Mathematical Foundations of Second Quantization
Winter Semester 2026/27, FAU

Abstract. This course provides a rigorous introduction to the mathematical formalism of second quantization, the framework underlying quantum field theories and many–body physics. We begin with a review of key results on self–adjoint operators, including the Kato–Rellich theorem. The quantum harmonic oscillator is revisited as a guiding example, leading naturally to the construction of Fock spaces. From there, we develop the general theory of bosonic and fermionic Fock spaces, introduce creation and annihilation operators, and explore the second quantization of single–particle operators. We will finally use this knowledge to introduce models of light–matter interaction, such as the spin–boson model.

The course is self–contained and is intended for students with a background in quantum mechanics and an interest in deepening their understanding of the mathematical tools that support quantum many–body and field theories.

Schedule (unless otherwise specified):

• Main lectures: Wednesdays 10:15–11:45, Seminarraum 307, old ECAP building.

• Exercise classes: Fridays 10:15–11:45, Raum 308, old ECAP building.

References: I will provide slides and lecture notes. Useful references for the course are:

• G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators. American Mathematical Society, Graduate Studies in Mathematics, Volume 99, 2009. The book can be freely downloaded here.

• M Reed, B. Simon. Methods of Modern Mathematical Physics II: Fourier analysis, self-adjointness. Elsevier, 1975.

• O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II: Equilibrium States, Models in Statistical Mechanics. Springer, 1997.

• K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Springer Science & Business Media, 2012.

Slides: Here is the latest version (14.10.2026).