# Axioms for the category of Hilbert spaces

@article{Heunen2021AxiomsFT, title={Axioms for the category of Hilbert spaces}, author={Chris Heunen and Andre Kornell}, journal={Proceedings of the National Academy of Sciences of the United States of America}, year={2021}, volume={119} }

Significance Hilbert spaces and their operators are the mathematical foundation of quantum mechanics. The problem of reconstructing this foundation from first principles has been open for nearly a century. What is mathematically special about Hilbert spaces and their operators? This paper gives a categorical axiomatization. Unlike previous partial results of this type, the axioms do not presuppose probabilities, complex amplitudes, or continuity and are not limited to finite dimension.

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## References

SHOWING 1-10 OF 48 REFERENCES

### Orthogonality Spaces Arising from Infinite-Dimensional Complex Hilbert Spaces

- Mathematics
- 2020

The collection of one-dimensional subspaces of an anisotropic Hermitian space is naturally endowed with an orthogonality relation and represents the typical example of what is called an orthogonality…

### Is quantum mechanics exact

- Physics
- 2013

We formulate physically motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to quantum mechanics as the only nontrivial consistent…

### An embedding theorem for Hilbert categories

- Mathematics
- 2008

We axiomatically dene (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally…

### Division Algebras and Quantum Theory

- Mathematics
- 2012

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three…

### Hilbert lattices: New results and unsolved problems

- Mathematics
- 1990

The class of Hilbert lattices that derive from orthomodular spaces containing infinite orthonormal sets (normal Hilbert lattices) is investigated. Relevant open problems are listed. Comments on…

### Completeness of dagger-categories and the complex numbers

- Mathematics
- 2008

The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that…

### Categories for Quantum Theory

- Physics
- 2019

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to…

### Quantum Logic in Dagger Kernel Categories

- MathematicsElectron. Notes Theor. Comput. Sci.
- 2010

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels, which have interesting categorical/logical/order-theoretic properties, in terms of kernel fibrations.

### On characterizing the standard quantum logics

- Mathematics
- 1977

ABsTRAcr. Let e be a complete projective logic. Then e has a natural representation as the lattice of -closed subspaces of a left vector space V over a division ring D, where is a definite 0-bilinear…

### On the Foundations of Quantum Physics

- Physics
- 1976

The interpretation of quantum theory has always been a source of difficulties, especially with regard to the theory of measurement. We do not intend to enter here into the details of the polemic…