The Heisenberg group in mathematics and physics (MAGIC076)
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The relations, which define the Heisenberg group or its Lie algebra, are of a fundamental nature and appeared in very different areas. For example, the basic operators of differentiation and multiplication by an independent variable in analysis satisfy to the same commutation relations as observables of momentum and coordinate in quantum mechanics. It is very easy to oversee those common structures. In his paper "On the role of the Heisenberg group in harmonic analysis", Roger Howe said: "An investigator might be able to get what he wanted out of a situation while overlooking the extra structure imposed by the Heisenberg group, structure which might enable him to get much more." In this course we will touch many (but not all!) occurrences of the Heisenberg group, mainly from analysis and quantum mechanics. We will see how to derive important results from the general properties the Heisenberg group and its representations. We will discuss also some cross-fertilisation of different fields through their common ingredient-the Heisenberg group. The lectures will be given in a survey mode with many technicalities to be omitted.
Autumn 2019 (Monday, October 7 to Friday, December 13)
The prerequisites include elementary group theory, linear algebra, analysis and introductory Hilbert spaces. Some knowledge of Lie groups and quantum mechanics would be an advantage, however, this is not a strict requirement.
* Origins of the Heisenberg group and its Lie algebra in analysis and physics; Heisenberg commutation relations; structure of the Heisenberg groups, its automorphisms. * Unitary representations of the Heisenberg group; orbit methods of Kirillov. * Stone-von Neumann theorem; Schroedinger and Fock-Segal-Bargmann representations: their equivalence and intertwining operator (Bargmann integral transform). * Fourier inversion theorem, Schwartz space and Plancherel theorem. * Time-frequencies analysis and wavelets. * Theta-function, wavelet transform and the Gaussian. * Calculus of pseudo-differential operators and quantisation; analysis in the phase space and the Moyal bracket. * Connection between classic and quantum mechanics.
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