It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instan...It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equiv展开更多
Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the stand...Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.展开更多
The least massive fermion generation is attributed to an analogue of Weyl curvature which occurs when a closed, spin-string sweeps out a closed world tube: , where S represents string length. A second order tube and c...The least massive fermion generation is attributed to an analogue of Weyl curvature which occurs when a closed, spin-string sweeps out a closed world tube: , where S represents string length. A second order tube and consequent second order fermion mass are associated with a closed tube which circulates and itself sweeps out a closed tube: . Finally a Kth order tube and kth order fermion generation are associated with the general expression . By hypothesis six world tube orders establish an SU(3) symmetry and each closed tube interacts with a SUGRA connection of spin-. Such connections can either be photon-fermion composites or composites that consist strictly of fermions. Interactions that involve no photons are, by hypothesis unobserved and are therefore associated with closed world tubes that are classified as dark mass-energy. It is demonstrated that interactions involving ordinary mass-energy are identities (e.g. interactions that are incapable of generating the proposed SU(3) symmetry). It is therefore concluded that dark mass-energy is a necessary condition for the SU(3) symmetry that by hypothesis characterizes the proposed model. Since 95% of the mass-energy in the universe is regarded as dark, the total mass-energy that constitutes the proposed SU(3) symmetry can be calculated as , where QL is a left-handed quark, where ΨL is a left-handed spin particle and where is a right-handed anti-lepton. Thus the mass-energy that is associated with the wave is about 1067 GeV/c2 (the approximate mass of a typical galaxy). This wave is regarded by hypothesis as a single galactic unit and as the ground state of a large-scale quantization;i.e. as the ground state of a series of abstract waves which mimic de Broglie waves in the sense that the ground state is a wave of one anti-node which oscillates about a wave length that parallels a geodesic on a smallest abstract spherical shell. The first excited state is a wave of two anti-nodes that oscillate about a wave length that parallels a geodesic on the second smalles展开更多
Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms. They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the wel...Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms. They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schr?dinger Picture. These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality. For particles passing through nonvanishing magnetic or electric fields they lead to cancellations of Aharonov-Bohm phases at the observation point, generalizing earlier semiclassical experimental observations (of Werner & Brill) to delocalized (spread-out) quantum states. This leads to a correction of previously unnoticed sign-errors in the literature, and to a natural explanation of the deeper reason why certain time-dependent semiclassical arguments are consistent with static results in purely quantal Aharonov-Bohm configurations. These nonlocalities also provide a remedy for misleading results propagating in the literature (concerning an uncritical use of Dirac phase factors, that persists since the time of Feynman’s work on path integrals). They are shown to conspire in such a way as to exactly cancel the instantaneous Aharonov-Bohm phase and recover Relativistic Causality in earlier “paradoxes” (such as the van Kampen thought-experiment), and to also complete Peshkin’s discussion of the electric Aharonov-Bohm effect in a causal manner. The present formulation offers a direct way to address time-dependent single- vs double-slit experiments and the associated causal issues—issues that have recently attracted attention, with respect to the inability of current theories to address them.展开更多
We extend the linear quantum transformation theory to the case of quantum field operators. The corresponding general transformation expressions of CPT transformations and gauge field transformations are considered as ...We extend the linear quantum transformation theory to the case of quantum field operators. The corresponding general transformation expressions of CPT transformations and gauge field transformations are considered as its applications.展开更多
文摘It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equiv
基金the National Natural Science Foundation of China(Grant Nos.11601312,11631007,and 11875040)
文摘Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.
文摘The least massive fermion generation is attributed to an analogue of Weyl curvature which occurs when a closed, spin-string sweeps out a closed world tube: , where S represents string length. A second order tube and consequent second order fermion mass are associated with a closed tube which circulates and itself sweeps out a closed tube: . Finally a Kth order tube and kth order fermion generation are associated with the general expression . By hypothesis six world tube orders establish an SU(3) symmetry and each closed tube interacts with a SUGRA connection of spin-. Such connections can either be photon-fermion composites or composites that consist strictly of fermions. Interactions that involve no photons are, by hypothesis unobserved and are therefore associated with closed world tubes that are classified as dark mass-energy. It is demonstrated that interactions involving ordinary mass-energy are identities (e.g. interactions that are incapable of generating the proposed SU(3) symmetry). It is therefore concluded that dark mass-energy is a necessary condition for the SU(3) symmetry that by hypothesis characterizes the proposed model. Since 95% of the mass-energy in the universe is regarded as dark, the total mass-energy that constitutes the proposed SU(3) symmetry can be calculated as , where QL is a left-handed quark, where ΨL is a left-handed spin particle and where is a right-handed anti-lepton. Thus the mass-energy that is associated with the wave is about 1067 GeV/c2 (the approximate mass of a typical galaxy). This wave is regarded by hypothesis as a single galactic unit and as the ground state of a large-scale quantization;i.e. as the ground state of a series of abstract waves which mimic de Broglie waves in the sense that the ground state is a wave of one anti-node which oscillates about a wave length that parallels a geodesic on a smallest abstract spherical shell. The first excited state is a wave of two anti-nodes that oscillate about a wave length that parallels a geodesic on the second smalles
文摘Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms. They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schr?dinger Picture. These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality. For particles passing through nonvanishing magnetic or electric fields they lead to cancellations of Aharonov-Bohm phases at the observation point, generalizing earlier semiclassical experimental observations (of Werner & Brill) to delocalized (spread-out) quantum states. This leads to a correction of previously unnoticed sign-errors in the literature, and to a natural explanation of the deeper reason why certain time-dependent semiclassical arguments are consistent with static results in purely quantal Aharonov-Bohm configurations. These nonlocalities also provide a remedy for misleading results propagating in the literature (concerning an uncritical use of Dirac phase factors, that persists since the time of Feynman’s work on path integrals). They are shown to conspire in such a way as to exactly cancel the instantaneous Aharonov-Bohm phase and recover Relativistic Causality in earlier “paradoxes” (such as the van Kampen thought-experiment), and to also complete Peshkin’s discussion of the electric Aharonov-Bohm effect in a causal manner. The present formulation offers a direct way to address time-dependent single- vs double-slit experiments and the associated causal issues—issues that have recently attracted attention, with respect to the inability of current theories to address them.
文摘We extend the linear quantum transformation theory to the case of quantum field operators. The corresponding general transformation expressions of CPT transformations and gauge field transformations are considered as its applications.
