The propagator for the step potential using the path decomposition expansion

We present a direct path integral derivation of the propagator in the presence of a step potential. The derivation makes use of the Path Decomposition Expansion (PDX), and also of the definition of the propagator as a limit of lattice paths.Comment: To appear in DICE 2008 conference proceeding

Quantum cognition and decision theories: A tutorial

Models of cognition and decision making based on quantum theory have been the subject of much interest recently. Quantum theory provides an alternative probabilistic framework for modelling decision making compared with classical probability theory, and has been successfully used to address behaviour considered paradoxical or irrational from a classical point of view. The purpose of this tutorial is to give an introduction to quantum models, with a particular emphasis on how to build these models in practice. Examples are provided by the study of order effects on judgements, and we will show how order effects arise from the structure of the theory. In particular, we show how to derive the recent discovery of a particular constraint on order effects implied by quantum models, called the Quantum Question (QQ) Equality, which does not appear to be derivable from alternative accounts, and which has been experimentally verified to high precision. However the general theory and methods of model construction we will describe are applicable to any quantum cognitive model. Our hope is that this tutorial will give researchers the confidence to construct simple quantum models of their own, particularly with a view to testing these against existing cognitive theories

On the Relationship Between Complex Potentials and Strings of Projection Operators

It is of interest in a variety of contexts, and in particular in the arrival time problem, to consider the quantum state obtained through unitary evolution of an initial state regularly interspersed with periodic projections onto the positive $x$-axis (pulsed measurements). Echanobe, del Campo and Muga have given a compelling but heuristic argument that the state thus obtained is approximately equivalent to the state obtained by evolving in the presence of a certain complex potential of step-function form. In this paper, with the help of the path decomposition expansion of the associated propagators, we give a detailed derivation of this approximate equivalence. The propagator for the complex potential is known so the bulk of the derivation consists of an approximate evaluation of the propagator for the free particle interspersed with periodic position projections. This approximate equivalence may be used to show that to produce significant reflection, the projections must act at time spacing less than 1/E, where E is the energy scale of the initial state.Comment: 29 pages, LaTex, 4 figures. Substantial revision

Dissociating visuo-spatial and verbal working memory: It’s all in the features

Echoing many of the themes of the seminal work of Atkinson and Shiffrin (1968), this paper uses the Feature Model (Nairne, 1988, 1990; Neath & Nairne, 1995) to account for performance in working memory tasks. The Brooks verbal and visuo-spatial matrix tasks were performed alone, with articulatory suppression, or with a spatial suppression task; the results produced the expected dissociation. We used Approximate Bayesian Computation techniques to fit the Feature Model to the data and showed that the similarity-based interference process implemented in the model accounted for the data patterns well. We then fit the model to data from Guérard and Tremblay (2008); the latter study produced a double dissociation while calling upon more typical order reconstruction tasks. Again, the model performed well. The findings show that a double dissociation can be modelled without appealing to separate systems for verbal and visuo-spatial processing. The latter findings are significant as the Feature Model had not been used to model this type of dissociation before; importantly, this is also the first time the model is quantitatively fit to data. For the demonstration provided here, modularity was unnecessary if two assumptions were made: (1) the main difference between spatial and verbal working memory tasks is the features that are encoded; (2) secondary tasks selectively interfere with primary tasks to the extent that both tasks involve similar features. It is argued that a feature-based view is more parsimonious (see Morey, 2018) and offers flexibility in accounting for multiple benchmark effects in the field

Challenging the classical notion of time in cognition: a quantum perspective

All mental representations change with time. A baseline intuition is that mental representations have specific values at different time points, which may be more or less accessible, depending on noise, forgetting processes etc. We present a radically alternative, motivated by recent research using the mathematics from quantum theory for cognitive modelling. Such cognitive models raise the possibility that certain possibilities or events may be incompatible, so that perfect knowledge of one necessitates uncertainty for the others. In the context of time dependence, in physics, this issue is explored with the so-called temporal Bell (TB) or Leggett-Garg inequalities. We consider in detail the theoretical and empirical challenges involved in exploring the TB inequalities in the context of cognitive systems. One interesting conclusion is that we believe the study of the TB inequalities to be empirically more constrained in psychology, than in physics. Specifically, we show how the TB inequalities, as applied to cognitive systems, can be derived from two simple assumptions, Cognitive Realism and Cognitive Completeness. We discuss possible implications of putative violations of the TB inequalities for cognitive models and our understanding of time in cognition in general. Overall, the paper provides a surprising, novel direction, in relation to how time should be conceptualized in cognition