We present a contextualist statistical realistic model for quantum-like representations in physics, cognitive science, and psychology. We apply this model to describe cognitive experiments to check quantum-like structures of mental processes. The crucial role is played by interference of probabilities for mental observables. Recently one such experiment based on recognition of images was performed. This experiment confirmed our prediction on the quantum-like behavior of mind. In our approach “quantumness of mind” has no direct relation to the fact that the brain (as any physical body) is composed of quantum particles. We invented a new terminology “quantum-like (QL) mind.” Cognitive QL-behavior is characterized by a nonzero coefficient of interference λ . This coefficient can be found on the basis of statistical data. There are predicted not only cos ⁡ θ -interference of probabilities, but also hyperbolic cosh ⁡ θ -interference. This interference was never observed for physical systems, but we could not exclude this possibility for cognitive systems. We propose a model of brain functioning as a QL-computer (there is a discussion on the difference between quantum and QL computers).

Quantum-like brain: "Interference of minds".

By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of C-linear space probabilistic calculus to describe non-quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse the superposition principle in this framework.

Linear representations of probabilistic transformations induced by context transitions

We study the problem of correspondence between classical and quantum statistical models. We show that (contrary to a rather common opinion) it is possible to construct a natural pre-quantum classical statistical model. The crucial point is that such a pre-quantum classical statistical model is not the conventional classical statistical mechanics on the phase space R2n, but its infinite-dimensional analogue. Here the phase space Ω = H × H, where H is the (real separable) Hilbert space. The classical → quantum correspondence is based on the Taylor expansion of classical physical variables—maps f:Ω → R. The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and dispersion ≈h. The quantum statistical model is obtained as the limh→0 of the classical one. All quantum states including so-called 'pure states' (wavefunctions) are simply Gaussian fluctuations of the 'vacuum field', ω = 0 Ω, having dispersions of the Planck magnitude.

http://iopscience.iop.org/article/10.1088/0305-4470/38/41/015/pdf

A pre-quantum classical statistical model with infinite-dimensional phase space

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.

On Quantum-Like Probabilistic Structure of Mental Information

We study the role of context, complex of physical conditions, in quantum as well as classical experiments. It is shown that by taking into account contextual dependence of experimental probabilities we can derive the quantum rule for the addition of probabilities of alternatives. Thus we obtain quantum interference without applying the wave or Hilbert space approach. The Hilbert space representation of contextual probabilities is obtained as a consequence of the elementary geometric fact: cos-theorem. By using another fact from elementary algebra we obtain complex-amplitude representation of probabilities. Finally, we found contextual origin of noncommutativity of incompatible observables.

L. E. Ballentine

Papers

Interpretations of Probability and Quantum Theory