Table of contents

This is a call for contributions to a special issue ofJournal of Physics A: Mathematical and General entitled `Singular Interactions in Quantum Mechanics: Solvable Models'. This issue should be a repository for high quality original work. We are interested in having the topic interpreted broadly, that is, to include contributions dealing with point-interaction models, one- and many-body, quantum graphs, including graph-like structures coupling different dimensions, interactions supported by curves, manifolds, and more complicated sets, random and nonlinear couplings, etc., as well as approximations helping us to understand the meaning of singular couplings and applications of such models on different parts of quantum mechanics. We believe that when the second printing of the `bible' of the field, the book Solvable Models in Quantum Mechanics by S Albeverio, F Gesztesy, the late R Høegh-Krohn and H Holden, appears it is the right moment to review new developments in this area, with the hope of stimulating further development of these extremely useful techniques.

The Editorial Board has invited G Dell'Antonio, P Exner and V Geyler to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are as follows:

• The subject of the paper should relate to singular interactions in quantum mechanics in the sense described above.

• Contributions will be refereed and processed according to the usual procedure of the journal.

• Papers should be original; reviews of a work published elsewhere will not be accepted.

The guidelines for the preparation of contributions are as follows:

• The DEADLINE for submission of contributions is 31 October 2004. This deadline will allow the special issue to appear in about April 2005.

• There is a nominal page limit of 15 printed pages (approximately 9000 words) per contribution. Papers exceeding these limits may be accepted at the discretion of the Guest Editors. Further advice on publishing your work in Journal of Physics A: Mathematical and General may be found at www.iop.org/Journals/jphysa.

• Contributions to the Special Issue should if possible be submitted electronically by web upload at {www.iop.org/Journals/jphysa or by e-mail to jphysa@iop.org, quoting `JPhysA Special Issue-Quantum Mechanics: Solvable Models'. Submissions should ideally be in standard LaTeX form; we are, however, able to accept most formats including Microsoft Word. Please see the web site for further information on electronic submissions.

• Authors unable to submit electronically may send hard copy contributions to: Publishing Administrators, Journal of Physics A, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK, enclosing the electronic code on floppy disk if available and quoting `JPhysA Special Issue-Quantum Mechanics: Solvable Models'.

• All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address.

This special issue will be published in the paper and online version of the journal. The corresponding author of each contribution will receive a complimentary copy of the issue.

G Dell'Antonio, P Exner and V Geyler

Guest Editors

We focus on a reaction–diffusion approach proposed recently for experiments on combustion processes, where the heat released by combustion follows first-order reaction kinetics. This case allows us to perform an exhaustive analytical study. Specifically, we obtain the exact expressions for the speed of the thermal pulses, their maximum temperature and the condition of self-sustenance. Finally, we propose two generalizations of the model, namely, the case of several reactants burning together, and that of time-delayed heat conduction. We find an excellent agreement between our analytical results and simulations.

We describe a family of quantum spin models which are generators of a discrete Markovian process. We show that there exists an explicit expression for the ground state of such models and give a simple argument for the existence of two types of long-range order in such systems. Two special examples of these systems are analysed in detail.

We consider vortices in the nonlocal two-dimensional Gross–Pitaevskii equation with the interaction potential having Lorentz-shaped dependence on the relative momentum. It is shown that in the Fourier series expansion with respect to the polar angle, the unstable modes of the axial n-fold vortex have orbital numbers l satisfying 0 < |l| < 2|n|, as in the local model. Numerical simulations show that nonlocality slightly decreases the threshold rotation frequency above which the nonvortex state ceases to be the global energy minimum and decreases the frequency of the anomalous mode of the 1-vortex. In the case of higher axial vortices, nonlocality leads to instability against splitting with the creation of antivortices and gives rise to additional anomalous modes with higher orbital numbers. Despite new instability channels with the creation of antivortices, for a stationary solution comprised of vortices and antivortices there always exists another vortex solution, composed solely of vortices, with the same total vorticity but with a lower energy.

The resistance between two arbitrary nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulae for two-point resistances are deduced for regular lattices in one, two and three dimensions under various boundary conditions including that of a Möbius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyse large-size expansions in two and higher dimensions.

We introduce the concept of regular quantum graphs and construct connected quantum graphs with discrete symmetries. The method is based on a decomposition of the quantum propagator in terms of permutation matrices which control the way incoming and outgoing channels at vertex scattering processes are connected. Symmetry properties of the quantum graph as well as its spectral statistics depend on the particular choice of permutation matrices, also called connectivity matrices, and can now be controlled easily. The method may find applications in the study of quantum random walks and may also prove to be useful in analysing universality in spectral statistics.

We show how a large class of sufficient conditions for the existence of bound states, in non-positive central potentials, can be constructed. These sufficient conditions yield upper limits on the critical value, g^(ℓ)_c, of the coupling constant (strength), g, and of the potential, V(r) = −gv(r), for which a first ℓ-wave bound state appears. These upper limits are significantly more stringent than hitherto known results.

We study the ⊙-product of Bracken [1], which is the Weyl quantized version of the pointwise product of functions in phase space. We prove that it is not compatible with the algebras of finite rank and Hilbert–Schmidt operators. By solving the linearization problem for the special Hermite functions, we are able to express the ⊙-product in terms of the component operators, mediated by the linearization coefficients. This is applied to finite rank operators and their matrices, and operators whose symbols are radial and angular distributions.

We describe a Nambu–Jacobi structure as a 'Nambu–Poisson' structure on a certain Jacobi algebroid. It is shown that the matched pair of Leibniz algebroids for a Nambu–Jacobi structure is the Leibniz algebroid associated with this 'Nambu–Poisson' structure. We also see that a different Leibniz algebroid is associated with a Nambu–Jacobi structure, which is a natural generalization of the Lie algebroid associated with a Jacobi structure.

Irreducible representations (irreps) of a compact Lie group G, of class one w.r.t. a Lie subgroup H are those that contain the identity irrep of H once in their decompositions w.r.t. H. In the case of G = SU(4) and H = S(U(2) ⊗ U(2)), the class one irreps are identified, and for them a general formula for their decomposition into irreps of H is given. This admits a useful graphical presentation. The relevance of these results to the solution of the Schrödinger equation of SU(4)/S(U(2) ⊗ U(2)) and the state labelling difficulties encountered in implementing this solution, are discussed. For G = SU(n + 1) and H = S(U(n − 1) ⊗ U(2)), n ⩾ 4, the class one irreps of G, and hence the spectrum of the corresponding Schrödinger equation, have also been determined.

The dynamics of split fields in one dimension are extended to three dimensions using Clifford algebra. The solutions of the resulting equations provide a unique insight into wave splitting and allow the construction of wave splittings in three dimensions that may be useful in solving the three-dimensional inverse scattering problem in the time domain.

The time-dependent su(3) mean field equations are solved for a particular energy function relevant to nuclear structure. The model energy in the su(3) enveloping algebra is the sum of two terms which are the squared length of the angular momentum vector and the cubic rotational scalar X₃ = tr(lql). The mean field solutions for this energy have constant intrinsic quadrupole moments q. In the three-dimensional space of all angular momentum components in the rotating principal axis frame, a trajectory is defined by the intersection of a sphere and a hyperboloid. This conclusion is similar to the classical rigid rotor for which a solution is the intersection of a sphere and the inertia ellipsoid.

Many types of point singularity have a topological index, or 'charge', associated with them. For example, the phase of a complex field depending on two variables can either increase or decrease on making a clockwise circuit around a simple zero, enabling the zeros to be assigned charges of ±1. In random fields we can define a correlation function for the charge-weighted density of singularities. For many types of topologically charged singularity, this correlation function satisfies an identity which shows that the singularities 'screen' each other perfectly: a positive singularity is surrounded by an excess of concentration of negatives which exactly cancel its charge, and vice versa. This paper gives a simple and widely applicable derivation of this result. A counterexample where screening is incomplete is also exhibited.

We study the quantization of many-body systems in three dimensions in rotating coordinate frames using a gauge invariant formulation of the dynamics. We consider reference frames defined by linear gauge conditions, and discuss their Gribov ambiguities and commutator algebra. We construct the momentum operators, inner product and Hamiltonian in those gauges, for systems with and without translation invariance. The analogy with the quantization of non-Abelian Yang–Mills theories in non-covariant gauges is emphasized. Our results are applied to quasi-rigid systems in the Eckart frame.

The entanglement between occupation numbers of different single particle basis states depends on coupling between different single particle basis states in the second-quantized Hamiltonian. Thus, in principle, interaction is not necessary for occupation-number entanglement to appear. However, in order to characterize quantum correlation caused by interaction, we use the eigenstates of the single-particle Hamiltonian as the single particle basis upon which the occupation-number entanglement is defined. Using this so-called proper single particle basis, if there is no interaction, the many-particle second-quantized Hamiltonian is diagonalized and thus cannot generate entanglement, while its eigenstates can always be chosen to be non-entangled. If there is interaction, entanglement in the proper single particle basis arises in energy eigenstates and can be dynamically generated. Using the proper single particle basis, we discuss occupation-number entanglement in important eigenstates, especially ground states, of systems of many identical particles, in exploring insights the notion of entanglement sheds on many-particle physics. The discussions on Fermi systems start with Fermi gas, the Hartree–Fock approximation and the electron–hole entanglement in excitations. In the ground state of a Fermi liquid, in terms of the Landau quasiparticles, entanglement becomes negligible. The entanglement in a quantum Hall state is quantified as −fln f − (1 − f)ln(1 − f), where f is the proper fractional part of the filling factor. For BCS superconductivity, the entanglement is a function of the relative momentum wavefunction of the Cooper pair g_k, and is thus directly related to the superconducting energy gap, and vanishes if and only if superconductivity vanishes. For a spinless Bose system, entanglement does not appear in the Hartree–Gross–Pitaevskii approximation, but becomes important in the Bogoliubov theory, as a characterization of two-particle correlation caused by the weak interaction. In these examples, the interaction-induced entanglement as calculated is directly related to the macroscopic physical properties.

Consider random matrices A, of dimension m × (m + n), drawn from an ensemble with probability density f(tr AA†), with f(x) a given appropriate function. Break A = (B, X) into an m × m block B and the complementary m × n block X, and define the random matrix Z = B⁻¹X. We calculate the probability density function P(Z) of the random matrix Z and find that it is a universal function, independent of f(x). The universal probability distribution P(Z) is a spherically symmetric matrix-variate t-distribution. Universality of P(Z) is, essentially, a consequence of rotational invariance of the probability ensembles we study. As an application, we study the distribution of solutions of systems of linear equations with random coefficients, and extend a classic result due to Girko.

We have investigated the characteristics of stimulated electromagnetic shock radiation (SESR) by using classical, second-order, relativistic calculations. We have derived very compact analytical expressions specifying the electric field components of SESR, which are quite suitable for numerical estimation. We have used, here, a more exact method for solving Lorentz force equations. We have evaluated all the frequency integrals by explicitly imposing the conditions contained in them. Hence we have estimated the SESR effect in different possible physical situations. We have studied, in detail, the important characteristics of SESR, such as frequency up-shift, amplification, energy output and tunability. We have calculated the numerical values of its electric field components and also its output power and frequency. We have shown that very near to the threshold of superphase motion SESR contains two components of frequency 2Ω and 4Ω, which we have named, respectively, SESR-2Ω and SESR-4Ω. The SESR-2Ω is found to be stronger than the SESR-4Ω, with power output ∼10⁶ times that of SESR-4Ω. Each of these components is seen to be monochromatic, highly up-shifted in frequency as compared to the incident laser-frequency ω₀ (10³ < Ω/ω₀ < 10⁹), highly directional, enormously amplified giving power amplification ∼(10²⁷ to 10⁴⁴) as compared to the Cherenkov radiation (that may be emitted in the absence of the laser, but under the same other conditions), coherent electromagnetic radiation which is also tunable. Because of all these interesting characteristics, SESR may be of use for the generation of high frequency coherent electromagnetic radiation such as x-ray or gamma-ray.

The authors have noticed that the result obtained in appendix C.2 and stated in section 6 of this paper is incorrect. Please see pdf for details.

Quantum mechanics is usually defined in terms of some loosely connected axioms and rules. Such a foundation is far from the beauty of, e.g., the `principles' underlying classical mechanics. Motivated, in addition, by notorious interpretation problems, there have been numerous attempts to modify or `complete' quantum mechanics. A first attempt was based on so-called hidden variables; its proponents essentially tried to expel the non-classical nature of quantum mechanics. More recent proposals intend to complete quantum mechanics not within mechanics proper but on a `higher (synthetic) level'; by means of a combination with gravitation theory (R Penrose), with quantum information theory (C M Caves, C A Fuchs) or with psychology and brain science (H P Stapp). I think it is fair to say that in each case the combination is with a subject that, per se, suffers from a very limited understanding that is even more severe than that of quantum mechanics. This was acceptable, though, if it could convincingly be argued that scientific progress desperately needs to join forces.

Quantum mechanics of a closed system was a beautiful and well understood theory with its respective state being presented as a point on a deterministic trajectory in Liouville space---not unlike the motion of a classical N-particle system in its 6N-dimensional phase-space. Unfortunately, we need an inside and an outside view, we need an external reference frame, we need an observer. This unavoidable partition is the origin of most of the troubles we have with quantum mechanics. A pragmatic solution is introduced in the form of so-called measurement postulates: one of the various incompatible properties of the system under consideration is supposed to be realized (i.e. to become a fact, to be defined without fundamental dispersion) based on `instantaneous' projections within some externally selected measurement basis. As a result, the theory becomes essentially statistical rather than deterministic; furthermore there is an asymmetry between the observed and the observing. This is the point where consciousness may come in.

Complemented by an introduction and several appendices, Henry Stapp's book consists essentially of three parts: theory, implications, and new developments. The theory part gives a very readable account of the Copenhagen interpretation, some aspects of a psychophysical theory, and, eventually, hints towards a quantum foundation of the brain--mind connection. The next part, `implications', summarizes some previous attempts to bridge the gap between the working rules of quantum mechanics and their possible consequences for our understanding of this world (Pauli, Everett, Bohm, Heisenberg). The last section, `new developments', dwells on some ideas about the conscious brain and its possible foundation on quantum mechanics.

The book is an interesting and, in part, fascinating contribution to a field that continues to be a companion to `practical' quantum mechanics since its very beginning. It is doubtful whether such types of `quantum ontologies' will ever become (empirically) testable; right now one can hardly expect more than to be offered some consistent `grand picture', which the reader may find more or less acceptable or even rewarding. Many practicing quantum physicists, though, will remain unimpressed.

The shift from synthetic ontology to analytic ontology is the foundation of the present work. This means that fundamental wholes are being partitioned into their ontologically subordinate components by means of `events'. The actual event, in turn, is an abrupt change in the Heisenberg state describing the quantum universe. The new state then defines the tendencies associated with the next actual event. To avoid infinite regression in terms of going from one state of tendencies to the next, consciousness is there to give these events a special `feel', to provide a status of `intrinsic actuality'. The brain of an alert human observer is similar in an important way to a quantum detection device: it can amplify small signals to large macroscopic effects.

On the other hand, actual events are not postulated to occur exclusively in brains. They are more generally associated with the formation of records. Records are necessarily part of the total state of the universe: it is obvious that the state of the universe cannot undergo a Schrödinger dynamics and at the same time record its own history. `The full universe consists therefore of an exceedingly thin veneer of relatively sluggish, directly observable properties resting on a vast ocean or rapidly fluctuating unobservable ones.'

The present ideas also bear on how the world should be seen to develop. While conventional cosmology encounters problems as to how to define the intial conditions, which would enter the governing equations of motion, here `the boundary conditions are set not at some initial time, but gradually by a sequence of acts that imposes a sequence of constraints. After any sequence of acts there remains a collection of possible worlds, some of which will be eliminated by the next act.' Connected with those acts is `meaning': there has always been some speculation about the special significance of local properties in our understanding of the world. One could argue that correlations (even the quantum correlations found, e.g., in the EPR-experiments) were as real as anything else. But also Stapp stresses the special role of locality: the `local observable properties, or properties similar to them are the natural, and perhaps exclusive, carriers of meaning in the quantum universe. From this point of view the quantum universe tends to create meaning.' This sounds like an absolute concept: meaning not with respect to something else, but defined intrinsically---not easy to digest.

The role of consciousness in the developing quantum universe requires more attention. `The causal irrelevance of our thoughts within classical physics constitutes a serious deficiency of that theory, construed as a description of reality.' This is taken to be entirely different within quantum mechanics. `The core idea of quantum mechanics is to describe our activities as knowledge-seeking and knowledge-using agents.' `21st century science does not reduce human beings to mechanical automata. Rather it elevates human beings to agents whose free choices can, according to the known laws, actually influence their behaviour.' An example with respect to perception is discussed: `Why, when we look at a triangle, do we experience three lines joined at three points and not some pattern of neuron firings?' The brain `does not convert an actual whole triangle into some jumbled set of particle motions; rather it converts a concatenation of separate external events into the actualization of some single integrated pattern of neural activity that is congruent to the perceived whole triangle.'

How convincing is this proposal? It is hard to tell. I think Henry Stapp did a good job, but there are tight limitations to any such endeavour. Quantum mechanics is often strange indeed, but it also gives rise to our classical world around us. For the emergence of classicality jumps and measurement projections (the basic phenomena connected with those fundamental events of choice) are not needed. Therefore, I doubt whether the explanation of the evolution of our world really allows (or requires) that much free choice. On the other hand, most scientist will agree that empirical science was not possible without free will: we could not ask independent questions if this asking was part of a deterministic trajectory. The fact that the result of a quantum measurement is indeterminate (within given probabilities) does certainly not explain free will. How about the type of measurment? The experimentalist will have to assume that he can select the pertinent observable within some limits. But given a certain design the so-called pointer basis (producing stable measurement results) is no longer a matter of free choice.

`The main theme of classical physics is that we live in a clocklike universe.' Today it is often assumed that the universe was a big (quantum-) computer or a cellular automaton. Many would be all too happy to leave that rather restrictive picture behind. But where to go? Stapp suggests giving consciousness a prominent role: `The most profound alteration of the fundamental principles was to bring consciousness of human beings into the basic structure of the physical theory.' How far we are able to go in this direction will depend on the amount of concrete research results becoming available to support this view.