arXiv subjects

Daniel W. Hook

Publications and source records attributed to Daniel W. Hook.

At least 19 recordsLinked to original sources

arXiv2026

Market Dynamics, Governance and Open Research Metadata in the AI Era

The debate about scholarly knowledge infrastructure has long been framed as a contest between openness and commercial enclosure. This framing distorts both policy and practice. The real tension lies between the persistent cost of producing and refining structured metadata under deep technological friction, and the differentiated demands distinct communities place on data quality, focus and granularity. We introduce the innovation annulus: the zone between freely available structured data and the advancing frontier of commercially refined knowledge products. This zone is a permanent, functional feature of the ecosystem -- not a pathology to eliminate. By analogy with the efficient market hypothesis, its width measures production inefficiency, set by the interplay of friction and demand. Artificial intelligence reshapes the annulus, lowering barriers to basic structuring, raising the threshold at which refinement adds value, and introducing systemic risks through unprovenanced AI-derived metadata. CRediT contributions, funding acknowledgements and AI disclosure statements illustrate the annulus lifecycle. Governance should calibrate the annulus, not abolish it: thin enough to serve research efficiently, wide enough to sustain innovation. A formal welfare framework, analogous to the Nordhaus optimal patent life, characterises the trade-offs and yields testable predictions. The Barcelona Declaration offers a promising forum for boundary governance.

cs.DL↗

arXiv2025

Understanding the importance of SHAPE to the UK research ecosystem

The UK has a long-established reputation for excellence in research across a broad range of fields, but in recent years, there has been greater emphasis on STEM investment and greater recognition of the UK's success in STEM. This paper examines the relative strengths of SHAPE disciplines and demonstrates that the UK's SHAPE research portfolio outperforms the UK's STEM research, for each international benchmark considered in this work. It is argued that SHAPE research is becoming increasingly important as a partner to STEM as the widespread use of technology creates societal challenges. It is also argued that the strength of UK SHAPE is the basis of a strategic advantage for UK research.

physics.soc-ph↗

arXiv2024

Complex phases in quantum mechanics

Hamilton's equations of motion are local differential equations and boundary conditions are required to determine the solution uniquely. Depending on the choice of boundary conditions, a Hamiltonian may thereby describe several different physically observable phases, each exhibiting its own characteristic global symmetry.

quant-ph↗

arXiv2023

PT-symmetric quantum mechanics

It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.

quant-ph↗

arXiv2021

$PT$-symmetric classical mechanics

This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian $PT$-symmetric Hamiltonians $H=p^2+ x^2(ix)^\varepsilon$ ($\varepsilon\geq0$). A variety of phenomena, heretofore overlooked, have been discovered such as the existence of infinitely many separatrix trajectories, sequences of critical initial values associated with limiting classical orbits, regions of broken $PT$-symmetric classical trajectories, and a remarkable topological transition at $\varepsilon=2$. This investigation is a work in progress and it is not complete; many features of complex trajectories are still under study.

math-ph↗

arXiv2019

The Price of Gold: Curiosity?

Gold open access as characterised by the payment of an article processing charge (APC) has become one of the dominant models in open access publication. This paper examines an extreme hypothetical case in which the APC model is the only model and the systematic issues that could develop in such a scenario.

cs.DL↗

arXiv2017

Behavior of eigenvalues in a region of broken-PT symmetry

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. When $\varepsilon\geq0$, the eigenvalues of this non-Hermitian Hamiltonian are discrete, real, and positive. This portion of parameter space is known as the region of unbroken PT symmetry. In the region of broken PT symmetry $\varepsilon<0$ only a finite number of eigenvalues are real and the remaining eigenvalues appear as complex-conjugate pairs. The region of unbroken PT symmetry has been studied but the region of broken PT symmetry has thus far been unexplored. This paper presents a detailed numerical and analytical examination of the behavior of the eigenvalues for $-4<\varepsilon<0$. In particular, it reports the discovery of an infinite-order exceptional point at $\varepsilon=-1$, a transition from a discrete spectrum to a partially continuous spectrum at $\varepsilon=-2$, a transition at the Coulomb value $\varepsilon=-3$, and the behavior of the eigenvalues as $\varepsilon$ approaches the conformal limit $\varepsilon=-4$.

math-ph↗

arXiv2015

PT-symmetric interpretation of unstable effective potentials

The conventional interpretation of the one-loop effective potentials of the Higgs field in the Standard Model and the gravitino condensate in dynamically broken supergravity is that these theories are unstable at large field values. A PT-symmetric reinterpretation of these models at a quantum-mechanical level eliminates these instabilities and suggests that these instabilities may also be tamed at the quantum-field-theory level.

hep-th↗

arXiv2014

Infinitely many inequivalent field theories from one Lagrangian

Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field $ϕ$. In Euclidean space the Lagrangian of such a theory, $L=\frac{1}{2}(\nablaϕ)^2-igϕ\exp(iaϕ)$, is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the mth energy level in the nth sector is given by $E_{m,n}\sim(m+1/2)^2a^2/(16n^2)$.

hep-th↗

arXiv2014

Complex classical motion in potentials with poles and turning points

Complex trajectories for Hamiltonians of the form H=p^n+V(x) are studied. For n=2 time-reversal symmetry prevents trajectories from crossing. However, for n>2 trajectories may indeed cross, and as a result, the complex trajectories for such Hamiltonians have a rich and elaborate structure. In past work on complex classical trajectories it has been observed that turning points act as attractors; they pull on complex trajectories and make them veer towards the turning point. In this paper it is shown that the poles of V(x) have the opposite effect --- they deflect and repel trajectories. Moreover, poles shield and screen the effect of turning points.

math-ph↗

arXiv2012

Universal spectral behavior of $x^2(ix)^ε$ potentials

The PT-symmetric Hamiltonian $H=p^2+x^2(ix)^ε$ ($ε$ real) exhibits a phase transition at $ε=0$. When $ε\geq0$, the eigenvalues are all real, positive, discrete, and grow as $ε$ increases. However, when $ε<0$ there are only a finite number of real eigenvalues. As $ε$ approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at $ε=-1$. In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Hamiltonians $H^{(2n)}=p^{2n}+x^2(ix)^ε$ ($ε$ real, n=1, 2, 3, ...). The complex classical behaviors of these Hamiltonians are also examined.

hep-th↗

arXiv2012

Negative-energy PT-symmetric Hamiltonians

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian $H=p^2+x^2(ix)^ε$ has real, positive, and discrete eigenvalues for all $ε\geq 0$. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues $E_n=2n+1$ (n=0, 1, 2, 3, ...) at $ε=0$. However, the harmonic oscillator also has negative eigenvalues $E_n=-2n-1$ (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian $H=p^2+x^2(ix)^ε$ also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, ...). For the Nth class of eigenvalues, $ε$ lies in the range $(4N-6)/3<ε<4N-2$. At the low and high ends of this range, the eigenvalues are all infinite. At the special intermediate value $ε=2N-2$ the eigenvalues are the negatives of those of the conventional Hermitian Hamiltonian $H=p^2+x^{2N}$. However, when $ε\neq 2N-2$, there are infinitely many complex eigenvalues. Thus, while the positive-spectrum sector of the Hamiltonian $H=p^2+x^2(ix)^ε$ has an unbroken PT symmetry (the eigenvalues are all real), the negative-spectrum sector of $H=p^2+x^2(ix)^ε$ has a broken PT symmetry (only some of the eigenvalues are real).

hep-th↗

arXiv2011

Quantum tunneling as a classical anomaly

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: Complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im(E) -> 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as ReE increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.

hep-th↗

arXiv2010

Probability Density in the Complex Plane

The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density $ρ(z)$ in the complex plane. It is shown that there exist infinitely many complex contours $C$ of infinite length on which $ρ(z) dz$ is real and positive. Furthermore, the probability integral $\int_Cρ(z) dz$ is finite. Demonstrating the existence of such contours is the essential element in establishing the correspondence between complex quantum and classical mechanics. The mathematics needed to analyze these contours is subtle and involves the use of asymptotics beyond all orders.

hep-th↗

arXiv2010

Classical Particle in a Complex Elliptic Potential

This paper reports a numerical study of complex classical trajectories of a particle in an elliptic potential. This study of doubly-periodic potentials is a natural sequel to earlier work on complex classical trajectories in trigonometric potentials. For elliptic potentials there is a two-dimensional array of identical cells in the complex plane, and each cell contains a pair of turning points. The particle can travel both horizontally and vertically as it visits these cells, and sometimes the particle is captured temporarily by a pair of turning points. If the particle's energy lies in a conduction band, the particle drifts through the lattice of cells and is never captured by the same pair of turning points more than once. However, if the energy of the particle is not in a conduction band, the particle can return to previously visited cells.

hep-th↗

arXiv2009

Complex Elliptic Pendulum

This paper briefly summarizes previous work on complex classical mechanics and its relation to quantum mechanics. It then introduces a previously unstudied area of research involving the complex particle trajectories associated with elliptic potentials.

hep-th↗

arXiv2009

Complex Correspondence Principle

Quantum mechanics and classical mechanics are two very different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on extending both quantum mechanics and classical mechanics into the complex domain. This letter shows that these complex extensions continue to exhibit a correspondence, and that this correspondence becomes more pronounced in the complex domain. The association between complex quantum mechanics and complex classical mechanics is subtle and demonstrating this relationship prequires the use of asymptotics beyond all orders.

hep-th↗

arXiv2008

Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.

hep-th↗