In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more fundamental quantities which may be physically mysterious and may not even be observable by themselves. Such `mysterious square roots' underpin supersymmetry, and Kapranov explains how they all arise from actions of the first levels of the sphere spectrum $\pi_1^{\mathrm{st}}= \mathbb{Z}/2$ and $\pi_2^{\mathrm{st}}= \mathbb{Z}/2$.

This made me curious, because the only other non-contrived cases of square roots that I can think of in mathematics are distances. For example, the $1/\sqrt{n}$ in the Central Limit Theorem (note that $n$ has immediate meaning as a number of samples but $\sqrt{n}$ doesn't) comes from a standard deviation which is a distance.

Question: Is every square root in mathematics, in which an object of immediate interest is replaced by its square root which is perhaps more fundamental but less readily interpreted, either a distance or an action of $\pi_1^{\mathrm{st}}$ or of $\pi_2^{\mathrm{st}}$ or as one of Kapranov's examples? Where else can square roots come from?

These are Kapranov's basic examples (the last is a simple case of supersymmetry):

- The wave function $\psi(x)$ of a particle, which cannot be measured, although $\left\vert \psi(x)\right\vert^2= \bar{\psi}(x)\cdot \psi(x)$ represents the probability density of the particle which is real, non-negative, and measurable.
- The Laplace operator on forms on a smooth Riemannian manifold is non-negative definite and real (self-adjoint), but is defined as $\Delta= d \circ d^\ast + d^\ast \circ d$ where $d$ is the exterior derivative and $d^\ast$ is its adjoint with respect to the Riemannian metric.
- Spinors as square roots of vectors.
- For $X$ be a smooth projective curve over a finite field $\mathbb{F}_q$, the étale cohomology group $H^1(X \otimes \bar{\mathbb{F}}_q,\mathbb{Q}_l)$ is acted upon by the Frobenius element $Fr$ generating $\mathrm{Gal}(\bar{\mathbb{F}}_q/\mathbb{F}_q)$. The image of each eigenvalue of $Fr$ in each complex embedding has absolute value a square root of $q$. This example motivated the Weil conjectures.
- The differential operator $Q =\frac{\partial}{\partial\xi}+\xi \frac{\partial}{\partial t}$ in $\mathbb{C}[t] \otimes \Lambda[\xi]$ is the square root of $\frac{\partial }{\partial t}$.

UPDATE: Thank you for the nice answers, which came up with quite a few examples! Are some of these manifestations of one another, or are they distances in disguise? I will list the ones I understand, together with what I don't understand about them:

- Square roots of line bundles.
- Volume elements, which are square roots of determinants of metric tensors.
- The $\sqrt{\pi}$ term in Gaussian integrals/ the Stirling approximation.
- The size of a square array with $n$ elements, as in the longest monotone subsequence of a sequence of length n.
- The square root of a two qubit operation such as SWAP. Is this a manifestation of one of the other examples?
- As a balancing term for errors behaving like $y$ and like $1/y$ (although, as pointed out by Elkies, this might be a distance in disguise).
- Values of periodic regular continued fractions.
- Grover's quadratic speedup and Tsirelson's bound.
- In the isoperimetric inequality, bounding the length of the boundary of planar regions of given area.