Erlangen-Nürnberg lectures lay mathematical foundations for quantum systems

Mathematical Foundations for Quantum Systems: Highlights from the Erlangen‑Nürnberg Lectures

By Rusty Flint, November 2 2026

The winter‑semester 2025/26 course at Friedrich‑Alexander‑Universität Erlangen‑Nürnberg, taught by Davide Lonigro, has produced a comprehensive set of lecture notes that bridge the gap between abstract operator theory and concrete quantum‑physics models. Aimed at both physics and mathematics students, the material delivers a rigorous treatment of second quantization, unbounded operators, and the spin‑boson family of models—key tools for modern quantum‑technology research.

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1. From Hilbert Spaces to Fock Spaces

The notes begin with a careful construction of direct sums and tensor products of Hilbert spaces, laying the groundwork for multi‑particle quantum systems. By defining the Fock space \(\Gamma(\mathfrak h)\) as a countable direct sum of symmetric tensor powers of a single‑particle space \(\mathfrak h\), the authors obtain a vacuum state \(\varepsilon_0\) and a systematic way to describe any number of bosonic excitations. This abstract formulation reproduces the familiar quantum harmonic oscillator while remaining flexible enough for arbitrary separable Hilbert spaces.

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2. Unbounded Operators and Core Theorems

A central theme is the treatment of unbounded operators, which are unavoidable when representing physical observables such as momentum or field operators. The lectures devote substantial space to two cornerstone results:

• Kato‑Rellich theorem – guarantees self‑adjointness of a sum \(A+B\) when \(A\) is self‑adjoint and \(B\) is relatively bounded with respect to \(A\) with bound less than one.
• KLMN theorem – provides a quadratic‑form approach to constructing self‑adjoint operators from closed, semi‑bounded forms, essential for handling singular interaction terms.

By applying these theorems, the notes establish rigorous criteria for the domains of ladder and number operators, ensuring that subsequent physical models rest on a solid mathematical foundation.

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3. Spin‑Boson Models as a Test Bed

The spin‑boson model— a two‑level system coupled to a bosonic field—serves as the primary example throughout the course. The authors:

• Prove self‑adjointness of the full Hamiltonian, even when the form factor \(f\) is not square‑integrable, by invoking the KLMN theorem.
• Derive explicit spectra for the pure‑dephasing and Jaynes‑Cummings limits, illustrating how the abstract machinery translates into concrete energy levels and qubit channels.
• Analyze multimode extensions, showing that irreversible decoherence emerges naturally from the mathematical structure, a result of direct relevance to open‑quantum‑system engineering.

These achievements demonstrate that the rigorous framework is not merely academic; it yields tangible predictions for realistic quantum devices.

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4. Error Bounds for the Rotating‑Wave Approximation

A standout contribution is the derivation of an explicit error bound for the rotating‑wave approximation (RWA) within the quantum Rabi model. By comparing the full Hamiltonian with its RWA counterpart and employing norm estimates rooted in the Kato‑Rellich theorem, the notes quantify how the approximation deviates as a function of coupling strength and detuning. This result offers practitioners a clear guideline for when the RWA is justified, a question that often arises in circuit‑QED and trapped‑ion experiments.

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5. Practical Takeaways for Researchers and Students

• Unified language: The course equips physicists with the operator‑theoretic vocabulary traditionally reserved for mathematicians, fostering clearer communication across disciplines.
• Worked exercises: Each chapter includes fully solved problems, allowing readers to practice domain calculations, form‑factor manipulations, and spectral analysis.
• Open‑access notes: The lecture material is publicly available on arXiv (arXiv:2610.41657), encouraging adoption in graduate curricula worldwide.

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Closing Thoughts

The Erlangen‑Nürnberg lecture notes represent a rare synthesis of mathematical rigor and physical insight. By proving self‑adjointness for spin‑boson Hamiltonians and providing quantitative error bounds for common approximations, they set a new standard for how quantum‑mechanical models should be constructed and validated. For anyone developing quantum hardware, algorithms, or theory, these notes are an indispensable reference that clarifies the limits of our models and points the way toward more reliable quantum‑technology design.