International Journal of Theoretical Physics

Orthogonality Spaces Arising from Infinite-Dimensional Complex Hilbert Spaces

Thomas Vetterlein

Published Mar 10, 2020• 17 citations

Abstract

Abstract 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 space: a set endowed with a symmetric, irreflexive binary relation. We investigate in this paper symmetry properties of orthogonality spaces. We show that two conditions concerning the existence of automorphisms of orthogonality spaces are essentially sufficient to characterise the basic model of quantum physics, the countably infinite dimensional complex Hilbert space.

DOI: 10.1007/s10773-019-04230-4

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International Journal of Theoretical Physics

Orthogonality Spaces Arising from Infinite-Dimensional Complex Hilbert Spaces

Thomas Vetterlein

Published Mar 10, 2020• 17 citations

Publisher

Abstract

DOI: 10.1007/s10773-019-04230-4

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Let

H

be an infinite-dimensional complex Hilbert space and let

{\mathcal L}(H)

be the logic formed by all closed subspaces of

H

. For every natural

k

we denote by

{\mathcal G}_{k}(H)

the Grassmannian consisting of

k

-dimensional subspaces. An orthogonal apartment of

{\mathcal G}_{k}(H)

is the set consisting of all

k

-dimensional subspaces spanned by subsets of a certain orthogonal base of

H

. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of

{\mathcal G}_{k}(H)

. We show that every bijective transformation

f

{\mathcal G}_{k}(H)

such that

f

and

f^{-1}

send orthogonal apartments to orthogonal apartments (in other words,

f

preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of

{\mathcal L}(H)

Orthogonality preserving transformations on the set of

n

-dimensional subspaces of a Hilbert space

We characterize bijective transformations on the set of all

n

-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.