Definition of the principal symbol of a differential operator on a real vector bundle.

Solution 1:

Let $E\rightarrow M$ and $F\rightarrow M$ be vector bundles with spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections. Consider a linear partial differential operator of order $k$, which is a map \begin{align} L=\sum_{|\alpha|\leq k}\ell_\alpha\partial^\alpha:\Gamma(E)&\rightarrow\Gamma(F)\\ S&\mapsto L(S)=\sum_{|\alpha|\leq k}\ell_\alpha\partial^\alpha S. \end{align} Here $\alpha=(\alpha_1,...,\alpha_m)$ is a multi-index and each $\ell_\alpha:E\rightarrow F$ is a bundle homomorphism. Now let $\omega=\omega_i\text{d}x^i\in\Gamma(T^*M)$ be a covector field (a $1$-form). The total symbol of a linear partial differential operator $L$ in the direction of the covector field $\omega$ is the bundle homomorphism: \begin{align} \sigma_L(\omega)=\sum_{|\alpha|\leq k}\omega^\alpha\ell_\alpha:E&\rightarrow F\\ e&\mapsto\sigma_L(\omega)e=\sum_{|\alpha|\leq k}\omega^\alpha\ell_\alpha e. \end{align} Here $\omega^\alpha=\omega_1^{\alpha_1}\cdots\omega_m^{\alpha_m}$. The principal symbol simply takes the highest-order partial derivative terms of the symbol, and is the bundle homomorphism: \begin{align} \hat{\sigma}_L(\omega)=\sum_{|\alpha|=k}\omega^\alpha\ell_\alpha:E&\rightarrow F\\ e&\mapsto\hat{\sigma}_L(\omega)e=\sum_{|\alpha|=k}\omega^\alpha\ell_\alpha e. \end{align} Hence, the principal symbol captures the properties of the linear partial differential operator which are held in the highest-order partial derivative terms. A linear partial differential operator is elliptic if it's principal symbol is a linear-space isomorphism for all nonzero covector fields $\omega\neq0\in\Gamma(T^*M)$.

These notions also hold for nonlinear partial differential operators between spaces of sections of vector bundles, by considering the operator's linearisation. The linearisation of a nonlinear partial differential operator is a linear partial differential operator. The symbol (principal symbol) of a nonlinear partial differential operator is the symbol (principal symbol) of its linearisation. A nonlinear partial differential operator is elliptic if the principal symbol of its linearisation is a linear space ismorphism for all nonzero covector fields $\omega\neq0\in\Gamma(T^*M)$.

Solution 2:

The principle symbol arise naturally when you take the Fourier transform, where the symbol appears the top order multiplier. So the choice of including $i$ is immaterial since then $i^{m}$ is a constant. The important thing is the property of $\sigma(D)$ (like whether it is elliptic, hyperbolic, invertible, etc), and that would not be changed by multiplying a constant.

I do not have Spin geometry with me, but I think what you wrote is incorrect. Here the $\sigma(D)$ should only include the top order term $|\alpha|=m$. What you wrote is the definition of the symbol instead.