nLab
symbol of a differential operator

Contents

Definition

For XX a smooth manifold, EXE \to X a vector bundle and D:Γ(E)Γ(E)D : \Gamma(E) \to \Gamma(E) a differential operator on sections of EE, its symbol is the bundle morphism

σ(D):T *X XEE \sigma(D) \;:\; T^* X \otimes_X E \to E

from the tensor product of vector bundles with the cotangent bundle which is given at any point xXx \in X on a cotangent vector of the form (df) xΓ(T *X) x(\mathbf{d}f)_x \in \Gamma(T^* X)_x by

σ(D) x:df x[D,f] x, \sigma(D)_x \;\colon\; \mathbf{d}f_x \mapsto [D,f]_x \,,

where in the commutator on the right we regard multiplication by ff as an endomorphism of Γ(E)\Gamma(E).

The symbol may naturally be thought of as an element in the K-theory of XX (Freed).

Examples

Definition

(symbol map)

For X= nX = \mathbb{R}^n a Cartesian space and DD the Dirac operator of the flat connection, the symbol of DD reproduces the symbol map between differential forms and Clifford algebra elements.

Example

(bicharacteristic flow)

For EE a real trivial line bundle then the principal symbol is equivalently just a real-valued smooth function on the cotangent bundle. Since any cotangent bundle is canonically a symplectic manifold, in this case the symbol may be regarded as a Hamiltonian funtion. The corresponding Hamiltonian flow is called the bicharacteristic flow of the given differential operator.

References

For instance chapter 2.5 of

  • Nigel Higson, John Roe, Lectures on operator K-theory and the Atiyah-Singer Index Theorem (pdf)

  • Dan Freed, Geometry of Dirac operators (pdf)

Revised on August 4, 2017 15:19:41 by Urs Schreiber (94.220.75.2)