The quantization of the magnetic flux in superconducting rings is studied in the frame of a topological model of electromagnetism that gives a topological formulation of electric charge quantization. It turns out that the model also embodies a topological mechanism for the quantization of the magnetic flux with the same relation between the fundamental units of magnetic charge and flux as there is between the Dirac monopole and the fluxoid.

Electromagnetic gauge as an integration condition was my wording in previous publications. I argue here, on the examples of the Möllenstaedt-Bayh and Tonomura tests of the Ahraronov–Bohm (AB) effect, that not only the trapped flux Φ but also, under the integration condition A ≡ 0 if Φ = 0, the local value of the vector potential is measured.

Electromagnetic gauge as an integration condition was my wording in previous publications. I argue here, on the examples of the Möllenstaedt-Bayh and Tonomura tests of the Ahraronov–Bohm (AB) effect, that not only the trapped flux Φ but also, under the integration conditionA ≡ 0 if Φ = 0, the local value of the vector potential is measured.

The relation is developed between rotation generators of the Lorentz group and the magnetic fields of free-space electromagnetism. Using these classical relations, it is shown that in the quantum field theory there exists a longitudinal photomagneton, a quantized magnetic flux density operator which is directly proportional to the photon spin angular momentum. Commutation relations are given in the quantum field between the longitudinal photomagneton and the usual transverse magnetic components of quantized electromagnetism. The longitudinal component is (...) phase free, but the transverse components are phase dependent. All three components can magnetize material in general, but only the transverse components contribute to Planck's law. The photon therefore has three, not two, relativistically invariant degrees of polarization, an axial, longitudinal, polarization, and the usual right and left circular transverse polarizations. Since the longitudinal polarization is axial, it is a phase- free magnetic field. (shrink)

The explanation of inertia based on “Mach's principle” is briefly revisited and an experiment whereby the gravitational origin of inertia can be tested is described. The test consists of detecting a small stationary force with a sensitive force sensor. The force is presumably induced when a periodic transient Mach effect mass fluctuation is driven in high voltage, high energy density capacitors that are subjected to 50 kHz, 1.3 kV amplitude voltage signal, and threaded by an alternating magnetic flux (...) of the same frequency. An effect of the sort predicted is shown to be present in the device tested. It has the expected magnitude and depends on the relative phase of the Mach effect mass fluctuation and the alternating magnetic flux as expected. The observed effect also displays scaling behaviors that are unique to Mach effects. Other tests for spurious signals suggest that the observed effect is real. (shrink)

In elementary derivations of the quantization of azimuthal angular momentum the eigenfunction is determined to be exp(im φ), which is “oversensitive” to the rotation φ → φ+2π, unlessm is an integer. In a recent paper Kerner examined the classical system of charge and magnetic pole, and expressed Π, a vector constant of motion for the system, in terms of a physical angle ψ, to deduce a remarkable paradox. Kerner pointed out that Π(ψ) is “oversensitive” to ψ → ψ+2π (...) unless a certain charge quantization condition is met. Our explicandum of this paradox highlights the distinction between coordinates in classical and quantum physics. It is shown why the single-valuedness requirement on Π(ψ) is devoid of physical significance. We are finally led to examine the classical analog of the quantum mechanical argument that demonstrates the quantization of magnetic charge, to show that there is “no hope” of a classical quantization condition. (shrink)

It is shown that, in some non-QCD theories, there are effects shared by QCD: (i) in SU(2) Yang–Mills theory containing a nonlinear spinor field, there is a mass gap; (ii) in SU(3) Proca–Higgs theory, there are flux tube solutions with a longitudinal electric field required for producing a force binding quarks; (iii) in non-Abelian Proca–Higgs theories, there exist flux tube solutions with a momentum directed along the tube axis and particlelike solutions with a nonvanishing total angular momentum created (...) by crossed color electric and magnetic fields; in QCD, such configurations may contribute to the proton spin. We discuss the conjecture that such non-QCD theories might be a consequence of approximate solution of an infinite set of Dyson–Schwinger equations describing the procedure of nonperturbative quantization. The phenomenon of dimensional transmutation for nonperturbative quantization and the analogy between nonperturbative quantization and turbulence modeling are also discussed. (shrink)

The development of a coherent field-emission electron beam has made it possible to observe microscopic magnetic lines of force by detecting the electron-wave phase shifts that are due to vector potentials. Electron-holographic interference microscopy has been used to observe magnetic lines of force of magnetic vortices in superconductors, and Lorentz microscopy has been used to observe the dynamics of the vortices. The observation of vortices not only helps us understand the microscopic mechanism of flux pinning, which (...) holds the key to practical applications of superconductors, but also clarifies fundamental phenomena of superconductivity which have implications for more general physical phenomena. (shrink)

We revisit via a path-integral approach the magnetic top proposed recently by Barut, Božić, and Marić. We point out that the magnetic top has the SU(2) symmetry and that it can be viewed as a free top seen from a rotating frame. We present an alternative path-integral quantization of the magnetic top on the basis of the symmetry, and show that the magnetic coupling does not participate in altering the spin quantum numbers.

The observable phase factor is taken as a basic concept for the description of electromagnetism. Generalization of this concept toSU(2) andSU(2) × U(1) groups is carried out in such a way that the monopoles with quantized charges appear naturally and that the symmetry between the electric and magnetic phenomena is preserved. Some physical implications are discussed.

Writing about loneliness has been a struggle in the midst of the pandemic. Characterized by loneliness, isolation, anxiety, and fear, the COVID-19 pandemic is an exceptionally challenging time. At various points while navigating this loneliness project amid a particularly lonely time, we lamented the seeming futility of it all. A main goal of developing a Feminist Loneliness Studies in this introduction is to understand the ways that systems of oppression – white supremacy, settler colonialism, anti-queer bias, misogyny, neoliberal capitalism, and (...) so on – create our lonely world. To date, there remains no comprehensive feminist analysis of the structural conditions that both produce and intensify experiences of loneliness. We aim to remedy this gap. That is, we seek to address what a Feminist Loneliness Studies can contribute to understanding the complexities of this complicated emotion. For example, what is the unique loneliness of the feminist killjoy who calls out, or calls in, existing forms of queerphobia, racism, and sexism? What does it mean to be a politicized person and how does that result in both alienation and isolation? What might the relationship be between white supremacy and loneliness? How is loneliness both individual and systemic, and what is the relationship between the two? What distinctive forms of loneliness are created by ableism, sanism, neoliberalism, capitalism, globalization, and the gig economy? Ought loneliness be avoided at all costs? What are the ethics of loneliness? In our introduction to this special issue, we unpack and theorize the potential perils and generative possibilities offered up by this profound emotion. Establishing a Feminist Loneliness Studies provides us with the space we need to begin addressing and comprehending loneliness. (shrink)

Loneliness is intimately related to the ongoing epidemics of systemic forms of oppression, including white supremacy, capitalism, heteropatriarchy and settler colonialism. The epidemic of loneliness has only intensified and grown during the isolation engendered by the COVID-19 pandemic. In this article, we aim to think about how children's picturebooks wrestle with explaining loneliness and its antidotes and how these picturebooks are themselves manifestations of ongoing conversations related to Emotional Justice. We conclude by reviewing a number of children's books in order (...) to think about how the picturebook might itself be an artifact that helps to fight feelings of loneliness as well as teaching children and adults alike about the importance of connection. (shrink)

En el particular contexto filosófico de la Alemania de entreguerras, el joven H. Marcuse llevó a cabo una original recepción en clave marxista de la fenomenología existencial y de la Lebensphilosophie (I). En sus primeros artículos, «Contribuciones a una fenomenología del materialismo histórico» (1928) y «Sobre filosofía concreta» (1929), esa recepción se orientó al proyecto de elaboración de una fenomenología dialéctica y, vinculado con ello, al intento de fundamentar la acción radical en el concepto ontológico de historicidad (Geschichtlichkeit) sin descuidar (...) su componente material (II, III, IV). Asimismo, en la situación de caída (Verfallen) del Dasein, Marcuse va a otorgar una misión primordial a la filosofía concreta (V). (shrink)

Con la obra de los jóvenes hegelianos (Junghegelianer) concluye el período clásico de la filosofía alemana. Bruno Bauer (1809-1882) fue uno de sus máximos exponentes, considerado en muchos aspectos como mentor del movimiento. En el primer apartado del presente artículo, se contextualiza la figura de Bauer en el trasfondo de las disputas generadas en el seno de la escuela hegeliana. A continuación, se analizan con mayor detenimiento dos nociones centrales en su pensamiento, especialmente durante el período comprendido entre 1829 y (...) 1848: su dialéctica de la autoconciencia infinita (unendlichen Selbstbewußtsein) y la forma adoptada por su crítica pura. (shrink)

Pure quantum phenomena are characterized by intrinsic recurrences in space and time. We use this intrinsic periodicity as a quantization condition to derive a heuristic description of the essential quantum phenomenology of superconductivity. The resulting description is based on fundamental quantum dynamics and geometrical considerations, rather than on microscopical characteristics of the superconducting materials. This allows us to investigate the related gauge symmetry breaking in terms of the competition between quantum recurrence and thermal noise. We also test the validity (...) of this approach to describe the case of carbon nanotubes. (shrink)

Removing a black hole conic singularity by means of Kruskal representation is equivalent to imposing extensibility on the Kasner–Fronsdal local isometric embedding of the corresponding black hole geometry. Allowing for globally non-trivial embeddings, living in Kaluza–Klein-like M 5 × S 1 (rather than in standard Minkowski M 6 ) and parametrized by some wave number k, extensibility can be achieved for apparently “forbidden” frequencies ω in the range ω 1 (k) ≤ ω ≤ ω 2 (k). As k → 0, (...) ω 1, 2 (0) → ωH (e.g., ωH = 1/4M in the Schwarzschild case) such that the Hawking–Gibbons limit is fully recovered. The various Kruskal sheets are then viewed as slices of the Kaluza–Klein background. Euclidean k discreteness, dictated by imaginary time periodicity, is correlated with flux quantization of the underlying embedding gauge field. (shrink)

We explore the intimate connection between spacetime geometry and electrodynamics. This link is already implicit in the constitutive relations between the field strengths and excitations, which are an essential part of the axiomatic structure of electromagnetism, clearly formulated via integration theory and differential forms. We review the foundations of classical electromagnetism based on charge and magnetic flux conservation, the Lorentz force and the constitutive relations. These relations introduce the conformal part of the metric and allow the study of (...) electrodynamics for specific spacetime geometries. At the foundational level, we discuss the possibility of generalizing the vacuum constitutive relations, by relaxing the fixed conditions of homogeneity and isotropy, and by assuming that the symmetry properties of the electro-vacuum follow the spacetime isometries. The implications of this extension are briefly discussed in the context of the intimate connection between electromagnetism and the geometry of spacetime. (shrink)

I argue that the Holonomy Interpretation, at least as it has been presented in Richard Healey’s Gauging What’s Real, faces serious problems. These problems are revealed when certain approximations and idealizations that are innate in the original formulation of the Aharonov-Bohm effect are thrust aside; in particular, when the temporal dimension is taken into account. There are two ways in which time re-appears in the picture: by considering complete solutions to the original problem, where the magnetic flux is (...) static, and by examining the effects of time dependent magnetic fluxes. Both cases expose explanatory gaps in the Interpretation, as well as conflicts between it and customary ideas about relativistic locality and local action on which the Interpretation depends. (shrink)

The article illustrates an alternative topology by taking advantage of the possibilities for new forms of perception in the realm of online virtual worlds. It is based on the outcomes of the ‘Noetic Grace’ project that was held in Linden Lab’s Second Life™ (SL) platform as a part of Yoshikaze project, supported form Humlab, Umeå University. During three months of virtual residency in Yoshikaze studio, the author experimented with different kinds of spatiotemporal awareness formed from the notions of virtuality as (...) expressed during the times of late antiquity. Saint John of Damascus (c. 676–749) was speaking about ‘virtual vision’ when he referred to the concept of ‘noetic grace’. Such an approach raises some fundamental questions about whether virtual worlds can have the potential to alter visual mechanisms in the whole spectrum of imagery; both in its creation as well as in its vision. Technologies of computing can be considered as the most outstanding practice of Descartes’ theories. However even if Cartesian resonance can be regarded as an abstract Logos and as such it is clearly distinguished from the class of the world, its principal effect to western thought was that of its dualism as a deterministic organization. Hence, Euclidian plane with a chosen Cartesian system, the so-called Cartesian plane, became the standard spatial environment. In the dual process model of thinking the forward and backward reasoning can be used to express Euclidian transformations in which the points are detached. Their interaction exists solely to define themselves in relation to the other, rather than to embed themselves in an aggregate space. Nevertheless, scientific progress has demonstrated a source that spurts many kinds of phenomena that are by far distinct from those of a systematic space. Anisotropic and heterogeneous topologies, magnetic flux, non-linear, dynamic systems and emergence are some of the findings that can alter the whole spectrum of human awareness. Consequently, in order to create a being-to-be into virtual worlds, one has to overcome the notion of single-self objectiveness and its systematic space and act as a potential entity released from any references that may set barriers to a formation of a different perception and imagery. (shrink)

For time-independent fields the Aharonov-Bohm effect has been obtained by idealizing the coordinate space as multiply-connected and using representations of its fundamental homotopy group to provide information on what is physically identified as the magnetic flux. With a time-dependent field, multiple-connectedness introduces the same degree of ambiguity; by taking into account electromagnetic fields induced by the time dependence, full physical behavior is again recovered once a representation is selected. The selection depends on a single arbitrary time (hence the (...) so-called holonomies are not unique), although no physical effects depend on the value of that particular time. These features can also be phrased in terms of the selection of self-adjoint extensions, thereby involving yet another question that has come up in this context, namely, boundary conditions for the wave function. (shrink)

We propose a simple situation in which the magnetic Aharonov–Bohm potential influences the values of the deficiency indices of the initial Schrödinger operator, so determining whether the particle interacts with the solenoid or not. Even with the particle excluded from the magnetic field, the number of self-adjoint extensions of the initial Hamiltonian depends on the magnetic flux. This is a new point of view of the Aharonov–Bohm effect.

It is shown, in a large variety of manifestations, that the Aharonov—Bohm effect has classical counterparts in aspects concerning energy and momentum balance. No counterexamples are found in the cases considered, although whenever image charges shield the magnetic field region from the electric field of the passing electron the classical momentum effects, while present, would not be observable. Similarly, if the magnetic flux is maintained by superconductors, magnetic shielding will also render the classical energy effect unobservable. (...) Partial shieldings of either type will reduce but not totally eliminate the corresponding observable classical manifestations of these effects. (shrink)

We consider QED processes in the presence of an infinitely thin and infinitely long straight string with a magnetic flux inside it. The bremsstrahlung from an electron passing by the magnetic string and the electron-positron pair production by a single photon are reviewed. Based on the exact electron and positron solutions of the Dirac equation in the external Aharonov-Bohm potential we present matrix elements for these processes. The dependence of the resulting cross sections on energies, directions, and (...) polarizations of the involved particles is discussed for low energies. (shrink)

One of the most efficient nondestructive methods for pipeline in-line inspection is magnetic flux leakage inspection. Estimating the size of the defect from MFL signal is one of the key problems of MFL inspection. As the inspection signal is usually contaminated by noise, sizing the defect is an ill-posed inverse problem, especially when sizing the depth as a complex shape. An actor-critic structure-based algorithm is proposed in this paper for sizing complex depth profiles. By learning with more information (...) from the depth profile without knowing the corresponding MFL signal, the algorithm proposed saves computational costs and is robust. A pinning strategy is embedded in the reconstruction process, which highly reduces the dimension of action space. The pinning actor-critic structure helps to make the reward for critic network more efficient when reconstructing the depth profiles with high degrees of freedom. A nonlinear FEM model is used to test the effectiveness of algorithm proposed under 20 dB noise. The results show that the algorithm reconstructs the depth profile of defects with good accuracy and is robust against noise. (shrink)

We present a description of the Stern–Gerlach type experiments using only the concepts of classical electrodynamics and the Newton’s equations of motion. The quantization of the projections of the spin (or the projections of the magnetic dipole) is not introduced in our calculations. The main characteristic of our approach is a quantitative analysis of the motion of the magnetic atoms at the entrance of the magnetic field region. This study reveals a mechanism which modifies continuously the (...) orientation of the magnetic dipole of the atom in a very short time interval, at the entrance of the magnetic field region. The mechanism is based on the conservation of the total energy associated with a magnetic dipole which moves in a non uniform magnetic field generated by an electromagnet. A detailed quantitative comparison with the (1922) Stern–Gerlach experiment and the didactical (1967) experiment by J.R. Zacharias is presented. We conclude, contrary to the original Stern–Gerlach statement, that the classical explanations are not ruled out by the experimental data. (shrink)

We discuss the intertwined topics of Fulling non‐uniqueness and the Unruh effect. The Fulling quantization, which is in some sense the natural one for an observer uniformly accelerated through Minkowski spacetime to adopt, is often heralded as a quantization of the Klein‐Gordon field which is both physically relevant and unitarily inequivalent to the standard Minkowski quantization. We argue that the Fulling and Minkowski quantizations do not constitute a satisfactory example of physically relevant, unitarily inequivalent quantizations, and indicate (...) what it would take to settle the open question of whether a satisfactory example exists. A popular gloss on the Unruh effect has it that an observer uniformly accelerated through the Minkowski vacuum experiences a thermal flux of Rindler quanta. Taking the Unruh effect, so glossed, to establish that the notion of particle must be relativized to a reference frame, some would use it to demote the particle concept from fundamental status. We explain why technical results do not support the popular gloss and why the attempted demotion of the particle concept is both unsuccessful and unnecessary. Fulling non‐uniqueness and the Unruh effect merit attention despite these negative verdicts because they provide excellent vehicles for illustrating key concepts of quantum field theory and for probing foundational issues of considerable philosophical interest. (shrink)

We discuss the intertwined topics of Fulling non-uniqueness and the Unruh effect. The Fulling quantization, which is in some sense the natural one for an observer uniformly accelerated through Minkowski spacetime to adopt, is often heralded as a quantization of the Klein-Gordon field which is both physically relevant and unitarily inequivalent to the standard Minkowski quantization. We argue that the Fulling and Minkowski quantizations do not constitute a satisfactory example of physically relevant, unitarily inequivalent quantizations, and indicate (...) what it would take to settle the open question of whether a satisfactory example exists. A popular gloss on the Unruh effect has it that an observer uniformly accelerated through the Minkowski vacuum experiences a thermal flux of Rindler quanta. Taking the Unruh effect, so glossed, to establish that the notion of particle must be relativized to a reference frame, some would use it to demote the particle concept from fundamental status. We explain why technical results do not support the popular gloss and why the attempted demotion of the particle concept is both unsuccessful and unnecessary. Fulling non-uniqueness and the Unruh effect merit attention despite these negative verdicts because they provide excellent vehicles for illustrating key concepts of quantum field theory and for probing foundational issues of considerable philosophical interest. (shrink)

Gauge invariance of a manifestly covariant relativistic quantum theory with evolution according to an invariant time τ implies the existence of five gauge compensation fields, which we shall call pre-Maxwell fields. A Lagrangian which generates the equations of motion for the matter field (coinciding with the Schrödinger type quantum evolution equation) as well as equations, on a five-dimensional manifold, for the gauge fields, is written. It is shown that τ integration of the equations for the pre-Maxwell fields results in the (...) usual Maxwell equations with conserved current source. The analog of the O (3, 1) symmetry of the usual Maxwell theory is found to be O (3, 2) or O (4, 1), depending on the space-time Fourier spectrum of the field. We argue that the structure that is relevant to the description of radiation in interaction with matter evolving in a timelike sense is that of O (3, 2). The noncovariant form of the field equations is given; there are two fields of electric type and one (divergenceless) magnetic type field. The Noether currents are studied, and some remarks are made on second quantization. (shrink)

The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with structural features inherent in the Dirac equation. This paper develops a self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung and electromagnetic interactions. The model admits periodic solutions with quantized energy, and the correct magnetic moment is generated by charge circulation. It attributes to the electron an electric dipole moment rotating with ultrahigh frequency, and (...) the possibility of observing this directly as a resonance in electron channeling is analyzed in detail. Correspondence with the Dirac equation is discussed. A modification of the Dirac equation is suggested to incorporate the rotating dipole moment. (shrink)

A novel photon-based proton model is developed. A proton’s ground state is assumed to be coherent to the degree that all of its mass-energy precipitates into a single uncharged spherical structure. A quantum vortex, initiated by the strong force, but sustained in the proton’s ground state by the circular Unruh effect and a spherical Rindler horizon, is proposed to confine the proton’s mass-energy in its ground state. A direct connection between the circular Unruh effect, the zitterbewegung effect, spin, and general (...) relativity is proposed. Such a structure acts as an uncharged zitterbewegung fermion, and may explain neutrino mass. A ground-state proton’s central zitterbewegung fermion is assumed to be surrounded by a halo of charge shells of both signs. Virtual photon standing waves are assumed to synchronize the inner shell with the central zitterbewegung fermion. The charge shells are proposed to be associated with isospin and proton g-factor. There are only two model inputs—proton mass and quantized electronic charge—and just one adjustable parameter. The adjustable parameter, reduced only by about 0.4% from an initial estimate, provides the proton’s experimentally determined magnetic moment to arbitrary precision. The resulting modeled proton charge radius agrees very well with the 2018 CODATA value. Magnetic moment and charge radius are calculated algebraically in a manner easily understood by undergraduate physics students. This proposed ground-state proton model could be considered a low-energy approximation to a full quantum chromodynamical proton model. (shrink)

Different or conflicting accounts of the same episode in the history of science may arise from viewing that episode from different perspectives. The metaphor suggests that conflicting accounts can be seen as complementary, constructing a multi-dimensional understanding, if the different perspectives can be coordinated. As an example, I discuss different perspectives on the Stern-Gerlach experiment. In a static interpretation, the SGE has been viewed as an experiment that allows the determination of the magnetic moment of silver atoms. Based on (...) the concept of magnetic momentum arising from orbital angular momentum, the original experiment was designed in 1922 as an experimentum crucis to decide between Bohr’s quantum theory and classical electromagnetic theory, and its outcome was interpreted as a confirmation of the Bohr-Sommerfeld quantum postulates. After the advent of quantum mechanics, the SGE was reinterpreted in terms of magnetic moment arising from the electron’s spin angular momentum. In a dynamical interpretation, physicists have asked for the physical mechanism responsible for the quantization of the angular momentum with respect to the direction of the magnetic field. Although different suggestions were explored, none was ever accepted as fully satisfactory. Today this difficulty is seen as a paradigmatic instance of the unsolved quantum measurement problem. (shrink)

Using a framework of Dirac algebra, the Clifford algebra appropriate for Minkowski space-time, the formulation of classical electromagnetism including both electric and magnetic charge is explored. Employing the two-potential approach of Cabibbo and Ferrari, a Lagrangian is obtained that is dyality invariant and from which it is possible to derive by Hamilton's principle both the symmetrized Maxwell's equations and the equations of motion for both electrically and magnetically charged particles. This latter result is achieved by defining the variation of (...) the action associated with the cross terms of the interaction Lagrangian in terms of a surface integral. The surface integral has an equivalent path-integral form, showing that the contribution of the cross terms is local in nature. The form of these cross terms derives in a natural way from a Dirac algebraic formulation, and, in fact, the use of the geometric product of Dirac algebra is an essential aspect of this derivation. No kinematic restrictions are associated with the derivation, and no relationship between magnetic and electric charge evolves from the (classical) formulation. However, it is indicated that in bound states quantum mechanical considerations will lead to a version of Dirac's quantization condition. A discussion of parity violation of the generalized electromagnetic theory is given, and a new approach to the incorporation of this violation into the formalism is suggested. Possibilities for extensions are mentioned. (shrink)

In this paper, we report on a new approach to relativistic quantum theory. The classical theory is derived from a new implementation of the first two postulates of Einstein, which fixes the proper-time of the physical system of interest for all observers. This approach leads to a new group that we call the proper-time group. We then construct a canonical contact transformation on extended phase space to identify the canonical Hamiltonian associated with the proper-time variable. On quantization we get (...) a new relativistic wave equation for spin 1/2 particles that generalizes the Dirac theory. The Hamiltonian is positive definite so we naturally interpret antiparticles as particles with their proper-time reversed. We show that for the hydrogen atom problem, we get the same fine structure separation. When the proton spin magnetic moment is taken into account, we get the standard hyperfine splitting terms of the Pauli approximation and two additional terms. The first term is small in p-states. It diverges in s-states, and provides more than enough to account for the Lamb-shift when the proton radius is used as a cut off. The last term promises to provide a correction to the hyperfine splitting term. Although incomplete, the general approach offers hope of completely accounting for the hydrogen spectrum as an eigenvalue problem. (shrink)