symbol of a differential operator



For XX a smooth manifold, EXE \to X a vector bundle and D:Γ(E)Γ(E)D : \Gamma(E) \to \Gamma(E) a linear 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).

Generally, the symbol of a possibly non-linear differential operator is similarly the map on the cotangent bundle given by “replacing partial derivatives by covectors” in the definition of the differential operator. In this case the principal symbol is the highest degree homogeneous component of the symbol.



(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.


(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.


  • Nigel Higson, John Roe, chapter 2.5 of Lectures on operator K-theory and the Atiyah-Singer Index Theorem (pdf)

  • Dan Freed, Geometry of Dirac operators (pdf)

See also

Revised on November 23, 2017 10:55:50 by Urs Schreiber (