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Hilbert Space Modelling with Applications in Classical Optics, Human Cognition, and Game Theory

Partha Ghose, Sudip Patra

Published Jan 1, 2023• 4 citations
Publisher
DOI
10.1007/978-3-031-38833-0_3

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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 existing computational cognitive models in cyber security risk management have major limitations in the analysis of cyber security behaviors. To address this issue, we introduce Hilbert space and quantum cognition to cyber security risk management. We compare some key axioms and definitions of classical cognition and quantum cognition. We provide examples on how some unique principles of Hilbert space and quantum cognition such as compatibility can help with the event representation, and capturing cognitive biases that are not possible with classical probability. We shed light on how the mathematical formalism of quantum probability can model the observed deviations from the classicality in human reasoning. We also shed light on how quantum cognition can contribute to the science and practice of cyber security. We argue that Hilbert space and quantum cognition can reconcile violations of classical probability theory in cyber security risk management with a formal theory, and examine whether it is possible to express formally some of the key heuristics in cyber security behaviors.