TITLE: Examples of naturally occurring Quadratic forms or quadrics.
QUESTION [9 upvotes]: I am always fascinated when a quadratic form (or a quadric) arises naturally. I have
some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too vague for MO.  Most forms I list are really
elementary, and all are finite dimensional.
I got most of the following examples from M.Berger, Geometry I & II, and from the truly beautiful book "Eléments de géométrie : actions de groupes" by french author Rached Meinmné.
$(0)$ the discriminant on the affine space of unitary degre 2 polynomials
$(i)$ the determinant on endomorphisms of a 2 dimensional vector space, and
$\mathrm{Tr}^2-4\mathrm{det}$
$(ii)$ the radical on the space of quadratic forms on a 2 dimensional vector space,
and the isotrope cone (not sure about the name, degenerate cone?). 
$(iii)$ the family of hermitian forms (built from the Wronskian) on the solution
space of the discrete Schroedinger equation that allow one to show the existence of
right and left side $L^2$ solutions, and the Weyl m function.
$(iv)$ If $\Delta$ is any $2$ dimensional complex vector space, then
$\mathrm{Herm}(\Delta)$, the real vector space of hermitian forms on $\Delta$,
carries a natural quadratic form obtained by constructing an essentially unique
morphism $\rho$ from $\mathrm{Herm}(\Delta)$ to
$\mathrm{Hom}(\Delta\oplus\overline{\Delta})$ such that for all
$h\in\mathrm{Herm}(\Delta),~\rho(h)^2$ is proportional to $\mathrm{Id}$, the proportionality defining the quadratic form. Here, $\rho$ only depends on a choice of
a nonzero element $\omega\in\Lambda^2\Delta^*$.
$(v)$ If $V$ is a 4 dimensional vector space, then $\Lambda^2 V$ carries the natural
quadric $Q(v)=v\wedge v$ where $\Lambda^4 V$ is identified with the underlying
field, which vanishes exactly when $v$ comes from the canonical map
$\mathrm{Gr}(2,V)\rightarrow P\Lambda^2V$.
I remember reading about one on the space of circles, but I forgot the details. What other examples of natural quadratic forms are there?

REPLY [2 votes]: Binary quadratic forms arise in nature as norm forms for a quadratic field. This point of view has various consequences in number theory.

For a fixed negative discriminant (the definite case), Gauss discovered that the quadratic forms (or their $\mathrm{SL}_2(\mathbb{Z})$ equivalence classes) can be composed. This led him to the phenomenon of the ideal class group before ideals were invented by Kummer and Dedekind. Besides in the ideal class group for more general number fields, Gauss's composition law has found an extension in Bhargava's higher composition laws. These are based on the representation theory of arithmetic groups ($\mathrm{SL}_2(\mathbb{Z})$ and its generalizations), in which regard they are natural structures in themselves. They have striking applications to old problems regarding mean asymptotics of Selmer ranks of elliptic curves, the $3$-parts of class groups of quadratic fields, etc.
The Epstein zeta function takes the shape $\zeta_Q(s) := \sum_{\mathbf{n} \neq \mathbf{0}} Q(\mathbf{n})^{-s}$, for a given signature $(d,0)$ quadratic form $Q$. It has all the right analytical properties (meromorphic with simple pole at $s = 1$ and a functional equation relating $s \leftrightarrow 1-s$), allowing to decompose the zeta function of an imaginary quadratic field over a set of representatives $Q$ for the class group. This has consequences for the arithmetic of these fields, beautifully developed in Siegel's Lectures on Advanced Analytic Number Theory (Tata Institute lecture series, 1961). 
For $d = 2$, $\zeta_Q(s)$ is in effect an Eisenstein series ($|mz+n|^2$ being a binary quadratic form in $m,n$), which is a natural structure all over mathematics, being a continuum of modular forms in the spectral resolution of the hyperbolic Laplacian. Siegel apparently had much interest in the conceptual role played in number theory by the higher rank quadratic forms and their Epstein zeta function. Much of his work was put on representation theoretic footing in Weil's 1964 paper Sur certains groupes d'operateurs unitaires. Michael Berg's book, The Fourier-Analytic Proof of Quadratic Reciprocity, is a terrific introduction to these ideas.