Mathematical foundations of second quantization: an introduction through spin-boson models

Abstract

These lecture notes originate from a one-semester course on the mathematical foundations of second quantization, first given at the Department of Physics of the Friedrich-Alexander-Universität Erlangen-Nürnberg in the Winter Semester 2025/26 to a mixed audience of physics and mathematics students. Their guiding thread is a class of physical models: spin-boson models, describing a two-level system coupled to a boson field. After summarizing the necessary background on unbounded operators, up to the Kato-Rellich and KLMN theorems, and developing direct sums and tensor products of Hilbert spaces and of unbounded operators, we treat single-mode boson fields, constructing the ladder and number operators and characterizing their domains, before moving to Fock spaces, second quantization, bosonic and fermionic annihilation and creation operators, and the canonical (anti)commutation relations. Throughout, spin-boson models provide both the motivation and the test bench for the formalism: we rigorously establish their self-adjointness, solve the pure dephasing and Jaynes-Cummings models explicitly, deriving their spectra and the qubit channels they induce, show how their multimode counterparts display irreversible decoherence and decay, and, in an appendix, prove the rotating-wave approximation for the quantum Rabi model with an explicit error bound. Finally, we use the KLMN theorem to construct spin-boson models with non-normalizable form factors, and briefly discuss stronger ultraviolet divergences and the need for renormalization. Exercises with full solutions are included.

These lecture notes address how to formulate second quantization and spin–boson dynamics rigorously, including operator domains, self-adjointness, and the justification of approximations. Intended for readers with basic functional analysis and quantum mechanics, they develop direct sums and tensor products of unbounded operators, single-mode ladder operators, Fock spaces, second quantization, and bosonic and fermionic canonical relations. Spin–boson models organize the applications. Kato–Rellich perturbation theory establishes self-adjointness for regular couplings. The single-mode pure-dephasing and Jaynes–Cummings models are solved explicitly, including spectra, eigenvectors, and reduced qubit channels; multimode versions illustrate irreversible decoherence and decay. An appendix proves the rotating-wave approximation for the quantum Rabi model with an explicit error bound. Exercises and full solutions accompany the development. For non-square-integrable form factors satisfying $\omega^{-1/2}f\in\mathfrak h$, annihilation operators remain controlled on the free-energy form domain, while creation terms are interpreted as forms. The KLMN theorem constructs a semibounded self-adjoint Hamiltonian and yields ultraviolet-cutoff convergence in norm resolvent, with error bounded by $2\sqrt{2}\|B\|\|\omega^{-1/2}(f-f_\Lambda)\|$ at a specified negative spectral parameter. The notes explicitly claim neither originality nor completeness. Stronger ultraviolet singularities require renormalization methods only briefly surveyed, and the form-sum construction generally does not identify the operator domain. Multimode exponential decay is ordinarily approximate; exact all-time exponential decay requires idealized singular spectral densities, with a Hamiltonian unbounded below in the spontaneous-emission example.