Two-valued states on Baer ∗^*-semigroups

In this paper we develop an algebraic framework that allows us to extend families of two-valued states on orthomodular lattices to Baer $^*$-semigroups. We apply this general approach to study the full class of two-valued states and the subclass of Jauch-Piron two-valued states on Baer $^*$-semigroups.Comment: Reports on mathematical physics (accepted 2013

Decompositions of Measures on Pseudo Effect Algebras

Recently in \cite{Dvu3} it was shown that if a pseudo effect algebra satisfies a kind of the Riesz Decomposition Property ((RDP) for short), then its state space is either empty or a nonempty simplex. This will allow us to prove a Yosida-Hewitt type and a Lebesgue type decomposition for measures on pseudo effect algebra with (RDP). The simplex structure of the state space will entail not only the existence of such a decomposition but also its uniqueness

Rotations in the Space of Split Octonions

The geometrical application of split octonions is considered. The modified Fano graphic, which represents products of the basis units of split octonionic, having David's Star shape, is presented. It is shown that active and passive transformations of coordinates in octonionic '8-space' are not equivalent. The group of passive transformations that leave invariant the norm of split octonions is SO(4,4), while active rotations is done by the direct product of O(3,4)-boosts and real non-compact form of the exceptional group $G_2$. In classical limit these transformations reduce to the standard Lorentz group.Comment: 10 pages, 1 figur

A universe of processes and some of its guises

Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can then be converted into mathematical structure. With very little structural effort (i.e. in very abstract terms) and in a very short time span the categorical quantum mechanics (CQM) research program has reproduced a surprisingly large fragment of quantum theory. It also provides new insights both in quantum foundations and in quantum information, and has even resulted in automated reasoning software called `quantomatic' which exploits the deductive power of CQM. In this paper we complement the available material by not requiring prior knowledge of category theory, and by pointing at connections to previous and current developments in the foundations of physics. This research program is also in close synergy with developments elsewhere, for example in representation theory, quantum algebra, knot theory, topological quantum field theory and several other areas.Comment: Invited chapter in: "Deep Beauty: Understanding the Quantum World through Mathematical Innovation", H. Halvorson, ed., Cambridge University Press, forthcoming. (as usual, many pictures