arXiv subjects

Peter Holland

Publications and source records attributed to Peter Holland.

At least 19 recordsLinked to original sources

arXiv2022

Eliminating the wavefunction from quantum dynamics: the bi-Hamilton-Jacobi theory, trajectories and time reversal

We observe that the Schrodinger equation may be written as two real coupled Hamilton-Jacobi (HJ)-like equations, each involving a quantum potential. Developing our established programme of representing the quantum state through exact free-standing deterministic trajectory models, it is shown how quantum evolution may be treated as the autonomous propagation of two coupled congruences. The wavefunction at a point is derived from two action functions, each generated by a single trajectory. The model shows that conservation as expressed through a continuity equation is not a necessary component of a trajectory theory of state. Probability is determined by the difference in the action functions, not by the congruence densities. The theory also illustrates how time-reversal symmetry may be implemented through the collective behaviour of elements that individually disobey the conventional transformation (T) of displacement (scalar) and velocity (reversal). We prove that an integral curve of the linear superposition of two vectors can be derived algebraically from the integral curves of one of the constituent vectors labelled by integral curves associated with the other constituent. A corollary establishes relations between displacement functions in diverse trajectory models, including where the functions obey different symmetry transformations. This is illustrated by showing that a (T-obeying) de Broglie-Bohm trajectory is a sequence of points on the (non-T) HJ trajectories, and vice versa.

quant-ph↗

arXiv2020

Unification of the wave and guidance equations for spin 1/2

We generalize our previous unification of the Schrodinger and guidance equations in a single inhomogeneous Schrodinger equation to a Riemannian manifold with an external vector potential. A special case yields the unified theory for a spin 1/2 rigid rotator. The theory is proved to be symmetrical under the Galileo group, the unified field that integrates the particle and guiding wave being a 2-spinor.

quant-ph↗

arXiv2020

Trajectory construction of Dirac evolution

We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wavefunction, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the relativity of the trajectory label. We show how these results extend to the inclusion of external potentials. We further show that the angular version of the Dirac equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.

quant-ph↗

arXiv2020

Uniting the wave and the particle in quantum mechanics

We present a unified field theory of wave and particle in quantum mechanics. This emerges from an investigation of three weaknesses in the de Broglie-Bohm (deBB) theory: its reliance on the quantum probability formula to justify the particle law; its insouciance regarding the absence of reciprocal action of the particle on the guiding wave; and its lack of a unified model to represent its inseparable components. These problems are resolved within an analytical framework by requiring that the wave-particle composite exhibits no observable differences with a quantum system. This scheme is implemented by appealing to symmetries (global gauge and spacetime translations) and imposing equality of the corresponding conserved Noether densities (matter, momentum and energy) with their Schrodinger counterparts. In conjunction with the condition of time reversal covariance this implies the deBB law for the particle where the quantum potential mediates the wave-particle interaction (we also show how the time reversal assumption may be replaced by a statistical condition). The method clarifies the nature of the mass of the composite, and its energy-momentum conservation laws. Our principal result is the unification of the Schrodinger equation and the deBB law in an inhomogeneous equation whose solution amalgamates the wavefunction and a singular soliton model of the particle in a unified spacetime field. The wavefunction suffers no reaction from the particle since it is the homogeneous part of the unified field to whose source the particle contributes. The theory applies to many-body systems. We review the objections of de Broglie to the pilot-wave theory and suggest that our field-theoretic model provides a realization of his hitherto unfulfilled double-solution programme. A revised set of postulates for the deBB theory is given in which the unified field is taken as the basic descriptive element.

quant-ph↗

arXiv2019

Trajectory-state theory of the Klein-Gordon field

We develop a trajectory construction of solutions to the massless wave equation in n+1 dimensions and hence show that the quantum state of a massive relativistic system in 3+1 dimensions may be represented by a stand-alone four-dimensional congruence comprising a continuum of 3-trajectories coupled to an internal scalar time coordinate. A real Klein-Gordon amplitude is the current density generated by the temporal gradient of the internal time. Complex amplitudes are generated by a two-phase flow. The Lorentz covariance of the trajectory model is established.

quant-ph↗

arXiv2018

Three-dimensional representation of the many-body quantum state

Using the trajectory conception of state we give a simple demonstration that the quantum state of a many-body system may be expressed as a set of states in three-dimensional space, one associated with each particle. It follows that the many-body wavefunction may be derived from a set of waves in 3-space. Entanglement is represented in the trajectory picture by the mutual dependence of the 3-states on the trajectory labels.

quant-ph↗

arXiv2016

The quantum state as spatial displacement

We give a simple demonstration that the Schrödinger equation may be recast as a self-contained second-order Newtonian law for a congruence of spacetime trajectories. This provides a pictorial representation of the quantum state as the displacement function of the collective whereby quantum evolution is represented as the deterministic unfolding of a continuous coordinate transformation. Introducing gauge potentials for the density and current density it is shown that the wave-mechanical and trajectory pictures are connected by a canonical transformation. The canonical trajectory theory is shown to provide an alternative basis for the quantum operator calculus and the issue of the observability of the quantum state is examined within this context. The construction illuminates some of the problems involved in connecting the quantum and classical descriptions.

quant-ph↗

arXiv2015

Towards Real-time Customer Experience Prediction for Telecommunication Operators

Telecommunications operators (telcos) traditional sources of income, voice and SMS, are shrinking due to customers using over-the-top (OTT) applications such as WhatsApp or Viber. In this challenging environment it is critical for telcos to maintain or grow their market share, by providing users with as good an experience as possible on their network. But the task of extracting customer insights from the vast amounts of data collected by telcos is growing in complexity and scale everey day. How can we measure and predict the quality of a user's experience on a telco network in real-time? That is the problem that we address in this paper. We present an approach to capture, in (near) real-time, the mobile customer experience in order to assess which conditions lead the user to place a call to a telco's customer care center. To this end, we follow a supervised learning approach for prediction and train our 'Restricted Random Forest' model using, as a proxy for bad experience, the observed customer transactions in the telco data feed before the user places a call to a customer care center. We evaluate our approach using a rich dataset provided by a major African telecommunication's company and a novel big data architecture for both the training and scoring of predictive models. Our empirical study shows our solution to be effective at predicting user experience by inferring if a customer will place a call based on his current context. These promising results open new possibilities for improved customer service, which will help telcos to reduce churn rates and improve customer experience, both factors that directly impact their revenue growth.

cs.CY↗

arXiv2014

Quantum potential energy as concealed motion

It is known that the Schroedinger equation may be derived from a hydrodynamic model in which the Lagrangian position coordinates of a continuum of particles represent the quantum state. Using Routh\s method of ignorable coordinates it is shown that the quantum potential energy of particle interaction that represents quantum effects in this model may be regarded as the kinetic energy of additional concealed freedoms. The method brings an alternative perspective to Planck\s constant, which plays the role of a hidden variable, and to the canonical quantization procedure, since what is termed kinetic energy in quantum mechanics may be regarded literally as energy due to motion.

quant-ph↗

arXiv2014

On systems having Poincaré and Galileo symmetry

Using the wave equation in d > or = 1 space dimensions it is illustrated how dynamical equations may be simultaneously Poincaré and Galileo covariant with respect to different sets of independent variables. This provides a method to obtain dynamics-dependent representations of the kinematical symmetries. When the field is a displacement function both symmetries have a physical interpretation. For d = 1 the Lorentz structure is utilized to reveal hitherto unnoticed features of the non-relativistic Chaplygin gas, including a relativistic structure with a limiting case that exhibits the Carroll group, and field-dependent symmetries and associated Noether charges. The Lorentz transformations of the potentials naturally associated with the Chaplygin system are given. These results prompt the search for further symmetries and it is shown that the Chaplygin equations support a nonlinear superposition principle. A known spacetime mixing symmetry is shown to decompose into label-time and superposition symmetries. It is shown that a quantum mechanical system in a stationary state behaves as a Chaplygin gas. The extension to d > 1 is used to illustrate how the physical significance of the dual symmetries is contingent on the context by showing that Maxwells equations exhibit an exact Galileo covariant formulation where Lorentz and gauge transformations are represented by field-dependent symmetries. A natural conceptual and formal framework is provided by the Lagrangian and Eulerian pictures of continuum mechanics.

physics.class-ph↗

arXiv2014

Hydrodynamic construction of the electromagnetic field

We present an alternative Eulerian hydrodynamic model for the electromagnetic field in which the discrete vector indices in Maxwell\s equations are replaced by continuous angular freedoms, and develop the corresponding Lagrangian picture in which the fluid particles have rotational and translational freedoms. This enables us to extend to the electromagnetic field the exact method of state construction proposed previously for spin 0 systems, in which the time-dependent wavefunction is computed from a single-valued continuum of deterministic trajectories where two spacetime points are linked by at most a single orbit. The deduction of Maxwell\s equations from continuum mechanics is achieved by generalizing the spin 0 theory to a general Riemannian manifold from which the electromagnetic construction is extracted as a special case. In particular, the flat-space Maxwell equations are represented as a curved-space Schrödinger equation for a massive system. The Lorentz covariance of the Eulerian field theory is obtained from the non-covariant Lagrangian-coordinate model as a kind of collective effect. The method makes manifest the electromagnetic analogue of the quantum potential that is tacit in Maxwell\s equations. This implies a novel definition of the \classical limit\ of Maxwell\s equations that differs from geometrical optics. It is shown that Maxwell\s equations may be obtained by canonical quantization of the classical model. Using the classical trajectories a novel expression is derived for the propagator of the electromagnetic field in the Eulerian picture. The trajectory and propagator methods of solution are illustrated for the case of a light wave.

quant-ph↗

arXiv2014

A quantum of history

With reference to primary sources it is shown that key claims made regarding the history of the pilot wave theory in Quantum Theory at the Crossroads are not supported by the historical record. It is also argued that the association of de Broglie with just a first-order law of particle motion, and Bohm with a second-order one, has no historical basis.

physics.hist-ph↗

arXiv2014

The roads not taken: empty waves, wavefunction collapse and protective measurement in quantum theory

Within the class of ontological interpretations of quantum theory where a physical system comprises a particle and a field (wavefunction) guiding it, an empty wave is a segment of the wavefunction not containing the particle. We examine the impact of this concept on the debate between the epistemological and ontological viewpoints. The theoretical merits of the empty wave in avoiding the wavefunction collapse hypothesis, and in supplying conceptual precision in the application of quantum mechanics, are emphasized. Particular reference is made to path detection in interferometry and it is shown how a consistent application of the particle and empty wave concepts dispels the suggestion that this is a theory. The problem of how the reality of an empty wave might be demonstrated by its effect on other systems is examined and general arguments advanced against this possibility. However, these arguments are not conclusive and we describe how the technique of protective measurement suggests a method to probe the empirical implications of empty waves and potentially distinguish between the epistemological and ontological perspectives.

quant-ph↗

arXiv2014

Dynamics-dependent symmetries in Newtonian mechanics

We exhibit two symmetries of one-dimensional Newtonian mechanics whereby a solution is built from the history of another solution via a generally nonlinear and complex potential-dependent transformation of the time. One symmetry intertwines the square roots of the kinetic and potential energies and connects solutions of the same dynamical problem (the potential is an invariant function). The other symmetry connects solutions of different dynamical problems (the potential is a scalar function). The existence of corresponding conserved quantities is examined using Noethers theorem and it is shown that the invariant-potential symmetry is correlated with energy conservation. In the Hamilton-Jacobi picture the invariant-potential transformation provides an example of a field-dependent symmetry in point mechanics. It is shown that this transformation is not a symmetry of the Schroedinger equation.

physics.class-ph↗

arXiv2012

Symmetries and conservation laws in the Lagrangian picture of quantum hydrodynamics

Quantum hydrodynamics represents quantum mechanics through two complementary models: the Eulerian picture, a direct transcription of wave mechanics, and the Lagrangian picture, in which the quantum state is represented by the collective motion of a continuum of fluid particles, the latter being obtained by continuously varying the particle label. The Lagrangian picture thus adds variables to the quantum formalism and exhibits a corresponding new quantum symmetry viz. a continuous particle-relabelling covariance group. Using Noether's theorem, the relabelling symmetry is derived as a component of a general investigation of symmetries and conservation laws in the quantum Lagrangian picture. Relations with symmetries and conservation laws in the Eulerian picture are explored in detail. Alongside the infinite relabelling group, the 12-parameter kinematical covariance group of the Schrödinger equation is derived. It is stressed that the role of label transformations extends beyond the class corresponding to Eulerian invariance. As an example, it is shown that the linear superposition of waves, a fundamental symmetry of the quantum Eulerian picture, can be generated by a deformation-dependent label transformation in the Lagrangian picture.

quant-ph↗

arXiv2011

Hydrodynamics, particle relabelling and relativity

Using the wave equation as an example, it is shown how to extend the hydrodynamic Lagrangian-picture method of constructing field evolution using a continuum of trajectories to second-order theories. The wave equation is represented through Eulerian-picture models that are distinguished by their Lorentz transformation properties. Introducing the idea of the relativity of the particle label, it is demonstrated how the corresponding trajectory models are compatible with the relativity principle. It is also shown how the Eulerian variational formulation may be obtained by canonical transformation from the Lagrangian picture, and how symmetries in the Lagrangian picture may be used to generate Eulerian conserved charges.

physics.flu-dyn↗

arXiv2009

Spin-like current from phase space distributions

The spin 0 generalized phase space approach provides a general expression for local current which depends on the choice of distribution function and generally deviates from the Schrodinger current. It is shown that the continuity equation restricts the admissible bilinear distributions such that the current has a unique dependence on the wavefunction and coincides with the non-relativistic limit of the relativistic spin 1/2 current for a spin eigenstate, up to a constant vector. Examples of non-bilinear distributions that have the latter property are given.

quant-ph↗

arXiv2008

Schrodinger dynamics as a two-phase conserved flow: an alternative trajectory construction of quantum propagation

It is shown that the Schrodinger equation can be cast in the form of two coupled real conservation equations, in Euclidean spacetime in the free case and in a five-dimensional Eisenhart geometry in the presence of an external potential. This implies a novel two-phase quantum hydrodynamic model whose Lagrangian picture provides an exact scheme to calculate the time-dependent wavefunction from a continuum of deterministic trajectories where two points are linked by at most two orbits. Properties of the model are examined, including the appearance of entangled trajectories in separable states. Wavefunction constructions employing alternative two-phase models are proposed.

quant-ph↗