Let $\mathcal{N}(\mu,\sigma^2)$ denote the Gaussian distribution on $\mathbb{R}$:
$$ \mathcal{N}(\mu,\sigma^2)(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$
A Gaussian distribution is defined by its mean $\mu\in\mathbb{R}$ and its standard deviation $\sigma>0$. Thus the upper-half plane is the parameter space for the normal family of distributions. Let $z = \mu+i\sigma$ denote an element of the upper-half plane, to be interpreted as the parameters for the Gaussian $\mathcal{N}(\mu,\sigma^2)$.
Let $G\in SL(2,\mathbb{R})$, the group of two by two real matrices with unit determinant. Let $g = \begin{pmatrix}a & b\\ c&d\end{pmatrix}$ be an element of $G$. This group acts on the upper-half plane by fractional linear transformations:
$$g\cdot z = \frac{az+b}{cz+d}.$$
Thus $G$ acts on Gaussians via $$ g\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}(Re(g\cdot z),\, (Im(g\cdot z))^2).$$
We now describe this action in more detail. If $g = \begin{pmatrix}a & b\\ 0&d\end{pmatrix}$, then $g$ acts on a Gaussian as follows: $$ g\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(\frac a d \mu + \frac b d,\, \frac {a^2} {d^2} \sigma^2\right).$$
For an arbitrary $g$, it follows that
$$\begin{align*} Re(g\cdot z) &= \frac{ac(\mu^2+\sigma^2) + bd + (ad+bc)\mu}{(c\mu+d)^2+c^2\sigma^2}\\ Im(g\cdot z) &= \frac{\sigma}{(c\mu+d)^2+c^2\sigma^2}, \end{align*} $$
and the action is $$ g\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(\frac{ac(\mu^2+\sigma^2) + bd + (ad+bc)\mu}{(c\mu+d)^2+c^2\sigma^2},\left(\frac{\sigma}{(c\mu+d)^2+c^2\sigma^2}\right)^2\right).$$
$G$ has the Iwasawa decomposition $KAN$. Thus $g = k(g)a(g)n(g)$, where
$$\begin{align*} k(g) &= \begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}\end{align*}\\ a(g) = \begin{pmatrix}e^t & 0\\ 0&e^{-t}\end{pmatrix}\\ n(g) = \begin{pmatrix}1 & u\\ 0&1\end{pmatrix}, $$
where $\theta$, $t$, and $u$ are analytic functions of the coefficients of $g$ (see Keith Conrad's notes).
It follows that $N$ acts on a Gaussian by moving its mean:
$$ n(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left( \mu + u\,, \sigma^2\right),$$
and $A$ acts on a Gaussian by dilating the mean and standard deviation:
$$ a(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left( e^{2t}\mu\,, e^{4t}\sigma^2\right).$$
$K$'s action is much more interesting:
$$ k(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(\frac{\cos\theta\sin\theta(\mu^2+\sigma^2) – \cos\theta\sin\theta + (\cos 2\theta)\mu}{((\sin\theta)\mu+\cos\theta)^2+(\sin^2\theta)\sigma^2}, \left(\frac{\sigma}{((\sin\theta)\mu+\cos\theta)^2+(\sin^2\theta)\sigma^2}\right)^2\right).$$
The standard normal $\mathcal{N}(0,1)$ is fixed by $K$. For any other Gaussian $\mathcal{N}(\mu,\sigma^2)$, moving through values of $\theta$ will cause oscillations of the mean and standard deviation, creating a "wobbling" of the Gaussian.
In particular, if $\mu=0$ and $\theta = \pi/2$, then $$ k(g)\cdot \mathcal{N}(0,\sigma^2) = \mathcal{N}\left(0,\frac 1 {\sigma^2}\right). $$
The right-hand side is the Fourier transform of $\mathcal{N}(0,\sigma^2)$. For an arbitrary $\mathcal{N}(\mu,\sigma^2)$, with $\theta = \pi/2$ we have: $$ k(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(-\frac \mu {\mu^2+\sigma^2},\frac 1 {\sigma^2}\right), $$
which means that for any $z=\mu+i\sigma$ on the upper semi-circle of radius $\rho$: $$ k(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(- \frac \mu {\rho^2} ,\frac 1 {\sigma^2}\right), $$
and again we have the Fourier transform, combined with reflection about the imaginary axis and dilation of the mean by the square of the curvature of the semi-circle.
Now define $$ g_\mu = \begin{pmatrix}1 & \mu\\ 0&1\end{pmatrix} \begin{pmatrix}0 & -1\\ 1&0\end{pmatrix}\begin{pmatrix}1 & -\mu\\ 0&1\end{pmatrix}. $$
Then $$g_\mu\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(\mu,\frac 1 {\sigma^2}\right).$$
Questions: What are some physical interpretations of this? What are some good resources for exploring similar links among probability and geometry?
Best Answer
The physics application I am aware of is not quite the one in the OP, but similar in spirit: in ray optics the SL(2,R) matrix $$g=\begin{pmatrix} A & B \\ C & D \end{pmatrix}$$ describes the effect of a lens on a Gaussian beam in the paraxial approximation. (The determinant of $g$ is unity if the refractive index remains unchanged.) The fractional linear transformation is called the "ABCD law" in that context.
The SL(2,R) matrix $g$ is called the "ABCD matrix" in the ray optics community, which here at MO sounds a bit silly.
The complex parameter $q$ that is transformed has real and imaginary parts given by $$\frac{1}{q}=\frac{1}{R}-\frac{i\lambda}{\pi w^2},$$ where $\lambda$ is the wave length, $w$ the spot size, and $R$ the radius of curvature of the beam. The corresponding wave profile is $$u(x,y,z)=\frac{1}{w(z)}\exp\left(-\frac{x^2+y^2}{w(z)^2}\right)\exp\left(\frac{i\pi(x^2+y^2)}{\lambda R(z)}\right).$$ As the beam propagates along the $z$-axis through a lens, $\lambda$ remains the same (if the refractive index does not vary), but $w$ and $R$ change according to the fractional linear transformation $$g\cdot q=\frac{Aq+B}{Cq+D}.$$ This Wiki lists examples of transfer matrices $g$ for various optical elements.