NOL Seminar with Åsa Hirvonen
published: 2024-10-14
event date:
2024-10-21
The Nordic Online Logic Seminar
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Date Monday, 21 October, 2024 at 16:00 (UTC+2) on Zoom
Speaker Åsa Hirvonen (University lecturer, University of Helsinki
Title Looking at quantum mechanics with model theoretic glasses
Abstract
The axioms of quantum mechanics present the state of a quantum system as a unit
vector in complex Hilbert space. However, when Dirac [1] presented his bra- and
ket-vectors, he had a more general space in mind. Schwartz [4] later gave a
rigorous account for Dirac’s “vectors” as distributions, but in elementary
physics books one still encounters presentations where ket-vectors are presented
just as elements of a Hilbert space, and treated with methods from
finite-dimensional linear algebra.
During the last years Tapani Hyttinen and I [2,3] have been looking at various models justifying the finite dimensional approaches from such textbooks. Our approaches are based on various embeddings of a Hilbert space into metric ultraproducts of finite-dimensional Hilbert spaces. In this talk I will present the basic ideas, their benefits and limitations. Time permitting, I will also contrast our approach to Boris Zilber’s work on the same questions, that was the original inspiration for us.
[1] P.A.M. Dirac. The principles of Quantum Mechanics, 3rd ed, Clarendon Press,
Oxford, 1947.
[2] Å. Hirvonen, T. Hyttinen, On eigenvectors, approximations and the Feynman
propagator, Ann. Pure Appl. Logic 170 (2019).
[3] Å. Hirvonen, T. Hyttinen, On Ultraproducts, the Spectral Theorem and Rigged
Hilbert Spaces, to appear in J. Symb. Log.
[4] L. Schwartz, Théory de Distributions, Hermann, Paris, 1950.