The macroscopic linear light scattering theory is extended to the semiclassical level starting with a general discussion of spatially nonlocal scattering. An inte- gral relation for the gauge invariant transverse vector potential is given, stressing similarities and differences between the so-called photon-eye and electron-eye picture. The linear constitutive relations and the causal response tensor are key elements in the semiclassical light scattering theory. Elastic scattering processes are discussed from first principles by making use of the transverse Feynman propagator and the causal response tensor. In scattering from mesoscopic me- dia, e.g, the causal response tensor is a two-point function in direct space. In the spatial dispersion subfield this tensor becomes a one-point function, depending only on the vectorial distance between given source and observation points. In the spatial dispersion approximation a simple relation can be obtained between the selfconsistent and causal response tensors in the wave-vector-frequency do- main. Next, we turn to the scattering matrix concept in quantum physics. The treatment describes the transition from an initial to a final object-field tensor product. These states are connected by the S -matrix operator. A general ex- pression for this is obtained in the Interaction Picture. The reduced S -matrix which relates in a direct manner to the scattering between selected initial and final photon states is introduced. The chapter is finished by the establishment and a discussion of the weakly relativistic interaction Hamiltonian, a key ele- ment in the S -matrix operator.
The macroscopic linear light scattering theory is extended to the semiclassical level starting with a general discussion of spatially nonlocal scattering. An inte- gral relation for the gauge invariant transverse vector potential is given, stressing similarities and differences between the so-called photon-eye and electron-eye picture. The linear constitutive relations and the causal response tensor are key elements in the semiclassical light scattering theory. Elastic scattering processes are discussed from first principles by making use of the transverse Feynman propagator and the causal response tensor. In scattering from mesoscopic me- dia, e.g, the causal response tensor is a two-point function in direct space. In the spatial dispersion subfield this tensor becomes a one-point function, depending only on the vectorial distance between given source and observation points. In the spatial dispersion approximation a simple relation can be obtained between the selfconsistent and causal response tensors in the wave-vector-frequency do- main. Next, we turn to the scattering matrix concept in quantum physics. The treatment describes the transition from an initial to a final object-field tensor product. These states are connected by the S -matrix operator. A general ex- pression for this is obtained in the Interaction Picture. The reduced S -matrix which relates in a direct manner to the scattering between selected initial and final photon states is introduced. The chapter is finished by the establishment and a discussion of the weakly relativistic interaction Hamiltonian, a key ele- ment in the S -matrix operator.
The theory of optical birefringence is discussed from the point of view of the scattering of polarized photons by molecules using the framework of formal scattering theory and quantum electrodynamics. In Part One, those features from noncovariant quantum electrodynamics relevant to the molecular scattering of photons are reviewed. Chapter One contains an outline of classical electrodynamics in the Coulomb gauge, including the Stokes parameter and density matrix formulations of polarization. Chapter Two deals with the quantization of the radiation field, and the occupation number representation is defined. The quantum mechanical description of polarized light is outlined in Chapter Three: operator equivalents of the density matrix and Stokes parameters are invoked, the parameters themselves being obtained from the expectation values of the Stokes operators with the state vector of the photon. The Stokes parameter and density matrix are then applied to the polarization of a general beam containing an arbitrary number of photons, and some new methods are developed. The interaction of the electromagnetic field with bound electrons is formulated in Chapter Four in terms of a canonically transformed interaction Hamiltonian in which the multipole terms are explicit. Since each term is then gauge-invariant, this Hamiltonian is particularly suited to the application of diagrammatic perturbation theory which follows from the iteration of the scattering matrix in the interaction representation (Chapter Five). Also the extension of the theory to multiphoton processes is facilitated "because the field strength is explicit in the transformed Hamiltonian. The S-matrix connects the initial and final asymptotic states of the scattering system, and in Chapter Six the state vector and density operator of the scattered photon beam are obtained from those of the incident beam. Part Two contains the application of the above methods to a variety of linear and nonlinear optical birefringence effects in the absence or presence of external fields. In Chapter Seven, general equations are obtained for the polarization changes imposed on a forward scattered photon in terms of matrix elements of the reaction operator between plane polarized photon base states; whereas Chapter Eight extends this treatment to an n-photon beam, thereby encompassing the hyperpolarizability contributions to the polarization changes of the forward scattered beam. It is then demonstrated that the fundamental process in optical birefringence can be interpreted as an infinitesimal unitary transformation of the polarization spinor. Also the S-matrix approach to birefringence is compared with the methods used in crystal optics, and with the classical treatments of the scattering of polarized light by molecules. The general equations obtained for the polarization changes of the forward scattered beam are subsequently interpreted in the light of the physical circumstances appropriate to the bire- fringence effect of interest. In Chapters Nine, Ten, and Eleven, diagrammatic perturbation theory is used to obtain explicit expressions, in terms of molecular polarizability and rotation tensors, for the linear and nonlinear contributions to the polarization changes involved in natural optical activity, and in birefringence phenomena induced by static and optical electric and magnetic fields. The nonforward scattering of polarized photons by molecules is discussed in Part Three. General equations are calculated in Chapter Twelve for the azimuth, ellipticity, and depolarization ratio of the Rayleigh beam at any scattering angle and for any polarization of the incident beam. Explicit expressions in terms of molecular quantities are then obtained for the polarization at scattering angles of 90° and almost zero from both optically inactive and optically active molecules. It is demonstrated that a determination of the precise polarization state of the scattered beam can provide useful molecular information additional to that obtained from depolarization ratios alone. This is because the depolarization ratio depends on only the diagonal elements of the polarization density matrix and so does not convey all the information about the system. Raman scattering processes are contained in the formulation and the polarization of the Raman light at any scattering angle can be obtained. This is shown to be particularly useful for predicting the polarization of stimulated Raman light for any Raman-active liquid placed within a laser cavity. By extending the nonforward scattering formulation to incorporate multiphoton processes, equations are obtained in Chapter Thirteen for the ellipticity, azimuth, and depolarization ratio of the second harmonic wave at any scattering angle as a function of the incident polarization. Several specific situations are subsequently investigated. Finally, it is shown in Chapter Fourteen that the near-forw- ard Rayleigh and Raman scattering formulations provide an explanation for the observed polarization, as a function of the initial polarization, of the self-trapped filaments that arise from the self-focusing of an intense laser beam.
Classical, semi-classical, and quantum-field descriptions for the interaction of light with matter are systematically discussed. Applications of interest include precise determinations of the linear and the non-linear electromagnetic response relevant to resonant pump-probe optical phenomena, such as electromagnetically induced transparency. In the quantum-mechanical description of matter systems, we introduce a general reduced-density-matrix framework. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified and self-consistent manner, using a Liouville-space operator representation. A preliminary semi-classical perturbation treatment of the electromagnetic interaction is adopted, in which the electromagnetic field is described as a classical field satisfying the Maxwell equations. Compact Liouville-space operator expressions are derived for the linear and the general (nth order) non-linear electromagnetic-response tensors describing moving many-electron systems. The tetradic matrix elements of the Liouville-space self-energy operators, which are introduced in the time-domain and frequency-domain formulations, are evaluated for environmental collisional and radiative interactions, in order to provide explicit forms for the quantum kinetic equations and the spectral-line shape formulas. It is emphasized that a quantized-field approach is essential for a fully self-consistent quantum-mechanical description of the interacting light-matter system.
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.
Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator M′ M ′ of the interacting time evolution has to be unitarily equivalent to the invariant mass M=P2 M = P 2 where P, acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest P0 P 0 generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian H′ H ′ to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity U=P/M U = P / M but not with P, else there is no scattering. Even though H′ H ′ does not commute with P, the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a Klong K long are seen by an inversely shifted observer as a shifted Klong K long with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted Kshort K short with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.
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 space-time propagator for multichannel scattering is cast in the form of a Feynman path integral where the action is a matrix and the interactions are nonlocal in time. The S matrix is evaluated in the semiclassical limit, leading to trajectories which naturally follow individual channel potentials in noninteracting regions and adiabatic-like ones in strongly interacting ones. A numerical example is provided for the curve-crossing problem, and the agreement with exact quantum results is good even when the interaction region is large and/or encompasses classical turning points.