Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving principal angles
TLDR
In this paper, the authors extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of a Hilbert space with n fixed elements.Abstract:
Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the following way: Every bijective transformation on the set L of all 1-dimensional subspaces of a Hilbert space H which preserves the angle between the elements of L is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n fixed.read more
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Orthogonality preserving transformations on indefinite inner product spaces: Generalization of Uhlhorn's version of Wigner's theorem
Abstract: We present an analogue of Uhlhorn's version of Wigner's theorem on symmetry transformations for the case of indefinite inner product spaces. This significantly generalizes a result of Van den Broek. The proof is based on our main theorem, which describes the form of all bijective transformations on the set of all rank-one idempotents of a Banach space which preserve zero products in both directions.
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TL;DR: In this article, the authors characterize bijective transformations on the set of all $n$-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions.
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Wigner's theorem and its generalizations
TL;DR: Uhlhorn's version of Wigner's theorem is a purely lattice theoretical result, close to the First Fundamental Theorem of projective geometry as discussed by the authors, and it is proved in a way which uses only the definition of an inner product space and elementary results of linear algebra and complex numbers.
References
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Relations Between Two Sets of Variates
TL;DR: The concept of correlation and regression may be applied not only to ordinary one-dimensional variates but also to variates of two or more dimensions as discussed by the authors, where the correlation of the horizontal components is ordinarily discussed, whereas the complex consisting of horizontal and vertical deviations may be even more interesting.
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Essai sur la géométrie à $n$ dimensions
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Journal ArticleDOI
Jordan homomorphisms of rings
Nathan Jacobson,C. E. Rickart +1 more
TL;DR: In this paper, the Jordan homomorphisms of rings that are additive and satisfy the conditions (1) and (2) were studied. But the main aim of this paper is to study mappings J of rings of the additive groups that satisfy (1), i.e., mappings that satisfy the condition that the additive group admits the operator 1/2 in the sense that 2x = a has a unique solution (1/2)a for every a.
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