James M. Yearsley

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 radical 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, this paper provides a surprising, novel direction in relation to how time should be conceptualized in cognition.

Arrival times, complex potentials, and decoherent histories

Abstract We address a number of aspects of the arrival time problem defined using a complex potential of step function form. We concentrate on the limit of a weak potential, in which the resulting arrival time distribution function is closely related to the quantum-mechanical current. We first consider the analagous classical arrival time problem involving an absorbing potential, and this sheds some light on certain aspects of the quantum case. In the quantum case, we review the path decomposition expansion (PDX), in which the propagator is factored across a surface of constant time, so is very useful for potentials of step function form. We use the PDX to derive the usual scattering wave functions and the arrival time distribution function. This method gives a direct and geometrically appealing account of known results (but also points the way to how they can be extended to more general complex potentials). We use these results to carry out a decoherent histories analysis of the arrival time problem, taking advantage of a recently demonstrated connection between pulsed measurements and complex potentials. We obtain very simple and plausible expressions for the class operators (describing the amplitudes for crossing the origin during intervals of time) and show that decoherence of histories is obtained for a wide class of initial states (such as simple wave packets and superpositions of wave packets). We find that the decoherent histories approach gives results with a sensible classical limit that are fully compatible with standard results on the arrival time problem. We also find some interesting connections between backflow and decoherence.

Negative probabilities, Fine's theorem, and linear positivity

Many situations in quantum theory and other areas of physics lead to quasi-probabilities which seem to be physically useful but can be negative. The interpretation of such objects is not at all clear. In this paper, we show that quasi-probabilities naturally fall into two qualitatively different types, according to whether their non-negative marginals can or cannot be matched to a non-negative probability. The former type, which we call viable, are qualitatively similar to true probabilities, but the latter type, which we call non-viable, may not have a sensible interpretation. Determining the existence of a probability matching given marginals is a non-trivial question in general. In simple examples, Fine’s theorem indicates that inequalities of the Bell and CHSH type provide criteria for its existence, and these examples are considered in detail. Our results have consequences for the linear positivity condition of Goldstein and Page in the context of the histories approach to quantum theory. Although it is a very weak condition for the assignment of probabilities it fails in some important cases where our results indicate that probabilities clearly exist. We speculate that our method, of matching probabilities to a given set of marginals, provides a general method of assigning probabilities to histories and we show that it passes the Diósi test for the statistical independence of subsystems. PACS numbers: 03.65.Yz, 03.65.Ta, 02.50.Cw Electronic address: j.halliwell@imperial.ac.uk Electronic address: jmy27@cam.ac.uk

Pitfalls of path integrals: Amplitudes for spacetime regions and the quantum Zeno effect

Path integrals appear to offer natural and intuitively appealing methods for defining quantummechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect – the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a “softer” way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale ǫ, describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as ǫ ≫ 1/E, where E is the energy scale of the incoming state. PACS numbers: 04.60.Gw, 03.65.Yz, 03.65Xp Electronic address: j.halliwell@imperial.ac.uk Electronic address: jmy27@cam.ac.uk

The propagator for the step potential and delta function potential using the path decomposition expansion

We present a derivation of the propagator for a particle in the presence of the step and delta function potentials. These propagators are known, but we present a direct path integral derivation, based on the path decomposition expansion and the Brownian motion definition of the path integral. The derivation exploits properties of the Catalan numbers, which enumerate certain classes of lattice paths.

Zeno’s paradox in decision making

Classical probability theory has been influential in modelling decision processes, despite empirical findings that have been persistently paradoxical from classical perspectives. For such findings, some researchers have been successfully pursuing decision models based on quantum theory (QT). One unique feature of QT is the collapse postulate, which entails that measurements (or in decision-making, judgements) reset the state to be consistent with the measured outcome. If there is quantum structure in cognition, then there has to be evidence for the collapse postulate. A striking, a priori prediction, is that opinion change will be slowed down (under idealized conditions frozen) by continuous judgements. In physics, this is the quantum Zeno effect. We demonstrate a quantum Zeno effect in decision-making in humans and so provide evidence that advocates the use of quantum principles in decision theory, at least in some cases.

Progress and current challenges with the quantum similarity model.

This opinion paper reviews progress with the quantum similarity model (QSM), which was proposed by Pothos et al. (2013). In the QSM, concepts are associated with subspaces, the mental state is a state vector in a Hilbert space, and similarity between two concepts is computed in terms of the sequential projection, between the corresponding subspaces. If there is a relevant context, this is incorporated as prior projections (e.g., Figure ​Figure11).

On the relationship between complex potentials and strings of projection operators

Abstract 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 stepfunction 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 ~/E, where E is the energy scale of the initial state.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 /E, where E is the energy scale of the initial state.

Quantum arrival time formula from decoherent histories

Abstract In the arrival time problem in quantum mechanics, a standard formula that frequently emerges as the probability for crossing the origin during a given time interval is the current integrated over that time interval. This is semiclassically correct but can be negative due to backflow. Here, we show that this formula naturally arises in a decoherent histories analysis of the arrival time problem. For a variety of initial states, we show that histories crossing during different time intervals are approximately decoherent. Probabilities may therefore be assigned and coincide with the standard formula (in a semiclassical approximation), which is therefore positive for these states. However, for initial states for which there is backflow, we show that there cannot be decoherence of histories, so probabilities may not be assigned.

Quantum arrival and dwell times via idealized clocks

Abstract A number of approaches to the problem of defining arrival and dwell time probabilities in quantum theory make use of idealised models of clocks. An interesting question is the extent to which the probabilities obtained in this way are related to standard semiclassical results. In this paper we explore this question using a reasonably general clock model, solved using path integral methods. We find that in the weak coupling regime where the energy of the clock is much less than the energy of the particle it is measuring, the probability for the clock pointer can be expressed in terms of the probability current in the case of arrival times, and the dwell time operator in the case of dwell times, the expected semiclassical results. In the regime of strong system-clock coupling, we find that the arrival time probability is proportional to the kinetic energy density, consistent with an earlier model involving a complex potential. We argue that, properly normalized, this may be the generically expected result in this regime. We show that these conclusions are largely independent of the form of the clock Hamiltonian.