A fundamental tenet of general relativity is geodesic motion of point particles. For extended objects, however, tidal forces make the trajectories deviate from geodesic form. In fact Mathisson, Papapetrou, and others have found that even in the limit of very small size there exists a residual curvature-spin force. Another important physical case is that of field theory. Here the ray (WKB) approximation may be used to obtain the equation of motion. In this article I consider an alternative procedure, the proper (...) time translation operator formalism, to obtain the covariant Heisenberg equations for the quantum velocity, momentum, and angular momentum operators for the case of spinor fields. I review the flat spacetime results for Dirac particles in Yang-Mills fields, where we recover the Lorentz force. For curved spacetime I find that the geodesic equation is modified by an additional term involving the spin tensor, and the parallel transport equation for the momentum is modified by an additional term involving the curvature tensor. This curvature term is the “Lorentz force” of the gravitational field. The main result of this article is that these equations are exactly the (symmetrized) Mathisson-Papapetrou equations for the quantum operators. Extension of these results to the case of spin-one fields may be possible by use of the KDP formalism. (shrink)

Relativistic quantum mechanics has been formulated as a theory of the evolution ofevents in spacetime; the wave functions are square-integrable functions on the four-dimensional spacetime, parametrized by a universal invariant world time τ. The representation of states with spin is induced with a little group that is the subgroup of O(3, 1) leaving invariant a timelike vector nμ; a positive definite invariant scalar product, for which matrix elements of tensor operators are covariant, emerges from this construction. In a previous study (...) a second-order equation was introduced similar to the second-order Dirac equation, based on a quadratic function of two operators which are the self-adjoint parts, in this new scalar product, of γμpμ. It is shown in this paper that one of these operators, in fact the one from which the gyromagnetic moment is obtained, can be used to construct a first-order equation. The corresponding quantum theory is somewhat analogous to Dirac's spinor form; the Hamilton equations appear to describe dynamical degrees of freedom in a spacelike hyperplane orthogonal to nμ (in Dirac's theory the motion appears to be lightlike). It is shown that the integration over nμ required by unitarity results in timelike motion (as in the expectation value of Dirac's α). Explicit forms are obtained for the wave functions and currents for free motion. The general form of the theory is written for the (five-dimensional) pre-Maxwell fields required by gauge invariance. (shrink)

We consider the relativistic statistical mechanics of an ensemble of N events with motion in space-time parametrized by an invariant “historical time” τ. We generalize the approach of Yang and Yao, based on the Wigner distribution functions and the Bogoliubov hypotheses to find approximate dynamical equations for the kinetic state of any nonequilibrium system, to the relativistic case, and obtain a manifestly covariant Boltzmann- type equation which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU) equation for indistinguishable particles. This (...) equation is then used to prove the H-theorem for evolution in τ. In the equilibrium limit, the covariant forms of the standard statistical mechanical distributions are obtained. We introduce two-body interactions by means of the direct action potential V(q), where q is an invariant distance in the Minkowski space-time. The two- body correlations are taken to have the support in a relative O(2, 1)-invariant subregion of the full spacelike region. The expressions for the energy density and pressure are obtained and shown to have the same forms (in terms of an invariant distance parameter) as those of the nonrelativistic theory and to provide the correct nonrelativistic limit. (shrink)

In this paper the Lorentz transformations (LT) and the standard transformations (ST) of the usual Maxwell equations (ME) with the three-dimensional (3D) vectors of the electric and magnetic fields, E and B, respectively, are examined using both the geometric algebra and tensor formalisms. Different 4D algebraic objects are used to represent the usual observer dependent and the new observer independent electric and magnetic fields. It is found that the ST of the ME differ from their LT and consequently that (...) the ME with the 3D E and B are not covariant upon the LT but upon the ST. The obtained results do not depend on the character of the 4D algebraic objects used to represent the electric and magnetic fields. The Lorentz invariant field equations are presented with 1-vectors E and B, bivectors EHv and BHv and the abstract tensors, the 4-vectors Ea and Ba. All these quantities are defined without reference frames, i.e., as absolute quantities. When some basis has been introduced, they are represented as coordinate-based geometric quantities comprising both components and a basis. It is explicitly shown that this geometric approach agrees with experiments, e.g., the Faraday disk, in all relatively moving inertial frames of reference, which is not the case with the usual approach with the 3D bf E and B and their ST. (shrink)

BackgroundThere is much overlap among the symptomology of autistic spectrum disorders, obsessive compulsive disorders, and alexithymia, which all typically involve impaired social interactions, repetitive impulsive behaviors, problems with communication, and mental health.AimThis study aimed to identify direct and indirect associations among alexithymia, OCD, cardiac interoception, psychological inflexibility, and self-as-context, with the DV ASD and depression, while controlling for vagal related aging.MethodologyThe data involved electrocardiogram heart rate variability and questionnaire data. In total, 1,089 participant's data of ECG recordings of healthy resting (...) state HRV were recorded and grouped into age categories. In addition to this, another 224 participants completed an online survey that included the following questionnaires: Yale-Brown Obsessive Compulsive Scale ; Toronto Alexithymia Scale 20 ; Acceptance and Action Questionnaire ; Depression, Anxiety, and Stress Scale 21 ; Multi-dimensional Assessment of Interoceptive Awareness Scale ; and the Self-as-Context Scale.ResultsHeart rate variability was shown to decrease with age when controlling for BMI and gender. In the two SEMs produced, it was found that OCD and alexithymia were causally associated with autism and depression indirectly through psychological inflexibility, SAC, and ISen interoception.ConclusionThe results are discussed in relation to the limitations of the DSM with its categorical focus of protocols for syndromes and provide support for more flexible ideographic approaches in diagnosing and treating mental health and autism within the Extended Evolutionary Meta-Model. Graph theory approaches are discussed in their capacity to depict the processes of change potentially even at the level of the relational frame. (shrink)

We construct an explicit covariant Majorana formulation of Maxwell electromagnetism which does not make use of vector 4-potential. This allows us to write a “Dirac” equation for the photon containing all the known properties of it. In particular, the spin and (intrinsic) boost matrices are derived and the helicity properties of the photon are studied.

The traditional approach to the covariance problem in quantum mechanics is inverted and the space-time transformations are assumed as the basicunknowns, according to the prescription that the correspondence principle and the commutation rules must becovariant. It is shown that the only solutions are either Galilean or Lorentzian (including the possibility of an imaginary light-velocity c2<0). The Dirac formalism for the wave-equation and the condition c2>0 are obtained simoultaneously as theunique solution, provided that the Hamiltonian is Hermitean (in the usual (...) sense), and the internal degrees of freedom allow for afinite-dimensional representation. Infinite-dimensional representations are introduced in order to extend the Hamiltonian formalism to other spinors. (shrink)

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrödinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner–Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.

In this paper the quantum covariant relativistic dynamics of many bodies is reconsidered. It is emphasized that this is an event dynamics. The events are quantum statistically correlated by the global parameter τ. The derivation of an event Boltzmann equation emphasizes this. It is shown that this Boltzmann equation may be viewed as exact in a dilute event limit ignoring three event correlations. A quantum entropy principle is obtained for the marginal Wigner distribution function. By means of event linking (concatenations) (...) particle properties such as the equation of state may be obtained. We further reconsider the generalized quantum equilibrium ensemble theory and the free event case of the Fermi-Dirac and Bose-Einstein distributions, and some consequences. The ultra-relativistic limit differs from the non-covariant theory and is a test of this point of view. (shrink)

This paper examines the Stark effect, as a first order perturbation of manifestly covariant hydrogen-like bound states. These bound states are solutions to a relativistic Schrödinger equation with invariant evolution parameter, and represent mass eigenstates whose eigenvalues correspond to the well-known energy spectrum of the nonrelativistic theory. In analogy to the nonrelativistic case, the off-diagonal perturbation leads to a lifting of the degeneracy in the mass spectrum. In the covariant case, not only do the spectral lines split, but they acquire (...) an imaginary part which is linear in the applied electric field, thus revealing induced bound state decay in first order perturbation theory. This imaginary part results from the coupling of the external field to the non-compact boost generator. In order to recover the conventional first order Stark splitting, we must include a scalar potential term. This term may be understood as a fifth gauge potential, which compensates for dependence of gauge transformations on the invariant evolution parameter. (shrink)

We present a simple first step toward a relativistically covariant generalization of the Bohm-Bub hidden-variable theory. The model is applicable to spin measurement on a single Dirac particle and describes the collapse of the state vector to a spin-up or spin-down state. The essential postulate is that the hidden-variable vector transforms in the same way as the state vector under a Lorentz transformation. This yields a covariant collapse equation, which reduces to the ordinary Bohm-Bub equation for an observer stationary with (...) respect to the particle and shows a time dilated collapse for a moving observer. The model yields the correct quantum transition probabilities for all observers. (shrink)

We give a review of recent progress on the application of the relativistic chiral SU(3) Lagrangian to meson–baryon scattering. It is shown that a combined chiral and 1/Nc expansion of the Bethe–Salpeter interaction kernel leads to a good description of the kaon–nucleon, antikaon–nucleon and pion–nucleon scattering data typically up to laboratory momenta of p lab ≃500 MeV. We solve the covariant coupled channel Bethe–Salpeter equation with the interaction kernel truncated to chiral order Q 3 where we include only those terms (...) which are leading in the large Nc limit of QCD. (shrink)

The objective of the study was to develop a structural model that explores the relationship between Mathematics Performance and students’ self-regulated learning skills, grit, and expectancy-value towards science, technology, engineering and mathematics (STEM). The research collected survey data from 664 senior high school students from 17 STEM high schools, and conducted a covariance-based structural equation modeling (SEM) analysis. The results of the SEM analysis indicate that the Re-specified Self-Regulated Learning Skill – Expectancy-Value towards STEM – Grit – Mathematics Performance (...) (Re-specified SRL-EV-GR-MP) model is the most parsimonious fit, offering the best empirical support for the theoretical model of the study. The research findings suggest that the mathematics performance of senior high school students in STEM curriculum is attributed to their high expectancies for success and perceived values of the STEM tasks, high grit, and high self-regulated learning skills. Moreover, the research also observed evidence of mediating and moderating grit effects in the concurrent effects of expectancy-values towards STEM and self-regulated learning skills towards students’ mathematics performance. (shrink)

This paper deals with a number of technical achievements that are instrumental for a dis-solution of the so-called "Hole Argument" in general relativity. Such achievements include: 1) the analysis of the "Hole" phenomenology in strict connection with the Hamiltonian treatment of the initial value problem. The work is carried through in metric gravity for the class of Christoudoulou-Klainermann space-times, in which the temporal evolution is ruled by the "weak" ADM energy; 2) a re-interpretation of "active" diffeomorphisms as "passive and metric-dependent" (...) dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose their (up to now unknown) connection to gauge transformations on-shell; understanding such connection also enlightens the real content of the Hole Argument or, better, dis-solves it together with its alleged "indeterminism"; 3) the utilization of the Bergmann-Komar "intrinsic pseudo-coordinates", defined as suitable functionals of the Weyl curvature scalars, as tools for a peculiar gauge-fixing to the super-hamiltonian and super-momentum constraints; 4) the consequent construction of a "physical atlas" of 4-coordinate systems for the 4-dimensional "mathematical" manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent "Dirac observables"). Such construction embodies the "physical individuation" of the points of space-time as "point-events", independently of the presence of matter, and associates a "non-commutative structure" to each gauge fixing or four-dimensional coordinate system; 5) a clarification of the multiple definition given by Peter Bergmann of the concept of "(Bergmann) observable" in general relativity. This clarification leads to the proposal of a "main conjecture" asserting the existence of i) special Dirac's observables which are also Bergmann's observables, ii) gauge variables that are coordinate independent (namely they behave like the tetradic scalar fields of the Newman-Penrose formalism). A by-product of this achievements is the falsification of a recently advanced argument asserting the absence of (any kind of) "change" in the observable quantities of general relativity. 6) a clarification of the physical role of Dirac and gauge variables as their being related to "tidal-like" and "inertial-like" effects, respectively. This clarification is mainly due to the fact that, unlike the standard formulations of the equivalence principle, the Hamiltonian formalism allows to define notion of "force" in general relativity in a natural way; 7) a proposal showing how the physical individuation of point-events could in principle be implemented as an experimental setup and protocol leading to a "standard of space-time" more or less like atomic clocks define standards of time. We conclude that, besides being operationally essential for building measuring apparatuses for the gravitational field, the role of matter in the non-vacuum gravitational case is also that of "participating directly" in the individuation process, being involved in the determination of the Dirac observables. This circumstance leads naturally to a peculiar new kind of "structuralist" view of the general-relativistic concept of space-time, a view that embodies some elements of both the traditional "absolutist" and "relational" conceptions. In the end, space-time point-events maintain a "peculiar sort of objectivity". Some hints following from our approach for the quantum gravity programme are also given. (shrink)

In his general theory of relativity (GR) Einstein sought to generalize the special-relativistic equivalence of inertial frames to a principle according to which all frames of reference are equivalent. He claimed to have achieved this aim through the general covariance of the equations of GR. There is broad consensus among philosophers of relativity that Einstein was mistaken in this. That equations can be made to look the same in different frames certainly does not imply in general that (...) such frames are physically equivalent. We shall argue, however, that Einstein's position is tenable. The equivalence of arbitrary frames in GR should not be equated with relativity of arbitrary motion, though. There certainly are observable differences between reference frames in GR (differences in the way particles move and fields evolve). The core of our defense of Einstein's position will be to argue that such differences should be seen as fact-like rather than law-like in GR. By contrast, in classical mechanics and in special relativity (SR) the differences between inertial systems and accelerated systems have a law-like status. The fact-like character of the differences between frames in GR justifies regarding them as equivalent in the same sense as inertial frames in SR. (shrink)

What is the meaning of general covariance? We learn something about it from the hole argument, due originally to Einstein. In his search for a theory of gravity, he noted that if the equations of motion are covariant under arbitrary coordinate transformations, then particle coordinates at a given time can be varied arbitrarily - they are underdetermined - even if their values at all earlier times are held fixed. It is the same for the values of fields. The (...) argument can also be made out in terms of transformations acting on the points of the manifold, rather than on the coordinates assigned to the points. So the equations of motion do not fix the particle positions, or the values of fields at manifold points, or particle coordinates, or fields as functions of the coordinates, even when they are specified at all earlier times. It is surely the business of physics to predict these sorts of quantities, given their values at earlier times. The principle of general covariance therefore seems untenable. (shrink)

In his general theory of relativity Einstein sought to generalize the special-relativistic equivalence of inertial frames to a principle according to which all frames of reference are equivalent. He claimed to have achieved this aim through the general covariance of the equations of GR. There is broad consensus among philosophers of relativity that Einstein was mistaken in this. That equations can be made to look the same in different frames certainly does not imply in general that such (...) frames are physically equivalent. We shall argue, however, that Einstein's position is tenable. The equivalence of arbitrary frames in GR should not be equated with relativity of arbitrary motion, though. There certainly are observable differences between reference frames in GR. The core of our defense of Einstein's position will be to argue that such differences should be seen as fact-like rather than law-like in GR. By contrast, in classical mechanics and in special relativity the differences between inertial systems and accelerated systems have a law-like status. The fact-like character of the differences between frames in GR justifies regarding them as equivalent in the same sense as inertial frames in SR. (shrink)

The Einstein equations can be written as Fierz-Pauli equations with self-interaction, $W\gamma _{ik} = - G_{ik} + \tfrac{1}{2}g_{ik} g^{mn} G_{mn} - k(T_{ik} - \tfrac{1}{2}g_{ik} g^{mn} T_{mn} )$ together with the covariant Hilbert-gauge condition, $(\gamma _i^h - \tfrac{1}{2}\delta _i^k g^{mn} \gamma _{mn} )_{;k} = 0$ where W denotes the covariant wave operator and G ik the Einstein tensor of the metric g ik collecting all nonlinear terms of Einstein's equations. As is known, there do not, however, exist plane-wave (...) solutions γ ik(z)with g ik Z,i Z,k=0of these equations such that what is essential to the introduction of gravitons is not satisfied in general relativity. This nonexistence corresponds with the uncertainty relation,Δp(Δg*)2(Δx)3≥h hG/ c 3 concerning the total nonlinear gravitational field g *ik =g k +γ k. (shrink)

It is common in the literature on classical electrodynamics and relativity theory that the transformation rules for the basic electrodynamical quantities are derived from the pre-assumption that the equations of electrodynamics are covariant against these---unknown---transformation rules. There are several problems to be raised concerning these derivations. This is, however, not our main concern in this paper. Even if these derivations were completely correct, they leave open the following fundamental question: Are the so-obtained transformation rules indeed identical with the true (...) transformation rules of the fundamental electrodynamical quantities? In other words, is it indeed the case that the values calculated from the quantities in one inertial frame by means of the transformation rules we derived are equal to the values of the same quantities obtained by the same operations with the same measuring equipments when they are co-moving with the other inertial frame? This is of course an empirical question. In this paper, we will investigate the problem in a purely theoretical framework by applying what J. S. Bell calls “Lorentzian pedagogy”---according to which the laws of physics in any one reference frame account for all physical phenomena. We will show that the transformation rules of the electrodynamical quantities are indeed identical with the ones obtained by presuming the covariance of the equations of electrodynamics, and that the covariance is indeed satisfied. Beforehand, however, we need to clarify the operational definitions of the fundamental electrodynamical quantities. As we will see, these semantic issues are not as trivial as one might think. (shrink)

Einstein and Hilbert both struggled to reconcile general covariance and causality in their early work on general relativity. In Einstein’s case, this first led to his infamous “hole argument”, a stumbling block that persuaded him early on that generally covariant field equations for gravitation could never be found. After his breakthrough to general covariance in the fall of 1915, the resolution came in form of the “point-coincidence argument.” Hilbert from the beginning took a different view of the (...) “causality problem,” though he shifted his position somewhat in the light of Einstein’s breakthrough in November 1915. Nevertheless, his aim was to establish initial conditions that would lead to a well-defined Cauchy problem in general relativity. Hilbert consistently advocated the use of coordinate conditions in order to obtain solutions of the field equations that would maintain the causal ordering of events. Einstein’s “causality problem” thus differs from that of Hilbert, and the latter was never a victim of Einstein’s “hole argument.”. (shrink)

We give picture-covariant formulations of the equations of motion for observables and states such that the Hamiltonian operator is transformed asH-0304;=U(t)HU † (t) under a time-dependent unitary transformationU(t). Next, we consider the explicit and implicit covariance of Heisenberg's equations of motion for observables with respect to general transformations of coordinate operators. Most of our representation is spread out over a number of textbooks and articles, where the subject has been considered with greater or lesser clarity from different (...) points of view. (shrink)

The Maxwell equations are shown to be the one-photon spin-one quantum equations. All Maxwell equations (without sources) are derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. The wavefunction is a linear combination of the electric and magnetic fields. The procedure is not unique, there are ambiguities of adding a scalar field. A quaternionic representation of the Maxwell equations (with sources) is constructed, a covariant reformulation of (...) which is presented. Whittaker potentials are analysed. Conservation laws are derived using a method of “pseudo-Lagrangians.”. (shrink)

In order to get to a geometrically based theory of gravitation and electromagnetism, a gauge covariant bimetric tetrad space-time is introduced. The Weylian connection vector is derived from the tetrads and it is identified with the electromagnetic potential vector. The formalism is simplified by the use of gauge-invariant quantities. The theory contains a contorsion tensor that is connected with spinning properties of matter. The electromagnetic field may be induced by conventional sources and by spinning matter. In absence of spinning matter, (...) the equations are identical with those of the gauge-covariant bimetric theory.(23). (shrink)

In this paper we study a classical and theoretical system which consists of an elastic medium carrying transverse waves and one point-like high elastic medium density, called concretion. We compute the equation of motion for the concretion as well as the wave equation of this system. Afterwards we always consider the case where the concretion is not the wave source any longer. Then the concretion obeys a general and covariant guidance formula, which leads in low-velocity approximation to an equivalent de (...) Broglie-Bohm guidance formula. The concretion moves then as if exists an equivalent quantum potential. A strictly equivalent free Schrödinger equation is retrieved, as well as the quantum stationary states in a linear or spherical cavity. We compute the energy of the concretion, naturally defined from the energy density of the vibrating elastic medium. Provided one condition about the amplitude of oscillation is fulfilled, it strikingly appears that the energy and momentum of the concretion not only are written in the same form as in quantum mechanics, but also encapsulate equivalent relativistic formulas. (shrink)

Structural equation model trees are data-driven tools for finding variables that predict group differences in SEM parameters. SEM trees build upon the decision tree paradigm by growing tree structures that divide a data set recursively into homogeneous subsets. In past research, SEM trees have been estimated predominantly with the R package semtree. The original algorithm in the semtree package selects split variables among covariates by calculating a likelihood ratio for each possible split of each covariate. Obtaining these likelihood ratios is (...) computationally demanding. As a remedy, we propose to guide the construction of SEM trees by a family of score-based tests that have recently been popularized in psychometrics. These score-based tests monitor fluctuations in case-wise derivatives of the likelihood function to detect parameter differences between groups. Compared to the likelihood-ratio approach, score-based tests are computationally efficient because they do not require refitting the model for every possible split. In this paper, we introduce score-guided SEM trees, implement them in semtree, and evaluate their performance by means of a Monte Carlo simulation. (shrink)

We show that particle-antiparticle exchange and covariant motion reversal are two physically different aspects of the same mathematical transformation, either in the prequantal relativistic equation of motion of a charged point particle, in the general scheme of second quantization, or in the spinning wave equations of Dirac and of Petiau-Duffin-Kemmer. While, classically, charge reversal and rest mass reversal are equivalent operations, in the wave mechanical case mass reversal must be supplemented by exchange of the two adjoint equations, implying (...) ψ ⇄ $\bar \psi$ .Denoting by M the rest mass reversal, P the parity reversal, T the Racah time reversal, and Z the ψ ⇄ $\bar \psi$ exchange, the connection with the usual scheme of charge conjugation, parity reversal, and Wigner motion reversal, is with, of course. (shrink)

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)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov (...) constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

An overlap between the general relativist and particle physicist views of Einstein gravity is uncovered. Noether's 1918 paper developed Hilbert's and Klein's reflections on the conservation laws. Energy-momentum is just a term proportional to the field equations and a "curl" term with identically zero divergence. Noether proved a \emph{converse} "Hilbertian assertion": such "improper" conservation laws imply a generally covariant action. Later and independently, particle physicists derived the nonlinear Einstein equations assuming the absence of negative-energy degrees of freedom for (...) stability, along with universal coupling: all energy-momentum including gravity's serves as a source for gravity. Those assumptions imply that the energy-momentum is only a term proportional to the field equations and a symmetric curl, which implies the coalescence of the flat background geometry and the gravitational potential into an effective curved geometry. The flat metric, though useful in Rosenfeld's stress-energy definition, disappears from the field equations. Thus the particle physics derivation uses a reinvented Noetherian converse Hilbertian assertion in Rosenfeld-tinged form. The Rosenfeld stress-energy is identically the canonical stress-energy plus a Belinfante curl and terms proportional to the field equations, so the flat metric is only a convenient mathematical trick without ontological commitment. Neither generalized relativity of motion, nor the identity of gravity and inertia, nor substantive general covariance is assumed. The more compelling criterion of lacking ghosts yields substantive general covariance as an output. Hence the particle physics derivation, though logically impressive, is neither as novel nor as ontologically laden as it has seemed. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov (...) constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (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 problem of two relativistically-moving pointlike particles of constant mass is undertaken in an arbitrary Lorentz frame using the classical Lagrangian mechanics of Stückelberg, Horwitz, and Piron. The particles are assumed to interact at events along their world lines at a common “world time,” an invariant dynamical parameter which is not in general synchronous with the particle proper time. The Lorentz-scalar interaction is assumed to be the Coulomb potential (i.e., the inverse square spacetime potential) of the spacetime event separation. The (...) classical orbit equations are found in 1 + 1 spacetime dimensions in the hyperbolic angle coordinates for the reduced problem. The solutions to the reduced motion in these coordinates are the spacetime generalizations of the nonrelativistic Kepler solutions. and they introduce an invariant eccentricity which is a function of other known constants of the motion for the reduced problem. Solutions compatible with physical scattering are obtained by the assumption that the eccentricity is a given function of the ratio of the particle masses. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov (...) constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

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)

We discuss, within the framework provided by a recently developed variational method, transposition-invariant field equations for unified field theories. Systems that are, in addition, invariant under Weyl-type gauge transformations or lambda transformations are derived. It is found that in a weak field limit two of the systems contain the equations of general relativity and the covariant Maxwell equations for a charge-free region.

In the early history of spinors it became evident that a single undotted covariant elementary spinor can represent a plane wave of light. Further study of that relation shows that plane electromagnetic waves satisfy the Weyl equation, in a way that indicates the correct spin angular momentum. On the subatomic scale the Weyl equation discloses more detail than the vector equations. The spinor and vector equations are equivalent when applied to plane waves, and more generally (in the absence (...) of sources) on the large scale when the spinors are incoherent. (shrink)

The spinor covariant derivative through which the equations of quantum fields are generalized to include gravitational coupling has a direct and simple geometric significance. The formula for the difference of two spinor covariant derivatives taken in different order is derived geometrically; and the geometric proof of the covariant constancy of the spin-1/2 γ-matrices in curved space is given.

A direct method showing the Thomas precession for an evolution of any vector quantity (a spatial part of a four-vector) is proposed. A useful application of this method is a possibility to trace correctly the presence of the Thomas precession in the Bargmann-Michel-Telegdi equation. It is pointed out that the Thomas precession is not incorporated in the kinematical term of the Bargmann-Michel-Telegdi equation, as it is commonly believed. When the Bargmann-Michel-Telegdi equation is interpreted in curved spacetimes, this term is shown (...) to be equivalent to the affine connection term in the covariant derivative of the spin four-vector evolving in a gravitational field. It then contributes to the geodetic precession. The described problem is an interesting and unexpected example showing that approximate methods used in special relativity, in this case to identify the Thomas precession, can distort the true meaning of physical laws. (shrink)

The matrix notation of paper I is extended to include first-rank spinors expressed as two-component spin-vectors. Well-known two-component and four-component spinor equations are expressed in this notation. In addition, it is shown how other covariant wave equations can easily be invented. A certain nonlinear equation is found to have only positive-energy solutions for particles and antiparticles.

Using covariant derivatives and the operator definitions of quantum mechanics, gauge invariant Proca and Lehnert equations are derived and the Lorenz condition is eliminated in U(1) invariant electrodynamics. It is shown that the structure of the gauge invariant Lehnert equation is the same in an O(3) invariant theory of electrodynamics.

Based on de Broglie’s wave hypothesis and the covariant ether, the Three Wave Hypothesis (TWH) has been proposed and developed in the last century. In 2007, the author found that the TWH may be attributed to a kinematical classical system of two perpendicular rolling circles. In 2012, the author showed that the position vector of a point in a model of two rolling circles in plane can be transformed to a complex vector under a proposed effect of partial observation. In (...) the present project, this concept of transformation is developed to be a lab observation concept. Under this transformation of the lab observer, it is found that velocity equation of the motion of the point is transformed to an equation analogising the relativistic quantum mechanics equation (Dirac equation). Many other analogies has been found, and are listed in a comparison table. The analogy tries to explain the entanglement within the scope of the transformation. These analogies may suggest that both quantum mechanics and special relativity are emergent, both of them are unified, and of the same origin. The similarities suggest analogies and propose questions of interpretation for the standard quantum theory, without any possible causal claims. (shrink)

In this work a calculation of the cluster decomposable formalism for relativistic scattering as developed by Lindesay, Markevich, Noyes, and Pastrana (LMNP) is made for an ultra-light quantum model. After highlighting areas of the theory vital for calculation, a description is made of the process to go from the general theory to an eigen-integral equation for bound state problems, and calculability is demonstrated. An ultra-light quantum exchange model is then developed to examine calculability.

In this work, a neo-classical relativistic mechanics theory is presented where the spin of an electron is an inherent part of its world space-time path as a point particle. The fourth-order equation of motion corresponds to the same covariant Lagrangian function in proper time as in special relativity except for an additional spin energy term. The theory provides a hidden-variable model of the electron where the dynamic variables give a complete description of its motion, giving a classical mechanics explanation of (...) the electron’s spin, its dipole moments, and Schrödinger’s zitterbewegung, These features are also described mathematically by quantum mechanics theory, of course, but without any physical picture of an underlying reality. The total motion of the electron can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called here the spin center. The global motion is sub-luminal and described by Newton’s Second Law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac’s velocity operators having magnitude c. The local spin motion is an inherent perpetual motion, which for a free electron is periodic at the ultra-high zitterbewegung frequency and its path is circular in a spin-center reference frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron’s point charge. The electric dipole energy corresponds to the spin-orbit coupling term involving the electric field that appears in the corrected Pauli non-relativistic Hamiltonian, which has long been used to explain the doublet structure of the spectral lines of the excited hydrogen atom. Pauli’s spin-orbit term is usually derived, however, from his magnetic dipole energy term, including also the effect of Thomas precession, which halves this energy. The magnetic dipole energy from Pauli’s and Dirac’s theory is twice that in the neo-classical theory, a discrepancy that has not been resolved. By defining a spin tensor as the angular momentum of the electron’s total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut–Zanghi electron theory. This allows the equations of motion to be expressed in an equivalent form involving operators applied to a state function of proper time satisfying a neo-classical Dirac–Schrödinger spinor equation. This state function produces the dynamic variables from the same operators as in Dirac’s theory for the electron but without any probability implications. It leads to a neo-classical wave function that satisfies Dirac’s relativistic wave equation for the free electron by applying the Lorentz transformation to express proper time in the state function in terms of an observer’s space-time coordinates, showing that there is a close connection between the neo-classical theory and quantum mechanics theory for the electron’s dynamics. (shrink)

Conventionally, multilevel analysis of covariance (ML-ANCOVA) has been the recommended approach for analyzing treatment effects in quasi-experimental multilevel designs with treatment application at the cluster-level. In this paper, we introduce the generalized ML-ANCOVA with linear effect functions that identifies average and conditional treatment effects in the presence of treatment-covariate interactions. We show how the generalized ML-ANCOVA model can be estimated with multigroup multilevel structural equation models that offer considerable advantages compared to traditional ML-ANCOVA. The proposed model takes into account (...) measurement error in the covariates, sampling error in contextual covariates, treatment-covariate interactions, and stochastic predictors. We illustrate the implementation of ML-ANCOVA with an example from educational effectiveness research where we estimate average and conditional effects of early transition to secondary schooling on reading comprehension. (shrink)

A covariant quantum mechanics for systems of finite-mass particles at finite energy follows from interpreting as Wick-Yukawa fluctuations in particle number the quantum fluctuations which are needed by Phipps to understand measurement theory and by Gyftopoulos to understand the second law of thermodynamics. The dynamical one-variable equations require as input the (N − 1)-particle transition matrices and an N-N vertex or coupling constants at three-particle vertices.

Linear structural equation models (SEMs) are widely used in sociology, econometrics, biology, and other sciences. A SEM (without free parameters) has two parts: a probability distribution (in the Normal case specified by a set of linear structural equations and a covariance matrix among the “error” or “disturbance” terms), and an associated path diagram corresponding to the functional composition of variables specified by the structural equations and the correlations among the error terms. It is often thought that the (...) path diagram is nothing more than a heuristic device for illustrating the assumptions of the model. However, in this paper, we will show how path diagrams can be used to solve a number of important problems in structural equation modelling. (shrink)

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as (...) those based on the maximum likelihood solution, e.g., output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters of underidentified models, as we illustrate on a simple errors-in-variables model. (shrink)