This Theme Issue is devoted to selected papers from work presented at the conference ‘Quantum probability and the mathematical modelling of decision-making’ held at the Fields Institute for Research in Mathematical Sciences, University of Toronto, in March of 2015. The meeting came about from a joint effort which included Andrei Khrennikov, Emmanuel Haven, Jerome Busemeyer, Emmanuel Pothos, Ehti Dzhafarov and Arkady Plotnitsky to apply for funding to the Fields Institute. The generous support from the Fields Institute made it possible for us to organize a meeting with a quite substantial number of talks.

This event was surely not the first one which hosted papers on using concepts from quantum mechanics to areas outside physics. One of the editors (Andrei Khrennikov) of this Theme Issue has been running the longest conference series on the foundations of quantum mechanics and probability, and numerous sessions at those conferences were devoted to precisely the topic of this very conference.

We may want to think that the current state of affairs in this new area of research can possibly be subdivided along the division that quantum theory itself is using in studying systems with a fixed number of particles (quantum mechanics) and with a varying number of particles represented as excitations of quantized fields (quantum field theory).

Hence, we have work developing along two major research ‘corridors’: first and second quantization.

The applications around ‘first quantization’ (in a nutshell, position and momentum are operators and there is a commutation relation) within decision-making have done very well. Several papers in this Theme Issue are written by protagonists in that very field. We need to stress though that the area of decision-making formalisms which is inspired from this research ‘corridor’ in quantum mechanics, is multi-faceted. This Theme Issue reflects those varied approaches. Just for argument's sake, compare for instance the paper by Aerts *et al.* [1] with the paper by Yukalov & Sornette [2].

In any case, it is reasonable to claim that the use of quantum mechanical concepts implies the use of non-classical logic. Such logic may indeed also lead to non-Boolean information processing by decision-makers. Because classical probability theory (Kolmogorov axiomatics 1933) is based on classical Boolean logic, modifications of logical reasoning imply modifications of the Kolmogorov probability model. One can speak about non-Kolmogorovean probability models and their applications to decision-making. The quantum theory of probability based on the representation of probabilities with the aid of complex probability amplitudes (or generally on the complex Hilbert space formalism) is one of the most well developed and powerful non-Kolmogorovean probability models. Therefore, its application to decision-making is very natural. At the same time, there are no reasons to expect that it will cover completely non-classical probabilistic reasoning. In principle, non-Kolmogorovean models different from quantum probability theory may play an important role in future studies.

In the second research corridor, namely second quantization, there is to date much less work so far. This orientation may be useful to model open systems (with baths of the quantum field type) and it is the latter approach which is quite useful to model decision-making within political systems for instance. The paper by Khrennikova & Haven [3] hints to the use of open systems. In their paper, the quantum field contribution is formally encoded in the coefficients of the quantum master equation.

The issue starts with a selection of three papers which were delivered by the keynote speakers. The first paper by Gustafson [4] discusses many important topics (including the complex notion of context). His paper really shows the distinction we need to keep in mind between formalizing decision-making and what is ‘really happening’ (see especially §4 in the paper). Can our modelling really inform us of what may occur at deeper levels in decision-making? The work by Brandenburger & LaMura [5] studies a crucially important problem which is encountered in practical situations in finance, where for instance direct communication is non-existent but where there exists what they call ‘a shared global environment’ from where coordination becomes very much possible. The paper contains an excellent example of a high-frequency trading situation where a quantum signal is considered relative to a classical signal. One of the many highlights in the paper by Saari [6] is his analysis of the QQ equalities. He develops an orthogonality condition (for no-cyclic effects variables) and it is this condition which informs us about the strength of the QQ relationship. The paper by Narens [7] discusses the links between quantum-like probability theories and other theories which do not have complementation operators. Yukalov & Sornette's [2] work considers necessary conditions from which the quantum approach can be validly used. They argue for a mathematical framework which allows one to consider, on an equal formal footing, quantum measurements and quantum decision-making. The work by Busemeyer & Wang [8] discusses how self versus other judgements are incompatible in a quantum sense. They also test sequential effects where they compare a classical probability approach versus a quantum probability approach. The work by Aerts *et al.* [1] considers the combination of concepts and they show in their paper that there are deviations from five classicality conditions (which assure use of using classical probability). Dzhfarov *et al.* [9] analyse several behavioural and social datasets and argue that relative to those specific datasets there is non-existence of contextuality. White *et al.* [10] provide three experiments which lend support for the use of the quantum approach in the setting of the so-called constructive effect well known in psychology. Khrennikova & Haven [3] consider the quantum master equation. They apply techniques from open systems to model the decision-making process of voters. The work by Plotnitsky [11] provides for a fresh look at (non)-realist mathematical models. Connections are made with the quantum-like models used in social science. The paper makes an essential claim that one actually may well experiment with mathematics in understanding the world. The work by Baaquie [12] provides for a reformulation of basic microeconomic theory via the use of potential functions. The market price dynamics are formulated with an action functional. The paper also provides for evidence of how potentials behave with real financial data. Khrennikov [13] studies the structure of the ‘social laser’. One can argue for a human gain medium and there is also a stimulation of emission from the gain medium.

In closing this Introduction, we would like to thank all the academics who presented papers at the conference. Bailey Fallon, the Commissioning Editor at the Royal Society, was really unfailing in his ever-present support. Bailey endured multiple phone calls and he always remained totally helpful. Heartfelt thanks also need to go to Matheus Grasselli, the Deputy Director of the Fields Institute for his wonderful support. We also would like to thank the fabulous staff at the Fields Institute for their superb input in having the conference run very smoothly. In particular, many thanks to Mimi Hao and Esther Berzunza.

## Footnotes

One contribution of 14 to a theme issue ‘Quantum probability and the mathematical modelling of decision making’.

- Accepted October 5, 2015.

- © 2015 The Author(s)