Davide Lonigro
Researcher in quantum mechanics
Davide Lonigro
Researcher in quantum mechanics
Mathematical Foundations of Second Quantization
Winter Semester 2025/26, 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, Raum 308, old ECAP building.
Exercise classes: Fridays 10:15–11:45, Raum 308, old ECAP building.
Reference: I will provide slides. 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 (26.11.25).
Exercises with solutions: Here is the latest version (24.11.25).
Diary of the lectures
22.10.25: 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; Resolvent and spectrum.
29.10.25: The Schwartz space; 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; harmonic oscillator with linear potential.
05.11.25: 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.
12.11.25: 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.
26.11.25: Hilbert space tensor product; Separability of the Hilbert space tensor product of separable spaces; 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.
Diary of the exercise classes
31.10.25: All separable Hilbert spaces are equivalent to l²; Sufficient criterion for essential self-adjointness; Adjoint of sum vs. sum of adjoints.
07.11.25: Eigenfunctions of the harmonic oscillator; Every operator is infinitesimally bounded with respect to its square; Equivalent characterizations of relative boundedness.
14.11.25: Quantum harmonic oscillator with linear potential; Resolvent estimates via the spectral theorem.
19.11.25: Direct sum of separable Hilbert spaces is separable; Direct sum of essentially self-adjoint operators; Direct sum of uniformly bounded operators is bounded.
21.11.25: Direct sum of uniformly bounded operators is bounded; Algebraic tensor product is complete if one space is finite-dimensional.
28.11.25: Tensor product of separable spaces is separable; Characterization of unitary maps between Hilbert spaces; Double orthogonal complement equals closure.