Gradual transitivity in orthogonality spaces of finite rank

Vetterlein, Thomas

doi:10.1007/s00010-020-00756-9

Cited by 7 publications

(2 citation statements)

References 22 publications

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“…We begin by showing those conditions which we know to ensure the representability of (X, ⊥) by means of a Hermitian space if (X, ⊥) has a finite rank [Vet2].…”

Section: A Further Pair Of Linearly Independent Vectors Then There Is...mentioning

confidence: 99%

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Transitivity and homogeneity of orthosets and the real Hilbert spaces

Vetterlein¹

2021

Preprint

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An orthoset (also called an orthogonality space) is a set X equipped with a symmetric and irreflexive binary relation ⊥, called the orthogonality relation. In quantum physics, orthosets play a central role. In fact, a Hilbert space gives rise to an orthoset in a canonical way and can be reconstructed from it.A complex Hilbert space can be seen as a real Hilbert space endowed with a complex structure. This fact motivates us to explore characteristic features of real Hilbert spaces by means of the abelian groups of rotations of a plane. Accordingly, we consider orthosets together with the groups of automorphisms that keep the orthogonal complement of a given pair of distinct elements fixed. We establish that, under a transitivity and a homogeneity assumption, an orthoset arises from a projective (anisotropic) Hermitian space.To find conditions under which the latter's scalar division ring is R is difficult in the present framework. However, restricting considerations to divisible automorphisms, we can narrow down the possibilities to positive definite quadratic spaces over an ordered field. The further requirement that the action of these automorphisms is quasiprimitive implies that the scalar field is a subfield of R.

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“…We begin by showing those conditions which we know to ensure the representability of (X, ⊥) by means of a Hermitian space if (X, ⊥) has a finite rank [Vet2].…”

Section: A Further Pair Of Linearly Independent Vectors Then There Is...mentioning

confidence: 99%

“…Proof. To show that {e} ⊥⊥ = {e} for any e ∈ X, we may argue as in case of in [Vet2,Lemma 3.2]; the proof applies also without the assumption of a finite rank.…”

Section: A Further Pair Of Linearly Independent Vectors Then There Is...mentioning

confidence: 99%

Transitivity and homogeneity of orthosets and the real Hilbert spaces

Vetterlein¹

2021

Preprint

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Transitivity and homogeneity of orthosets and inner-product spaces over subfields of $${{\mathbb {R}}}$$

Vetterlein

2022

Geom Dedicata

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An orthoset (also called an orthogonality space) is a set X equipped with a symmetric and irreflexive binary relation $$\perp $$ ⊥ , called the orthogonality relation. In quantum physics, orthosets play an elementary role. In particular, a Hilbert space gives rise to an orthoset in a canonical way and can be reconstructed from it. We investigate in this paper the question to which extent real Hilbert spaces can be characterised as orthosets possessing suitable types of symmetries. We establish that orthosets fulfilling a transitivity as well as a certain homogeneity property arise from (anisotropic) Hermitian spaces. Moreover, restricting considerations to divisible automorphisms, we narrow down the possibilities to positive definite quadratic spaces over an ordered field. We eventually show that, under the additional requirement that the action of these automorphisms is quasiprimitive, the scalar field embeds into $${{\mathbb {R}}}$$ R .

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A Characterisation of Orthomodular Spaces by Sasaki Maps

Lindenhovius

Vetterlein

2023

Int J Theor Phys

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Given a Hilbert space H, the set P(H) of one-dimensional subspaces of H becomes an orthoset when equipped with the orthogonality relation ⊥ induced by the inner product on H. Here, an orthoset is a pair (X,⊥) of a set X and a symmetric, irreflexive binary relation ⊥ on X. In this contribution, we investigate what conditions on an orthoset (X,⊥) are sufficient to conclude that the orthoset is isomorphic to (P(H),⊥) for some orthomodular space H, where orthomodular spaces are linear spaces that generalize Hilbert spaces. In order to achieve this goal, we introduce Sasaki maps on orthosets, which are strongly related to Sasaki projections on orthomodular lattices. We show that any orthoset (X,⊥) with sufficiently many Sasaki maps is isomorphic to (P(H),⊥) for some orthomodular space, and we give more conditions on (X,⊥) to assure that H is actually a Hilbert space over $\mathbb R$ ℝ , $\mathbb C$ ℂ or $\mathbb H$ ℍ .

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Gradual transitivity in orthogonality spaces of finite rank

Cited by 7 publications

References 22 publications

Transitivity and homogeneity of orthosets and the real Hilbert spaces

Transitivity and homogeneity of orthosets and the real Hilbert spaces

Transitivity and homogeneity of orthosets and inner-product spaces over subfields of $${{\mathbb {R}}}$$

A Characterisation of Orthomodular Spaces by Sasaki Maps

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