Abstract In this paper are introduced several novel techniques having as their objects, first, the gain of a new understanding of the divergence difficulties of orthodox quantum theory, and secondly, the systematic development of the quantum mechanics of fields in terms of the density matrix. There is included a new presentation of a form of perturbation theory which has the outstanding advantages over the usual one that it is much more quickly convergent, and leaves the density matrix normalized at every stage so that the diagonal elements representing occupational probabilities cannot diverge. Exact general formulae are derived for self-energies and cross-sections for the purpose of examining their convergence properties. The general theory of fields is developed abinitio, and it is shown that the use of the density matrix in place of the wave vector illuminates and simplifies the customary theory. The second quantization of the density matrix which follows throws unexpected light on the existence of particles with negative expectation values. Finally, the whole theory is applied to the crucial example of the electron or positron in an electromagnetic field. It is confirmed that the self-energy of the electron in the orthodox theory cannot be made finite without the introduction of negative energy photons.
Abstract In this paper are introduced several novel techniques having as their objects, first, the gain of a new understanding of the divergence difficulties of orthodox quantum theory, and secondly, the systematic development of the quantum mechanics of fields in terms of the density matrix. There is included a new presentation of a form of perturbation theory which has the outstanding advantages over the usual one that it is much more quickly convergent, and leaves the density matrix normalized at every stage so that the diagonal elements representing occupational probabilities cannot diverge. Exact general formulae are derived for self-energies and cross-sections for the purpose of examining their convergence properties. The general theory of fields is developed abinitio, and it is shown that the use of the density matrix in place of the wave vector illuminates and simplifies the customary theory. The second quantization of the density matrix which follows throws unexpected light on the existence of particles with negative expectation values. Finally, the whole theory is applied to the crucial example of the electron or positron in an electromagnetic field. It is confirmed that the self-energy of the electron in the orthodox theory cannot be made finite without the introduction of negative energy photons.
The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.
The present work is concerned with de Broglie's quantum theory of light. It is assumed that the photon is described by a Hermitian wave function with 16 components. Using this wave function it is shown that the 32 of de Broglie's equations are reduced to one set of 16 equations in the form: [equation] where the [alpha]'[rho] are Dirac matrices and [psi] is a matrix with 16 components. The electromagnetic quantities associated with the photon are described by means of the Dirac matrices operating on [psi] in a specified way. It is shown also that these electromagnetic quantities satisfy Maxwell's equations as a result of the equation the interaction between an electron and a photon is developed and the matrix elements for the radiation transitions are calculated. It is further shown that the above wave equation can be considered as the superposition of two similar wave equations, one for the positive energy photons and the other for the negative energy photons. To each of these states there corresponds electromagnetic quantities defined by the above method. It is the superposition of these fields which gives rise to the reality of the electromagnetic field, found in experience. The wave mechanics of the positive energy photon is discussed and the method of second quantization is applied to its wave function, from which we deduce the commutation relations for the complex fields.
The quantum mechanics of fields recently developed by us leads to a modification of statistical mechanics of elementary particles which seems to overcome some of the difficulties (divergence of integrals) occurring in the usual quantum theory of fields. The main difference between the new theory and the usual one is as follows. In the usual theory the wave-vector k is introduced classically and, so to speak, kinematically by the Fourier analysis of the field. The Fourier coefficients of the field components are then treated according to quantum mechanics as non-commuting quantities; those belonging to the wave-vector k describe the corresponding “model” mechanical system, namely the kth radiation oscillator. But the statement that the Fourier coefficients belonging to a certain k all vanish, which statement classically is significant, is now meaningless because there is a lowest state with zero-point energy for each radiation oscillator. The field is thus made to be equivalent to the assembly of radiation oscillators of all possible wave-vectors which, being necessarily infinite in number, contribute an infinite zero-point energy for the pure field and lead to other divergent integrals for the interaction between different fields.
The self-consistent set of equations that describes the interaction of an atom with the electromagnetic field is proposed. The set includes the equations for vector and scalar potentials of electromagnetic field vector A(vector r,t), (phi) (vector r,t) and electron current and charge density vector j(vector r,t), (rho) (vector r,t). The obtained equations are similar to the magnetic hydrodynamics equation and differ from them by the presence of the terms depending on the logarithmic gradient of charge density. The specific feature of the obtained equations is that the steady-state distribution of the electron charge and current density is determined by the real and imaginary parts of the Schorodinger equation. The obtained equations are generalized for the relativistic case. In the latter case the equations include the atomic variables of the two types. The atomic variables of the first type are the bilinear forms of the wave function and Dirac conjugated wave function. The atomic variables of second type are the Hermitian conjugated combinations of the wave function first derivative and the Dirac conjugated wave function. Another specific feature of the obtained equations is that the equation for the electron current density can be transformed to the classical Hamilton-Jacobi equation with the new Hamiltonian. The phase of the wave function plays the role of action and new Hamiltonian includes the term depending on the logarithmic gradient of the electron density and its divergence. The relativistic Hamilton-Jacobi equation includes the term depending on the electron spin as well as the terms depending on electron density logarithmic four- gradient and its four-dimensional derivative.
According to classical electromagnetic theory, quantum mechanics, and quantum field theory, particles are often associated with self-energy flow. In electromagnetic theory, the so-called self-energy flow corresponds to the Poynting vector of energy flow, and in quantum mechanics, it represents the probability current density. The author believes this is likely incorrect. The author finds that self-energy flow can be replaced by mutual energy flow. Mutual energy flow consists of retarded waves emitted by the source and advanced waves emitted by the sink. The source includes the primary coil of a transformer, the transmitting antenna, radiation atoms, and the cathode emitting electrons. The sink includes the secondary coil of the transformer, the receiving antenna, absorbing atoms, and the anode receiving electrons. Originally, all energy flows are generated by the source. If we assume energy flow is generated jointly by the source and the sink, we first need to compress the field produced by the source to half its original size. In this way, the source produces half of the field, and the sink produces the other half. The mutual energy flow generated by the source and sink together will exactly match the original self-energy flow produced by the source. However, there is still a problem. If the energy flow is mutual energy flow rather than self-energy flow, then self-energy flow should not transfer energy. However, if we do not alter the original electromagnetic theory or quantum theory, self-energy flow still transmits energy. This creates an absurd situation where mutual energy flow particles and self-energy flow particles are two distinct particles, both transferring energy. The author finds that both self-energy flows in electromagnetic theory and quantum theory should be reactive power and therefore do not transfer energy. The author adds a magnetic field, source, and sink to the Schrödinger equation and extends the mutual energy flow law from electromagnetic theory to quantum theory. The author then corrects the phase of the magnetic field in Maxwell’s electromagnetic theory and quantum theory by 90 degrees. After the correction, both self-energy flows in the two theories become reactive power. Thus, particles, including photons, electrons, and all other particles, are mutual energy flows, not self-energy flows. For electromagnetic theory, the author also proves that the composite energy flow of many photon mutual energy flows coincides with the self-energy flow corresponding to the Poynting vector calculated from Maxwell’s electromagnetic theory. Thus, electromagnetic waves have three different modes: (1) broadcasting mode, where the source and sink simultaneously emit spherical waves at random, which decay with propagation distance. (2) Photon mode, where the retarded wave emitted by the source and the advanced wave emitted by the sink are perfectly synchronized. At this point, the advanced wave forms the waveguide of the retarded wave, and the retarded wave forms the waveguide of the advanced wave. Due to the interference between the retarded and advanced waves, the two waves interfere constructively along the line connecting the source and the sink and weaken in other directions, ultimately turning into quasi-planar waves along the connection line between the source and sink. These retarded and advanced waves constitute the mutual energy flow, which is the photon. (3) The average energy flow consisting of countless photons is consistent with the energy flow calculated according to the Poynting vector. Therefore, Maxwell’s electromagnetic theory is still valid for calculating average radiation electromagnetic energy flow. However, calculating photons requires the author’s electromagnetic theory. The situation for electrons or other particles is roughly similar.
Abstract This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a survey of classical probability theory and with an introduction to the foundations of quantum mechanics, with particular emphasis on its statistical interpretation and on the formulation of generalized measurement theory through quantum operations and effects. The fundamentals of density matrix theory, quantum Markov processes, and completely positive dynamical semigroups are developed. The most important master equations used in quantum optics and condensed matter theory are derived and applied to the study of many examples. Special attention is paid to the Markovian and non-Markovian theory of environment induced decoherence, its role in the dynamical description of the measurement process, and to the experimental observation of decohering electromagnetic field states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics. The fundamentals of the treatment of non-Markovian quantum processes in open systems are developed on the basis of various mathematical techniques, such as projection superoperator methods and influence functional techniques. In addition, the book expounds the relativistic theory of quantum measurements and the density matrix theory of relativistic quantum electrodynamics.