1. Introduction Dirac's (Dirac 1928) theory of the electron was the first widely accepted relativistic quantum theory, and it later provided the basis for constructing the modern electromagnetic theory of quantum electrodynamics (QED). Whereas Dirac's theory in its simplest form describes relativistic freely-propagating massive non-chiral particles of spin-½, QED describes how such particles interact with one another electromagnetically, via a dynamical quantum field.
Introduction Dirac's (Dirac 1928) theory of the electron was the first widely accepted relativistic quantum theory, and it later provided the basis for constructing the modern electromagnetic theory of quantum electrodynamics (QED). Whereas Dirac's theory in its simplest form describes relativistic freely-propagating massive non-chiral particles of spin-½, QED describes how such particles interact with one another electromagnetically, via a dynamical quantum field.
Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.
Chapter 19 introduces relativistic quantum mechanics, with the aim of providing the student with a basic knowledge of the subject and a link to more advanced courses in relativistic quantum theory. It begins by showing that when the Schrödinger equation is modified to make it consistent with special relativity, the result is the Klein–Gordon equation, which gives better qualitative agreement with experiment. Introducing the electron spin into the Schrödinger equation leads to the Pauli equation, which gives rise to important spin effects. By considering both special relativity and spin, one obtains the Dirac and the van der Waerden equations. The main properties and implications of Dirac’s equation are discussed, including the electron’s zitterbewegung. The solution of Dirac’s equation for the free particle is presented in detail, the hydrogen-like atom is shown to be exactly solvable, and the particle in an external electromagnetic field is included as a complementary material.
This book is an attempt to build a consistent relativistic quantum theory of interacting particles. In the first part of the book "Quantum electrodynamics" we follow rather traditional approach to particle physics. Our discussion proceeds systematically from the principle of relativity and postulates of quantum measurements to the renormalization in quantum electrodynamics. In the second part of the book "Quantum theory of particles" this traditional approach is reexamined. We find that formulas of special relativity should be modified to take into account particle interactions. We also suggest reinterpreting quantum field theory in the language of physical "dressed" particles. This formulation eliminates the need for renormalization and opens up a new way for studying dynamical and bound state properties of quantum interacting systems. The developed theory is applied to realistic physical objects and processes including the energy spectrum of the hydrogen atom, the decay law of moving unstable particles, and the electric field of relativistic electron beams. These results force us to take a fresh look at some core issues of modern particle theories, in particular, the Minkowski space-time unification, the role of quantum fields and renormalization as well as the alleged impossibility of action-at-a-distance. A new perspective on these issues is suggested. It can help to solve the old problem of theoretical physics -- a consistent unification of relativity and quantum mechanics.
We describe here the coherent formulation of electromagnetism in the nonrelativistic quantummechanical many-body theory. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because of the manifold of misinterpretations emerging from the hystorical development of quantum mechanics, starting from the Schrödinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body theories of many charged identical particles having just Coulomb interactions inspired from the classical electromagnetic theory of point-like charges. However, this later is known to be inconsistent due to the self-interaction. This way could not be continued further to include properly the magnetic forces between the charged particles and lead to a lot of confusion about the interpretation of the magnetic field in the Hamiltonian, as well as about the gauge invariance. We emphasize the importance of the distinction between the applied (external fields) and the field in the matter. All these problems are length properly solved within the non-relativistic QED, nevertheless the confusion dominates in all the problems related to the magnetic properties of the solid state.
We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schrodinger equation. We define a relativistic quantum principle asserting that a quantum particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of coordinates. From the time dependent phase invariance, the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2) an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation, Ampere-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.
Maxwell's equations describe the interactions of the electromagnetic field at a macroscopic level. In the 1920s, Louis DeBroglie demonstrated that every moving particle (including an electron) has a wave nature, and we know from Einstein that every wave has a particle nature, which we call the photon. Later in the 1930s, Paul Dirac's development of the famous Dirac equation showed the quantum nature of the electron at relativistic speeds. Then in 1948 Richard Feynman and Julian Schwinger extended these concepts in the development of quantum electrodynamics which gives a full accounting (although a very strange one) of how an electron can borrow energy from the vacuum of space and return it legally as long it does so within limits of the uncertainty principle.