HILBERT SPACE OVER COMPLEX HYPERBOLIC NUMBERS AND HYPER-TRIGONOMETRIC INTERFERENCE

This note is devoted to extension of quantum probability calculus to generalizations of complex Hilbert space. Starting with Hilbert space over complex hyperbolic numbers, we derive general hyper-trigonometric interference of probabilities.

On Quantum-Like Probabilistic Structure of Mental Information

We describe methodology of cognitive experiments (based on interference of probabilities for mental observables) which could verify quantum-like structure of mental information, namely, interference of probabilities for incompatible observables. In principle, such experiments can be performed in psychology, cognitive, and social sciences. In fact, the general contextual probability theory predicts not only quantum-like trigonometric (cos θ) interference of probabilities, but also hyperbolic (cosh θ) interference of probabilities (as well as hyper-trigonometric). In principle, statistical data obtained in experiments with cognitive systems can produce hyperbolic (cosh θ) interference of probabilities. We introduce a wave function of (e.g., human) population. In general, we should not reject the possibility that cognitive functioning is neither quantum nor classical. We discuss the structure of state spaces for cognitive systems.