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I find Wikipedia's discussion 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 $\mathrm{U}L\to\mathrm{S}L$.

I find Wikipedia's discussion 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 $\mathrm{U}L\to\mathrm{S}L$.

Background

I think I understand the basic idea on ℝⁿ, so for readers who know as little as I do, I will provide some ideas. Any differential operator on ℝⁿ is (uniquely) of the form Σ p_i1,...,ik(x) ∂^k/(∂x_i1...∂x_ik), where x₁,...,x_n are the canonical coordinate functions on ℝⁿ, the p_i1,...,ik(x) are smooth functions, and the sum ranges over (finitely many) possible indexes (of varying length). Then the symbol of such an operator is Σ p_i1,...,ik(x) ξ^i₁...ξ^i_k, where ξ¹,...,ξⁿ are new variables; the symbol is a polynomial in the variables {ξ¹,...,ξⁿ} with coefficients in the algebra of smooth functions on ℝⁿ.

Ok, great. So symbols are well-defined for ℝⁿ. But most spaces are not ℝⁿ — most spaces are formed by gluing together copies of (open sets in) ℝⁿ 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 + Σ_i Y_jⁱx_i for some vector (a₁,...,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,...,n, and think of them as keeping track of tensor contraction; then everything transforms as tensors under affine coordinate changes, e.g. the variables ξⁱ transform as coordinates on the cotangent bundle.

On the other hand, consider the operator D = ∂²/∂x² on ℝ, with symbol ξ²; and consider the change of coordinates y = f(x). By the chain rule, the operator D transforms to (f'(y))² ∂²/∂y² + f''(y) ∂/∂y, with symbol (f'(y))² ψ² + f''(y) ψ. 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*X, then the inverse of this Symbol map would determine a "quantization" of the functions on T*X, corresponding to the QP quantization of ℝⁿ.

Let L be a Lie algebra over ℝ or ℂ, G a group integrating the Lie algebra, UL the universal enveloping algebra of L and SL the symmetric algebra of the vector space L. Then UL is naturally the space of left-invariant differential operators on G, and SL 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 UL → SL.

###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 $\mathbb{R}^n$.

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 $\mathrm{U}L\to\mathrm{S}L$.