# The Theory of Positrons

ArticleinPhysical Review 76(6) · January 1949with 348 Reads
Abstract
The problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the solutions of the Dirac equation. It is possible to write down a complete solution of the problem in terms of boundary conditions on the wave function, and this solution contains automatically all the possibilities of virtual (and real) pair formation and annihilation together with the ordinary scattering processes, including the correct relative signs of the various terms.
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The classical evolution of the universe can be seen as a parametrised worldline of the minisuperspace, with the time variable t being the parameter that parametrises the worldline. The time reversal symmetry of the field equations implies that for any positive oriented solution there can be a symmetric negative oriented one that, in terms of the same time variable, respectively represent an expanding and a contracting universe. However, the choice of the time variable induced by the correct value of the Schrödinger equation in the two universes makes it so that their physical time variables can be reversely related. In that case, the two universes would both be expanding universes from the perspective of their internal inhabitants, who identify matter with the particles that move in their spacetimes and antimatter with the particles that move in the time reversely symmetric universe. If the assumptions considered are consistent with a realistic scenario of our universe, the creation of a universe–antiuniverse pair might explain two main and related problems in cosmology: the time asymmetry and the primordial matter–antimatter asymmetry of our universe.
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Resonance energy transfer (RET), the transport of electronic energy from one atom or molecule to another, has significant importance to a number of diverse areas of science. Since the pioneering experiments on RET by Cario and Franck in 1922, the theoretical understanding of the process has been continually refined. This review presents a historical account of the post-Förster outlook on RET, based on quantum electrodynamics, up to the present-day viewpoint. It is through this quantum framework that the short-range, R−6 distance dependence of Förster theory was unified with the long-range, radiative transfer governed by the inverse-square law. Crucial to the theoretical knowledge of RET is the electric dipole-electric dipole coupling tensor; we outline its mathematical derivation with a view to explaining some key physical concepts of RET. The higher order interactions that involve magnetic dipoles and electric quadrupoles are also discussed. To conclude, a survey is provided on the latest research, which includes transfer between nanomaterials, enhancement due to surface plasmons, possibilities outside the usual ultraviolet or visible range and RET within a cavity.
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Quantum tunneling, a phenomenon which has no counterpart in classical physics, is the quantum-mechanical process by which a microscopic particle can transition through a potential barrier even when the energy of the incident particle is lower than the height of the potential barrier. In this work, a mechanism based on electron/positron annihilation and creation with the participation of virtual photons is proposed as an alternative to explain quantum tunneling processes.
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This monograph identifies the essential characteristics of the objects described by current quantum theory and considers their relationship to space-time. In the process, it explicates the senses in which quantum objects may be consistently considered to have parts of which they may be composed or into which they may be decomposed. The book also demonstrates the degree to which reduction is possible in quantum mechanics, showing it to be related to the objective indefiniteness of quantum properties and the strong non-local correlations that can occur between the physical quantities of quantum subsystems. Careful attention is paid to the relationships among such property correlations, physical causation, probability, and symmetry in quantum theory. In this way, the text identifies and clarifies the conceptual grounds underlying the unique nature of many quantum phenomena.
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In this paper, we point out that interactions with time delay can be described at the quantum level using a multi-time wave function $\psi(x_1,...,x_N)$, i.e., a wave function depending on one spacetime variable $x_i = (t_i,\mathbf{x}_i) \in \mathbb{R}^4$ per particle. In particular, such wave functions (first suggested by Dirac in 1932) make it possible to implement direct interaction along light cones (not mediated by fields), as in the Wheeler-Feynman (WF) formulation of electrodynamics. Our results are as follows. (1) We derive a covariant two-particle integral equation and discuss it in detail. (2) It is shown how this integral equation as well as an equivalent system of integro-differential equations can be understood as time evolution equations. As an important step, we extract a condition to extend solutions in time: the "super consistency condition". (3) Two different ways how to extend the two-particle equation to $N$ particles are presented. The first is based on the super consistency condition, the second on an analogy of the two-particle equations with classical WF electrodynamics. This analogy is interesting in its own right as it suggests a possible new quantization of WF electrodynamics. (4) Finally, we demonstrate that both $N$-particle equations reduce to the usual Schr\"odinger equation with Coulomb pair potentials if time delay effects are neglected. The equations therefore have the correct limiting behavior.
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The purpose of this study, mainly historical and pedagogical, is to investigate the physical-mathematical similitudes of the spectroscopic and beta decay Fermi theories. Both theories were formulated using quantum perturbative theory that allowed obtaining equations whose algebraic structure and physical interpretation suggest that the two phenomena occur according to the same mechanism. Fermi, therefore, could have guessed well in advance of the times that the two theories could be unified into a single physical-mathematical model that led to different results depending on the considered energy. The electroweak unification found its full realization only in the 1960s within the Standard Model that, however, is mainly based on a mathematical approach. Retracing the reasoning made by Fermi facilitates the understanding of the physical foundations that underlie the unification of the electromagnetic and weak forces.
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We investigate the idea that the universe before the Big Bang is the CPT reflection of the universe after the bang, so that the state of the universe does {\it not} spontaneously violate CPT. The universe before the bang and the universe after the bang may be viewed as a universe/anti-universe pair, created from nothing. The early universe is radiation dominated and inflationary energy is not required. We show how CPT selects a preferred vacuum state for quantum fields on such a cosmological spacetime. This, in turn, leads to a new view of the cosmological matter/anti-matter asymmetry, and a novel and economical explanation of the dark matter abundance. If we assume that the matter fields in the universe are described by the standard model of particle physics (including right-handed neutrinos), it is natural for one of the heavy neutrinos to be stable, and we show that in order to match the observed dark matter density, its mass must be $4.8\times10^{8}~{\rm GeV}$. We also obtain further predictions, including: (i) that the three light neutrinos are majorana; (ii) that the lightest of these is exactly massless; and (iii) that there are no primordial, long-wavelength gravitational waves.
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In this work, we study the magnetic effects of gravity in the framework of special relativity. Imposing covariance of the gravitational force with respect to the Lorentz transformations, we show from a thought experiment that a magnetic-like force must be present whenever two or more bodies are in motion. The exact expression for this gravitomagnetic force is then derived purely from special relativity and the consequences of such a covariant theory are developed. For instance, we show that the gravitomagnetic fields satisfy a system of differential equations similar to the Maxwell equations of electrodynamics. This implies that the gravitational waves spread out with the speed of light in a flat spacetime, which is in agreement with the recent results concerning the gravitational waves detection. We also propose that the vector potential can be associated with the interaction momentum in the same way as the scalar potential is usually associated with the interaction energy. Other topics are also discussed, for example, the transformation laws for the fields, the energy and momentum stored in the gravitomagnetic fields, the invariance of the gravitational mass and so on. We remark that is not our intention here to propose an alternative theory of gravitation but, rather, only a first approximation for the gravitational phenomena, so that it can be applied whenever the gravitational force can be regarded as an ordinary effective force field and special relativity can be used with safety. To make this point clear we present briefly a comparison between our approach and that based on the (linearized) Einstein's theory. Finally, we remark that although we have assumed nothing from the electromagnetic theory, we found that gravity and electricity share many properties in common-these similarities, in fact, are just a requirement of special relativity that must apply to any physically acceptable force field.
• Thesis
The experiments at the LHC are studying the fundamental interactions of the elementary constituents of matter, in particular in order to understand how the electroweak symmetry breaking mechanism is realized in nature. The hope is to find traces of new physics by observing deviations from the Standard Model (SM) expectations. For the interpretation of the experimental data, precise theoretical predictions are needed, both within the SM and its extensions. In this thesis, we consider a Singlet Extension of the SM with a Z2 -symmetric singlet sector in the Lagrangian and non-vanishing vacuum expectation values (vevs) both for the scalar doublet and singlet fields. The model predicts the presence of two Higgs bosons, and the non-vanishing vev for the scalar singlet leads to mixing between the two bosons. We parametrize the scalar sector of the model by a mixing angle α, a scalar self-coupling λ12 and the masses of the two Higgs bosons of the model. Considering the lightest of the two scalars of the theory to be the Higgs boson with mass about 125 GeV discovered at the LHC, we discuss theoretical constraints from perturbativity of the couplings and vacuum stability in order to find valid input values for the unknown parameters of the theory. We renormalize the theory adopting two schemes, which includes MS renormalization conditions for the parameters of the theory that are not directly accessible by experiments and on-shell renormalization conditions for all the other parameters. We investigate the conversion of the input parameters between the two schemes, finding potentially large effects. For some scenarios proposed in the literature, we compute next-to-leading-order electroweak and QCD corrections to the decays h → WW/ZZ → 4 fermions of the light Higgs boson and compare the results obtained in the two renormalization schemes. We solve the renormalization-group equations for the parameters α and λ12 , which are defined by MS renormalization conditions, and study the dependence of the h→4f decay widths on the renormalization scale. The inclusion of next-to-leading-order corrections reduces the (unphysical) scale dependence of the results drastically. For each scenario, the total decay width for the process h→4f is computed as a function of the mixing angle and compared to the SM result, finding relative deviations below 10%. Differential distributions do not show significant distortions by effects beyond the SM. The calculation are implemented in the Monte Carlo generator Prophecy4f, which is ready for further applications in data analyses in the singlet extension.
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This year is the 100th birth anniversary of Richard Philips Feynman. This article commemorates his scientific contributions and lasting legacy.
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Much of the current problems in organisations are emanating from the worldviews of the stakeholders involved in those organisations. The construct of worldview hails from the German word 'Weltanschauung'. It refers to the belief framework through which an individual or community (group, organisation, nation, civilisation) conceptualises reality and interprets the world and interacts with it. A closer look at the world around us reveals that most of our lives are being impacted by business organisations especially from the last few hundred years. Business entities are getting bigger and stronger day by day, especially with the power to take decisions which impact one and all. Therefore it builds a case for analysis of organisational worldviews and their consequences on people, profit and the planet. One way of observing a worldview is through language. Applying it to organisations brings us to the concept of metaphor. This paper addresses how languages are manifestations of worldviews and what metaphor suits the elements of a new worldview representing the call for balance between people, profit and planet. It proposes the Banyan tree as a suitable metaphor for a natural worldview.
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An overview is given of the molecular quantum electrodynamical (QED) theory of resonance energy transfer (RET). In this quantized radiation field description, RET arises from the exchange of a single virtual photon between excited donor and unexcited acceptor species. Diagrammatic time-dependent perturbation theory is employed to calculate the transfer matrix element, from which the migration rate is obtained via the Fermi golden rule. Rate formulae for oriented and isotropic systems hold for all pair separation distances, R, beyond wave function overlap. The two well-known mechanisms associated with migration of energy, namely the R−6 radiationless transfer rate due to Förster and the R−2 radiative exchange, correspond to near- and far-zone asymptotes of the general result. Discriminatory pair transfer rates are also presented. The influence of an environment is accounted for by invoking the polariton, which mediates exchange and by introducing a complex refractive index to describe local field and screening effects. This macroscopic treatment is compared and contrasted with a microscopic analysis in which the role of a neutral, polarizable and passive third-particle in mediating transfer of energy is considered. Three possible coupling mechanisms arise, each requiring summation over 24 time-ordered diagrams at fourth-order of perturbation theory with the total rate being a sum of two- and various three-body terms.
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There is a formal analogy between the evolution of the universe, when this is seen as a trajectory in the minisuperspace, and the worldline followed by a test particle in a curved spacetime. The analogy can be extended to the quantum realm, where the trajectories are transformed into wave functions that give us the probabilities of finding the universe or the particle in a given point of their respective spaces: the spacetime in the case of the particle and the minisuperspace in the case of the universe. The wave function of the spacetime and the matter fields, all together, can then be seen as a super-field that propagates in the minisuperspace and the so-called third quantisation procedure can be applied in a parallel way as the second quantisation procedure is performed with a matter field that propagates in the spacetime. The super-field can thus be interpreted as made up of universes propagating, i.e. evolving, in the minisuperspace. The corresponding Fock space for the quantum state of the multiverse is then presented. The analogy can also be used in the opposite direction. The way in which the semiclassical state of the universe is obtained in quantum cosmology from the quantum state of the universe allows us to obtain, from the quantum state of a field that propagates in the spacetime, the geodesics of the underlying spacetime as well as their quantum uncertainties or dispersions. This might settle a new starting point for a different quantisation of the spacetime coordinates.
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The approach in which the electron and positron are treated as ordinary, different particles, each being characterized by the complete set of the Dirac plane waves, is examined. This completely symmetric representation that is beyond the standard QED, makes it necessary to choose another solution of the Dirac equation for the free particle propagator as compared to that used currently. The Bethe-Salpeter equation with these particle propagators is studied in the ladder approximation. A new branch of the massless composite bosons formed by the coupled electron-positron pairs with the coupling equal to the fine structure constant, has been found. It has been obtained that: 1) the massless bosons have the normalized complex wave functions, which are transversely compressed plane waves; 2) the transverse radius of the wave functions diverges as the boson energy goes to zero that is, the composite bosons cannot be at rest; 3) the extension of the transverse wave function in the momentum space and the compression of the coordinate wave function are continuously occurred with increasing the boson energy. The reaction $e^{-}e^{+}\to B\gamma\gamma$ products of which are the massless composite boson and two photons, is investigated. The cross-section of this reaction is derived for the non-relativistic colliding beams of the spin-polarized electrons and positrons. In this case the $2\gamma$ angular correlation spectrum is characterized by a narrow peak with the full-width-at-half-maximum not exceeding 0.2 mrad. It is shown that to establish whether there is the conventional annihilation of singlet electron-positron pair with the two-photon emission or the investigated reaction, products of which are the three particles, experiments with the extremely non-relativistic colliding beams should be conducted.
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Abstract. In this paper a more detailed consideration in framework of the “The Information as Absolute” conception of the Space and Time problems “is presented. In this file two versions of the paper are presented: English, pages 1-14, and Russian, pages 15-32
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A hyperunified field theory is built in detail based on the postulates of gauge invariance and coordinate independence along with the conformal scaling symmetry. All elementary particles are merged into a single hyper-spinor field and all basic forces are unified into a fundamental interaction governed by the hyper-spin gauge symmetry SP(1,$D_h$-1). The dimension $D_h$ of hyper-spacetime is conjectured to have a physical origin in correlation with the hyper-spin charge of elementary particles. The hyper-gravifield fiber bundle structure of biframe hyper-spacetime appears naturally with the globally flat Minkowski hyper-spacetime as a base spacetime and the locally flat hyper-gravifield spacetime as a fiber that is viewed as a dynamically emerged hyper-spacetime characterized by a non-commutative geometry. The gravitational origin of gauge symmetry is revealed with the hyper-gravifield that plays an essential role as a Goldstone-like field. The gauge gravity and gravity geometry correspondences bring on the gravitational gauge geometry duality. The basic properties of hyperunified field theory and the issue on the fundamental scale are analyzed within the framework of quantum field theory, which allows us to describe the laws of nature in deriving the gauge gravitational equation with the conserved current and the geometric gravitational equations of Einstein-like and beyond.
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The decay rate of the Bunch-Davies state of a massive scalar field in the expanding flat spatial sections of de Sitter space is determined by an analysis of the particle pair creation process in real time. The Feynman definition of particle and antiparticle Fourier mode solutions of the scalar wave equation, and their adiabatic phase analytically continued to the complexified time domain, show conclusively that the Bunch-Davies state is not the vacuum state at late times. The closely analogous creation of charged particle pairs in a uniform electric field is reviewed and Schwinger's result for the vacuum decay rate is recovered by the real time analysis. The vacuum decay rate in each case is also calculated by switching the background field on adiabatically, allowing it to act for a very long time, and then adiabatically switching it off again. In both the uniform electric field and de Sitter cases the particles created while the field is switched on are verified to be real, in the sense that they persist in the final asymptotic flat zero-field region. In the de Sitter case there is an interesting residual dependence of the rate on how the de Sitter phase is ended, indicating a greater sensitivity to spatial boundary conditions. The electric current of the created particles in the E-field case and their energy density and pressure in the de Sitter case are also computed, and the magnitude of their backreaction effects on the background field estimated. Possible consequences of the Hubble scale instability of the de Sitter vacuum for cosmology, vacuum dark energy, and the cosmological `constant' problem are discussed.
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Cambridge Core - Theoretical Physics and Mathematical Physics - Mass Dimension One Fermions - by Dharam Ahluwalia
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It is shown that a Dirac(-type) equation for a rank-two bi-spinor field on Minkowski (configuration) spacetime furnishes a Lorentz-covariant quantum-mechanical wave equation in position-space representation for a single free photon. This equation does not encounter any of the roadblocks that have obstructed previous attempts (by various authors) to formulate a {quantum-mechanical} photon wave equation. In particular, it implies that the photon wave function yields conserved non-negative Born-rule-type quantum probabilities, and that its probability current density four-vector transforms properly under Lorentz transformations. Moreover, the eigenvalues of the pertinent photon Dirac Hamiltonian and the vector eigenvalues of the photon momentum operator yield the familiar Einstein relations $E=\hbar\omega$ and ${\bf p}=\hbar{\bf k}$, respectively. Furthermore, these spin-1 wave modes are automatically transversal without the need of an additional constraint on the initial data. Some comments on other proposals to set up a photon wave equation are supplied as well.
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In this paper we describe in detail a graphical method allowing the computation of field operator transformations in quantum optics (QO). Its applications include beam splitters (BS), Mach-Zehnder interferometers (MZI), optical resonators (Fabry–Perot etc) as well as non-linear crystals featuring the process of spontaneous parametric down-conversion (SPDC). Its main advantage compared to the traditional computation step-by-step method is its visual and intuitive approach, somehow similar to Feynman's diagrammatic approach in Quantum Field Theory. It also seems adapted to computer-based implementations since calculations mainly consist on complex additions and multiplications, not matrix operations.
• Chapter
The step from the Gell-Mann and Low (gml) formulation of the pt expansion of the electron propagator toward a diagramatic representation is enabled by Wick’s theorem. According to this theorem, the expectation values of time-ordered fermion operator products arising in the gml expression can be evaluated in terms of contractions of operator pairs (Sect. 5.1). The contractions are related to free electron propagators and can be represented graphically by a directed line between the respective fermion operators. Together with the wiggly lines as graphical symbols for the interaction integrals, this allows one to replace the original analytic terms with diagrams (Sect. 5.2). The contraction concept allows one to distinguish linked and unlinked contributions (or diagrams) to the propagator pt expansion. The linked-cluster theorem, discussed in Sect. 5.3, states that the denominator in the gml expressions exactly cancels all unlinked parts so that only linked contributions (or diagrams) need to be considered in the pt expansion of the electron propagator.
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We consider a new approach to the quantum field theory description of processes passing at finite space-time intervals. The formalism is based on the Feynman diagram technique in the coordinate representation supplemented with modified rules for the transition to the momentum representation reflecting specific experimental situations. This effectively implies that only the particle propagators in the momentum representation are modified, while the standard Feynman rules in the momentum representation remain unchanged. No wave packets are used in the approach, i.e., the initial and final states of particles are described by plane waves, which significantly simplifies the calculations. Three processes—neutrino oscillations, unstable particle decay, and neutral kaon oscillations—are used as examples to show that the proposed approach correctly reproduces the well-known results.
• Article
A detailed analysis of electron–positron pair creation induced by a spatially non-uniform and static electric field from vacuum is presented. A typical example is provided by the Sauter potential. For this potential, we derive the analytic expressions for vacuum decay and pair production rate accounted for the entire range of spatial variations. In the limit of a sharp step, we recover the divergent result due to the singular electric field at the origin. The limit of a constant field reproduces the classical result of Euler, Heisenberg and Schwinger, if the latter is properly averaged over the width of a spatial variation. The pair production by the Sauter potential is described for different regimes from weak to strong fields. For all these regimes, the locally constant-field rate is shown to be the upper limit.
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In quantum mechanics the time dimension is treated as a parameter, while the three space dimensions are treated as observables. This assumption is both untested and inconsistent with relativity. From dimensional analysis, we expect quantum effects along the time axis to be of order an attosecond. Such effects are not ruled out by current experiments. But they are large enough to be detected with current technology, if sufficiently specific predictions can be made. To supply such we use path integrals. The only change required is to generalize the usual three dimensional paths to four. We predict a large variety of testable effects. The principal effects are additional dispersion in time and full equivalence of the time/energy uncertainty principle to the space/momentum one. Additional effects include interference, diffraction, and entanglement in time. The usual ultraviolet divergences do not appear: they are suppressed by a combination of dispersion in time and entanglement in time. The approach here has no free parameters; it is therefore falsifiable. As it treats time and space with complete symmetry and does not suffer from the ultraviolet divergences, it may provide a useful starting point for attacks on quantum gravity.
• Article
My analysis of temporal direction begins by establishing that time-reversal scenarios, scenarios in which the direction of time itself is reversed, whether locally or globally, are incoherent. Building on this conclusion, I argue that temporal directionality cannot be defined or explicated in terms of processes in time, such as the movements of celestial bodies, biological evolution or radioactive decay. In other words, while it is easy to imagine any process occurring in reverse, one cannot define the "earlier"/"later" relation by appeal to the stages of a process, for example, the consecutive stages of radioactive decay. The second section of the paper turns to a study of our firsthand experiences, and qualifies this result by arguing that experiences constitute the one class of events that cannot so much as be conceived as reversed in time. In the third section I suggested that our firsthand experiences figure as a standard by means of which temporal direction is given to us. I further argue that, despite the pivotal role played by experience in this scheme, we are fully warranted in being realists with respect to temporal directionality.
• In this paper, I take up the following puzzle: If Feynman diagrams represent states of affairs, but do not do so truthfully what can their epistemic value be? I argue that Feynman diagrams have been epistemically powerful (at least in part) because, as pictorial representations, they facilitate an understanding of quantum electrodynamics, and quantum field theories more generally. Drawing on Richard Feynman’s own remarks and Catherine Z. Elgin’s account of the role of understanding in science, I tease out what it might mean to have an understanding of something that is not factive. Although my approach allows for a thin sense of substantively non-factive epistemic success, it is continuous with a factive sense of understanding that is more familiar in the sciences.
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The paper aimed to analyze the decay mode of 253 kinds of electron particles and one kind of collision reaction in order to discover the objective existence of the decay state of the electron particles. Meanwhile, the paper also intended to research the cause of the magnetic moment anomaly of the electron to obtain the theoretical value of g, and this value was compared with the experimental value: the two values have 12 significant figures as the same, and the error is within 5.3E-13. Furthermore, the proposed method was compared with QED method for advantage and disadvantage analysis in the aspects of action type, theoretical accuracy of magnetic moment anomaly, particle state, basic reaction type, potential energy form, formula for magnetic moment anomaly of electron and seven major items of detail. The proposed method is superior in all above aspects and can effectively avoid the three problems exposed in QED method, namely: intrinsic property explanation by external factors, inconsistence between μ theoretical value and experimental value and point state difficulty, so the proposed method becomes the most reasonable theory for explaining the magnetic moment anomaly of electron.
• Chapter
In the first part of this chapter, a brief introduction into the Standard Model of particle physics and its interactions is given. This is followed by a discussion of the formalism which is needed to describe proton–proton (pp) collisions. Also the extraction of the needed ingredients to predict the outcome of these collisions is described, followed by a discussion of the Drell–Yan and photon induced process. Finally, the limitations and problems of the Standard Model are discussed and some theories which aim to solve these limitations are presented. The chapter ends with a discussion of models predicting new physics in the final state of a charged lepton and neutrino. The discussion follows to large parts the discussion in (Zinser M, Double differential cross section for Drell–Yan production of high-mass e$$^{+}$$e$$^{-}$$-pairs in pp collisions at $$\sqrt{s} =$$ 8 TeV with the ATLAS experiment, 2013)
• Article
A consistent (off-shell) canonical classical and quantum dynamics in the framework of special relativity was formulated by Stueckelberg in 1941, and generalized to many-body theory by Horwitz and Piron in 1973 (SHP). In this paper, this theory is embedded into the framework of general relativity (GR), here denoted by SHPGR. The canonical Poisson brackets of the SHP theory remain valid (invariant under local coordinate transformations) on the manifold of GR, and provide the basis for formulating a canonical quantum theory. A scalar product is defined for constructing the Hilbert space and a Hermitian momentum operator defined. The Fourier transform is defined, connecting momentum and coordinate representations. The potential which may occur in the SHP theory emerges as a spacetime scalar mass distribution in GR, and electromagnetism corresponds to a gauge field on the quantum mechanical SHPGR Hilbert space in both the single particle and many-body theory. A diffeomorphism covariant form of Newton’s law is found as an immediate consequence of the canonical formulation of SHPGR. We compute the classical evolution of the off shell mass on the orbit of a particle and the force on a particle and its energy at the Schwarzschild horizon. The propagator for evolution of the one-body quantum state is studied and a scattering theory on the manifold is worked out.
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