HOLIK Federico Hernan
congresos y reuniones científicas
Why algebras of operators are so important in physics?
Workshop; Workshop: Quantum Contextuality in Quantum Mechanics and Beyond (QCQMB); 2019
Why algebras of operators are so important in physics?In this talk, we propose a new approach to the problem of explaining why algebras of operatorsplay a key role in physics and other probasbilistic theories as well. Instead of trying to explain thesingularities of Hilbert-space quantum mechanics, we assume that quantum theory is one among ahuge family of models of contextual probabilistic theories. This move is motivated by the fact that,in the last decades, many models of non-Kolmogorovian probability have been considered outsidethe quantum domain, and found applications in biology, psychology, social sciences, economics andlinguistics. While many of these approaches use Hilbert spaces, it has became evident that thereis no reason to expect that the Hilbert space description will be the optimal one for every modelof interest. Here, it is assumed that any system (understood as a domain of phenomena in whichwe are interested) can be represented by a propositional structure that acquires meaning in anempirical/operational way. We will assume that the propositional structures can be represented bybounded orthomodular lattices. The generality of this assumption is justified by the operationalapproach to physical theories, but also by the fact that, most important theories of physics fall intothis setting: states of classical, quantum, quantum relativistic and quantum statistical theories canbe described as measures over bounded orthomodular lattices. Our contribution to the problem ofexplaining why algebras of operators are so important is as follows: assuming that the propositionalstructure associated to our system of interest can be represented by an orthomodular lattice, we showthat an algebra of operators acting on a suitably defined space can be constructed in a natural way.It turns out that, if the lattices are Boolean (distributive), the algebra of operators will be Abelian,while it will fail to be Abelian for non-distributive lattices. Thus, one of the most important featuresof quantum theory, namely, that observables must be represented by non-commutative algebras ofoperators, is not as strange as its seems: it is a common feature of many contextual probabilisticmodels. When models are non-contextual, the algebras are still there, but since they are Abelian,they find a natural representation as functions over a suitably defined phase space.