# Overview

To accomodate an intuitive notion of a “function of differential operator” there is a simple trick used: consider the Fourier transform. Then the differential operators become polynomials. This correspondence of operators and their symbols, is by the definition, under some analytic care can be extended to define generalizations of differential operators by suitably extending a notion of symbols. Thus 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’s 4-volume book on analysis of linear PDEs.

## Relevance for QFT

When considering AQFT on curved spacetimes 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.

# References

• A. Kaneko, Microlocal analysis, Springer Online Enc. Of Math.

• J.J. Duistermaat, Fourier integral operators, Progress in Mathematics, Birkhäuser 1995.

• V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84, 327–402 (1986), MR87j:32030, doi

• Lars Hörmander, The analysis of linear partial differential operators, in 4 vols.: I. Distribution theory and Fourier analysis, II. Differential operators with constant coefficients, III. Pseudo-differential operators, IV. Fourier integral operators.

• A. Grigis, J. Sjöstrand, Microlocal analysis for differential operators: an introduction, Cambridge U.P. 1994.

• R. Melrose, Introduction to microlocal analysis, pdf

• Yu. Safarov, Distributions, Fourier transforms and microlocal analysis (course online notes) pdf

• Masaki Sato, Restriction, localization and microlocalization, (expository paper) pp. 195–205 in collection “Quadrature domains and their application” (Operator theory: advances and applications 156), doi

• M. Kashiwara, P. Schapira, F. Ivorra, I. Waschkies, Microlocalization of ind-sheaves, in “Studies in Lie theory”, Progress in Math. 243, Birkhäuser 2006.

• C. Bardos, L. Boutet de Monvel, From atomic hypothesis to microlocal analysis (lecture notes) pdf

• Dmitry Tamarkin, Microlocal condition for non-displaceablility, arxiv/0809.1584(application of microlocal analysis to symplectic/Lagrangean geometry).

• Goro Kato, Daniele Carlo Struppa, Fundamentals of algebraic microlocal analysis, M. Dekker 1999, googB

• V. Guillemin, Masaki Kashiwara, Takahiro Kawai, Seminar on micro-local analysis, Ann. of Math. Studies 93 (1979), googlebooks

• Ю. В. Егоров, Микролокальный анализ, Дифференциальные уравнения с частными производными – 4, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 33, ВИНИТИ, М., 1988, 5-–156, pdf, MR93e:35002

• Masaki Kashiwara, Systems of microdifferential equations, Birkhäuser 1983, 87 pp.

• Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of algebraic analysis, Transl. from Japanese by Goro Kato. Princeton Mathematical Series 37, 1986. xii+255 pp. MR87m:58156; J. L. Brylinski?, Book Review: Foundations of algebraic analysis. Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 104–108, doi

• Masaki Kashiwara, Pierre Schapira, Hochschild homology and microlocal Euler classes, arxiv/1203.4869

Revised on March 23, 2012 08:41:49 by Zoran Škoda (161.53.130.104)