Quantum Framework for Cross-Domain Applications: From Infinite-Dimensional Operator Theory to Quantum Cognition and Quantum Game Theory

Abstract

Infinite-dimensional operator theory provides a powerful mathematical foundation for modern physics and cognitive science. This paper begins with the distinction between linear operators in finite- and infinite-dimensional spaces, then explores their applications in quantum field theory, quantum probability decision theory, quantum cognition, and quantum game theory. Through concepts such as Hilbert space, projection operators, quantum interference, and entanglement, this paper demonstrates how the quantum framework surpasses classical models in accurately describing uncertainty, contextuality, and strategic interactions in complex systems. Finally, the latest real-world developments at the research frontier are discussed.

1. Introduction: From Operators to Quantum Framework

In finite-dimensional vector spaces, linear operators can be fully represented by matrices. However, when systems enter infinite-dimensional Hilbert spaces, operator theory must address unbounded operators such as differential and multiplication operators. This transition is not merely mathematical but is key to understanding quantum field theory and quantum cognitive phenomena.

Quantum field theory posits that the fundamental constituents of the universe are quantum fields, rather than particles or waves. This perspective further inspires quantum cognition and quantum game theory, applying quantum probability, superposition, and interference to human decision-making, semantic processing, and strategic interactions.

2. Foundations of Infinite-Dimensional Operator Theory

Infinite-dimensional operator theory studies linear operators defined on infinite-dimensional Hilbert spaces H (e.g., L2(R3)). Compared to finite-dimensional matrices, these operators are divided into bounded and unbounded operators (such as the position operator x^ and momentum operator p^=−iℏddx).

Figure 1: Comparison between Finite-Dimensional and Infinite-Dimensional Operators


Key concepts include self-adjoint operators and spectral theory. The canonical commutation relation [x^,p^]=iℏ lays the foundation for quantization.

3. Applications in Quantum Field Theory

Quantum field theory extends infinite-dimensional operators to their fullest extent. Quantum fields ϕ^(x) are operator-valued distributions satisfying canonical commutation relations and describing particle excitations via creation and annihilation operators.

Figure 2: Conceptual Illustration of Quantum Field Theory


4. Quantum Probability Decision Theory

Quantum Probability Decision Theory (QPDT) represents the decision-maker’s mental state as a vector ∣ψ⟩ in Hilbert space. Probabilities follow the Born rule and include interference terms.

Figure 3: Interference Effect in Quantum Decision Theory


This framework effectively explains the Allais paradox, Ellsberg paradox, and order effects.

5. Quantum Cognition and Quantum Semantic Vector Space Models

Quantum cognition applies the quantum framework to conceptual combination, memory, and perception.

The Guppy Effect (Pet-Fish Problem) serves as a classic example. The quantum model explains the non-compositional increase in typicality through constructive interference.

Figure 4: Quantum Interference Illustration of the Guppy Effect


The Quantum Semantic Vector Space Model represents word meanings as quantum states in Hilbert space, using quantum inner products and contextual projections to handle semantic ambiguity and non-compositionality, showing advantages in information retrieval and natural language processing.

6. Quantum Game Theory and Equilibrium Strategies

In quantum game theory, player strategies are unitary operators U^i, with initial states possibly entangled. Quantum Nash equilibrium is defined as a stable strategy profile where no player can unilaterally improve their payoff.

In the Quantum Prisoner’s Dilemma, a new quantum Nash equilibrium enables both players to stably achieve Pareto-optimal cooperation.

Figure 5: Quantum Prisoner’s Dilemma and Equilibrium Strategies


7. Current Real-World Developments at the Research Frontier (2024–2026)

The year 2025 was declared by the United Nations as the “International Year of Quantum Science and Technology,” marking an accelerated phase for the field. Quantum cognition and quantum game theory have moved from pure theory toward quantum-inspired practical applications, particularly in combination with artificial intelligence, behavioral economics, and complex systems optimization.

  • Quantum Cognition: Quantum-inspired neural networks are used to model optical illusions, opinion polarization, and preference uncertainty, and are increasingly integrated into large language models (LLMs) to improve semantic understanding and contextual processing.
  • Quantum Probability Decision Theory: Applied in financial risk assessment, showing 15–20% better risk-adjusted return predictions.
  • Quantum Game Theory: Used in cloud resource allocation, traffic management, and routing in quantum networks.
  • Overall Trend: Hybrid quantum-classical systems are demonstrating early advantages in optimization, simulation, and machine learning tasks, though challenges remain in computational complexity and cross-cultural validation.

8. Conclusion and Outlook

Infinite-dimensional operator theory provides the foundation for quantum field theory and, through quantum probability, interference, and entanglement, enriches decision theory, cognitive science, and game theory. This cross-domain quantum framework not only enhances the descriptive power of human behavior but also opens new directions for AI systems, behavioral economics, and complex social system modeling.

With the impetus of the 2025 International Quantum Year, future research may focus on the deep integration of quantum cognitive models with large language models, the practical deployment of quantum games in multi-agent systems, and further extensions of quantum field concepts to social science field modeling. This cross-domain quantum framework is poised to drive a paradigm shift from classical reductionism toward a holistic, context-dependent understanding.

References (selected)

  • Eisert, J., et al. (1999). Quantum Games and Quantum Strategies.
  • Busemeyer, J. R., & Bruza, P. D. (2012). Quantum Models of Cognition and Decision.
  • Aerts, D., & Gabora, L. (2005). A Theory of Concepts and Their Combinations.

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