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.