The idea of the systematic Weisskopf-Wigner approximation as used sporadically in atomic physics and quantum optics is extended here to the interaction of a field of non-relativistic fermions with a field of relativistic bosons. It is shown that the usual (non-existing) interaction Hamiltonian of this system can be written as a sum of a countable number of self-adjoint and bounded partial Hamiltonians. The system of these Hamiltonians defines the order hierarchy of the approximation scheme. To demonstrate its physical utility the author shows that in a certain order it provides a satisfactory quantum theory of the 'self-energy' of the fermions. This is defined as the binding energy of bosons bound to the fermions and building up the latter's 'individual Coulomb or Yukawa fields' in the sense of expectation values of the corresponding field operator. In states of more than one fermion the bound photons act as a mediating agent between the fermions; this mechanism closely resembles the Coulomb or Yukawa 'forces' used in conventional non-relativistic quantum mechanics.
The idea of the systematic Weisskopf-Wigner approximation as used sporadically in atomic physics and quantum optics is extended here to the interaction of a field of non-relativistic fermions with a field of relativistic bosons. It is shown that the usual (non-existing) interaction Hamiltonian of this system can be written as a sum of a countable number of self-adjoint and bounded partial Hamiltonians. The system of these Hamiltonians defines the order hierarchy of the approximation scheme. To demonstrate its physical utility the author shows that in a certain order it provides a satisfactory quantum theory of the 'self-energy' of the fermions. This is defined as the binding energy of bosons bound to the fermions and building up the latter's 'individual Coulomb or Yukawa fields' in the sense of expectation values of the corresponding field operator. In states of more than one fermion the bound photons act as a mediating agent between the fermions; this mechanism closely resembles the Coulomb or Yukawa 'forces' used in conventional non-relativistic quantum mechanics.
The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c)2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2e2/3c3)(d〈v〉/dt)2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.
We derive a functional form for the energy of interacting many-particle systems from first principles. The effective zero-point energy of a Coulomb-interacting quantum system can be considered in terms of fluctuations around its classical electrostatic interaction and fluctuations of momentum. This perspective provides a controlled and intuitive way to realize the Hohenberg-Kohn theorem. Specifically, an uncertainty equation that relates the interaction energy, kinetic energy and local charge density is derived; it is then combined with the Lieb-Thirring bound on kinetic energy. The combination of these two constraints results in the functional form for total energy of interacting many particle systems that applies for bosons as well as fermions, with appropriate consideration for (anti)symmetry of wavefunctions under coordinate exchange. In the uniform density limit, the functional form is consistent with Quantum Monte Carlo (QMC) data for the bosonic fluid, as well as fermionic systems -- paramagnetic, ferromagnetic and BCC Wigner crystalline.
The principle of minimal energy, which has been set up in the preceding papers for systems of non-identical particles (e.g. positronium), is now generalized to include also identical particles. Since the latter kind of particles feels also the exchange forces (besides the usual electromagnetic forces), one has to deal with non-zero exchange potentials which render the theory nonlinear, according to the non-Abelian character of Relativistic Schroedinger Theory (RST). However, the present extension of the variational principle refers only to the linearized version of RST in order to keep the calculations sufficiently simple. It is also demonstrated that in RST the Dirac particles can occur in fermionic and bosonic quantum states; and the mathematical and physical consistency of the variational principle is validated for both types of states (concretely the fermionic hydrogen state 2p_3/2 and the bosonic positronium state 2^1P_1).
We give here a field-theoretical derivation of the Hamiltonian of the non-relativistic quantum electrodynamics in the Coulomb gauge using the Lagrange formalism. It leads to the same result as the usual derivation, where one just replaces the classical vector potential in the minimal coupling of the second quantized electron Hamiltonian by the quantized one and adds the photon energy. This approach illustrates however the proper use of the Euler-Lagrange equations and the canonical formalism that fail if one tries to build up the many-body quantum theory starting from the classical theory of point-like particles interacting with electromagnetic fields.
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.
Relativistic quantum theory poses a number of conceptual problems over and above the non-relativistic mechanics. From a purely mathematical point of view it is also much more sophisticated. Even the kinematic theory poses considerable difficulties: locality, antimatter, negative energy, and charge are already systematically linked in ways that cannot simply be paraphrased. A further complication is that in the midst of this structure we have what is usually called the Newton-Wigner representation (or, in the case of the Dirac theory, the Foldy-Wouthuysen representation), in which fields and states are no longer covariantly described and where the “local” self-adjoint quantities (NW-local observables) do not obey microcausality, but only satisfy equal-time commutators with respect to a particular inertial frame. If we pass to this represenation, we obtain at a stroke the basic structure of non-relativistic quantum field theory; it is not too hard to descend from that to a many-particle mechanics, and to recover the usual definitions of localization.