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.
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.
The theory of infinite-dimensional operators provides a powerful mathematical foundation for modern physics and cognitive science. This paper starts by exploring the differences between linear operators in finite-dimensional and infinite-dimensional spaces, and then discusses their applications in quantum field theory, quantum probabilistic decision theory, quantum cognitive science, and quantum game theory. Through concepts such as Hilbert spaces, projection operators, quantum interference, and entanglement, this paper demonstrates how the quantum framework can transcend classical models and more accurately describe uncertainty, context dependence, and policy interactions in complex systems. Finally, the latest real-world developments in this research are discussed.
The increasing complexity of modern decision-making environments has challenged the effectiveness of classical mathematical models. Traditional probabilistic frameworks often struggle to represent uncertainty, ambiguity, contextual dependence, and nonlinear interactions among decision variables. Quantum-inspired mathematical models have emerged as a promising interdisciplinary approach that leverages mathematical principles derived from quantum theory without requiring quantum computing hardware. These models utilize concepts such as superposition, interference, entanglement, and contextuality to represent complex decision processes more accurately. This paper explores the theoretical foundations, mathematical structures, and applications of quantum-inspired decision models in complex systems. The study examines how quantum probability theory provides a more flexible representation of human cognition, organizational decision-making, financial forecasting, and artificial intelligence systems. Furthermore, the paper discusses mathematical formulations, implementation strategies, advantages, challenges, and future research directions. The findings suggest that quantum-inspired models offer significant potential for addressing uncertainty and dynamic interactions in complex decision environments, thereby contributing to the advancement of intelligent decision support systems.
The current chapter explores philosophical and mathematically inspired concepts of quantum cognition paradigm. Quantum cognition is an interdisciplinary domain of overlaps between quantum foundations or quantum information theory concepts and cognitive decision-making processes. The chapter explores frontier works which demonstrate that various quantum-like concepts can describe, explain and even predict human cognitive phenomena better than the standard cognitive science models. Again, no reductionism is implied, or we necessarily do not hold that mind is quantum physical for sure.
Abstract In the current context, where the productivity paradox coexists with growing concerns about inequality, quantum economics and quantum game theory are emerging as novel frameworks to address inherent uncertainty and complex strategic interactions. Quantum game theory applies the principles of quantum mechanics to model strategic interactions, providing new insights into decision-making, optimization and conflict resolution. In this article, we conduct a systematic review following the PRISMA methodology and a bibliometric analysis of the literature on quantum game theory to assess its development, methodologies and applications. We examine the growing interest in quantum technologies such as quantum computing, quantum communication and quantum simulation, which have had a significant impact on both theoretical models and practical economic and social systems. By mapping trends in the quantum game theory research landscape, this article aims to identify key advances, highlight important milestones and propose future directions for research in this emerging field. This comprehensive mapping not only elucidates the dynamic evolution and interdisciplinary convergence of quantum and evolutionary game theory, but also establishes a foundation for future research, fostering innovative approaches to complex decision-making in increasingly uncertain and interconnected environments.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.