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001-es BibID:	BIBFORM103581
035-os BibID:	(WOS)000563836200012 (Scopus)85084470398
Első szerző:	Gehér György Pál (matematikus)
Cím:	Symmetries of projective spaces and spheres / György Pál Gehér
Dátum:	2020
ISSN:	1073-7928
Megjegyzések:	Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an alpha is an element of(0, pi/2). The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle alpha (or Fubini-Study distance alpha) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when H = C-2 we do this for every alpha, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for alpha <= pi/4. Our result immediately gives an Uhlhorn-type generalisation of Wigner's theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability cos(2) alpha in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when H = C-2 and alpha = pi/4 where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the alpha = pi/2 case). Usually in the literature, results which are connected to Wigner's theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.
Tárgyszavak:	Természettudományok Matematika- és számítástudományok idegen nyelvű folyóiratközlemény külföldi lapban folyóiratcikk
Megjelenés:	International Mathematics Research Notices. - 2020 : 7 (2020), p. 2205-2240. -
Pályázati támogatás:	Lendület-LP-2012-46/2012 Egyéb NKFIH-K-115383 Egyéb
Internet cím:	Szerző által megadott URL DOI Intézményi repozitóriumban (DEA) tárolt változat
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