Symplectic microgeometry, IV: Quantization
Symplectic microgeometry, IV: Quantization
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辛微观几何，IV：量化
DOI：
10.2140/pjm.2021.312.355
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发表时间：
2020
影响因子：
0.6
通讯作者：
A. Weinstein
中科院分区：
文献类型：
--
作者：
A. Cattaneo;B. Dherin;A. Weinstein
我们构造了一类特殊的半经典傅里叶积分算子，其波前是辛微态。这些算子具有非常好的性质：它们形成一个范畴，在该范畴上波前图成为余切微束范畴的函子，并且它们允许用半密度胚增强的辛微态方面的全符号演算。这个新的算子类别涵盖了半经典伪微分微积分，并为薛定谔方程的半经典分析提供了函数框架。我们还评论了经典力学和量子力学的应用，以及泊松流形量化的函数和几何方法。
We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the cotangent microbundle category, and they admit a total symbol calculus in terms of symplectic micromorphisms enhanced with half-density germs. This new operator category encompasses the semi-classical pseudo-differential calculus and offers a functorial framework for the semi-classical analysis of the Schrodinger equation. We also comment on applications to classical and quantum mechanics as well as to a functorial and geometrical approach to the quantization of Poisson manifolds.