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This is a 131-page set of lecture notes on the mathematical foundations of second quantization, using spin-boson models (a two-level system coupled to a boson field) as the running example. It covers rigorous self-adjointness, exactly solvable Jaynes-Cummings and pure dephasing models, and decoherence in multimode systems. It is pedagogical mathematical physics rather than a technology or product announcement, but the qubit channels and decoherence results are relevant background for quantum information work.
Mathematical foundations of second quantization: an introduction through spin-boson models
arXiv:2610.41657v1 Announce Type: cross 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-Universit\"at Erlangen-N\"urnberg 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.
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.
131 pages. Lecture notes with exercises
arXiv:2610.41657 [math-ph]
(or arXiv:2610.41657v1 [math-ph] for this version)
https://doi.org/10.48550/arXiv.2610.41657 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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