

Purgy wrote:

Orientation is inherently bound to coordinate systems

Just to restate what I think you meant, "Orientation is inherently bound to ORDERED coordinate systems"

In the special theory of relativity, Maxwell's equations are expressed in tensor form and they are beautiful, and they don't have a little asterisk next to them saying "be sure and use the right hand rule". You won't find a single cross product in the theory of relativity because it strongly stresses that all physical equations must be invariant to coordinate transformations, mirror image or otherwise, and all the asterisks would be a needless complication.

I'm not happy with the reversion, but I ask that you try to understand what I am saying and maybe we can come up with an agreement on what to say. PAR (talk) 22:52, 20 March 2019 (UTC)

PAR, I think what you're saying is that pseudovectors are "not physical" because their sign depends on the (arbitrary) choice of right-hand-rule vs left-hand-rule. I understand the sentiment, and I agree that we should make sure the article gets across the notion that the right-hand rule is an arbitrary convention, and if we were to switch that convention, pseudovectors would switch signs.

I disagree with describing these sentiments using words like "not physical" or "not a physical quantity". For one thing, I think you are using the word "physical" in an unusual way, e.g. note that L, B, and loads of other pseudovectors are at list of physical quantities on wikipedia, and I'm very confident that textbooks will back up this usage of terminology. I mean, ask anyone whether "the Earth's magnetic field points north" is a true fact about physics and physical quantities. I think they'll say "yes". But it's not just following common usage, I think it's really appropriate to not treat pseudovectors as second-class citizens. If you really want to say that physical quantities must be independent of arbitrary conventions, there would be no physical quantities left! It's an arbitrary convention, the whim of Ben Franklin, that the charge of an electron is negative not positive. Is electric charge "not physical"? It's an arbitrary convention, the whim of Newton, that force is defined F=ma not F=-ma. Is force "not physical"? If there are magnetic monopoles, their charges would be pseudoscalars. If I have two magnetic monopoles in my hands, and I throw them in the air, and one goes flying north and the other goes flying south, isn't it a bit weird and confusing to say that the signs of their magnetic charges are "not physical"?

Anyway, it's not like we actually live in a mirror-symmetric universe anyway. Thanks to the weak force, "right" and "left" are physically distinguishable :-P --Steve (talk) 01:18, 21 March 2019 (UTC)

I just cannot look at the diagram above and reconcile myself to the idea that the pseudovector B field has physical reality, when its direction is dependent upon what goes on my mind (i.e. choice of rule). But then, as you say, we live in a let's say, right-handed world due to weak force interactions. But then the vector B field would only have what I call "reality" if it were defined in terms of those weak force phenomena. The charge on an electron is not a great example. The question is not "what have we named it" (Ben says lets use the word "negative") but rather do our physics equations carry warnings (use the right hand rule) when using them. With regard to the electron, we can start by saying no, it is a "true" scalar, not dependent on our coordinate system. Nevertheless there is still a convention here, namely "assume a matter universe, not an antimatter universe". And I guess weak force phenomena will tell us which one we live in. But that means that the electron's charge must be defined in terms of weak force phenomena to have any "true" validity, which can be arranged, so we would then not be dealing with convention. That also says that your "build a mirror image" explanation is invalid, you cannot build a "mirror image" of a physical phenomenon due to weak force phenomena. The same is true of Newton' law, force, negative or positive, does not flip sign when I change my mind. It's funny, when I read your magnetic monopole example, I thought to myself "so THAT's why there are no magnetic monopoles!". But seriously, changing to a left hand rule would not make them behave differently. Just as the B-tensor is dual to the B-pseudovector (Hodge dual, I think), the dual of a pseudoscalar is a 3-tensor, and I would be prone to think in those term.

I don't think that my notion of physical reality is at all unusual. It's what Einstein used over and over again in pondering the theory of relativity. Once he realized that different inertial systems are, like, rotated with respect to each other in spacetime, he realized that physics equations, in order to express reality, must be invariant with respect to these Lorentz transformations. They cannot be inextricably bound to the conventions of what constitutes space and what constitutes time. Likewise, the B field cannot be bound to the conventions of what constitutes a cross product.

I have to admit to a certain amount of prejudice here. I am a bit dyslexic, I don't have a immediate gut understanding of left and right, its something I have to think about for a second, its an intellectual concept to me. This is an asset when considering questions of symmetry, etc., but when writing physics equations I just got tired of scratching my head over the cross product, when it's so much nicer in my mind to write an equation about a physical phenomenon which has no intrinsic handedness in terms of intermediate user-defined objects which have no intrinsic handedness (like the B-tensor). When I saw Maxwell's equations in tensor form, and none of the terms requiring knowledge of left and right, I was happy. Same with the special theory of relativity's take on electromagnetic phenomena.

Let me ask you, what would you propose to address these ideas? PAR (talk) 02:23, 21 March 2019 (UTC)

I think we all agree on the physics, and on the dependency of the ×-product on the ordering of the bases (we might, perhaps, disagree on the order being inherent part of a coordinate system). Maybe we also agree on a "polar vector" being something different than an "axial vector", especially when considering the physical behavior of associated phenomena. The problem arises, imho, when a possible representation of both via just three numbers is presented as "vectors", because then the independence of bases is violated in the axial case, when insisting on the ×-product in the formulations.

Applying 2-tensors (also specified by three numbers and symmetry, finally involving Hodge duality) is for the time being a topic in advanced courses, and doing away with the problem with right hand waving is state of the art in not-top physics education (imho), and therefore WP-adequate.

Denying a 3-number object "physical importance", just because it does not transform like e.g. a spatial vector, is not justified, calling it an axial "vector" might cause misconceptions, and tensors might exceed the intened scope (imho). Yes, there are things beyond 3-vectors and ×-products, it's all about math not being physics, but always only delivers a possible description. Purgy (talk) 09:28, 21 March 2019 (UTC)

I propose to switch the order of section 2 (details) and section 3 (the right-hand rule), and add another sentence or two to the right-hand rule section, e.g. "There are various right-hand rules in physics that determine the directions of vectors. For example, when a wheel is spinning, there is a right-hand rule to determine what direction the angular momentum vector points, and when current flows in a wire, there is a right-hand rule to determine what direction the magnetic field points. If all these "right-hand rules" were simultaneously switched to "left-hand rules", the laws of physics would still work (apart from parity-violating phenomena, see below), but the orientation of some vectors would flip. The vectors that would flip orientation are called pseudovectors." Or something like that..? --Steve (talk) 10:53, 21 March 2019 (UTC)

I would prefer not to use the phrase "right hand rules" since there is only one right-hand rule applied to multiple cases. Also, while parity-violating phenomena may mean that you can't simply replace the phrase "right-hand rule" with "left-hand rule", nevertheless an entirely consistent body of physics could be expressed using the left-hand rule convention. Humans would not be at a loss if they happened to have a majority of their population left-handed rather than right-handed and decided to use a left-handed rule convention. With this in mind, how about:

In physics, the right-hand rule determines the directions of pseudovectors. For example, when a wheel is spinning, the right-hand rule determines in what direction the angular momentum pseudovector points, and when current flows in a wire, it determines in what direction the pseudovectorial magnetic field points. The entire body of physics expressed using the right-hand rule could be replaced with expressions using a left-hand rule. The directions of the pseudovectors so defined would be opposite in direction to those defined by the right-hand rule.

PAR (talk) 08:02, 23 March 2019 (UTC)

P.S. - also include a sentence explaining that the cross product would be defined in the second case according to the left hand rule.

What about Fleming's left-hand rule for motors? I think I understand where our perspectives are different on this. I am thinking of geometric relations first and foremost: "Angular momentum of a solid body is by definition a vector with magnitude ∫r×p, pointed along the axis of rotation, in a direction given by the right-hand rule". You are thinking of formulas: "Angular momentum is r×p by definition, end of story". Leaving aside which mode of thought is better, do you see how from my point of view, there are multiple "right-hand rules" and "left-hand rules" sprinkled throughout physics? And how, from my perspective, "the cross product would be defined in the second case according to the left hand rule" is not necessarily true? After all, I could equally well sneak in during the night, replace "right-hand rule" with "left-hand rule" in the diagrams and descriptions of physical relationships but not in the definition of cross product, and instead change the formulas so that they describe the revised physical relationships, e.g. L=-r×p.

Again, we can describe things however we want, I just want you to understand that this is a perspective that many readers probably have when they think of right hand rule(s) in physics, especially (but not exclusively) in high school physics textbooks which may not even assume students know what a cross product is. Then we can write text that is maximally clear to everyone. I think we're converging though, this is very good progress. :-D --Steve (talk) 13:30, 24 March 2019 (UTC)

We are making progress, I mean, after thinking about the point you have made I realize the flaws in my original statement.

• A right or left hand rule can be defined without reference to a coordinate system therefore, a pseudovector can be defined without reference to a coordinate system. Nevertheless it is a "convention".
• Current is defined by convention, B field is defined by convention, and if there's a substantial difference between the two (I think there is), it has to be explained.
• The idea of what constitutes "physically real" is arguable. In any case, the distinction between a physical vector and its representation as a triplet of real numbers referred to a coordinate system has to be made. (In a sense, a coordinate system is a convention)

Nevertheless, I cannot come to the viewpoint that there are multiple right hand rules rather than multiple applications of one right-hand rule, and I am still not comfortable with the viewpoint that the B field defined as a pseudovector is as "real" as the tensor B field.

How about, after reviewing the above arguments, I carefully rewrite, here on the talk page, the section that was reverted, making the points I think need to be made. I think you will still have objections, so then you state your objections, perhaps write your own version of what needs to be said, and then we can submit a Request For Comment RFD and see what happens. As you say, we agree on the physics, we disagree on interpretation and semantics. If the RFD creates a consensus, I'll ultimately go along with it, whatever it is. PAR (talk) 20:50, 25 March 2019 (UTC)

I would like to add this to the "physical examples" section, let me know what you think. I've tried to avoid philosophical issues.

In physics, pseudovectors are generally the result of taking the cross product of two vectors. The cross product is defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule. The entire body of physics that deals with (right) pseudovectors and the right hand rule could be replaced by using (left) pseudovectors and the left hand rule without issue. The (left) pseudovectors so defined would be opposite in direction to those defined by the right-hand rule.

While vector relationships in physics can be expressed in a coordinate-free manner, a coordinate system is required in order to express vectors and pseudovectors as numerical quantities. Vectors are represented as ordered triplets of numbers: e.g. $\mathbf {a} =(ax,ay,az)$, as are pseudovectors. When transforming between left and right-handed coordinate systems, representations of pseudovectors do not transform as vectors, and treating them as vector representations will cause an incorrect sign change, so that care must be taken to keep track of which ordered triplets represent vectors, and which represent pseudovectors. This problem does not exist if the cross product of two vectors is replaced by the exterior product of the two vectors, which yields a bivector which is a 2nd rank tensor and is represented by a 3x3 matrix. This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.

Also, please take a look at https://en.wikipedia.org/wiki/Pseudoscalar#Pseudoscalars_in_physics PAR (talk) 01:56, 27 March 2019 (UTC)

I think your first paragraph is nice, but I don't see how it belongs in the "Physical examples" section. I would propose to insert it into the "Right-hand rule" section (and then optionally bring that section up so that it's before "Details" and after "Physical examples").

For the second paragraph: I don't understand why it's helpful to bring up coordinate systems at all. The important thing, the thing in your first paragraph, is:

(A) "Pseudovectors flip when you switch from left-hand cross product to right-hand cross product."

I gather that, in the second paragraph, you're trying to say the same thing in a less direct way:

(B1) "We use the left-hand cross product in a left-hand coordinate system and the right-hand cross product in a right-hand coordinate system."

(B2) "Pseudovectors flip when you switch from left-hand coordinate system to right-hand coordinate system."

So (B1) here is your unstated assumption, and it's a dubious assumption, or at least controversial. I vaguely recall that years ago, I dug through a bunch of textbooks to try to figure out how they defined cross products in left-hand coordinate systems, and found that half the textbooks treated εijk Aj Bk as the universal definition of cross product (which means left-hand cross product in a left-hand coordinate system), and the other half used the geometric definition (so right-hand cross product regardless of coordinate system). (See various discussions earlier on this talk page, if memory serves.)

I also take issue with the suggestion that there is a "problem" with using pseudovectors, and this problem is solved by using bivectors. I open my quantum field theory textbook and they talk about pseudovectors (and pseudoscalars etc.) on (what feels like) practically every page without any hint of a suggestion that there is anything problematic or tricky about them. That said, it is absolutely useful to readers to say that pseudovectors have a 1-to-1 correspondence (Hodge dual) to bivectors, that bivectors are represented in coordinates as 3x3 skew-symmetric matrices, and that this 3x3 matrix transforms like an ordinary 2-tensor and not like a pseudotensor, i.e. no extra sign-flip on improper rotation. I just think this information should be conveyed in a neutral way, and that we should not suggest that coordinatizing the magnetic field as a 3x3 skew-symmetric matrix is better, and coordinatizing it as a 3-element pseudovector is worse. Let's just leave it as: these are two mathematically equivalent representations, and experts ought to be familiar with both.

So that stuff about bivectors is worth saying, and it already is said, in the "Geometric algebra" section (also, redundantly, in the "Formalization" section), although I think those sections have room for improvement. There's also a mention of geometric algebra in the intro, although again it could be edited.

I am thinking that this is a good organization scheme, i.e. that we should tell a whole detailed story about pseudovectors before we start talking about bivectors. That seems to be the universal approach in physics textbooks (at least the ones I've seen), and I think there's good reason for that, both pedagogically and in terms of prioritizing the things that come up most often in physics. So I don't like the idea of putting your second proposed paragraph into the "Physical examples" section. I think it should be merged into the "Geometric algebra" section instead. --Steve (talk) 01:32, 28 March 2019 (UTC)

The first paragraph says nothing about coordinate systems. The right-cross or left-cross product does not need a coordinate system in order to be defined, you have pointed that out, and I see it. This means I am not saying that we use a left-cross product with a left-hand coordinate system, right with right. If we follow convention, we use a right-cross product no matter what the coordinate system. If we were to use the convention of a left-hand rule to define the left-cross product, the pseudovectors we obtain would be opposite in sign to those that we defined using a right-cross product. None of this has anything to do with coordinate systems. So, I am not saying that pseudovectors flip when you transform from a right-handed to a left-handed coordinate system. Since their definition is free of coordinate systems, that could not possibly be the case.

There is a difference between a vector or pseudovector and its *representation* with respect to some coordinate system. A vector is *represented* with respect to a coordinate system by an ordered triplet of numbers. It is *represented* in a different coordinate system by a different triplet of numbers. The numerical way of obtaining one triplet from the other is by a 3x3 transformation matrix. If both coordinate systems have the same handedness, the 3x3 matrix will have unit determinant and its transpose will be its inverse. In other words it is a "proper" transformation. If the coordinate systems have different handedness, the determinant of the matrix will be -1, and it will be an "improper" transformation.

If you use proper transformation matrix on an ordered triplet *representation* of a vector or pseudovector, you will obtain an ordered triplet that DOES correctly specify the vector or pseudovector. If you use an improper transformation matrix on an ordered triplet *representation* of a vector or pseudovector, you will obtain an ordered triplet that DOES correctly specify the vector but DOES NOT correctly specify the pseudovector; it will be the wrong sign.

I have a problem with the idea that pseudovectors are as "real" as vectors, but I agree with you for the moment, that we should leave that out, even though that point is made in https://en.wikipedia.org/wiki/Pseudoscalar#Pseudoscalars_in_physics . Also, the scenario of the mirror image of a physical system, using the right hand rule for the cross product in both the system and its image, shows that the pseudovectors are not mirror images of each other, while vectors are. It's a nice device, but I think it can cause confusion. In the real world, pseudovectors don't "flip". Their *representations* flip if they are incorrectly transformed as if they were vectors. (Again, using bivectors and forgetting about pseudo-anything, you don't have to keep track of which is a tensor/vector and which is a pseudotensor/vector. Every mathematical object is "true")

Regarding εijk Aj Bk, I always think of εijk as *representing* a 3rd rank pseudotensor. According to the article Levi-Civita symbol this is the case. Accordingly, εijk Aj Bk is the *representation* of a pseudovector. If εijk *represented* a true tensor, then it would change sign under an improper transformation, and it doesn't. Please read the discussion in pseudoscalar mentioned above. PAR (talk) 05:18, 30 March 2019 (UTC)

Not adequate

Our arguments.

Pseudo means "fake, false". The cross product of two vectors is still a vector. If it has some additional properties it should be overqualified but not deemed fake.

Someone has infested the whole dictionary re-referencing vectors as pseudo-vectors in articles about physics. It is unnecessary and absolutely misleading for the newcomers to the theory and even for experts.

This concept seems to reflect current research more than encyclopaedic contents. The references to this entry in physics' articles should be carefully reviewed and re-addressed to a more meaningful concept.

Vectors as seen in a Moniod Algebraic context.

In the traditional physics that I learned during the mid-20th century under the United States federally-funded "Nation Defense Education Act" (of 1958), all vectors were like arrows, with a direction and a length. Moreover they could be, "connected head-to-toe" in single-link chains as we may please. As this is how vectors were, and this is how they would ever be. Algebraically, this required nothing more abstract than a monoid.

My next class was, "Introduction to Linear Algebra". Newton was right, but he thought only in the large scale of the planets that orbit Sol, our sun. Linear Algebra is The Rock of practical mathematics in this era. Computers can do a thing a human will never do: multiply matrices, the basic blocks of linear algebra, at very high rates. This allows practical matrix math. The truth is in mathematical vector spaces; physicists call them, "pseudo".

Vector Spaces will never do the head-to-toe thing like they should, *sigh*. And they don't look much like arrows do mostly, neither. These realities are plainly seen with simply opened eyes.

After the Einstein-Bohr debates (1925), and I see Bohr as Newtonian-minded, Einstein worked with Elie Cartan on the fundamentals of a radial field that originated at a point, such as starlight for a fundamental example. This lead to the center of the star, and what would it look like? Suppose you collapse the radius of a sphere toward zero: what will the center look like? "A point!" is what the Newtonian would exclaim, "That is plain to see, most obviously."

The kernel found by Einstein and Cartan was not a traditional point at all. Cartan's original 1913 paper on Spinors was difficult to read, but correct. CFjohnny1955 (talk) 07:15, 21 April 2024 (UTC)