Davide Lonigro
Researcher in quantum mechanics
Davide Lonigro
Researcher in quantum mechanics
⚠️ No lecture on Wednesday 18 November. The exercise class on 20 November will take place as usual.
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: The course is based on my lecture notes (with exercises and solutions):
D. Lonigro, Mathematical foundations of second quantization: an introduction through spin–boson models, arXiv:2610.41657 (2026).
Further useful references 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 (30.10.2026).
Diary of the lectures
14.10.2026: Motivation: describing fields in quantum mechanics; Hilbert spaces, orthonormal bases, l² and L² spaces; Unbounded linear operators, extensions and restrictions, symmetric and self-adjoint operators; Examples: multiplication operators, momentum, Laplacian; Essential self-adjointness and criteria of (essential) self-adjointness.
21.10.2026: Resolvent and spectrum; The quantum harmonic oscillator; Spectral theorem and Stone theorem; The axioms of quantum mechanics; Relatively bounded and infinitesimally relatively bounded operators; The Kato–Rellich theorem: statement; Examples: Laplace operators with L² potentials.
28.10.2026: Quantum harmonic oscillator with linear potential; Properties of relative bound for A closed and B closable; Properties of relative bound for A self-adjoint and B symmetric; Proof of the Kato–Rellich theorem; Direct sum of Hilbert spaces; Completeness and separability of the direct sum of countably many Hilbert spaces.
04.11.2026: Denseness of sequences with finitely many nonzero entries; Direct sum of operators; Denseness of the domain of the direct sum; Adjoint of the direct sum; (Un)boundedness of the direct sum of bounded operators; Example: kinetic energy of spinful particle with spin-dependent and spin-independent boundary conditions; Algebraic tensor product of Hilbert spaces; Well-definedness of the scalar product; Uncompleteness of the algebraic tensor product; Hilbert space tensor product; Separability of the Hilbert space tensor product of separable spaces.
11.11.2026: Tensor product of L² spaces; Algebraic tensor product of operators; Well-definedness of the algebraic tensor product of operators; Algebraic tensor product of densely defined operators; Adjoint of the algebraic tensor product of operators; Algebraic tensor product of multiplication operators; Hilbert space tensor product of operators; Essential self-adjointness of tensor product polynomials of self-adjoint operators.
Diary of the exercise classes
23.10.2026: All separable Hilbert spaces are equivalent to l²; Sufficient criterion for essential self-adjointness .
30.10.2026: Adjoint of sum vs. sum of adjoints; Eigenfunctions of the harmonic oscillator; Every operator is infinitesimally bounded with respect to its square.
06.11.2026: Equivalent characterizations of relative boundedness; Supremum norm of Dirichlet functions in H2(0,1); Quantum harmonic oscillator with linear potential.