The concepts of complementarity and entanglement are considered with respect to their significance in and beyond physics. A formally generalized, weak version of quantum theory, more general than ordinary quantum theory of physical systems, is outlined and tentatively applied to two examples.

A fundamental link between system theory and statistical mechanics has been found to be established by the Kolmogorov entropy K. By this quantity the temporal evolution of dynamical systems can be classified into regular, chaotic, and stochastic processes. Since K represents a measure for the internal information creation rate of dynamical systems, it provides an approach to irreversibility. The formal relationship to statistical mechanics is derived by means of an operator formalism originally introduced by Prigogine. For a Liouville operator L (...) and an information operator $\tilde M$ acting on a distribution in phase space, it is shown that i[L, $\tilde M$ ]≡KI (I=identity operator). As a first consequence of this equivalence, a relation is obtained between the chaotic correlation time of a system and Prigogine's concept of a “finite duration of presence.” Finally, the existence of chaos in quantum systems is discussed with respect to the existence of a quantum mechanical time operator. (shrink)

Quantum theory has provoked intense discussions about its interpretation since its pioneer days. One of the few scientists who have been continuously engaged in this development from both physical and philosophical perspectives is Carl Friedrich von Weizsaecker. The questions he posed were and are inspiring for many, including the authors of this contribution. Weizsaecker developed Bohr's view of quantum theory as a theory of knowledge. We show that such an epistemic perspective can be consistently complemented by Einstein's ontically oriented position.

We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both; more specifically, that the set of macrostates forms the unique maximal partition of phase space which 1) is consistent with our observations (a subjective fact about our ability to observe the system) and 2) obeys a Markov process (an objective fact about the system's dynamics). We review the ideas of computational mechanics, (...) an information-theoretic method for finding optimal causal models of stochastic processes, and argue that macrostates coincide with the ``causal states'' of computational mechanics. Defining a set of macrostates thus consists of an inductive process where we start with a given set of observables, and then refine our partition of phase space until we reach a set of states which predict their own future, i.e. which are Markovian. Macrostates arrived at in this way are provably optimal statistical predictors of the future values of our observables. (shrink)

The context of an operational description is given by the distinction between what we consider as relevant and what as irrelevant for a particular experiment or observation. A rigorous description of a context in terms of a mathematically formulated context-independent fundamental theory is possible by the restriction of the domain of the basic theory and the introduction of a new coarser topology. Such a new topology is never given by first principles, but depends in a crucial way on the abstractions (...) made by the cognitive apparatus or the pattern recognition devices used by the experimentalist. A consistent mathematical formulation of a higher-level theory requires the closure of the restriction of the basic theory in the new contextual topology. The validity domain of the so constructed higher-level theory intersects nontrivially with the validity domain of the basic theory: neither domain is contained in the other. Therefore, higher-level theories cannot be totally ordered and theory reduction is not transitive. The emergence of qualitatively new properties is a necessary consequence of such a formulation of theory reduction (which does not correspond to the traditional one). Emergent properties are not manifest on the level of the basic theory, but they can be derived rigorously by imposing new, contextually selected topologies upon context-independent first principles. Most intertheoretical relations are mathematically describable as singular asymptotic expansions which do not converge in the topology of the primary theory, or by choosing one of the infinitely many possible, physically inequivalent representations of the primary theory (Gelfand–Naimark–Segal-construction of algebraic quantum mechanics). As examples we discuss the emergence of shadows, inductors, capacitors and resistors from Maxwell’s electrodynamics, the emergence of order parameters in statistical mechanics, the emergence of mass as a classical observable in Galilei-relativistic theories, the emergence of the shape of molecules in quantum mechanics, the emergence of temperature and other classical observables in algebraic quantum mechanics. (shrink)

We present a toy theory that is based on a simple principle: the number of questions about the physical state of a system that are answered must always be equal to the number that are unanswered in a state of maximal knowledge. Many quantum phenomena are found to have analogues within this toy theory. These include the noncommutativity of measurements, interference, the multiplicity of convex decompositions of a mixed state, the impossibility of discriminating nonorthogonal states, the impossibility of a universal (...) state inverter, the distinction between bipartite and tripartite entanglement, the monogamy of pure entanglement, no cloning, no broadcasting, remote steering, teleportation, entanglement swapping, dense coding, mutually unbiased bases, and many others. The diversity and quality of these analogies is taken as evidence for the view that quantum states are states of incomplete knowledge rather than states of reality. A consideration of the phenomena that the toy theory fails to reproduce, notably, violations of Bell inequalities and the existence of a Kochen-Specker theorem, provides clues for how to proceed with this research program. (shrink)

Classical cognitive science assumes that intelligently behaving systems must be symbol processors that are implemented in physical systems such as brains or digital computers. By contrast, connectionists suppose that symbol manipulating systems could be approximations of neural networks dynamics. Both classicists and connectionists argue that symbolic computation and subsymbolic dynamics are incompatible, though on different grounds. While classicists say that connectionist architectures and symbol processors are either incompatible or the former are mere implementations of the latter, connectionists reply that neural (...) networks might be incompatible with symbol processors because the latter cannot be implementations of the former. In this contribution, the notions of 'incompatibility' and 'implementation' will be criticized to show that they must be revised in the context of the dynamical system approach to cognitive science. Examples for implementations of symbol processors that are incompatible with respect to contextual topologies will be discussed. (shrink)