I've come across two slightly different conventions for the symbol of a differential operator $D$ (let's say on $\mathbb{R}^n$) and haven't thought much about the motivation behind them until now.
The first convention is to replace each instance of $\frac{\partial}{\partial x_j}$ by $i\xi_j$, where $\xi_j$ is a real variable for each $j=1,\dotsc,n$.
The second is to replace $\frac{\partial}{\partial x_j}$ by $\xi_j$.
Some of the advantages of the first convention seem to be:
- It fits exactly with the way differentiation and multiplication correspond under the Fourier transform;
- Self-adjoint operators have self-adjoint principal symbols;
- Positive operators have positive principal symbols.
On the other hand, I've sometimes come across the second convention as well. For example it is used in Notes on the Atiyah–Singer Index Theorem by Liviu Nicolaescu; see Example 2.1.6 (p. 49) and Cor 2.1.15 (p. 52). I've also seen other people use this convention as well.
Are there certain advantages to using the second convention also?