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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.

J. M. ain't a mathematician
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KCd
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2 Answers2

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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.

r.e.s.
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    Very nice and interesting find! The Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (1899) has an entry devoted to the Aronhold process (p.366f) and calls an instance of it "Polarenprozess". They refer to geometry texts for a geometric explanation of the name: Clebsch-Lindemann (see 3 Abtheilung, p.167ff, especially part II, p.183ff) and also Thieme. – t.b. Nov 15 '11 at 21:35
  • I don't read German. If someone can look at those two references t.b. mentions at the end and indicate how the term polarization came from a specific geometric idea, please add a comment or additional answer. – KCd Nov 16 '11 at 03:59
  • I just added a paragraph to my answer re pole and polar as used in projective geometry. The first link has some nice illustrations, and the second link has (in the History section) some interesting comments in relation to the development of invariant theory and quantum mechanics. – r.e.s. Nov 17 '11 at 15:45
  • @t.b. just in case you didn't see KCd's request. – Willie Wong Nov 17 '11 at 15:48
  • @KCd: It seems to me that the beginning of Dolgachev's book contains a modern rendition of these ideas. The Aronhold process in its original form really is the same thing as the symbolic method described in remark 1.1.1 on p.3, while the polarization of a quadratic form $f$ via that process is given as Example 1.1.1 on p. 6. The polarization identity you ask about is another way of writing/interpreting that same formula for the polar bilinear form $g$. – t.b. Nov 18 '11 at 07:40
  • @t.b.: Thanks for the further information and link. – KCd Nov 18 '11 at 09:01
  • You've assumed that Hilbert uses "polarization" (or its modern German equivalent) in the original. The word he used in German may well have been anachronistically translated to the modern term. You would have to check the original text to be sure. – Chappers Oct 13 '15 at 09:02
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    we may now know who mentioned it first, but what does it mean? what does the polarization identity have to do with any poles? – peter May 14 '18 at 10:25
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    I second @peter's comment. The question asks "Where did the meaning [...] come from?" You answer "the meaning of polarization appears to derive from [...] projective geometry" but don't explain how. Is it just a general sense of duality? Bilinear forms are to quadratic forms as points are to lines? – mr_e_man Mar 03 '23 at 04:02
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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 ...

Scot Adams
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