The covariant Klein-Gordon equation requires twice the boundary conditions of the Schrödinger equation and does not have an accepted single-particle interpretation. Instead of interpreting its solution as a probability wave determined by an initial boundary condition, this paper considers the possibility that the solutions are determined by both an initial and a final boundary condition. By constructing an invariant joint probability distribution from the size of the solution space, it is shown that the usual measurement probabilities can nearly (...) be recovered in the non-relativistic limit, provided that neither boundary constrains the energy to a precision near ℏ/t 0 (where t 0 is the time duration between the boundary conditions). Otherwise, deviations from standard quantum mechanics are predicted. (shrink)

The potential term in the Schrödinger equation can be eliminated by means of a conformal transformation, reducing it to an equation for a free particle in a conformally related fictitious configuration space. A conformal transformation can also be applied to the Klein–Gordon equation, which is reduced to an equation for a free massless field in an appropriate (conformally related) spacetime. These procedures arise from the observation that the Jacobi form of the least action principle and (...) the Hamilton–Jacobi equation of classical non-relativistic mechanics can be interpreted in terms of conformal transformations. (shrink)

We illustrate, using a simple model, that in the usual formulation the time-component of the Klein–Gordon current is not generally positive definite even if one restricts allowed solutions to those with positive frequencies. Since in de Broglie's theory of particle trajectories the particle follows the current this leads to difficulties of interpretation, with the appearance of trajectories which are closed loops in space-time and velocities not limited from above. We show that at least this pathology can be avoided if (...) one adapts in a covariant form the formulation of relativistic point particle dynamics proposed by Gitman and Tyutin. (shrink)

A consistent causal interpretation of the Klein-Gordon equation treated as a field equation has been developed, and leads to a model of entities described by the Klein-Gordon equation, i.e., spinless, massive bosons, as objectively existing fields. The question arises, however, as to whether a causal interpretation based on a particle ontology of the Klein-Gordon equation is also possible. Our purpose in this article will be to indicate, by making what we believe is a best possible attempt (...) at developing a particle interpretation of the Klein-Gordon equation, that such an interpretation is untenable. To resolve the nonpositive-definite probability density difficulties with the Klein-Gordon equation, we modify this equation by the introduction of an evolution parameter. We base our subsequent considerations on this modified Klein-Gordon equation. Partly to motivate the need for a relativistic causal interpretation and partly to give emphasis to aspects of the causal interpretation often overlooked, we begin our article with a brief historical survey of the causal interpretation. (shrink)

This paper presents a new Symmetrical Interpretation (SI) of relativistic quantum mechanics which postulates: quantum mechanics is a theory about complete experiments, not particles; a complete experiment is maximally described by a complex transition amplitude density; and this transition amplitude density never collapses. This SI is compared to the Copenhagen Interpretation (CI) for the analysis of Einstein’s bubble experiment. This SI makes several experimentally testable predictions that differ from the CI, solves one part of the measurement problem, resolves some inconsistencies (...) of the CI, and gives intuitive explanations of some previously mysterious quantum effects. (shrink)

Starting from a nonlinear relativistic Klein-Gordon equation derived from the stochastic interpretation of quantum mechanics (proposed by Bohm-Vigier, (1) Nelson, (2) de Broglie, (3) Guerra et al. (4) ), one can construct joint wave and particle, soliton-like solutions, which follow the average de Broglie-Bohm (5) real trajectories associated with linear solutions of the usual Schrödinger and Klein-Gordon equations.

An elementary particle is described as a spherically symmetric solution of the Klein-Gordon equation and the Einstein equations of general relativity. It is found that it has a mass of the order of the Planck mass. If one assumes that the motion of its center of mass is determined by the Dirac equations, then it has a spin of 1/2.

It is generally believed that the de Broglie-Bohm model does not admit a particle interpretation for massive relativistic spin-0 particles, on the basis that particle trajectories cannot be defined. We show this situation is due to the fact that in the standard representation of the Klein-Gordon equation the wavefunction systematically contains superpositions of particle and anti-particle contributions. We argue that by working in a Foldy-Wouthuysen type representation uncoupling the particle from the anti-particle evolutions, a positive conserved density for a (...) particle and associated density current can be defined. For the free Klein-Gordon equation the velocity field obtained from this current density appears to be well-behaved and sub-luminal in typical instances. As an illustration, Bohmian trajectories for a spin-0 boson distribution are computed numerically for free propagation in situations in which the standard velocity field would take arbitrarily high positive and negative values. (shrink)

The formalism of (±)-frequency parts , previously applied to solution of the D'Alembert equation in the case of the electromagnetic field, is applied to solution of the Klein-Gordon equation for the K-G field in the presence of sources. Retarded and advanced field operators are obtained as solutions, whose frequency parts satisfy a complex inhomogeneous K-G equation. Fourier transforms of these frequency parts are solutions of the central equation, which determines the time dependence of the destruction/creation operators (...) of the field. The retarded field operator is resolved into kinetic and dissipative components. Correspondingly, the energy/stress tensor is resolved into three components; the power/force density, into two—a kinetic and a dissipative component. As in the analogous electromagnetic case the dissipation theorem is derived according to which work done by the dissipative power/force is negative: energy/momentum is dissipated from the sources to the K-G field. Boson quantization conditions are satisfied by the kinetic component but not by the dissipative component of the retarded K-G field. (shrink)

A Klein-Gordon-type equation onR×S 3 topology is derived, and its nonrelativistic Schrödinger equation is given. The equation is obtained with a Laplacian defined onS 3 topology instead of the ordinary Laplacian. A discussion of the solutions and the physical interpretation of the equation are subsequently given, and the most general solution to the equation is presented.

Darwinian Evolution and Classical Liberalism brings together a collection of new essays that examine the multifaceted ferment between Darwinian biology and classical liberalism.

This paper exploits the modified simple equation and dynamical system schemes to integrate the Klein–Gordon model amid quadratic nonlinearity arising in nonlinear optics, quantum theories, and solid state physics. By implementing the modified simple equation technique, we develop some disguise adaptation of analytical solutions in terms of hyperbolic, exponential, and trigonometric functions with some special parameters. We apply the dynamical system to bifurcate the model and draw distinct phase portraits on unlike parametric constraints. Following each orbit of (...) all phase portraits, we originate bounded and unbounded solitary, periodic, and periodic rogue-type wave solutions of the KG model. These two schemes extract widespread classes of solitary, periodic, and periodic rogue-type wave solutions for the KG model jointly due to restrictions on parameters. We also analyze the effect of parameters on the obtained wave solutions and discuss why and when it changes its nature. We illustrate some dynamical features of the acquired solutions via the 3D, 2D, and contour graphics. (shrink)

BackgroundAn increasing number of studies utilize intracranial electrophysiology in human subjects to advance basic neuroscience knowledge. However, the use of neurosurgical patients as human research subjects raises important ethical considerations, particularly regarding informed consent and undue influence, as well as subjects’ motivations for participation. Yet a thorough empirical examination of these issues in a participant population has been lacking. The present study therefore aimed to empirically investigate ethical concerns regarding informed consent and voluntariness in Parkinson’s disease patients undergoing deep brain (...) stimulator (DBS) placement who participated in an intraoperative neuroscience study.MethodsTwo semi-structured 30-minute interviews were conducted preoperatively and postoperatively via telephone. Interviews assessed participants’ motivations for participation in the parent intraoperative study, recall of information presented during the informed consent process, and participants’ postoperative reflections on the research study.ResultsTwenty-two participants (mean age = 60.9) completed preoperative interviews at a mean of 7.8 days following informed consent and a mean of 5.2 days prior to DBS surgery. Twenty participants completed postoperative interviews at a mean of 5 weeks following surgery. All participants cited altruism or advancing medical science as “very important” or “important” in their decision to participate in the study. Only 22.7% (n = 5) correctly recalled one of the two risks of the study. Correct recall of other aspects of the informed consent was poor (36.4% for study purpose; 50.0% for study protocol; 36.4% for study benefits). All correctly understood that the study would not confer a direct therapeutic benefit to them.ConclusionEven though research coordinators were properly trained and the informed consent was administered according to protocol, participants demonstrated poor retention of study information. While intraoperative studies that aim to advance neuroscience knowledge represent a unique opportunity to gain fundamental scientific knowledge, improved standards for the informed consent process can help facilitate their ethical implementation. (shrink)

We present an exact solution of the one-dimensional modified Klein Gordon and Duffin Kemmer Petiau equations with a step potential in the presence of minimal length in the uncertainty relation, where the expressions of the new transmission and reflection coefficients are determined for all cases. As an application, the Klein paradox in the presence of minimal length is discussed for all equations.

Intuitively, a classical field theory is background-in- dependent if the structure required to make sense of its equations is itself subject to dynamical evolution, rather than being imposed ab initio. The aim of this paper is to provide an explication of this intuitive notion. Background-independence is not a not formal property of theories: the question whether a theory is background-independent depends upon how the theory is interpreted. Under the approach proposed here, a theory is fully background-independent relative to an interpretation (...) if each physical possibility corresponds to a distinct spacetime geometry; and it falls short of full background-independence to the extent that this condition fails. (shrink)

Through the explicit introduction of hyperplane dependence as a form of relativistic dynamical evolution, we construct a manifestly covariant description of a single positive energy particle interacting with any one of a large class of “moving” external potentials. In1+1 dimensions, the simplified mathematics allows us to display a number of general properties of solutions to the equations of motion for evolution on hyperplanes.

Since 1949, the year that Kurt Gödel presented his solutions to Einstein’s field equations, there has been much discussion of time travel within the philosophical literature. Whereas theorizing about time travel had theretofore been relegated to the realm of science fiction, the imprimatur of Gödel’s work elevated the legitimacy of such discussion. It finally appeared that travel into the past might be a physical—if not yet technological—possibility. For the past few decades, philosophical inquiry into backward time travel and the closely (...) related topic of backward causation has centered on the logical possibility and conceptual coherence of these phenomena. In this paper, various conceptual problems associated with time travel concerning the nature of agency and nomological structure will be explored. However, the existence of such conceptual difficulties does not threaten the logical or physical possibility of time travel. Ultimately, the most rational stance to take regarding time travel may ultimately be one of epistemic humility, by which is meant a stance involving the admission that conceptual analysis may not constitute a very fruitful methodology for understanding the physical nature of our world. (shrink)

The long established but infrequently discussed dependence of Lorentz boost generators on the presence and nature of interactions is reviewed in this tutorial note. The last third of the note presents a discussion of the covariant transformation and evolution equations for the non-conserved partial generators of the inhomogeneous Lorentz group for interacting subsystems.

An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems, which defines a torsionless connection. One such class is that of the locally-geodesic systems, corresponding to the Levi-Civita connection. Another class, thus another connection, emerges if a preferred reference frame is available. From the classical Hamiltonian that rules geodesic motion, the correspondence yields two distinct Klein-Gordon equations and two distinct Dirac-type equations in a general metric, depending on the connection used. Each of (...) these two equations is generally-covariant, transforms the wave function as a four-vector, and differs from the Fock-Weyl gravitational Dirac equation (DFW equation). One obeys the equivalence principle in an often-accepted sense, whereas the DFW equation obeys that principle only in an extended sense. (shrink)

Accelerating Expansion explores some of the philosophical implications of modern cosmology, focused on the significance that the discovery of the accelerating expansion of the Universe has for our understanding of time, geometry, and physics. The appearance of the cosmological constant in the equations of general relativity allows one to model universes in which space has an inherent tendency towards expansion. This constant, introduced by Einstein but subsequently abandoned by him, returned to centre stage with the discovery of the accelerating expansion. (...) This pedagogically-oriented essay begins with a study of the most basic and elegant relativistic world that involves a positive cosmological constant, de Sitter spacetime. It then turns to the relatives of de Sitter spacetime that dominate modern relativistic cosmology. Some of the topics considered include: the nature of time and simultaneity in de Sitter worlds; the sense in which de Sitter spacetime is a powerful dynamical attractor; the limited extent to which observation can give us information about the topology of space in a world undergoing accelerated expansion; and cosmologists' favourite sceptical worry about the reliability of evidence and the possibility of knowledge, the problem of Boltzmann brains. (shrink)

Almost two decades ago, Prahalad and Hammond [Harv Bus Rev, 80(9):48–59, 2002] introduced the base/bottom of the pyramid (BOP) approach to profitably serving the poor with business models adapted from developed markets while alleviating poverty. In response to disappointing results and ethical criticism, the BOP approach evolved from a just-for-profit approach with a passive role of the poor to an inclusive development approach that integrates the principles of the triple bottom line. A recent review of the BOP literature [Dembek et (...) al., J Bus Ethics 165(3):365–382, 2020], however, reveals a lack of empirical evidence to support the sustainable BOP approach. In this paper, we specify the assumptions underlying the sustainable BOP approach and test them using structural equation modeling with clustered robust standard errors on a unique dataset of 212 firms. Our findings show that BOP business model involvement and adaptive capacity are significant drivers of the triple bottom line at the BOP; however, business model adaptive capacity does not guarantee an ecologically sustainable performance at the BOP. We find that there is a need for further extension of the ethical foundations of the sustainable BOP approach. (shrink)

_ Source: _Volume 39, Issue 2, pp 118 - 137 Most of the empirical research in trauma psychology focuses on posttraumatic reactions, rather than on positive outcomes besides increased attention to contributing factors to the concept of posttraumatic growth. The study presented in this paper investigates the role of religious commitment on the one hand and religious and non-religious coping on the other as contributing factors to posttraumatic growth among Internally Displaced Persons in Georgia. Based on data from _N_ = (...) 190 IDP s, Structural Equation Modeling was used to understand posttraumatic growth and the role of religiosity as a coping resource and strategy. Results indicate that religious commitment is not directly related to areas of posttraumatic growth but it has a positive impact on posttraumatic growth when it fosters religious and non-religious ways of coping. Each and every component has shown to have unique contribution to the outcome variable. Finally, implications of the practical value and for the field in general are discussed. (shrink)

The purpose of this paper is to discuss the various types of physical universe which could exist according to modern mathematical physics. The paper begins with an introduction that approaches the question from the viewpoint of ontic structural realism. Section 2 takes the case of the `multiverse' of spatially homogeneous universes, and analyses the famous Collins-Hawking argument, which purports to show that our own universe is a very special member of this collection. Section 3 considers the multiverse of all solutions (...) to the Einstein field equations, and continues the discussion of whether the notions of special and typical can be defined within such a collection. (shrink)

A quantization procedure without Hamiltonian is reported which starts from a statistical ensemble of particles of mass m and an associated continuity equation. The basic variables of this theory are a probability density ρ, and a scalar field S which defines a probability current j=ρ ∇ S/m. A first equation for ρ and S is given by the continuity equation. We further assume that this system may be described by a linear differential equation for a complex-valued (...) state variable χ. Using these assumptions and the simplest possible Ansatz χ(ρ,S), for the relation between χ and ρ,S, Schrödinger’s equation for a particle of mass m in a mechanical potential V(q,t) is deduced. For simplicity the calculations are performed for a single spatial dimension (variable q). Using a second Ansatz χ(ρ,S,q,t), which allows for an explicit q,t-dependence of χ, one obtains a generalized Schrödinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrödinger’ equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for a non-unique external q,t-dependence of χ, one obtains Schrödinger’s equation with electrodynamic potentials A,φ in the familiar gauge coupling form. This derivation shows a deep connection between non-uniqueness, quantum mechanics and the form of the gauge coupling. A possible source of the non-uniqueness is pointed out. (shrink)

Quantum Mechanics is typically divided into two parts: the unobserved amplitude given by the equations of quantum field theory and the observed measurement aspect. We argue that a better approach is insert a probability realm in the middle. The reason is that every measurement involves interactions with a complex environment where massive decoherence transforms the amplitudes into standard probabilities. The probabilities eliminate complex superpositions so that quantum states A AND B become classical states A OR B. Thus the measurement process (...) becomes a simpler and more familiar process of the observer selecting from classical type probabilities. This is close to the approach recommended by Henry Stapp. We anticipate that by using this approach many of the different interpretations of quantum mechanics become more similar to each other. (shrink)

Throughout the twentieth century, women's magazines in the United States have socialized their readers to the “proper” expression of love and anger in marriage. Our analysis of a random sample of marital advice articles from 1900 to 1979 examines this cultural convergence of gender, marriage, and emotion. A qualitative analysis identifies techniques for socializing readers to the emotional culture of marriage and shows a historical change toward equating love with self-fulfillment and advocating the expression of anger. A quantitative analysis then (...) specifies the timing of changes in emotion norms and showed an oscillation between modern and traditional norms that is related to waves of political liberation versus oppression. We discuss the relation between emotion norms and behavior and explore the effect of group conflict on the production of emotional culture. Emotion norms have become less rigid and more tolerant of diversity; but gender differences persist, and women are still responsible for maintaining intimate relationships. Historical trends in love and anger norms are nonlinear, not a continuous shift toward individualism, self-development, and free expression, as suggested by recent cultural theories. (shrink)

The following proposition appears in the context of a discussion about Melanie Klein: By seeing the unconscious as a site of sexual or verbal free fall, the humanities have aestheticized psychoanal...

This investigation examined age-related differences in auditory-visual integration as reflected on perceptual judgments of temporally misaligned AV English sentences spoken by native English and native Spanish talkers. In the detection task, it was expected that slowed auditory temporal processing of older participants, relative to younger participants, would be manifest as a shift in the range over which participants would judge asynchronous stimuli as synchronous. The older participants were also expected to exhibit greater declines in speech recognition for asynchronous AV stimuli (...) than younger participants. Talker accent was hypothesized to influence listener performance, with older listeners exhibiting a greater narrowing of the AV simultaneity window and much poorer recognition of asynchronous AV foreign-accented speech compared to younger listeners. Participant groups included younger and older participants with normal hearing and older participants with hearing loss. Stimuli were video recordings of sentences produced by native English and native Spanish talkers. The video recordings were altered in 50 ms steps by delaying either the audio or video onset. Participants performed a detection task in which they judged whether the sentences were synchronous or asynchronous, and performed a recognition task for multiple synchronous and asynchronous conditions. Both the detection and recognition tasks were conducted at the individualized signal-to-noise ratio corresponding to approximately 70% correct speech recognition performance for synchronous AV sentences. Older listeners with and without hearing loss generally showed wider AV simultaneity windows than younger listeners, possibly reflecting slowed auditory temporal processing in auditory lead conditions and reduced sensitivity to asynchrony in auditory lag conditions. However, older and younger listeners were affected similarly by misalignment of auditory and visual signal onsets on the speech recognition task. This suggests that older listeners are negatively impacted by temporal misalignments for speech recognition, even when they do not notice that the stimuli are asynchronous. Overall, the findings show that when listener performance is equated for simultaneous AV speech signals, age effects are apparent in detection judgments but not in recognition of asynchronous speech. (shrink)

This essay asks what Karl Barth meant by ‘speculation’ in volume two of the Church Dogmatics. Rather than equating speculative theology with metaphysical theology in general, Barth views speculation not as a monolithic act but as a conglomeration of modes of theological speech that undermine God's revelation in Jesus Christ. This essay argues that Barth's views of speculation, rather than undercutting the use of metaphysics in theology, pave the way for a responsible Christian use of metaphysics by tying one's use (...) of categories and concepts in theology closely to the text of Scripture. (shrink)

In some recent papers (G. ‘t Hooft and others), it has been argued that quantum mechanics can arise from classical cellular automata. Nonetheless, G. Shpenkov has proved that the classical wave equation makes it possible to derive a periodic table of elements, which is very close to Mendeleyev’s one, and describe also other phenomena related to the structure of molecules. Hence the classical wave equation complements Schrödinger’s equation, which implies the appearance of a cellular automaton molecular model (...) starting from classical wave equation. The other studies show that the microworld is constituted as a tessellation of primary topological balls. The tessellattice becomes the origin of a submicrospic mechanics in which a quantum system is subdivided to two subsystems: the particle and its inerton cloud, which appears due to the interaction of the moving particle with oncoming cells of the tessellattice. The particle and its inerton cloud periodically change the momentum and hence move like a wave. The new approach allows us to correlate the Klein-Gordon equation with the deformation coat that is formed in the tessellatice around the particle. The submicroscopic approach shows that the source of any type of wave movements including the Klein-Gordon, Schrödinger, and classical wave equations is hidden in the tessellattice and its basic exciations – inertons, carriers of mass and inert properies of matter. We also discuss possible correspondence with Konrad Zuse’s calculating space. (shrink)

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms (...) are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations. (shrink)

The Schrödinger equation for a particle of rest mass $m$ and electrical charge $ne$ interacting with a four-vector potential $A_i$ can be derived as the non-relativistic limit of the Klein–Gordon equation $\left( \Box '+m^2\right) \varPsi =0$ for the wave function $\varPsi $ , where $\Box '=\eta ^{jk}\partial '_j\partial '_k$ and $\partial '_j=\partial _j -\mathrm {i}n e A_j$ , or equivalently from the one-dimensional action $S_1=-\int m ds +\int neA_i dx^i$ for the corresponding point particle in the semi-classical (...) approximation $\varPsi \sim \exp {(\mathrm {i}S_1)}$ , both methods yielding the equation $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi $ in Minkowski space–time , where $\alpha ,\beta =1,2,3$ and $\phi =-A_0$ . We show that these two methods generally yield equations that differ in a curved background space–time $g_{ij}$ , although they coincide when $g_{0\alpha }=0$ if $m$ is replaced by the effective mass $\mathcal{M}\equiv \sqrt{m^2-\xi R}$ in both the Klein–Gordon action $S$ and $S_1$ , allowing for non-minimal coupling to the gravitational field, where $R$ is the Ricci scalar and $\xi $ is a constant. In this case $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi $ , where $\phi ^{(\mathrm g)} =\sqrt{g_{00}}$ and $\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} $ , the correctness of the gravitational contribution to the potential having been verified to linear order $m\phi ^{(\mathrm g)} $ in the thermal-neutron beam interferometry experiment due to Colella et al. Setting $n=2$ and regarding $\varPsi $ as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div ${{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0$ , where ${{\varvec{A}}}^{\alpha }=-A^{\alpha }$ and ${{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }$ . The quantum-cosmological Schrödinger (Wheeler–DeWitt) equation is also discussed in the $\mathcal{D}$ -dimensional mini-superspace idealization, with particular regard to the vacuum potential $\mathcal V$ and the characteristics of the ground state, assuming a gravitational Lagrangian $L_\mathcal{D}$ which contains higher-derivative terms up to order $\mathcal{R}^4$ . For the heterotic superstring theory , $L_\mathcal{D}$ consists of an infinite series in $\alpha '\mathcal{R}$ , where $\alpha '$ is the Regge slope parameter, and in the perturbative approximation $\alpha '|\mathcal{R}| \ll 1$ , $\mathcal V$ is positive semi-definite for $\mathcal{D} \ge 4$ . The maximally symmetric ground state satisfying the field equations is Minkowski space for $3\le {\mathcal {D}}\le 7$ and anti-de Sitter space for $8 \le \mathcal {D} \le 10$. (shrink)

Lagrangian formulation of quantum mechanical Schrödinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein–Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrödinger equation.

Abstract In this paper an algebraic method is presented to derive a 4 × 4 Hermitian Schrödinger equation from with and . The latter operator replacement is a common procedure in a quantum description of the total energy. In the derivation we don’t make use of Dirac’s method of four vectors. Moreover, the root operator isn’t squared either. Instead, use is made of the algebra of operators to derive a Hermitian matrix Schrödinger equation. We believe that new physics (...) can be obtained from an alternative quantization of the relativistic total energy. Note e.g. the pion physics behind the Klein–Gordon equation and the antimatter behind the Dirac quantization of the total relativistic energy. In this paper, for the sake of clarity, a time-only dependence of the electromagnetic potential vector is assumed. It is also demonstrated that a “latent” Lorentz invariance exists related to a derived expression for amplitude and phase. (shrink)

A completely Lorentz-invariant Bohmian model has been proposed recently for the case of a system of non-interacting spinless particles, obeying Klein-Gordon equations. It is based on a multi-temporal formalism and on the idea of treating the squared norm of the wave function as a space-time probability density. The particle’s configurations evolve in space-time in terms of a parameter σ with dimensions of time. In this work this model is further analyzed and extended to the case of an interaction with an (...) external electromagnetic field. The physical meaning of σ is explored. Two special situations are studied in depth: (1) the classical limit, where the Einsteinian Mechanics of Special Relativity is recovered and the parameter σ is shown to tend to the particle’s proper time; and (2) the non-relativistic limit, where it is obtained a model very similar to the usual non-relativistic Bohmian Mechanics but with the time of the frame of reference replaced by σ as the dynamical temporal parameter. (shrink)

Young-earth creationism is one of the more peculiar manifestations of broader evangelical culture. It continues to be the most common view of the relationship between science and Scripture held in the evangelical community and, unfortunately but understandably, the view of science most non-Christians associate with evangelicalism. For scientifically literate non-Christians, it presents an obstacle to Christian faith, and for young Christians who have been raised to equate YEC with the teaching of Scripture, it can destroy their faith altogether when its (...) falsity is discovered. With a view toward encouraging a culture of biblical and scientific literacy and overcoming the anti-intellectual legacy of fundamentalism that sustains this particular “scandal of the evangelical mind”, we offer a synoptic critique of young-earth creationism while developing and defending an evangelically acceptable alternative for understanding the relationship between God’s works and God’s words. (shrink)

A simple random walk model has been shown by Gaveauet al. to give rise to the Klein-Gordon equation under analytic continuation. This absolutely most probable path implies that the components of the Dirac wave function have a common phase; the influence of spin on the motion is neglected. There is a nonclassical path of relative maximum likelihood which satisfies the constraint that the probability density coincide with the quantum mechanical definition. In three space dimensions, and in the presence of (...) electromagnetic interaction, the Lagrangian for this optimal, nonclassical path coincides with the Lagrangian of the Dirac particle. The nonrelativistic, or diffusion, limit is shown to be a formal consequence of Einstein's dynamical equilibrium condition; the continuity equation reduces to the same diffusion equation derived from Schrödinger's equation. The relativistic, massless limit, which would describe a neutrino, is comparable (in the sense of analytic continuation) to a nonviscous liquid whose molecules possess internal degrees of freedom. (shrink)

It is shown in the present paper that the transformation relating a parallel transported vector in a Weyl space to the original one is the product of a multiplicative gauge transformation and a proper orthochronous Lorentz transformation. Such a Lorentz transformation admits a spinor representation, which is obtained and used to deduce the transportation properties of a Weyl spinor, which are then expressed in terms of a composite gauge group defined as the product of a multiplicative gauge group and the (...) spinor group. These properties render a spinor amenable to its treatment as a particle coupled to a multidimensional gauge field in the framework of the Kaluza–Klein formulation extended to multidimensional gauge fields. In this framework, a fiber bundle is constructed with a horizontal, base space and a vertical, gauge space, which is a Lie group manifold, termed its structure group. For the present, the base is the Minkowski spacetime and the vertical space is the composite gauge group mentioned above. The fiber bundle is equipped with a Riemannian structure, which is used to obtain the classical description of motion of a spinor. In its classical picture, a Weyl spinor is found to behave as a spinning charged particle in translational motion. The corresponding quantum description is deduced from the Klein–Gordon equation in the Riemann spaces obtained by the methods of path-integration. This equation in the present fiber bundle reduces to the equation for a Weyl spinor, which is close to but differs somewhat from the squared Dirac equation. (shrink)

A Dirac-type equation on R×S 3 topology is derived. It is a generalization of the previously obtained Klein-Gordon-type, Schrödinger-type, and Weyl-type equations, and reduces to the latter in the appropriate limit. The (discrete) energy spectrum is found and the corresponding complete set of solutions is given as expansions in terms of the matrix elements of the irreducible representations of the group SU 2 . Finally, the properties of the solutions are discussed.

A Weyl-type equation onR×S 3 topology is derived, as a generalization to previously obtained Klein-Gordon- and Schrödinger-type equations for the same topology. The general solution of the new equation is given as an expansion in the matrix elements of the irreducible representations of the groupSU 2. The properties of the solutions are discussed.

The Klein-Gordon equation for the stationary state of a charged particle in a spherically symmetric scalar field is partitioned into a continuity equation and an equation similar to the Hamilton-Jacobi equation. There exists a class of potentials for which the Hamilton-Jacobi equation is exactly obtained and examples of these potentials are given. The partitionAnsatz is then applied to the Dirac equation, where an exact partition into a continuity equation and a Hamilton-Jacobi equation (...) is obtained. (shrink)

The origin of the wave properties of matter is discussed from the point of view of stochastic electrodynamics. A nonrelativistic model of a charged particle with an effective structure embedded in the random zeropoint radiation field reveals that the field induces a high-frequency vibration on the particle; internal consistency of the theory fixes the frequency of this jittering at mc2/ħ. The particle is therefore assumed to interact intensely with stationary zeropoint waves of this frequency as seen from its proper frame (...) of reference; such waves, identified here as de Broglie's phase waves, give rise to a modulated wave in the laboratory frame, with de Broglie's wavelength and phase velocity equal to the particle velocity. The time-independent equation that describes this modulated wave is shown to be the stationary Schrödinger equation (or the Klein-Gordon equation in the relativistic version). In a heuristic analysis appled to simple periodic cases, the quantization rules are recovered from the assumption that for a particle in a stationary state there must correspond a stationary modulation.Along an independent and complementary line of reasoning, an equation for the probability amplitude in configuration space for a particle under a general potential V(x) is constructed, and it is shown that under conditions derived from stochastic electrodynamics it reduces to Schrödinger's equation. This equation reflects therefore the dual nature of the quantum particles, by describing simultaneously the corresponding modulated waveand the ensemble of particles. (shrink)

The solution of the Klein-Gordon equation for a complex scalar field in the presence of an electrostatic field orthogonal to a magnetostatic field is analyzed. Considerations concerning the quantum Hall-type evolution are presented also. Using the Hamiltonian with a self-interaction term, we obtain a critical value for the magnetic field in the case of the spontaneous symmetry breaking.

The Schrödinger and Klein-Gordon equations for free, structureless particles are derived classically from two different continuum approximations to a Boltzmann equation for the trace component of a mixture. The majority component is designated as the ether. Deviations from these continuum approximations (rarefied ether) yield deviations from the Schrödinger and Klein-Gordon equations which are shown explicitly.

What if gravity satisfied the Klein-Gordon equation? Both particle physics from the 1920s-30s and the 1890s Neumann-Seeliger modification of Newtonian gravity with exponential decay suggest considering a "graviton mass term" for gravity, which is _algebraic_ in the potential. Unlike Nordström's "massless" theory, massive scalar gravity is strictly special relativistic in the sense of being invariant under the Poincaré group but not the 15-parameter Bateman-Cunningham conformal group. It therefore exhibits the whole of Minkowski space-time structure, albeit only indirectly concerning volumes. (...) Massive scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein's principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity: matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric in the mass term. What is the 'true' geometry, one might wonder, in line with Poincaré's modal conventionality argument? Infinitely many theories exhibit this bimetric 'geometry,' all with the total stress-energy's trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. Much as Seeliger envisaged, the smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities---indeed an unconceived alternative. At least one version easily could have been developed before General Relativity; it then would have motivated thinking of Einstein's equations along the lines of Einstein's newly re-appreciated "physical strategy" and particle physics and would have suggested a rivalry from massive spin 2 variants of General Relativity. The Putnam-Grünbaum debate on conventionality is revisited with an emphasis on the broad modal scope of conventionalist views. Massive scalar gravity thus contributes to a historically plausible rational reconstruction of much of 20th-21st century space-time philosophy in the light of particle physics. An appendix reconsiders the Malament-Weatherall-Manchak conformal restriction of conventionality and constructs the 'universal force' influencing the causal structure. Subsequent works will discuss how massive gravity could have provided a template for a more Kant-friendly space-time theory that would have blocked Moritz Schlick's supposed refutation of synthetic _a priori_ knowledge, and how Einstein's false analogy between the Neumann-Seeliger-Einstein modification of Newtonian gravity and the cosmological constant \Lambda generated lasting confusion that obscured massive gravity as a conceptual possibility. (shrink)

Some problems relating to the probabilistic description of a free particle and of a charged particle moving in an electromagnetic field are discussed. A critical analysis of the Klein-Gordon equation and of the Dirac equation is given. It is also shown that there is no connection between commutativity of operators for physical quantities and the existence of their joint probability. It is demonstrated that the Heisenberg uncertainty relation is not universal and explained why this is so. A universal (...) uncertainty relation for canonically conjugate coordinates and momenta is suggested. (shrink)

Requiring covariance of Maxwell's equations without a priori imposing charge invariance allows for both spin-1 and spin-1/2 transformations of the complete Maxwell field and current. The spin-1/2 case yields new transformation rules, with new invariants, for all traditional Maxwell field and source quantities. The accompanying spin-1/2 representations of the Lorentz group employ the Minkowski metric, and consequently the primary spin-1/2 Maxwell invariants are also spin-1 invariants; for example, Φ2−A2, E2−B2+2iE⋅B−2. The associated Maxwell Lagrangian density is also the same for both (...) spin-1 and spin-1/2 fields. However, in the spin-1/2 case, standard field and source quantities are complex and both charge and gauge invariance are lost. Requiring the potentials to satisfy the Klein–Gordon equation equates the Maxwell and field-potential equations with two Dirac equations of the Klein–Gordon mass, and thus one complex Klein–Gordon Maxwell field describes either two real vector fields or two Dirac fields, all of the same mass. (shrink)

It has been known for long time that the cosmic sound wave was there since the early epoch of the Universe. Signatures of its existence are abound. However, such an acoustic model of cosmology is rarely developed fully into a complete framework from the notion of space, cancer therapy up to the sky. This paper may be the first attempt towards such a complete description of the Universe based on classical wave equation of sound. It is argued that one (...) can arrive at a consistent description of space, elementary particles, Sachs-Wolfe acoustic theorem, up to a novel approach for cancer therapy, starting from this simple classical wave equation of sound. We also discuss a plausible extension of Acoustic Sachs-Wolfe theorem based on its analogue with Klein-Gordon equation to become Acoustic Sachs-Wolfe-Christianto-Smarandache-Umniyati (ASWoCSU) equation. It is our hope that the new proposed equation can be verified with observation data. But we admit that our model is still in its infancy, more researches are needed to fill all the missing details. (shrink)

The special and general relativity theories are used to demonstrate that the velocity of an unradiative particle in a Schwarzschild metric background, and in an electrostatic field, is the group velocity of a wave that we call a “particle wave,” which is a monochromatic solution of a standard equation of wave motion and possesses the following properties. It generalizes the de Broglie wave. The rays of a particle wave are the possible particle trajectories, and the motion equation of (...) a particle can be obtained from the ray equation. The standing particle wave equation generalizes the Schrödinger equation of wave amplitudes. The particle wave motion equation generalizes the Klein–Gordon equation; this result enables us to analyze the essence of the particle wave frequency. The equation of the eikonal of a particle wave generalizes the Hamilton–Jacobi equation; this result enables us to deduce the general expression for the linear momentum. The Heisenberg uncertainty relation expresses the diffraction of the particle wave, and the uncertainty relation connecting the particle instant of presence and energy results from the fact that the group velocity of the particle wave is the particle velocity. A single classical particle may be considered as constituted of geometrical particle wave; reciprocally, a geometrical particle wave may be considered as constituted of classical particles. The expression for a particle wave and the motion equation of the particle wave remain valid when the particle mass is zero. In that case, the particle is a photon, the particle wave is a component a classical electromagnetic wave that is embedded in a Schwarzschild metric background, and the motion equation of the wave particle is the motion equation of an electromagnetic wave in a Schwarzschild metric background. It follows that a particle wave possesses the same physical reality as a classical electromagnetic wave. This last result and the fact that the particle velocity is the group velocity of its wave are in accordance with the opinions of de Broglie and of Schrödinger. We extend these results to the particle subjected to any static field of forces in any gravitational metric background. Therefore we have achieved a synthesis of undulatory mechanics, classical electromagnetism, and gravitation for the case where the field of forces and the gravitational metric background are static, and this synthesis is based only on special and general relativity. (shrink)