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.
- P.A.M. Dirac. The principles of Quantum Mechanics, 3rd ed, Clarendon Press, Oxford, 1947.
- Å. Hirvonen, T. Hyttinen, On eigenvectors, approximations and the Feynman propagator, Ann. Pure Appl. Logic 170 (2019).
- Å. Hirvonen, T. Hyttinen, On Ultraproducts, the Spectral Theorem and Rigged Hilbert Spaces, to appear in J. Symb. Log.
- L. Schwartz, Théory de Distributions, Hermann, Paris, 1950.
