The relevance of exchange effects for the stability of superheavy nuclei is examined within a linear QHD-II model by comparing Hartree-Fock with meanfield results. To allow a scan of the complete superheavy regime the recently developed local density approximation (LDA) for the exchange potential is employed for the Hartree-Fock level calculations. It turns out that, while many nuclear properties obtained with the LDA approach differ significantly from the corresponding mean-field results, the predictions of the two methods for shell closures (...) are very similar. Furthermore, a comparison with a nonlinear variant of QHD-II shows that many nuclear properties obtained with the LDA in the framework of linear QHD-II are somewhere in-between the corresponding linear and nonlinear mean-field results. This indicates that the LDA exchange partially includes nonlinear contributions, which supports the interpretation of the meson self-coupling as a parametrization of many-body effects. (shrink)

The relativistic three-particle systems are studied within the framework of Relativistic Schrödinger Theory, with emphasis on the determination of the energy functional for the stationary bound states. The phenomenon of entanglement shows up here in form of the exchange energy which is a significant part of the relativistic field energy. The electromagnetic interactions become unified with the exchange interactions into a relativistic U gauge theory, which has the Hartree–Fock equations as its non-relativistic limit. This yields a general framework for (...) treating entangled states of relativistic many-particle systems, e.g., the N-electron atoms. (shrink)

The Hartree–Fock–Popov theory of interacting Bose particles is generalized to the Cooper-pair system with a screened Coulomb repulsive interaction in high-temperature superconductors. At zero temperature, it is found that the condensate density \\) of Cooper pairs is of the order \\simeq 10^{18}\) cm\, consistently with the fact that a small fraction of the total p holes participate in pairing. We find that the phonon velocity c at zero temperature is of the order \\simeq 10\) km s\. The computation shows (...) that the transition temperature \ is a dome-shaped function of the p hole concentration \, which is consistent with experiments. At finite temperature, we find that the condensate fraction \/n\) decreases continuously from \/n\) to zero as the temperature increases from zero to the transition temperature \. For higher temperatures, we find that the repulsive interaction between Cooper pairs drives more Cooper pairs into the condensate. The computation reveals that the phonon velocity c decreases continuously from c to zero as the temperature increases from zero to the transition temperature \. The Cooper-pair system undergoes a first-order phase transition from the normal state to the BEC state. (shrink)

Within the framework of Relativistic Schrödinger Theory (RST), the scalar two-particle systems with electromagnetic interactions are treated on the basis of a non-Abelian gauge group U(2) which is broken down to the Abelian subgroup U(1)×U(1). In order that the RST dynamics be consistent with the (non-Abelian) Maxwell equations, there arises a compatibility condition which yields cross relationships for the links between the field strengths and currents of both particles such that self-interactions are eliminated. In the non-relativistic limit, the RST dynamics (...) becomes identical to the well-known Hartree–Fock equations (for spinless particles). Consequently the original RST field equations may be considered as the relativistic generalization of the Hartree–Fock equations, and the “exchange interactions” of the conventional theory (induced by the anti-symmetrization postulate) do reappear here as ordinary gauge interactions due to a broken symmetry. (shrink)

The mean-field theory with the use of the slave-boson functional method has been generalized to take account of the spin- and/or orbital-ordered state in the doubly degenerate Hubbard model. Numerical calculations are presented of the antiferromagnetic orbital-ordered state in the half-filled simple-cubic model. The orbital order in the present theory is much reduced compared with that in the Hartree–Fock approximation because of the large orbital fluctuations. From a comparison of the ground-state energy, the antiferromagnetic orbital state is shown (...) to be unstable against the antiferromagnetic spin state, although the situation becomes reversed when the exchange interaction is negative. (shrink)

The characteristic features of ortho- and para-helium are investigated within the framework of Relativistic Schrödinger Theory (RST). The emphasis lies on the conceptual level, where the geometric and physical properties of both RST field configurations are inspected in detail. From the geometric point of view, the striking feature consists in the splitting of the $\mathfrak{u}(2)$ -valued bundle connection $\mathcal{A}_{\mu}$ into an abelian electromagnetic part (organizing the electromagnetic interactions between the two electrons) and an exchange part, which is responsible for their (...) exchange interactions. The electromagnetic interactions are mediated by the usual four-potentials A μ and thus are essentially the same for both types of field configurations, where naturally the electrostatic forces (described by the time component A 0 of A μ) dominate their magnetostatic counterparts (described by the space part A of A μ). Quite analogously to this, the exchange forces are as well described in terms of a certain vector potential (B μ), again along the gauge principles of minimal coupling, so that also the exchange forces split up into an “electric” type ( $\rightsquigarrow B_{0}$ ) and a “magnetic” type ( $\rightsquigarrow {\bf B}$ ). The physical difference of ortho- and para-helium is now that the first (ortho-) type is governed mainly by the “electric” kind of exchange forces and therefore is subject to a stronger influence of the exchange phenomenon; whereas the second (para-) type has vanishing “electric” exchange potential (B 0 ≡ 0) and therefore realizes exclusively the “magnetic” kind of interactions ( $\rightsquigarrow {\bf B}$ ), which, however, in general are smaller than their “electric” counterparts. The corresponding ortho/para splitting of the helium energy levels is inspected merely in the lowest order of approximation, where it coincides with the Hartree–Fock (HF) approximation. Thus RST may be conceived as a relativistic generalization of the HF approach where the fluid-dynamic character of RST implies many similarities with the density functional theory. (shrink)

Vladimir Fock est un scientifique russo-soviétique connu pour diverses contributions à la physique moderne. Il en fut aussi un interprète, développant une position critique face à l’orthodoxie incarnée par Niels Bohr en mécanique quantique et Albert Einstein en relativité générale. Fock ayant adhéré au matérialisme dialectique, l’historiographie sur ses contributions aux débats d’interprétation met généralement l’accent sur sa défense d’une position réaliste, en faveur de l’objectivité du monde extérieur. Le présent article complète cette observation en s’appuyant sur un texte du (...) physicien soviétique jusqu’alors peu connu : « La signification fondamentale des méthodes d’approximation en physique théorique ». En effet, il révèle une forme d’antiréductionnisme essentielle à la compréhension du discours de Fock sur l’interprétation des théories de la physique moderne. (shrink)

Douglas Hartree and Hilleth Thomas were graduate students together at Cambridge University in the mid-1920s. Each developed an important approximation method to calculate the electronic structure of atoms. Each went on to make significant contributions to numerical analysis and to the development of scientific computing. Their early efforts were fused in the mid-1960s with the development of an approach to the many-particle problem in quantum mechanics called density functional theory. This paper discusses the experiences which led Hartree and Thomas (...) to their approximations, outlines the similarities in their subsequent careers, and highlights the essential role their work played in the foundational papers of modern density functional theory. (shrink)

Given the preponderance of corporate social responsibility initiatives across the corporate landscape and the correspondingly escalating demand for volunteers who participate in these initiatives, a need exists to better understand how to effectively motivate their voluntary engagement with tasks. Against this backdrop, this study argues the need to enhance their volunteer work meanings. We hypothesize that pride in volunteer work and volunteering as a calling are determinants of perceptions of the meaningfulness of volunteer work. In addition, we reveal that an (...) organization’s social responsibility climate (SRC) is a key moderator in these relationships. Interestingly, an SRC is a double-edged sword such that it strengthens the relationship between meaning and pride, yet weakens the relationship between meaning and calling. Findings are discussed, along with managerial implications and future research directions. (shrink)

Within quantum chemistry, the electron clouds that surround nuclei in atoms and molecules are sometimes treated as clouds of probability and sometimes as clouds of charge. These two roles, tracing back to Schrödinger and Born, are in tension with one another but are not incompatible. Schrödinger’s idea that the nucleus of an atom is surrounded by a spread-out electron charge density is supported by a variety of evidence from quantum chemistry, including two methods that are used to determine atomic and (...) molecular structure: the Hartree-Fock method and density functional theory. Taking this evidence as a clue to the foundations of quantum physics, Schrödinger’s electron charge density can be incorporated into many different interpretations of quantum mechanics. (shrink)

In the present work we explore the effects that an H impurity produces upon the geometry and electronic structure of the CaTiO 3 crystal considering the cubic and orthorhombic crystallographic lattices of the material. A quantum-chemical method based on the Hartree-Fock formalism and the periodic large-unit-cell model is used throughout the work. The analysis of the results shows that the interstitial H impurity binds to one of the O atoms forming the so-called O-H group. At equilibrium, the O-H distances are (...) found to be equal to 0.89 and 1.04 Å for cubic and orthorhombic lattices respectively. Atomic displacements and relaxation energies are analysed comparing the obtained results in the cubic lattice with those in the orthorhombic lattice. In the cubic phase the computed relaxation energy of vicinity of the O-H group is found to be equal to 1.1 eV and the atomic displacements generally obey the Coulomb law. So, the negatively charged O atoms move outwards from the defective region by about 0.09 Å while the positively charged Ti and Ca atoms move towards the defective region by about 0.05 and 0.01 Å respectively. A similar effect is observed in the orthorhombic lattice of CaTiO 3 doped with an H atom. It is necessary to mention that different O positions in the orthorhombic structure are considered for the O-H bond creation. The computed relaxation energies of the atomic displacements in this structure are found to be equal to 2.3 and 2.1 eV depending on the crystallographic type of the bonding O atom. (shrink)

Mathematical methods and physical ideas developed by Bogolubov are intensively used in the theory of the atomic nucleus. The variational Hartree-Fock-Bogolubov method and the self-consistent field method found wide application in calculating nuclear spectra and nucleus-nucleus interactions. The development of these methods led to the formulation of the quasiparticle-phonon nuclear model for calculating the properties of nuclear states at low, intermediate, and high excitation energies.

In a recent paper in this Journal, one of us argued against placing He above Be in Mendeleiev’s system of the elements. In it the goal was to dispute the notion that in Mendeleiev’s system of the elements the location of He should in fact lie above Be, which has a very similar electronic configuration, rather than above the noble gas column. That paper was based on rather old, Hartree–Fock limit studies on the strikingly limited non-additive contributions in the (...) He3 and He4 systems in contrast with the much larger non-additivity obtained for the Be3, Be4 and Be5 oligomers. In a recent benchmark multireference Averaged Quadratic Coupled Cluster results on Be2 and Be3 we showed that the delocalized non-additive contribution comprises 94 % of the binding energy of Be3. Here we use this and other pertinent information (drawn from the same paper) to conclude that He may not be associated with Be in Mendeleiev’s Table, despite their quite similar spectroscopic ground states. Furthermore, we use the new results to show that the large non-additivity implies that less than 2 % of the Be3 binding is located in each Be pair contained within the Be trimer. The rest of the interaction energy is necessarily delocalized over all three Be atoms. This might actually announce the bulk properties (i.e. “the electron gas”) that in solid-state physics explain the large electric and heat conduction for the solid Be metal. Thus, in the case of beryllium the metallic characteristics are already evident in Be3, a far cry from the monoatomic helium gas. (shrink)

This manuscript explores the orthogonality constraints on configurations and orbitals subject to the property that states are mutually orthogonal. The orthogonality constraints lead to properties that affect the description of chemical systems. When states are described as linear combinations of configurations, the coefficient matrix diagonalises S−1H. Therefore, single-configuration states are only possible in one-electron systems: non-orthogonal configurations yield single-configuration states only if S−1H is diagonal, but this would violate the orthonormalisation constraint. Further, the coefficient matrix is not constrained to be (...) square. Similarly, the orbitals used to construct configurations may also be orthogonal or non-orthogonal; orbitals are only required to be mutually orthogonal at the one-electron limit. Orthogonal orbitals are generally preferred due to their mathematical and conceptual simplicity, leading to fictitious unoccupied orbitals. Since the Fock operator is orthogonality agnostic, non-orthogonal orbitals can be generated by solving the Fock equation independently for each electron; the virtual orbitals produced by this conception are true excitation orbitals as they are eigensolutions of the Fock operator. Additionally, we show that the number of molecular orbitals generated is not restricted to the number of atomic orbitals employed in the computation. This manuscript explores the mathematical relationships that need to be satisfied under the various orthogonality regimes. We also present mathematical relationships that provide results that are independent of the orthogonality approximation within a particular computational method. (shrink)

The idea that the mass m of an elementary particle is explained in the semi-classical approximation by quantum-mechanical zero-point vacuum fluctuations has been applied previously to spin-1/2 fermions to yield a real and positive constant value for m, expressed through the spinorial connection Γ i in the curved-space Dirac equation for the wave function ψ due to Fock. This conjecture is extended here to bosonic particles of spin 0 and spin 1, starting from the basic assumption that all fundamental (...) fields must be conformally invariant. As a result, in curved space-time there is an effective scalar mass-squared term $m_{0}^{2}=-R/6=2\varLambda_{\mathrm{b}}/3$ , where R is the Ricci scalar and Λ b is the cosmological constant, corresponding to the bosonic zero-point energy-density, which is positive, implying a real and positive constant value for m 0, through the positive-energy theorem. The Maxwell Lagrangian density $\mathcal{L} =- \sqrt{-g}F_{ij}F^{ij}/4$ for the Abelian vector field F ij ≡A j,i −A i,j is conformally invariant without modification, however, and the equation of motion for the four-vector potential A i contains no mass-like term in curved space. Therefore, according to our hypothesis, the free photon field A i must be massless, in agreement with both terrestrial experiment and the notion of gauge invariance. (shrink)

Hamiltonian light-front dynamics of quantum fields may provide a useful approach to systematic nonperturbative approximations to quantum field theories. We investigate inequivalent Hilbert-space representations of the light-front field algebra in which the stability group of the light front is implemented by unitary transformations. The Hilbert space representation of states is generated by the operator algebra from the vacuum state. There is a large class of vacuum states besides the Fock vacuum which meet all the invariance requirements. The light-front Hamiltonian must (...) annihilate the vacuum and have a positive spectrum. We exhibit relations of the Hamiltonian to the nontrivial vacuum structure. (shrink)

It is demonstrated that in quantized general relativity one is led to Jordan-Fock type uncertainty relations implying the occurrence of cut-off lengths. We argue that these lengths (i) represent limitations on the measurability of quantum effects of general relativity and (ii) provide a cut-off length of quantum divergences.

This paper highlights some connections between work on truth approximation and work in social epistemology, in particular work on peer disagreement. In some of the literature on truth approximation, questions have been addressed concerning the efficiency of research strategies for approximating the truth. So far, social aspects of research strategies have not received any attention in this context. Recent findings in the field of opinion dynamics suggest that this is a mistake. How scientists exchange and take into account (...) information about each others’ beliefs may greatly influence the accuracy and speed with which the scientific community as a whole approximates the truth. On the other hand, social epistemologists concerned with peer disagreement have so far neglected the question of how practices of responding to disagreements with peers fare with respect to the goal of approximating the truth. Again, work on opinion dynamics shows that this may be a mistake, and that how we ought to respond to disagreements with our peers may depend on the specific purposes of our investigations. (shrink)

The main question addressed in this paper is whether some false sentences can constitute evidence for the truth of other propositions. In this paper it is argued that there are good reasons to suspect that at least some false propositions can constitute evidence for the truth of certain other contingent propositions. The paper also introduces a novel condition concerning propositions that constitute evidence that explains a ubiquitous evidential practice and it contains a defense of a particular condition concerning the possession (...) of evidence. The core position adopted here then is that false propositions that are approximately true reports of measurements can constitute evidence for the truth of other propositions. So, it will be argued that evidence is only quasi-factive in this very specific sense. (shrink)

Suppose that scientific realists believe that a successful theory is approximately true, and that constructive empiricists believe that it is empirically adequate. Whose belief is more likely to be false? The problem of underdetermination does not yield an answer to this question one way or the other, but the pessimistic induction does. The pessimistic induction, if correct, indicates that successful theories, both past and current, are empirically inadequate. It is arguable, however, that they are approximately true. Therefore, scientific realists overall (...) take less epistemic risk than constructive empiricists. (shrink)

A new approach for a uniform classification of the computably approximable real numbers is introduced. This is an important class of reals, consisting of the limits of computable sequences of rationals, and it coincides with the 0'-computable reals. Unlike some of the existing approaches, this applies uniformly to all reals in this class: to each computably approximable real x we assign a degree structure, the structure of all possible ways available to approximate x. So the main criterion for such classification (...) is the variety of the effective ways we have to approximate a real number. We exhibit extreme cases of such approximation structures and prove a number of related results. (shrink)

We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in PV₁ + dWPHP(PV).

Traditional theories construe approximate truth or truthlikeness as a measure of closeness to facts, singular facts, and idealization as an act of either assuming zero of otherwise very small differences from facts or imagining ideal conditions under which scientific laws are either approximately true or will be so when the conditions are relaxed. I first explain the serious but not insurmountable difficulties for the theories of approximation, and then argue that more serious and perhaps insurmountable difficulties for the theory (...) of idealization force us to sever its close tie to approximation. This leads to an appreciation of lawlikeness as a measure of closeness to laws, which I argue is the real measure of idealization whose main purpose is to carve nature at its joints. (shrink)

We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.

It is proposed that we use the term “approximation” for inexact description of a target system and “idealization” for another system whose properties also provide an inexact description of the target system. Since systems generated by a limiting process can often have quite unexpected, even inconsistent properties, familiar limit systems used in statistical physics can fail to provide idealizations, but are merely approximations. A dominance argument suggests that the limiting idealizations of statistical physics should be demoted to approximations.

In this article, we offer a detailed study of two important episodes in the early history of high-energy physics, namely the development of the Chew and the Nambu-Jona-Lasinio models. Our study reveals that both models resulted from the combination of an old Hamiltonian, which had been introduced by earlier researchers, and two new approximation methods developed by Chew and by Nambu and Jona-Lasinio. These new approximation methods, furthermore, were the key component behind the models’ success. We take this (...) historical investigation to support two philosophical theses about the manner in which scientific modelling operates in high-energy physics. Both of these theses run counter to a view that is commonly accepted among philosophers of science: the view that all approximations can be embedded within an equivalent idealized system, and that whatever role the former might play in scientific modelling is therefore parasitic on the much more substantial work performed by the latter. Our first thesis, which we call “Distinctness,” states that approximation methods constitute an independent category of theoretical output from idealized systems. We thus believe that approximations and idealized systems constitute two independent types of objects, both of which are essential to the practice of modelling. Our second, more radical thesis is called “Content Determination.” Our claim here is that approximation methods can in fact be essential to assigning determinate physical content to the idealized systems with which they jointly operate. As we show, this is due to the fact that quantum field theory allows for a very thin characterization of idealized systems only, making the use of approximations necessary to supply additional content. We conclude the paper with a few reflections about the manner in which our two theses can be used to articulate David Kaiser’s views on the “vanishing of scientific theory” in physics after WWII. (shrink)

Aspects of a formal theory of approximate generalizations, according to which they have degrees of truth measurable by the proportions of their instances for which they are true, are discussed. The idealizability of laws in theories of fundamental measurement is considered: given that the laws of these theories are only approximately true "in the real world", does it follow that slight changes in the extensions of their predicates would make them exactly true?

This paper, which is based on recent empirical research at the University of Leeds, the University of Edinburgh, and the University of Bristol, presents two difficulties which arise when condensed matter physicists interact with molecular biologists: (1) the former use models which appear to be too coarse-grained, approximate and/or idealized to serve a useful scientific purpose to the latter; and (2) the latter have a rather narrower view of what counts as an experiment, particularly when it comes to computer simulations, (...) than the former. It argues that these findings are related; that computer simulations are considered to be undeserving of experimental status, by molecular biologists, precisely because of the idealizations and approximations that they involve. The complexity of biological systems is a key factor. The paper concludes by critically examining whether the new research programme of ‘systems biology’ offers a genuine alternative to the modelling strategies used by physicists. It argues that it does not. (shrink)

This paper, which is based on recent empirical research at the University of Leeds, the University of Edinburgh, and the University of Bristol, presents two difficulties which arise when condensed matter physicists interact with molecular biologists: the former use models which appear to be too coarse-grained, approximate and/or idealized to serve a useful scientific purpose to the latter; and the latter have a rather narrower view of what counts as an experiment, particularly when it comes to computer simulations, than the (...) former. It argues that these findings are related; that computer simulations are considered to be undeserving of experimental status, by molecular biologists, precisely because of the idealizations and approximations that they involve. The complexity of biological systems is a key factor. The paper concludes by critically examining whether the new research programme of ‘systems biology’ offers a genuine alternative to the modelling strategies used by physicists. It argues that it does not. (shrink)

Causality is the most important topic in the history of western science, and since the beginning of the statistical paradigm, its meaning has been reconceptualized many times. Causality entered into the realm of multi-causal and statistical scenarios some centuries ago. Despite widespread critics, today deep learning and machine learning advances are not weakening causality but are creating a new way of finding correlations between indirect factors. This process makes it possible for us to talk about approximate causality, as well as (...) about a situated causality. (shrink)

A real number is recursively approximable if there is a computable sequence of rational numbers converging to it. If some extra condition to the convergence is added, then the limit real number might have more effectivity. In this note we summarize some recent attempts to classify the recursively approximable real numbers by the convergence rates of the corresponding computable sequences ofr ational numbers.

This paper describes a theory of accuracy or approximate truth and applies it to problems in the realist interpretation of scientific theories. It argues not only that realism requires approximate truth, but that an adequate theory of approximation also presupposes some elements of a realist interpretation of theories. The paper distinguishes approximate truth from vagueness, probability and verisimilitude, and applies it to problems of confirmation and deduction from inaccurate premises. Basic results are cited, but details appear elsewhere. Objections are (...) surveyed, including arguments by Miller, Laymon, and Laudan. Comparison is made with Niiniluoto's theory of verisimilitude, and the utility of his theory for realism assessed. (shrink)

We revive the idea that a deductive-nomological explanation of a scientific theory by its successor may be defensible, even in those common and troublesome cases where the theories concerned are mutually incompatible; and limiting, approximating and counterfactual assumptions may be required in order to define a logical relation between them. Our solution is based on a general characterization of limiting relations between physical theories using the method of nonstandard analysis.

In this paper, we investigate the problem of truth approximation via belief merging, i.e., we ask whether, and under what conditions, a group of inquirers merging together their beliefs makes progress toward the truth about the underlying domain. We answer this question by proving some formal results on how belief merging operators perform with respect to the task of truth approximation, construed as increasing verisimilitude or truthlikeness. Our results shed new light on the issue of how rational (dis)agreement (...) affects the inquirers’ quest for truth. In particular, they vindicate the intuition that scientific inquiry, and rational discussion in general, benefits from some heterogeneity in opinion and interaction among different viewpoints. The links between our approach and related analyses of truth tracking, judgment aggregation, and opinion dynamics, are also highlighted. (shrink)