Seminar Analysis: Quantization

Content

Schedule 
Thursday 13:15-15:00.

Organized by Michal Wrochna (Utrecht University, Mathematical Institute), m.wrochna@uu.nl.

Prerequisites
Bachelor-level knowledge of real and complex analysis, ordinary differential equations, as well as functional analysis, more specifically, knowledge of:

• Hilbert and Banach spaces, dual spaces, and convergence in these spaces
• Linear operators: definition, basic properties and compact operators
• Fourier transform, elements of distribution theory

A prior course on partial differential equations can be helpful but is not required, as long as the above prerequisites are met. For the first two items, see e.g. Bryan P. Rynne, Martin A. Youngson: "Linear Functional Analysis", ISBN 978-1-84800-004-9.  A concise catch up reference for the last item are Chapters 5, 7 and 8 of András Vasy: "Partial Differential Equations: An Accessible Route through Theory and Applications", ISBN 978-1-4704-6983-2.

Description and aim of the course
Quantization is the cornerstone of two great physical theories: Quantum Mechanics and Quantum Field Theory. It is also a robust mathematical tool widely used in spectral theory, partial differential equations and differential geometry which combines functional analysis and elements of symplectic geometry, giving rise to the subfields of semiclassical and microlocal analysis. The aim of this seminar is to provide an introduction to quantization and to semiclassical analysis in a variety of settings and present an overview of its applications. It covers topics including quantization on R^n, stationary phase, symbolic calculus and pseudo-differential operators, Schrödinger operators, quantum and classical dynamics, second quantization and quantum fields.

The first part provides a systematic introduction to the fundamentals of quantization, namely:

• Basic theory
• overview of classical vs. quantum mechanics
• symplectic structure on R^2n, Hamiltonian vector fields
• Fourier transform and stationary phase
• oscillatory integrals
• Quantization and pseudo-differential operators
• semiclassical calculus, quantization formulas
• symbol classes, composition, asymptotic expansions
• boundedness on L^2, compactness
• inverses, Gårding inequalities
• pseudo-differential calculus on manifolds
• Second quantization
• bosonic and fermionic Fock spaces
• bosonic and fermionic creation and annihilation operators, field operators

The second part focuses on projects on a choice of topics including for instance:

• resonances and scattering theory
• Weyl laws
• geometric quantization
• C^*-algebraic quantization
• hyperbolic partial differential equations
• microlocal analysis and wavefront sets
• Hamiltonians in non-relativistic QED
• quantum ergodicity
• quantum tunneling 
• microlocal analysis of Anosov flows (dynamical systems)

Main references 
The main reference is the publicly available manuscript

• S. Nonnenmacher: "An Introduction to Semiclassical Analysis"

They will be supplemented by additional literature for student projects. Useful references include in particular:

• M. Zworski: "An Introduction to Semiclassical Analysis"
• M. Reed, B. Simon: "Methods of Modern Mathematical Physics vol .2" ISBN 978-0125850025,
• V. Guillemin, S. Sternberg: "Semi-classical analysis" ISBN 978-1571462763
• M. Taylor: "Partial Differential Equations II: Qualitative Studies of Linear Equations" ISBN 978-3031336997 

Evaluation
50% written work, 50% presentation

In the first half of the semester, participants attend the introductory part of the seminar and present examples and exercises. In the second half of the semester, participants are expected to give two seminar talks (i.e., two presentations, 2 x 45 minutes each), possibly more or fewer depending on the number of participants, based on a project. They will study the material beforehand, hold a blackboard presentation about it, and make a handout. It is obligatory to attend all talks in this seminar (except in cases of force majeure). The final grade for the seminar is based on the average grade for preparation of (50%), and on your seminar presentations (50%).

Interactions with other courses
The course combines well with subjects in mathematics such as Functional Analysis, Symplectic/Poisson Geometry, Partial Differential Equations, Differential Geometry, Dynamical Systems, and with topics in physics including Quantum Mechanics and Quantum Field Theory.

Learning goals
After completing the course, the student will be able to:

• convert material from part of a graduate-level textbook or scientific paper into a coherent and comprehensible presentation for fellow students and mathematicians in general,
• choose appropriate means of communicating theoretical mathematics to fellow students and mathematicians, in written and oral form,
• formulate and correct exercises that maintain a balance between relevance, interest, and feasibility,
• explain specific topics from the content list of the seminar to fellow students, and put them into context as far as their relevance to wider mathematics is concerned.