Polarization: etymology question

Question

The polarization identity expresses a symmetric bilinear form on a vector space in terms of its associated quadratic form: $$ \langle v,w\rangle = \frac{1}{2}(Q(v+w) - Q(v) - Q(w)), $$ where $Q(v) = \langle v,v\rangle$. More generally (over fields of characteristic $0$), for any homogeneous polynomial $h(x_1,\dots,x_n)$ of degree $d$ in $n$ variables, there is a unique symmetric $d$-multilinear polynomial $F({\mathbf x}_1,\dots,{\mathbf x}_d)$, where each ${\mathbf x}_i$ consists of $n$ indeterminates, such that $h(x_1,\dots,x_n) = F({\mathbf x},\dots,{\mathbf x})$, where ${\mathbf x} = (x_1,\dots,x_n)$. There is a formula which expresses $F({\mathbf x}_1,\dots,{\mathbf x}_d)$ in terms of $h$, generalizing the above formula for a bilinear form in terms of a quadratic form, and it is also called a polarization identity.

Where did the meaning of "polarization", in this context, come from? Weyl uses it in his book The classical groups (see pp. 5 and 6 on Google books) but I don't know if this is the first place it appeared. Jeff Miller's extensive math etymology website doesn't include this term. See http://jeff560.tripod.com/p.html.

Answer 1

Where did the meaning of "polarization", in this context, come from? Weyl uses it in his book The classical groups (see pp. 5 and 6 on Google books) but I don't know if this is the first place it appeared.

A few things I've managed to find ...

The term polarization in this context did not originate with Weyl (1939). The book Hilbert's Invariant Theory Papers is an English translation of four papers by David Hilbert, and the term "polarization" appears in the first two of them (published 1885 and 1887), evidently in the sense you have in mind. In the fourth paper (published 1893), Hilbert uses an expression that translates as "Aronhold process" for what Hawkins' Emergence of the theory of Lie groups: an essay in the history of mathematics, 1869-1926 terms the "Aronhold polarization process". Also, Gordan (1885) refers to this same "Aronhold process", which was apparently published in 1838 (if not also earlier) by Aronhold.

In the above-cited works, the meaning of polarization appears to derive from that of the terms pole and polar as used in projective geometry. The entry for "POLE and POLAR" on the webpage by Jeff Miller, mentioned in the question, says the term pôle in this sense was introduced by François Joseph Servois in 1811, and that the term polar (polaire) was introduced by Joseph-Diez Gergonne in the modern geometric sense in 1813.

Answer 2

If you are looking at a single tangent space V to a Riemannian 2-manifold, then there is a positive definite quadratic form Q on V, and you can use that quadratic form to define a function r that represents lengths of the tangent vectors in V. Specifically, for any point v in V, r(v) will be the square root of Q(v). However, if you want to set up polar coordinates, you will also need to be able to compute angles, and the polarization B of Q allows you to do that. Specifically, the angle between two unit vectors v,w is the arccos of B(v,w). So polarization allows you to go from knowing which vectors are unit vectors to knowing how close two unit vectors are to pointing in the same direction. If you're traveling in some 2-dimensional Riemannian manifold, like the surface of the earth, it is useful to know how fast you're going, but it's also important to know whether you're pointed toward the north pole, the south pole, or in some other direction. The geometry of speed of travel comes from Q, but the second "polar" geometry comes from the polarization of Q. Something like that anyway ...