Davide Lonigro at Friedrich-Alexander-Universität Erlangen-Nürnberg has formalized the mathematical foundations of second quantization in newly released lecture notes. These notes document a one-semester course first delivered in the Winter Semester 2025/26, uniquely designed for students of both physics and mathematics. The material rigorously establishes the self-adjointness of spin-boson models, and explicitly proves an error bound for the rotating-wave approximation within the quantum Rabi model. As Lonigro explains, the course aims to bridge a gap between mathematical formalism and the physical models that use it, addressing questions of operator domains and the validity of approximations.

Erlangen-Nürnberg Course Introduces Second Quantization Framework

The Winter Semester 2025/26 at Friedrich-Alexander-Universität Erlangen-Nürnberg marked the inaugural delivery of a specialized course focused on the mathematical underpinnings of second quantization, attracting students from both physics and mathematics disciplines. This interdisciplinary approach suggests an effort to unify perspectives on a complex area of quantum mechanics, moving beyond traditional siloed instruction. Course materials include fully worked exercises designed to reinforce the theoretical concepts presented during lectures and practice classes.

Central to the course’s rigorous treatment of quantum systems are the Kato-Rellich and KLMN theorems, foundational elements for understanding unbounded operators, a mathematical necessity when describing quantum phenomena. These theorems provided essential background for constructing direct sums and tensor products of Hilbert spaces and unbounded operators, enabling a detailed exploration of single-mode boson fields. The lectures then progressed to constructing ladder and number operators, carefully defining their domains within the mathematical framework.

The course extends beyond theoretical development by explicitly proving the self-adjointness of spin-boson models, which is an important step for ensuring the physical validity of calculations. This focus on mathematical rigor is further demonstrated by the derivation of explicit error bounds for the rotating-wave approximation applied to the quantum Rabi model. As the notes state, the operator that multiplies by n the n-particle component of ψ is generalized and termed the number operator, with the notation N: = dΓ(I).

The denseness of $Γ^{}$ ± _0(h) is proven, taking into account that ± h^{(⊙ n)} is dense in ± h^{(⊗ n)}. These detailed mathematical treatments establish a firm foundation for applying second quantization techniques to physical models and assessing their limitations.

Spin-Boson Models Motivate Mathematical Foundations

Central to the course’s approach was the spin-boson model, a ubiquitous construct in the physics of open quantum systems, used not merely as a subject of study but as a consistent test case for the developed mathematical tools. This emphasis on application alongside theory distinguishes the course from more conventional treatments of second quantization. Explicit solutions were derived for these models, detailing their spectra and the qubit channels they induce, demonstrating a commitment to concrete results rather than purely abstract development.

The course also investigated multimode counterparts, revealing how they exhibit irreversible decoherence and decay, a phenomenon central to understanding the limitations of quantum systems. Detailed analysis of unbounded operators was a key element of the mathematical treatment, building upon foundational theorems like the Kato-Rellich and KLMN theorems.

Kato-Rellich and KLMN Theorems for Unbounded Operators

The lectures detailed in these notes establish a rigorous mathematical basis for understanding quantum systems, specifically through the application of the Kato-Rellich and KLMN theorems to unbounded operators, tools essential for defining the limits of physical accuracy in quantum models. These theorems formed core background material, indicating a mathematically demanding treatment extending beyond introductory quantum mechanics courses. The work addresses a critical need for precise criteria to determine when an operator, representing a physical observable, is truly self-adjoint, a condition ensuring physically meaningful results.

The notes summarize prerequisites including linear algebra, measure theory and spectral theorems, providing a foundation for advanced work with unbounded operators, and referencing existing texts for readers needing a refresher. The KLMN theorem is used to construct spin-boson models with non-normalizable form factors, and stronger ultraviolet divergences are briefly discussed, along with the need for renormalization.

Direct Sums and Tensor Products of Hilbert Spaces

These direct sums, denoted as, use the Cartesian product, defined as the set of all ordered pairs (u₁, u₂) where u₁ belongs to ℋ₁ and u₂ belongs to ℋ₂. The lectures further extend this concept to operators, defining direct sums of operators acting on these combined spaces, which is an important step for analyzing complex quantum systems. The notes summarize the necessary background on unbounded operators, up to the Kato–Rellich and KLMN theorems.

The work demonstrates that the direct sum of such operators results in a bounded operator on the combined Hilbert space ℋ, also bounded by M. A detailed solution to estimating the norm of this combined operator, |Av|² = for a vector v = ∈ ℕ, is provided in Appendix C.1. 4, highlighting the rigorous approach taken. This construction is fundamental to describing composite quantum systems where the state of the combined system is defined by the simultaneous state of its individual components. This careful treatment ensures mathematical consistency when applying these tools to physical models.

Single-Mode Boson Fields and Ladder Operators

The lectures detail a construction abstracting the quantum harmonic oscillator to an arbitrary separable Hilbert space, extending the principles governing single-mode boson fields beyond the electromagnetic field in a cavity. The treatment of single-mode boson fields, beginning with the normal mode expansion of the electromagnetic field, builds upon the foundation laid by earlier sections on unbounded operators and the Kato-Rellich and KLMN theorems. Establishing self-adjointness requires that the form factor f belong to h and that ω−1/2*f* be within 𝔥, but the lectures demonstrate that the first condition can be relaxed.

Construction and Properties of Fock Spaces

The construction of Fock spaces, detailed in lectures delivered at Friedrich-Alexander-Universität Erlangen-Nürnberg, establishes a mathematical framework for describing multi-particle quantum states. These spaces, denoted as Γ(h) where h represents a single-particle Hilbert space, are built through direct sums and tensor products, beginning with a foundational vacuum state ε₀ = (1,0,0,…). This approach allows for the systematic expansion of the Hilbert space to accommodate any number of particles or excitations.

Second Quantization: Bosonic and Fermionic Operators

Throughout, spin-boson models provide both the motivation and the test bench for the formalism. The notes rigorously establish the self-adjointness of spin-boson models, solve the pure dephasing and Jaynes-Cummings models explicitly, deriving their spectra and the qubit channels they induce, and show how their multimode counterparts display irreversible decoherence and decay. The KLMN theorem is used to construct spin-boson models with non-normalizable form factors.

Self-Adjointness of Spin-Boson Models and Explicit Solutions

The lectures define a single-mode spin-boson model, where a two-level system interacts with a single-mode boson field, with the total Hilbert space represented as H =. Establishing self-adjointness requires that the form factor f belong to h and that ω−1/2*f* be within 𝔥.

The lectures detailed a method for constructing spin-boson models even when form factors are not normalizable, extending the range of applicable quantum models and opening avenues for describing more complex interactions. The notes briefly discuss stronger ultraviolet divergences, hinting at the need for renormalization techniques to fully resolve them.

👉 More information
🗞 Mathematical foundations of second quantization: an introduction through spin-boson models
✍️ Davide Lonigro
🧠 ArXiv: https://arxiv.org/abs/2610.41657

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