# When was the pseudo-vector first defined?

In an answer to a question asked here on the history of vector notation, it's mentioned in an answer that W. Voigt in 1896 distinguished between "polar vectors" and axial vectors". Although a pseudo vector is an axial vector, emphasis is placed upon its transformation properties under a rotation, translation and reflection. Who was the first to define a pseudo-vector this way?

• I am not sure if he used "pseudovector", but the ideology of classifying vectors and tensors by their symmetries was developed by Weyl in Raum, Zeit, Materie (1921):"the character of a quantity is not in general described fully, if it is stated to be a tensor of such and such an order, but symmetrical characteristics have to be added." As Hawkins writes in From General Relativity to Group Representations, "it was he, who, drawing upon his Göttingen background, recast tensor calculus in its essentially modern form." Commented Jun 5 at 6:48

The history of nomenclature related to (multi)linear algebra, linear/affine geometry, and physics is complicated for many reasons, including work being overlooked and reinvented, often from a different starting points, or for different needs.

The earliest fitting use of the "pseudo" prefix known to me is the word German Pseudoskalar (pseudoscalar in english) as used in the German textbook:

• Max Abraham & August Föppl, Theorie der Elektrizität (1904)

Aus jedem Vektorfelde ist durch Berechnung der Divergenz ein Skalarfeld, das Quellenfeld, abzuleiten. Die Divergenz eines Vektors ist ein eigentlicher Skalar oder ein Pseudoskalar, je nachdem der betreffende Vektor ein polarer oder ein axialer ist.

(My translation: For every vector field, a scalar field can be derived via calculation of the divergence, a source field. The divergence of a vector is really a scalar or a pseudoscalar, depending on whether the vector in question is polar or axial.)

It's noteworthy to recognize that this substantially predates Weyl's observation in Raum-Zeit-Materie of 1921, though he uses the term "Pseudotensordichte" p. 312 (pseudo-tensor density). I think it is certainly very plausible that the grappling for building General Relativity on emerging tensorial methods played a role in sharpening the distinction of tensor types for phycisists. For an example of the awareness of the distinction not yet having a clear term for it see Einstein, A. "Note on E. Schrödinger’s Paper: The energy components of the gravitational field." Phys. Z 19 (1918): 115-116 where Einstein uses the wordage "not a tensor" and "energy components" in lieu of something like energy pseudo-tensor.

It seems to me that Abraham-Föppl was introduced among other pathways to the US via:

Gilbert N. Lewis, "On Four-Dimensional Vector Analysis, and Its Application in Electrical Theory" Proceedings of the American Academy of Arts and Sciences , Oct., 1910, Vol. 46, No. 7 (Oct., 1910), pp. 165-18.

The transition to a notion of pseudovector is complicated and my best hypothesis is that the development starts with Weyl and is molded through Schouten. To understand the evidence, we have already seen Weyl's use of the "pseudo" prefix.

Early in Schouten's book:

• Jan A. Schouten, Ricci-calculus: an introduction to tensor analysis and its geometrical applications. Springer (1924).

Schouten uses two notions: "pseudonormale Richtung" (pseudonormal direction) and "Pseudonormalvektor" (pseudo-normal vector) and he writes:

$$n^r$$ heiBt der Pseudonormalvektor. Der Pseudonormalvektor ist durch Angabe der pseudonormalen Richtung bis auf einen Zahlenfaktor bestimmt. Erst wenn auch die Normierung von $$t_\lambda$$, festgelegt wird, ist $$n^r$$ vollstandig bestimmt.

I.e. a pseudonormalvector is not a fully determined vector.

Five years later Shouten and Hlavaty use the pseudo-prefix is used as is customary today.

They use Pseudoskalare, Pseudovektor, Pseudotensor (pseudo-scalar,-vector,-tensor) along with less familiar concepts such as a Pseudoaffinor.

However by the time Schouten writes his Tensor analysis up in 1951, he distances himself from that nomenclature.

• Jan A. Schouten Tensor analysis for physicists. Oxford at the Calderon Press, 1951.

Schouten uses the nomenclature W-scalar, W-vector, W-tensor but footnotes on p.31 discussion his W-scalar nomenclature: "In physical publications it is sometimes called pseudoscalar." The index references this page under the entry "pseudo-vector" (see p. 274). On p. 117 he talks about "W-vectors (pseudo-vectors)".

There may well be other interactions of verbage and individual differences throughout. Nomenclature does not seem to be broadly stable until even as late as 1951 (see Schouten's last book). In some sense it might be wise to not consider these notion stable today, as there are a range of different nomenclatures for the same concepts (bivector vs pseudovector for example).

My personal take is that the pseudovector specifically in part serves to rescue of a type error made early on in vector calculus in defining the cross product, oft equivalently called vector product to be a map from vector to vector and obfuscating the problem that is inherent not being aware of the Hodge dual, a fact that certainly was understood by Grassmann at least as early as 1862.

• A proper mathematical definition had to wait until the notion of a vector bundle was formalized. Pseudovectors are sections of certain vector bundles. Commented Jun 5 at 19:24
• All one needs is multi-linear (exterior) algebra of an oriented finite dimensional vector space, where we provide a nondegenerate symmetric bilinear form. Or you can follow Grassmann in 1862 and define the Hodge dual first, and then derive the inner product from it. Nothing is stopping us having this live in a vector bundle (say cotangent bundle) but it's not required. Commented Jun 5 at 21:09
• Yes, it is very common in physics literature to say "scalar" when they mean "a function", "vector" when they mean "a vector field," etc. Mathematicians are also guilty of this as we tend to say "tensor" when we really mean "a tensor field." Commented Jun 5 at 21:54