Main Theorems

Peter Šemrl

Abstract We prove an optimal version of Uhlhorn’s generalization of Wigner’s unitary–antiunitary theorem. The main tool in our proof is Gleason’s theorem.

I. Mayer

Wojciech Florek

Simple formulation and a straightforward proof of the Lieb-Mattis theorem (LMT) do not indicate how powerful a tool it is.For more than fifty years, this theorem has been mainly discussed in its 'strong' form and applied mainly to many type of infinite spin lattices.It can be easily proved that in such cases, geometrically frustrated systems have to be excluded.However, it has been recently shown that the so-called 'weak'or general form of the LMT can be exploited considering some small, geometrically frustrated quantum spin systems.Moreover, many systems, which do not satisfy assumptions of the LMT, show identical features, including the level order characteristic for bipartite spin systems.It yields a question about possible generalizations or modifications of this theorem to involve a larger class of problems.To begin, algebraic aspects have to be understood with the invaluable role of the Perron-Frobenius theorem.The latter theorem is investigated, discussed, modified etc. in immense number of works.Many of them are important in physical applications and ways of reasoning they present can be exploited in different approaches to the LMT.

Edgard Elbaz

Mark S. Swanson

Ian M. Bassett

Liouville's theorem and its optical counterparts are extended into the wave domain, whereby some upper bounds, representing absolute limits on the performance of passive systems, are obtained. The equality of Liouville's theorem is degraded in general to an inequality, although the limiting equality sign is sometimes realizable. The main emphasis is on optics, and the main methods those of quantum optics.