orthogonality space
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Author(s):  
Thomas Vetterlein

Abstract An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the usual orthogonality relation is an orthogonality space. We present simple conditions to characterise the orthogonality spaces that arise in this way from finite-dimensional Hermitian spaces. Moreover, we investigate the consequences of the hypothesis that an orthogonality space allows gradual transitions between any pair of its elements. More precisely, given elements e and f, we require a homomorphism from a divisible subgroup of the circle group to the automorphism group of the orthogonality space to exist such that one of the automorphisms maps e to f, and any of the automorphisms leaves the elements orthogonal to e and f fixed. We show that our hypothesis leads us to positive definite quadratic spaces. By adding a certain simplicity condition, we furthermore find that the field of scalars is Archimedean and hence a subfield of the reals.


2019 ◽  
Vol 16 (05) ◽  
pp. 1950080 ◽  
Author(s):  
Thomas Vetterlein

An orthogonality space is a set endowed with a symmetric, irreflexive binary relation. By means of the usual orthogonality relation, each anisotropic quadratic space gives rise to such a structure. We investigate in this paper the question to which extent this strong abstraction suffices to characterize complex Hilbert spaces, which play a central role in quantum physics. To this end, we consider postulates concerning the nature and existence of symmetries. Together with a further postulate excluding the existence of nontrivial quotients, we establish a representation theorem for finite-dimensional orthomodular spaces over a dense subfield of [Formula: see text].


2013 ◽  
Vol 846-847 ◽  
pp. 1044-1047
Author(s):  
Chao Shao ◽  
Cun Yao Xu ◽  
Xin Shi

A column orthogonal space-time-frequency coding scheme is presented, its induced cost207 transmit environment channel matrix has greater determinant value, which will lead result in amplifying the signal-to noise ratio of system, and improving the performance of the system. Computer simulations confirm the theory of the article.


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