I find [Wikipedia's discussion](http://en.wikipedia.org/wiki/Symbol_of_a_differential_operator) of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion.


### Background


I think I understand the basic idea on $\mathbb{R}^n$, so for readers who know as little as I do, I will provide some ideas. Any differential operator on $\mathbb{R}^n$ is (uniquely) of the form $\sum p\_{i\_1,\dotsc,i\_k}(x)\frac{\partial^k}{\partial x\_{i\_1}\dots\partial x\_{i\_k}}$, where $x\_1,\dotsc,x\_n$ are the canonical coordinate functions on $\mathbb{R}^n$, the $p\_{i\_1,\dotsc,i\_k}(x)$ are smooth functions, and the sum ranges over (finitely many) possible indexes (of varying length). Then the **symbol** of such an operator is $\sum p\_{i\_1,\dotsc,i\_k}(x)\xi^{i\_1}\dotso\xi^{i\_k}$, where $\xi^1,\dotsc,\xi^n$ are new variables; the symbol is a polynomial in the variables $\{\xi^1,\dotsc,\xi^n\}$ with coefficients in the algebra of smooth functions on $\mathbb{R}^n$.


Ok, great. So symbols are well-defined for $\mathbb{R}^n$. But most spaces are not $\mathbb{R}^n$ — most spaces are formed by gluing together copies of (open sets in) $\mathbb{R}^n$ along smooth maps. So what happens to symbols under changes of coordinates? An **affine change of coordinates** is a map $y\_j(x)=a\_j+\sum\_jY\_j^ix\_i$ for some vector $(a\_1,\dotsc,a\_n)$ and some invertible matrix $Y$. It's straightforward to describe how the differential operators change under such a transformation, and thus how their symbols transform. In fact, you can forget about the fact that indices range $1,\dotsc,n$, and think of them as keeping track of tensor contraction; then everything transforms as tensors under affine coordinate changes, e.g. the variables $\xi^i$ transform as coordinates on the cotangent bundle.


On the other hand, consider the operator $D = \frac{\partial^2}{\partial x^2}$ on $\mathbb{R}$, with symbol $\xi^2$; and consider the change of coordinates $y = f(x)$. By the chain rule, the operator $D$ transforms to $(f'(y))^2\frac{\partial^2}{\partial y^2} + f''(y) \frac{\partial}{\partial y}$, with symbol $(f'(y))^2\psi^2 + f''(y)\psi$. In particular, the symbol did not transform as a function on the cotangent space. Which is to say that I don't actually understand where the symbol of a differential operator lives in a coordinate-free way.


### Why I care


One reason I care is because I'm interested in quantum mechanics. If the symbol of a differential operator on a space $X$ were canonically a function on the cotangent space $T^\ast X$, then the inverse of this Symbol map would determine a "quantization" of the functions on $T^\ast X$, corresponding to the QP quantization of $\mathbb{R}^n$.


But the main reason I was thinking about this is from Lie algebras. I'd like to understand the following proof of the PBW theorem:



> 
> Let $L$ be a Lie algebra over $\mathbb{R}$ or $\mathbb{C}$, $G$ a group integrating the Lie algebra, $\mathrm{U}L$ the universal enveloping algebra of $L$ and $\mathrm{S}L$ the symmetric algebra of the vector space $L$. Then $\mathrm{U}L$ is naturally the space of left-invariant differential operators on $G$, and $\mathrm{S}L$ is naturally the space of symbols of left-invariant differential operators on $G$. Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism $\mathrm{U}L\to\mathrm{S}L$.
> 
> 
> 



What is the symbol of a differential operator?
The **principal symbol** of a differential operator 
$\sum\_{|\alpha| \leq m} a\_\alpha(x) \partial\_x^\alpha$
is by definition the function $\sum\_{|\alpha| = m} a\_\alpha(x) (i\xi)^\alpha$
Here $\alpha$ is a multi-index (so $\partial\_x^\alpha$ denotes $\alpha\_1$ derivatives
with respect to $x\_1$, etc.)
At this point, the vector $\xi = (\xi\_1, \ldots, \xi\_n)$ is merely a formal variable.
The power of this definition is that if one interprets $(x,\xi)$ as variables in the
cotangent bundle in the usual way -- i.e. $x$ is any local coordinate chart, then
$\xi$ is the linear coordinate in each tangent space using the basis $dx^1, \ldots, dx^n$,
then the principal symbol is an invariantly defined function on $T^\*X$, where $X$ is
the manifold on which the operator is initially defined, which is homogeneous of degree
$m$ in the cotangent variables.


Here is a more invariant way of defining it: fix $(x\_0,\xi\_0)$ to be any point in $T^\*X$
and choose a function $\phi(x)$ so that $d\phi(x\_0) = \xi\_0$. If $L$ is the differential operator,
then $L( e^{i\lambda \phi})$ is some complicated sum of derivatives of $\phi$, multiplied together, but always with a common factor of $e^{i\lambda \phi}$. The `top order part' is the one which has a $\lambda^m$, and if we take only this, then its coefficient has only first derivatives of $\phi$ (lower order powers of $\lambda$ can be multiplied by higher derivatives of $\phi$).
Hence if we take the limit as $\lambda \to \infty$ of $\lambda^{-m} L( e^{i\lambda \phi})$
and evaluate at $x = x\_0$, we get something which turns out to be exactly the principal symbol of $L$ at the point $(x\_0, \xi\_0)$. 


There are many reasons the principal symbol is useful. There is indeed a `quantization map'
which takes a principal symbol to any operator of the correct order which has this as its principal symbol. This is not well defined, but is if we mod out by operators of one order lower. Hence the comment in a previous reply about this being an isomorphism between filtered algebras. 


In special situations, e.g. on a Riemannian manifold where one has preferred coordinate
charts (Riemann normal coordinates), one can define a total symbol in an invariant fashion
(albeit depending on the metric). There are also other ways to take the symbol, e.g. corresponding to the Weyl quantization, but that's another story. 


In microlocal analysis, the symbol captures some very strong properties of the operator $L$.
For example, $L$ is called **elliptic** if and only if the symbol is invertible (whenever $\xi \neq 0$). We can even talk about the operator being elliptic in certain directions if the principal symbol is nonvanishing in an open cone (in the $\xi$ variables) about those directions. Another interesting story is wave propagation: the characteristic set of the operator is the set of $(x,\xi)$ where the principal symbol $p(L)$ vanishes. If its differential (as a function on the cotangent bundle) is nonvanishing there, then the integral curves of the Hamiltonian flow associated to $p(L)$, i.e. for the Hamiltonian vector field determined by $p(L)$ using the standard symplectic structure on $T^\*X$, ``carries'' the singularities of solutions of $Lu = 0$. This is the generalization of the classical fact that singularities of solutions of the wave equation propagate along light rays.


