[Paper Review] Classical and quantum mechanics on information spaces with applications to cognitive, psychological, social and anomalous phenomena

Andrei Khrennikov|ArXiv.org|Mar 4, 2000

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TL;DR

This paper proposes a p-adic information space framework to model classical and quantum dynamics of information states, applying it to cognitive, psychological, and social systems. By using p-adic numbers to represent information with non-Archimedean metrics, it enables a natural description of associative thinking and quantum-like behavior in cognition, with key results showing that p-adic Hilbert spaces can model statistical ensembles of cognitive systems and measurement processes via quantum formalism.

ABSTRACT

We use the system of p-adic numbers for the description of information processes. Basic objects of our models are so called transformers of information, basic processes are information processes, the statistics are information statistics (thus we present a model of information reality). The classical and quantum mechanical formalisms on information p-adic spaces are developed. It seems that classical and quantum mechanical models on p-adic information spaces can be applied for the investigation of flows of information in cognitive and social systems, since a p-adic metric gives quite natural description of the ability to form associations.

Motivation & Objective

• To develop classical and quantum mechanical formalisms on information spaces using p-adic numbers.
• To model cognitive, psychological, and social processes as flows of information rather than matter.
• To provide a mathematical framework for anomalous phenomena and associative thinking through non-Archimedean metrics.
• To extend physical reality models beyond real-number-based spaces by introducing information-based p-adic structures.
• To apply quantum formalism to cognitive systems, treating students and professors as quantum-like information transformers.

Proposed method

• Uses p-adic numbers as a coding system for information, with each information vector represented as a p-adic integer.
• Introduces the 'association structure' (AS) where vectors with matching initial digits are considered close, enabling associative reasoning.
• Applies classical and quantum mechanics on p-adic information spaces, using p-adic Hilbert spaces for state representation.
• Models cognitive systems as statistical ensembles of students, with quantum states expressed as superpositions of book-reading states: $\phi = \sum c_j \phi_j$.
• Defines measurement as interaction between student and professor brains, modeled as $I$-transformers, with outcomes governed by probability amplitudes $d_k$.
• Uses diagonal operators $A$ on p-adic Hilbert space with spectrum corresponding to possible answers $a_k$, enabling quantum measurement description.

Experimental results

Research questions

• RQ1 Can p-adic information spaces provide a more natural model for associative thinking than real-number-based models?
• RQ2 How can classical and quantum mechanics be formalized on information spaces using p-adic structures?
• RQ3 To what extent can cognitive and social processes be modeled as information dynamics rather than material dynamics?
• RQ4 Can quantum-like formalism in p-adic Hilbert spaces describe statistical ensembles of cognitive systems and measurement outcomes?
• RQ5 How does the measurement process in cognitive systems differ from classical measurement, and can it be modeled via potentia and state reduction?

Key findings

• The p-adic metric naturally encodes associative similarity: two information vectors are close if they share many leading digits, enabling a mathematical foundation for cognitive associations.
• Information processes in cognitive systems can be modeled as quantum states in p-adic Hilbert spaces, with superpositions representing potential responses.
• The probability of a cognitive response $a_k$ is given by $u_k = d_k \bar{d}_k$, where $d_k$ are complex p-adic amplitudes, not classical frequencies.
• Measurement outcomes are not predetermined by prior states; observing a student’s answer can collapse the state, reflecting quantum-like contextuality.
• The model explains why additional measurements (e.g., checking if a student read a book) can alter the outcome, indicating non-objective properties in cognitive systems.
• The framework suggests that Bohm’s pilot wave theory can be improved by treating the $\psi$-field as an active information field in p-adic space, not just a physical field.