Talk:Pseudovector

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)

Formalization: Pseudovectors are not (n-1)-multivectors according to several authors

The page currently claims that pseudovectors are (n-1)-multivectors (elements of $\wedge ^{n-1}V$, where $n=\dim V$) but according to Schouten, Weinreich and Burke, they are not.

Schouten introduces pseudoscalars and pseudovectors on pp. 31-32 (he calls them W-scalars and W-vectors but in a footnote mentions the name pseudo). His pseudovectors transform with ${\tfrac {|\Delta |}{\Delta }}A_{k}^{j}$, where $A_{k}^{j}$ is the change of basis matrix and $\Delta =\det A_{k}^{j}$. The transformation matrix $A_{k}^{j}$ is not assumed to be orthogonal, hence he doesn't assume the existence of a scalar product. It is well known that the transformation rule for (n-1)-multivectors under general change of basis is different (he discusses multivectors starting p. 23).

Weinreich calls pseudovectors axial vectors (but mentions pseudo as a synonym in the index), and Burke calls them twisted vectors. The easiest way to see that all three are talking about the same objects is by looking at the pictures they draw for pseudovectors: line segments with an outer orientation (depicted as a little arrow twisting around the line segment); see Schouten p. 33, Weinreich p. 30, and Burke section 16. In these books also bivectors (in n=3) are depicted as pieces of planes with an inner orientation, which again makes the difference between pseudovectors and (3-1)-multivectors clear.

It is possible to give a direct geometrical definition of pseudovectors and pseudoscalars (not based on transformation rules under base change), but this is not the place for that, and I don't know references for such a construction.

Now, if we restrict to rigid (metric-preserving) transformations, pseudovectors and (n-1)-multivectors transform the same way, so in this sense the formalization of the article is not a contradiction to Schouten, Weinreich and Burke, but in the absence of a metric such an identification is not possible.

I suggest that Schoutens definition of pseudovectors (as elements of the tensor product between pseudoscalars and regular vectors) and it's difference to (n-1)-multivectors be mentioned in the article.

Also, pictures of pseudovectors (as in Schouten, Weinreich, Burke) might be helpful for readers.

References

Mbachtold (talk) 11:46, 29 December 2024 (UTC)