Model Theory in Quantum Foundations

Vikram Athalye, Emmanuel Haven

One of the main purposes of this chapter is to explain, albeit in an abstract manner, how quantum physics–like models of the economics-finance contexts would differ from quantum math-like (or simply, quantum-like) models. For this, the chapter begins by considering, what may be called, the “physical” foundations of quantum theory. These include the foundations pertaining to the theoretical, experimental, and interpretational aspects of quantum theory. With reference to the physical foundations, the chapter elaborates on certain expectations from agent-centric economics-finance models to qualify as “quantum physics–analogous”. Then, by briefly reviewing some of the prominent theories of analogical arguments and reasoning from the philosophy of science (for instance, Aristotle’s theory, Hesse’s theory, Gentner’s structure-mapping theory and Bartha’s articulation model), the chapter ends by proposing a strategy for the systematic construction of quantum physics–analogous models of economics and finance.

Peter Sidajaya

Diederik Aerts, Massimiliano Sassoli de Bianchi, Sandro Sozzo

The scientific community is becoming more and more interested in the research that applies the mathematical formalism of quantum theory to model human decision-making. In this paper, we provide the theoretical foundations of the quantum approach to cognition that we developed in Brussels. These foundations rest on the results of two decade studies on the axiomatic and operational-realistic approaches to the foundations of quantum physics. The deep analogies between the foundations of physics and cognition lead us to investigate the validity of quantum theory as a general and unitary framework for cognitive processes, and the empirical success of the Hilbert space models derived by such investigation provides a strong theoretical confirmation of this validity. However, two situations in the cognitive realm, `question order effects' and `response replicability', indicate that even the Hilbert space framework could be insufficient to reproduce the collected data. This does not mean that the mentioned operational-realistic approach would be incorrect, but simply that a larger class of measurements would be in force in human cognition, so that an extended quantum formalism may be needed to deal with all of them. As we will explain, the recently derived `extended Bloch representation' of quantum theory (and the associated `general tension-reduction' model) precisely provides such extended formalism, while remaining within the same unitary interpretative framework.

Lay Nam Chang, Z. LEWIS, D. MINIC +1 more

In this paper, we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.

D. I. Blokhint︠s︡ev