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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: $$ k(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(-\frac \mu {\mu^2+\sigma^2},\left(\frac \sigma {\mu^2+\sigma^2}\right)^2\right), $$ which means that for any $z=\mu+i\sigma$ on the upper semi-circle of radius $\rho$: $$ k(g)\cdot \mathcal{N}(\mu,\sigma^2) = \mathcal{N}\left(- \frac \mu {\rho^2} ,\frac {\sigma^2} {\rho^4}\right). $$

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?

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    $\begingroup$ I upvoted the question because I'd be interested in reading some informed answers/viewpoints but I don't know why one would expect that there is an interpretation of the SL(2,R) action on that particular space: after all, it acts on any space in bijection with $\mathbb{R}\times\mathbb{R}_{>0}$ just by transport of the action on the upper half plane $\mathfrak{H}$. And the space of pairs $(\mu,\sigma)$ doesn't seem to have anything to do with the complex structure on $\mathfrak{H}$ through which the SL(2,Z) action is defined. Of course, I'd be happy to be proved wrong. $\endgroup$
    – Qfwfq
    Commented Jul 24, 2018 at 15:18
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    $\begingroup$ There is a natural metric on this parameter space, namely the Fisher metric; the fesulting Riemannian manifold is the hyperbolic plane, but I don't know off the top of my head whether this particular action captures the isometry group of the Fisher metric. $\endgroup$ Commented Jul 24, 2018 at 15:47
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    $\begingroup$ @Qiaochu Yuan: well, SL(2,R) (actually PGL(2,R)) is the isometry group of the upper half plane with the standard hyperbolic Riemannian metric, so yes if SL(2,R) acts by Moebius transformations then it respects the Fisher metric (and my previous comment gets an 'answer'). $\endgroup$
    – Qfwfq
    Commented Jul 24, 2018 at 16:11
  • $\begingroup$ It seems weird to parameterise your elements $k$, $a$, and $n$ not by their natural parameters $\theta$, $t$, and $u$ but rather by an element $g$ that is never otherwise used. $\endgroup$
    – LSpice
    Commented Dec 31, 2018 at 17:27

1 Answer 1

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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.

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