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Cambridge University Press eBooks

Single-Particle Systems versus Agents – II

Vikram Athalye, Emmanuel Haven

Published Dec 19, 2025• 0 citations
Publisher

Abstract

This chapter provides, we believe, for the apogee of what we think will form the base for success of the quantum physics–like applications. Readers are invited in this chapter to carefully study the two-slit interference experiment with agents (and the agent two-preference interference) for a variety of real potential functions.
DOI
10.1017/9781108953177.013

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This chapter (together with the next one) introduces probably the highlight of the book, i.e. it attempts to answer the important question: what can we now do with the quantum-physics like stance? An immediate, almost obvious, discussion centres around the analogies with the famed double-slit experiment. We set ourselves the task of answering how we can begin to enumerate, quite precisely, analogies between electrons and agents. As the reader will find out, we will need to move over several (important) hurdles, one of them being the perennially difficult analogy we need to make with the Planck constant. We then proceed in shaping the idea of two-preference interference, a concept of paramount importance in our quest to properly define the quantum physics–like research direction.
In this last chapter of the book, we keep coming back to the potential function and we attempt to connect it to more precise ideas in finance, including that of the agent heterogeneities. We also initiate a discussion on agent behaviour and causality and nonlocality. Our last words in this book will be centred on what comes next. One of the key queries we have is whether we can consider more complicated real potentials in the two-slit interference experiment with agents (and the agent two-preference interference). The other one is centred around the investigation on the nonexistence of “spooky” free will of the individual agents.
In this chapter, we take a deeper look at a number of interesting and important models that contain one or two particles. We focus on examples, from a variety of settings and fields, with the goal of not only finding relevant physical phenomena occurring in these systems, but also giving the reader a place where they can actually practise much of what has been said previously throughout the book. In some cases, straight application of the methods explained earlier suffices, while in some others slightly different (yet related) methods are needed. Most of the models studied in this chapter are justified in chapters 4 and 5, sometimes in full. Other times, the justification from chapters 4 and 5 may be only partial, and the interested reader is encouraged to consult more specialized literature in the subject.
This chapter gives a summary of the mathematical formalism used in quantum mechanics, with a short bibliography put directly at its end. It should be seen as a complement, to be used by readers who wish to know more than what has been recalled about the mathematical tools in the other chapters. Some results are given without explicit proofs; they can be found for instance in Chapters II and IV of [1] . Many quantum mechanics textbooks introduce its general formalism in a more complete way, for instance Chapter VII of [2] , Chapter 3 of [3] , or Chapter 2 of [4] . We first summarize the general formalism of quantum mechanics for any physical system (§11.1), with the Dirac notation; we then study how this formalism treats the grouping of several physical systems into one single quantum system (§11.2); finally, we study a few simple special cases (§11.3), for instance a single particle in an external potential, with or without spin; the reader who prefers wave functions to more abstract reasonings in spaces of states may begin with this section. General physical system The general formalism of quantum mechanics applies to all physical systems, whether they contain a single particle, many particles of various sorts, one or several fields, etc. Quantum space of states The physical state of a system at each time is defined in quantum mechanics by a state vector which, in Dirac notation, is written ∣ ψ 〈 – or ∣ ψ(t) 〈 if one wishes to make the time dependence explicit.
The wave-particle duality mystery, present in the language of quantum physics since its historical origins 100 years ago, has no real correspondence with the actual behaviour of nature when it is observed. The dilemma of the so-called “single particle interference” experiments is not between particles versus waves: the interference in quantum physics is between probability amplitudes for different untagged alternatives in a definite transition between states. This article shows that the same conceptual basis is involved in a plethora of “two-state systems” as a consequence of the linear superposition principle, no matter whether the observables refer to either spatial or internal properties. Several examples, from the Young double-slit interference experiments to neutrino flavour oscillation experiments, are discussed in a common interpretation language.