Where can square roots come from when they are not distances? 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.

 A: Fixed points of fractional linear transformations $x \mapsto \frac{ax + b}{cx + d}$, where $a, b, c, d$ are integers, are usually quadratic surds. Equivalently, values of periodic regular continued fractions. I'm not convinced these are really best thought of as distances, even though geometric means are in the same ballpark. 
A: A third category would be counts of things. I think probability and computer science should have many examples. Noam Elkies already mentioned longest increasing subsequence here.


*

*Birthday paradox: If we throw $k$ balls uniformly into $n$ bins, and choose $k$ such that we expect one collision, then $k = \Theta(\sqrt{n})$. More precisely one chooses $n = {k \choose 2}$.

*More generally, to test whether a given distribution is uniform on $n$ coordinates or $\epsilon$-far from uniform, it is necessary and sufficient to draw order $\frac{\sqrt{n}}{\epsilon^2}$ samples.

*Grover's algorithm for quantum search inverts a function on domain of size $n$ with $O(\sqrt{n})$ queries (and this is the optimal order). I think there are probably many examples of algorithms along these lines once one starts looking. Pollard's rho algorithm perhaps.

*Along the lines of the above, $\sqrt{n}$ arises often when balancing the size of two terms. Noam Elkies again mentions similar reasoning here. I'm struggling to think up good examples on the spot, but here are two. If you're going to search through up to $n$ entries by restarting $k$ times and each time inspecting $n/k$ items, then you can minimize $\max\{k,n/k\}$ by choosing $k = \sqrt{n}$ and this is something you sometimes want to do.  In online learning theory, optimal regret bounds look like $\sqrt{T}$ when there are $T$ rounds. Interestingly the lower bound comes from the central limit theorem, but the upper arises from choosing $\eta > 0$ to minimize $\eta T + \frac{1}{\eta}$. I don't think regret is a "distance", whereas a key special case of regret is a count of number of mistakes.
A: Suppose a "geometry" consists on $n$ points and a collection of "blocks" all of size $k.$ Various conditions force $k \ge \sqrt{n}$ more or less. In the first two cases, suppose every pair of points is in at least one common block.


*

*If $k \le \sqrt{n}$ then some two blocks are disjoint. So one condition is no two blocks are disjoint. It is possible to have projective planes with $n=q^2+q+1$ and $k=q+1 \approx \sqrt{n}+\frac12$ for $q$ a prime power.

*Blocks may be disjoint. If $k \lt \sqrt{n}$ then there is a point $p$ and block $B$ so that two blocks $B'$ and $B''$ on $p$ are disjoint from  $B.$ So one condition is a point $p$ not on a block $B$ may belong to at most one disjoint block. Here affine planes with $n=q^2$ and $k=q=\sqrt{n}$ satisfying Playfair's axiom are possible.
Another example (not exactly the first in disguise, but close): Suppose $A \subset \{0,1,2,\cdots ,n-1\}$ is such that $|A| \le \sqrt{n}$, Then $A$ is  disjoint from at least one of the $n$ sets $A+i$  (considered $\mod n$). It is possible to have $n=q^2+q+1$ and $|A|=q+1$ so that the sets $A+i$ are the lines of a projective plane (hence each pair intersecting in a singleton.)
A: The Gauss sum attached to a primitive Dirichlet character mod $q$ has absolute value $\sqrt q$.
A: The solution to the optimization problem
$$
\underset{x\in(0,1)}{\text{minimize}} \quad
\frac{a}{x}+\frac{b}{1-x}
$$
is $x^*=\frac{\sqrt{a}}{\sqrt{a}+\sqrt{b}}$, with a minimum value of the objective function of $\left(\sqrt{a}+\sqrt{b}\right)^2$.
This does not involve any distances but rather follows from the derivative of $\frac1x$ being $\frac{-1}{x^2}$.
A: Square roots are common in the analysis of algorithms. A standard example is the Hopcroft–Karp bipartite matching algorithm (or its generalization to non-bipartite graphs by Micali and Vazirani) which takes time $O(m\sqrt n)$; here $n$ is the number of vertices and $m$ the number of edges in a bipartite graph.
Underlying some of these algorithm time bounds are corresponding combinatorial bounds in graph theory which also involve square roots (or half-integer powers). For instance, in a graph with $m$ vertices, the maximum number of triangles is $O(m^{3/2})$, which also forms a natural time bound for triangle-finding algorithms. Relatedly, the degeneracy (or arboricity) of an $m$-edge graph is always $O(\sqrt m)$, a bound that is sometimes tight.
Another important square root in graph theory comes from the planar separator theorem and its generalizations (the fact that you can evenly partition a planar graph into two subgraphs by removing $O(\sqrt n)$ vertices). This also has ramifications in the analysis of algorithms, e.g. exact algorthms for NP-hard problems on planar graphs tend to be exponential in $\sqrt n$ rather than exponential in $n$. However to some extent you could argue that this one is the same as the square root relation between area and perimeter of figures in the plane, so not unrelated to distances.
A: The most natural example I can think of is the Fibonacci numbers,
and the golden ratio, which I am sure most of you knows more about than me.
To recap: The golden ratio is the positive solution to $x^2=x+1$, $x=\frac12(\sqrt{5}+1)$, and gives the asymptotic growth rate of the Fibonacci numbers. This ratio appear in many places in nature and stuff with five-fold symmetry.
I am not sure if one can explain this square root using either distance, quantum physics or probability. 
A: Sometimes you can multiply two different numbers and get a square.  For instance if $f$ is a modular form and $\chi$ is a quadratic character attached to a quadratic field of absolute discriminant $d$, then the product of central values of $L$-functions $L(1/2, f, d) = L(1/2,f)L(1/2, f \otimes \chi)$ (a positive number) is essentially $b(d)^2$, where $b(d)$ is a Fourier coefficient of an associated form $g$ of half-integral weight.  So $b(d)$ is essentially the square root of $L(1/2, f, d)$.  Determining the sign of this square root is an interesting and nontrivial problem.
A: Polar decomposition of square matrix $A$ involves square-root of $A^*A$
A: There was a popular wall Street interview question about the least number of trials needed to test the maximum number of stairs a beer bottle can withstand falling through if you have two bottles. The final answer involves a square root. On a more serious note, see my conjecture about bounds on schur polynomials on the unit circle here.
A: $i=\sqrt{-1}$ has no apparent relation with any distance.
Also $\int_{-\infty}^\infty e^{-x^2}dx=\sqrt{\pi}.$
A: Besides in the calculation of distances, square roots appear naturally as the lower bound on the length of the boundary of planar regions of given area.
A: The Dirichlet-to-Neumann map on a half-space is the square root of the (positive definite) Laplacian on the boundary of the half-space (see the previous MathOverflow question Characterisation of the square root of the Laplacian as a Dirichlet to Neumann mapping ).  (This is related to the Hodge decomposition example in the OP.)
Dirac operators are also square roots of (negative definite) Laplacians.  (This is related to the spinor example in the OP, of course.)
(One could argue, though, that both these examples are measuring some sort of "distance" in frequency space.)
The Pfaffian of a skew-symmetric matrix is the square root of its determinant.  (Presumably this falls under the category of "supersymmetry" somehow.)
In geometric quantisation, it is convenient to work with half-densities - square roots of densities.  In particular, this allows one to work with the Hilbert space $L^2(M)$ on a manifold $M$ without needing a reference measure.  (This is of course related to the wave function example in the OP.)
In Bayesian probability, the Jeffreys prior is the square root of the (determinant of the) Fisher information. 
If one accepts a half-dimensional subspace as a "square root" of the space it is contained in, then Lagrangian subspaces can be viewed as square roots of symplectic vector spaces, and real vector spaces can be viewed as square roots of their complexifications.
Square roots of the canonical line bundle on an algebraic curve are known as theta characteristics.
UPDATE:  I found a table at this nlab page which lists three of the above examples and adds two more: metalinear structures (which seem connected both to theta characteristics and to spinors, the latter in that they both involve double covers of classical Lie groups, which I guess is the Lie group analogue of a square root) and the partition function from self-dual higher gauge theory (which I don't understand too well, but also seems linked to theta characteristics and spinors).
A: The longest monotone subsequence of a sequence of length $n$ has length at least $\lceil \sqrt n \, \rceil$, and this is sharp (special case of the Erdős–Szekeres theorem).  The longest increasing subsequence of a random permutation of $\{1,2,\ldots,n\}$ grows as $2\sqrt n$, the difference approaching a scaled Tracy-Widom distribution.
A: The square root of SWAP (which is the only two-qubit operation needed to realize a universal quantum computation) has no "distance" interpretation I can think of.
A: The number of partitions of $n$ is asymptotic to
$$
\frac1{4n\sqrt{3}} \exp \left( \pi \sqrt{\frac{2n}{3}} \right)
$$
as $n \to \infty$ [Hardy-Ramanujan 1918; refinements replace the $n$ in $\sqrt{2n/3}$ by $n-\frac1{24}$].
A: Computer Science:
In problems such as sensitivity conjecture and degree of approximation or representation of boolean functions ($n$ is number of variables) and such as find minimum number of bits in communication complexity ($n$ is rank of communication matrix) I have seen some stubborn upper bounds of order $\sqrt n$ while lower bounds turn out to be $\log^2 n$. I am not sure which of these is close to truth.
There is no intuitive way to metrize these quantities.
Information Theory:
$\sqrt5$ is zero-error capacity of the pentagon.
A: Most things using the Laplace domain.  For example, if I have a 2nd-order filter then I will often want to convert between the (s^2 + a*s + b) representation and the (s + m)*(s + n) representation.  Going from the former to the latter needs a square root.  This is not reliant on distance, but on the nature of the system's dynamic behaviour.
A: $X^2=X$ for infinite cardinal $X$ by the Axiom of Choice.  (It's about square not square root but it's far from being about distance.)
A: $P=I^2 \times R$ as in Power = The current squared times the resistance.
In this case the square root is the actual current flowing through the device. This is particularly useful in situations with low transconductance power mos-fets and the heat they would have to dissipate in high performance industrial situations, with 1000-10,000 amp welders and platers.
A: I think an abstract version of some of these examples is worth highlighting.
In a general algebraic system, one often has a basic or perhaps derived operation op() of arity two and given an element b one looks for an element a such that op(a,a)=b.  Sometimes a is considered as a square root of b.
Some examples are semigroups of functions, with the operation being composition: solve for f in f(f(x))=exp(x), for example.
In group and semigroup presentations, if a^2=b is part of a presentation, the natural question arises whether a^2=a as well.
Objects which are their own square roots are idempotents, and in some systems (rings especially) these are used in representing/decomposing the structure.
Many more examples of this are found in the other posts.
If one is using Cayley graphs or a time scale for a dynamical system, one can frame these as questions of distance.  However, I see the abstract version as motivated by distance but not necessarily interpreting or modeling distance.  It may model a different form which simpifies/complicates the present model, which may shorten/increase the distance from now to the point of understanding/enlightenment.
Gerhard "Is Humor Orthgonal To Enlightenment?" Paseman, 2016.03.16.
A: This sounds almost too obvious, but when you approximate a function locally by a parabola and then solve for its intersection with a line (which occurs in a lot of contexts, e.g. ODE solving), there would seem to really be no interpretation of the square-root except as the root of a 2nd-degree polynomial...
A: I feel quite silly in posting this triviality next to all this abstract advanced math, but I haven't seen it yet in the other answers.
Probably the easiest example of a square root that does not immediately relate to a distance is the good old
$$
\frac{-b \pm \sqrt{b^2-4ac}}{2a}.
$$
A: periodic continued fractions are irrational solutions of quadratic equations with rational coefficients, i.e. can be expressed as sums of rationals and square roots of rationals, but not without square roots. 
A: How about square root that pops up in the Laplace/stationary phase estimation? The asymptotics of an integral
$$
\int_{[0,1]} f^n(x) dx
$$
for a "generic" function $f$ is $CA^n n^{-\frac{1}{2}}$. I don't think it's trivial, and don't readily see how to interpret it as a distance. 
A: Other than itself, all of a number's prime factors will always be less than or equal to that number's square root.
