microlocal analysis



To accommodate an intuitive notion of a “function of differential operators” there is a simple trick used: consider the Fourier transform. Then the differential operators become polynomials. This correspondence of operators and their symbols may, with some analytic care, be extended to define generalizations of differential operators by suitably extending a notion of symbols. Thus the pseudodifferential operators of Kohn and Nirenberg appeared in 1965 with soon following revolution in harmonic analysis and analysis in PDE. This includes a further generalization, the Fourier integral operators of Lars Hörmander. A part of harmonic analysis involving geometric aspects in the cotangent bundles of such methods is called microlocal analysis. The geometric aspects include the support, wavefront set, characteristics…of distributions, pseudodifferential operators and their symbols. There are more technical definitions (involving wavefront sets, supports and filtrations on the algebras of symbols) of various “microlocal” properties of symbols: microlocalization, microhypoellipticity, microparametrix etc.). In addition to the analytic microlocalization there is a formal microlocalization; and a version of filtered localization theory in noncommutative algebra, so called algebraic microlocalization, which is however not used in operator theory. While local aspect of a differential operator is about its behaviour around a point in coordinate space, the microlocal aspect is about a point in the cotangent bundle, hence it also localizes around the fixed covector direction, hence “micro”.

This is clearly related to the general study of oscillating integrals, including the stationary phase method and WKB-method (and generalizations) in particular. These kind of approximations and related estimates are of importance to the study of the propagation of singularities of differential equations, wave fronts, eikonal equations, and so on.

As oscillating integrals are involved in the analysis of various Green functions like the heat kernel there is also a connection to index theorems for elliptic differential operators, see Hörmander 83.

Relevance for QFT

When considering AQFT on curved spacetimes and locally covariant perturbative quantum field theory one has to replace the axioms involving the representation of the Poincare group, since that is a speciality of the Minkowski spacetime. A general curved spacetime will not have any symmetries. In particular the axiom of the spectrum condition has to be replaced, which can be done by a condition on the wavefront set on two point functions of a state.


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Revised on August 14, 2017 04:43:18 by Urs Schreiber (