A distribution of electromagnetic fields presents a statistical assembly of a particular type, which is at scale h a quantum statistical assembly itself and has also been instrumental to concretisation of the basic probability assumption of quantum mechanics. Of specific concern in this discussion is an extensive train of radiation fields, of a total wave function ψ, which are continuously (re)emitted and (re)absorbed by an oscillatory (point) charge of a zero rest mass and yet a finite dynamical mass, with the waves and charge together making up an extensive undulatory IED particle. The IED particle will as any real particle be subject to interactions with the environmental fields and particles, hence to excitations, and therefore will explore all possible states over time; at scale $h$ the states are discrete. On the basis of the principles of statistics and statistical mechanics combined with first principles solutions for the IED particle, we derive for the IED particle the probability functions in position space, of a form |ψ|^2, and in dynamical-variable space.
A distribution of electromagnetic fields presents a statistical assembly of a particular type, which is at scale h a quantum statistical assembly itself and has also been instrumental to concretisation of the basic probability assumption of quantum mechanics. Of specific concern in this discussion is an extensive train of radiation fields, of a total wave function ψ, which are continuously (re)emitted and (re)absorbed by an oscillatory (point) charge of a zero rest mass and yet a finite dynamical mass, with the waves and charge together making up an extensive undulatory IED particle. The IED particle will as any real particle be subject to interactions with the environmental fields and particles, hence to excitations, and therefore will explore all possible states over time; at scale h the states are discrete. On the basis of the principles of statistics and statistical mechanics combined with first principles solutions for the IED particle, we derive for the IED particle the probability functions in position space, of a form |ψ|^2, and in dynamical-variable space.
An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics . In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics , deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.
An Ising-type classical statistical model is shown to describe quantum fermions. For a suitable time-evolution law for the probability distribution of the Ising-spins our model describes a quantum field theory for Dirac spinors in external electromagnetic fields, corresponding to a mean field approximation to quantum electrodynamics. All quantum features for the motion of an arbitrary number of electrons and positrons, including the characteristic interference effects for two-fermion states, are described by the classical statistical model. For one-particle states in the non-relativistic approximation we derive the Schrödinger equation for a particle in a potential from the time evolution law for the probability distribution of the Ising-spins. Thus all characteristic quantum features, as interference in a double slit experiment, tunneling or discrete energy levels for stationary states, are derived from a classical statistical ensemble. Concerning the particle-wave-duality of quantum mechanics, the discreteness of particles is traced back to the discreteness of occupation numbers or Ising-spins, while the continuity of the wave function reflects the continuity of the probability distribution for the Ising-spins.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
Abstract This paper is a study of a statistical ensemble of classical harmonic oscillators which is stationary in time, and whose position and momentum distribution functions are those of the corresponding quantum-mechanical oscillator in its ground state. If the oscillating particle is charged, then in order to maintain the distribution stationary a certain random electromagnetic field must be present. The intensity distribution of the radiation field is calculated, and it is found to be identical with the ‘photon vacuum’ of quantum electrodynamics. It is suggested, therefore, that this radiation field, which in quantum field theory is treated entirely formally, might exist in the classical sense. The method is extended to the excited states. It is found that the classical ensembles corresponding to these have probability distributions which may be negative. However, when attention is shifted to the quantum-mechanical ‘mixture’, this is no longer the case. Furthermore, the intensity distribution of the radiation field at temperature T is shown to be simply the sum of the Planck distribution and the previously obtained zero-temperature field. The application of these results to statistical mechanics is discussed. In this paper the treatment is non-relativistic throughout, but radiative corrections of the type which give rise to the Lamb shift are an integral part of the theory.
Abstract The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted principle of relativity, since a Hamiltonian function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic. This is the main object of the present paper. The theory is noil-relativistic only on account of the time being counted throughout as a c-number, instead of being treated symmetrically with the space co-ordinates. The relativity variation of mass with velocity is taken into account without difficulty. The underlying ideas of the theory are very simple. Consider an atom interacting with a field of radiation, which we may suppose for definiteness to be confined in an enclosure so as to have only a discrete set of degrees of freedom. Resolving the radiation into its Fourier components, we can consider the energy and phase of each of the components to be dynamical variables describing the radiation field. Thus if Er is the energy of a component labelled r and θr is the corresponding phase (defined as the time since the wave was in a standard phase), we can suppose each Er and θr to form a pair of canonically conjugate variables. In the absence of any interaction between the field and the atom, the whole system of field plus atom will be describable by the Hamiltonian H ═ ΣrEr + Ho equal to the total energy, Ho being the Hamiltonian for the atom alone, since the variables Er, θr obviously satisfy their canonical equations of motion Er ═ — ∂H/∂θr ═ 0, θr ═ ∂H/∂Er ═ 1.
It is by the long established fact in experiment and theory that electromagnetic waves, here as one component of an IED particle, passing a double slit will undergo self inference each, producing at a detector plane fringed intensities. The wave generating point charge of a zero rest mass, as the other component of the particle, is maintained a constant energy and speed by a repeated radiation reabsorption/reemission scheme, and in turn steered in direction in its linear motion by the reflected radiation field, and will thereby travel to the detector along (one of) the optical path(s) of the waves leading to a bright interference fringe. We elucidate the process formally based on first principles solutions for the IED particle and known principles for wave-matter interaction.