In the original sense, microlocal analysis (e.g. Strohmaier 09) is the study of the functional analysis of generalized functions/distributions with attention paid not just to their singular support, i.e. to the points around which they are singular as generalized functions, but also the directions of propagation of their singularities at each singular point, which is the set of covectors known as their wave front set. This extra directional (“microlocal”) information governs the basic operations on distributions, notably the pullback of distributions and the product of distributions.
Since the wave front set is the set of (co-)directions along which, locally, the Fourier transform of distributions is not rapidly decreasing (the set of “UV divergences” in applications to perturbative quantum field theory), much of microlocal analysis is concerned with constructions related to Fourier transformation, such as the discussion of pseudodifferential operators.
Microlocal analysis in this sense was introduced by (Sato 70), soon followed by (Hörmander 71, Hörmander 83) who both introduced the notion of wave front set. This microlocal point of view was then extended to sheaf theory by Kashiwara-Schapira 82, Kashiwara-Schapira 82) who introduced the notion of microsupport? of sheaves giving rise to microlocal sheaf theory (see Schapira 17). Beware that some authors still use the term “microlocal analysis” for this sheaf theoretic concept.
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 and V. Maslov. 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.
In perturbative quantum field theory microlocal analysis is used to define Wick algebras of quantum observables on free fields: The product on these algebras is a Moyal star-product induced from the Peierls-Poisson bracket, whose integral kernel is the causal propagator on the given globally hyperbolic spacetime. But the wave front set of this propagator is such that its UV-divergences in general collide with those of local functionals? (here). This is fixed by modifying the causal propagator to a Hadamard propagator. The resulting change of the algebra structure is known as normal ordering of quantum fields. It yields the properly defined Wick algebras of free quantum fields.
As the name suggests, normal ordering was originally an operation implementd simply by re-arranging the order of Fourier modes of quantum fields. The desire to generalize this procedure from Minkowski spacetime to general globally hyperbolic Lorentzian manifolds was what required use of tools from microlocal analysis. See at locally covariant perturbative AQFT and at S-matrix for more.
wave front set, microsupport?
Microlocal analysis of distributions in terms of wave front sets was introduced in
Mikio Sato, Regularity of hyperfunctions solutions of partial differential equations 2 (1970), 785–794.
Lars Hörmander, Fourier integral operators I. Acta Math. 127 (1971)
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, Grundlehren der mathematischen Wissenschaften 256, Springer 1983, 1990
Survey is in
Alexander Strohmaier, chapter Microlocal analysis (web) in Christian Bär, Klaus Fredenhagen Quantum Field Theory on Curved Spacetime, 2009 (web)
A. Kaneko, Microlocal analysis, Springer Online Enc. Of Math.
Comprehensive lecture notes are in
See also
C. Bardos, L. Boutet de Monvel, From atomic hypothesis to microlocal analysis (lecture notes) pdf
A. Grigis, J. Sjöstrand, Microlocal analysis for differential operators: an introduction, Cambridge U.P. 1994.
Hans Duistermaat, Fourier integral operators, Progress in Mathematics, Birkhäuser 1995.
Victor Guillemin, Masaki Kashiwara, Takahiro Kawai, Seminar on micro-local analysis, Ann. of Math. Studies 93 (1979), googlebooks
Yu. V. Egorov, Microlocal analysis, 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, Eng. translation in Partial Differential Equations IV (Egorov, Shubin eds.), Springer 1993 doi
Yu. Safarov, Distributions, Fourier transforms and microlocal analysis (course online notes), pdf
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