Abstract A local dynamical evolution equation for a two-spinor photon wave function is established. The vectorial spinor components are those of the two helicity components A T , ± ( + ) of the analytical part of the gauge invariant transverse vector potential. The theory’s Hamilton operator is spatially local and without singularities. Thereby previously established field-based formulations leading to various nonlocal integro-differential equations for the dynamical evolution of the photon wave function are simplified. The dynamical evolution equations for the spinor components can be written in the compact form F ± ( + ) ( r , t ) = 0 , where F ± ( + ) ( r , t ) are the analytical Riemann–Silberstein–Oppenheimer–Bialynicki vectors. The theory after extension to the QED level, allows one to describe the dynamical time evolution of single-photon wave packets. The bridge between the wave mechanical and the QED formulations is an operationally suitable mean position state | R ⟩ ( r , t ) for a transverse photon in Hilbert space. The inner product ⟨ R ( r , t ) ⟩ R ( r , t ) relates to our inability to localize a photon in space. The normalization condition for the two-spinor wave packet is established. A dynamical evolution equation describing the photon-source entanglement is established starting from the Heisenberg equation of motion for the annihilation operators associated to the two helicity species. We show how each released photon is labelled by the transverse current density operator dynamics of the charged source particles. The basic theory is applied to a study of single-photon emission from a rectlinear magnetic dipole tube (string) carrying a rotationally symmetric surface current. Numerical results are obtained for photon wave train emission. The transition from the photon wave mechanical formulation to the magnetostatic Aharonov–Bohm vector potential field is established. The generality of the A T , ± ( + ) -thoery is underlined linking it to the Minimal Coupling Principle, magnetic monopole electrodynamics, spatial photon localization, the perfe
Abstract A local dynamical evolution equation for a two-spinor photon wave function is established. The vectorial spinor components are those of the two helicity components A T , ± ( + ) of the analytical part of the gauge invariant transverse vector potential. The theory’s Hamilton operator is spatially local and without singularities. Thereby previously established field-based formulations leading to various nonlocal integro-differential equations for the dynamical evolution of the photon wave function are simplified. The dynamical evolution equations for the spinor components can be written in the compact form F ± ( + ) ( r , t ) = 0 , where F ± ( + ) ( r , t ) are the analytical Riemann–Silberstein–Oppenheimer–Bialynicki vectors. The theory after extension to the QED level, allows one to describe the dynamical time evolution of single-photon wave packets. The bridge between the wave mechanical and the QED formulations is an operationally suitable mean position state | R ⟩ ( r , t ) for a transverse photon in Hilbert space. The inner product ⟨ R ( r , t ) ⟩ R ( r , t ) relates to our inability to localize a photon in space. The normalization condition for the two-spinor wave packet is established. A dynamical evolution equation describing the photon-source entanglement is established starting from the Heisenberg equation of motion for the annihilation operators associated to the two helicity species. We show how each released photon is labelled by the transverse current density operator dynamics of the charged source particles. The basic theory is applied to a study of single-photon emission from a rectlinear magnetic dipole tube (string) carrying a rotationally symmetric surface current. Numerical results are obtained for photon wave train emission. The transition from the photon wave mechanical formulation to the magnetostatic Aharonov–Bohm vector potential field is established. The generality of the A T , ± ( + ) -thoery is underlined linking it to the Minimal Coupling Principle, magnetic monopole electrodynamics, spatial photon localization, the perfe
Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.
Elements of a theory which allows one to connect gauge-photon wave mechan- ics to the magnetostatic Aharonov-Bohm vector potential are outlined. In a first step a relativistic calculation gives one the correct identification of the field momentum and energy in classical electrodynamics. By working with manifest covariant expressions, the conservation law for the four-momentum is given on integral form. Only in the special Lorentz frame where the relativistic factor γ = [1 − ( v/c )2] − 1 / 2 equals γ = 1 the integral form reduce to the textbook integrals. In the rest frame of a charge we discuss aspects of photon wave mechanics based on the analytic part of the transverse vector potential, (A (+) ). Hamiltonian-like dynamical evolution equations for the two helicity species of A (+) represent together in bispinor form a theory which holds in a dynamical T (+)(+) structural vacuum, i.e. for E T ± ic B = 0. On the basis of the basic dynamical equations a dynmical version (extension) of the Aharonov-Bohm effect is derived for a linear magnetic tube (often called a string). From a microscopic point of view the string may be a ferrmagnet in which a running (or standing) spin wave is present on top of the time independent ferromagnetic magneti- zation. In the magnetostatic limit the well-known Aharonov-Bohm transverse vector potential is obtained. Chapters 9 – 12 constitute the books SECTION III , a section dealing with conservation laws, Lagrangians, geometric phases and topology.
Starting from the well-known plane-wave decomposition of the operator associ- ated with the transverse part of the vector potential (Aˆ T) , and by making use of the operator relations defining the relations between and magnetic (Bˆ [with ∇ · Bˆ = 0]) field operators, the plane-wave mode expansions for Eˆ T and Bˆ are established. Subsequently, the general operator relation ∇ × Eˆ T = −∂ Bˆ/ ∂t follows. From the Heisenberg equation of motion for the magnetic field operator the inhomogeneous Maxwell operator equation ∇ × Bˆ = µ 0 Jˆsym + c − 2 ∂ Eˆ T/∂t is established. Here the operator Jˆsym T is the symmetrized transverse current den- sity operator in the space-time domain. The transverse set of Maxwell operator equations are of utmost importance for the subsequent photon energy operator wave function formalism. For globally coherent quantum fields, where the par- ticle source current density distribution is quasi-classical, dynamical equations for the two helicity species of Riemann-Silberstien field operators are obtained. Neglect of the quantum field fluctuations in the coherent state makes the bridge to photon wave mechanics in the Oppenheimer-Riemann-Silberstein-Bialynicki approach. The chapter is closed with a discussion of response theory and Ein- stein causality.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.
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.
A brief account of the connection between the photon little group and gauge in- variance is given, considering the four-potential as the wave function of the photon. In most textbooks single photons are associated to the quanta of monochromatic plane-wave components of the electromagnetic field. However, polychromatic superposition of such states may also represent single photons. Having established the framework for polychromatic single-photon states in Hilbert space, and upon a discussion of the mean value and variance of the photon Hamilton operator for polychromatic states, we turn our attention to- wards the single-photon description in direct space. A photon wave function in direct space is obtained via the inner product of the photon mean position state (introduced in Chapter 3 ) and the Hilbert space polychromatic single-photon state. Noting that the photon is a relativistic object a rigorous polychromatic photon wave function is, in the continuum superposition of monochromatic plane-wave components, obtained noting the form of the relativistically invari- ant volume element in wave-vector space. We establish a wave-packet basis for single-photon states in Hilbert space, and the related photon wave functions. Thereafter, multi-photon wave packets are discussed, and the equal-time com- mutator relations between the wave packets are discussed, and the equal-time commutator relations between the wave-packet annihilation and creation oper- ators set up. As an example, we focus on two-photon wave packet states. We finish the chapter by a discussion of the two-photon concept and two-photon wave mechanics, analysing in particular the physical difference between two photons and bi-photons (an entangled state of two photons).