Similarities between the non-deterministic nature of quantum theory and the unpredictable patterns of human cognition and decision making have been observed and commented on many times since the invention of Quantum Mechanics in the first part of the 20th century. Niels Bohr himself took note of the parallels.[3] In fact, an entire field of study, Quantum Cognition, has been borne from the study of this analogy. [3] However, many of the attempts to model human behavior with quantum mechanics conflate the identity of a particle with its own wavefunction, which is incorrect and invalidates the analogy. In this paper, we seek to make explicit this error, make necessary corrections, and then deepen the analogy. We do this by creating a Quantum Decision Theory that directly parallels Richard Feynman's novel formulation of quantum mechanics published in 1948 at Cornell University: Space-Time Approach to Non-Relativistic Quantum Mechanics".
Similarities between the non-deterministic nature of quantum theory and the unpredictable patterns of human cognition and decision making have been observed and commented on many times since the invention of Quantum Mechanics in the first part of the 20th century. Niels Bohr himself took note of the parallels.[3] In fact, an entire field of study, Quantum Cognition, has been borne from the study of this analogy. [3] However, many of the attempts to model human behavior with quantum mechanics conflate the identity of a particle with its own wavefunction, which is incorrect and invalidates the analogy. In this paper, we seek to make explicit this error, make necessary corrections, and then deepen the analogy. We do this by creating a Quantum Decision Theory that directly parallels Richard Feynman's novel formulation of quantum mechanics published in 1948 at Cornell University: Space-Time Approach to Non-Relativistic Quantum Mechanics".
In 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent.\nAlthough Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams.\nIn his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists.\nThe papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist.\nExtensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved.\nIn this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space.\nWe give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.
In physics, at the beginning of the twentieth century it was recognized that some experiments could not be explained by the conventional classical mechanics, but the same could be explained by the newly discovered quantum theory. It resulted in a new mechanics called quantum mechanics that revolutionized scientific and technological developments. Again, at the beginning of the twenty-first century, it is being recognized that some experiments related to the human decision-making processes could not be explained by the conventional classical decision theory but the same could be explained by the models based on quantum mechanics. It is now recognized that we need quantum mechanics in psychology as well as in economics and finance. In this paper we attempt to advance and explain the present understanding of applicability of quantum mechanics to the human decision-making processes. Using the postulates analogous to the postulates of quantum mechanics, we show the derivation of the quantum interference equation to illustrate the quantum approach. The explanation of disjunction effect experiments of Tversky and Shafir (Tversky A, Shafir E (1992) The disjunction effect in choice under uncertainty. Psych. Sci. 3(5):305–309) has been chosen to demonstrate the necessity of a quantum model. Further, to suggest the possibility of application of the quantum theory to the business-related decisions, some terms such as price operator, state of mind of the acquiring firm, etc., are introduced and discussed in context of the merger/acquisition of business firms. The possibility of the development in areas such as quantum finance, quantum management, application of quantum mechanics to the human dynamics related to healthcare management, etc., is also indicated.
I offer some historical comments about the origins of Feynman's path integral approach, as an alternative approach to standard quantum mechanics. Looking at the interaction between Einstein and Feynman, which was mediated by Feynman's thesis supervisor John Wheeler, it is argued that, contrary to what one might expect, the significance of the interaction between Einstein and Feynman pertained to a critique of classical field theory, rather than to a direct critique of quantum mechanics itself. Nevertheless, the critical perspective on classical field theory became a motivation and point of departure for Feynman's space-time approach to non-relativistic quantum mechanics.
There have been numerous neurobiological theories positing that quantum mechanics is importantly responsible for certain psychological phenomena, such as consciousness and conscious decision-making. In this respect, such theories understand the relevant aspects of the brain to be “quantum mechanical.” Although it is highly questionable whether any of these theories have the requisite or sufficient empirical evidence strictly within the domain of neuroscience to appropriately justify their views, a different approach to arguing that quantum mechanics is importantly relevant to human decisionmaking will be discussed. By putting various puzzle pieces together of empirical evidence in psychology that quantum probability models describe certain human cognitive decision-making processes, I contend that human decision-making overall is best described by quantum mechanical processes. Finally, I at times provide a novel defense of this view from a number of objections.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
Why should you be interested in quantum theory applied to Cognition and decision? Perhaps you are a physicist who is curious whether or not quantum principles can be applied outside of physics. In fact, that is one purpose of this book. Perhaps you are a cognitive scientist who is interested in representing concepts by vectors in a multidimensional feature space. This is essentially the way quantum theory works too. Perhaps you are a decision scientist who is trying to understand how people make decisions under uncertainty. Quantum theory could provide some interesting new answers. Generally speaking, quantum theory is a new theory for constructing probabilistic and dynamic systems , and in this book we apply this new theory to topics in cognition and decision. Later in this chapter we will give some specific examples, but let us step back at this point and try to understand the more general principles that support a quantum approach to cognition and decision. Six reasons for a quantum approach to cognition and decision Quantum physics is arguably the most successful scientific theoretical achievement that humans have ever created. It was created to explain puzzling findings that were impossible to understand using the older classical physical theory, and it achieved this by introducing an entirely new set of revolutionary principles. The older classical physical theory is now seen as a special case of the more general quantum theory. In the process of creating quantum mechanics, physicists also created a new theory of probabilistic and dynamic systems that is more general than the previous classic theory (Pitowski, 1989).