Feynman propagator

The wave function is an irreducible entity completely defined by the Schrbdinger equation and this should be the cote of the message conveyed to students. It is not useful to introduce any hidden variables, not even Feynman paths. The wave function is an element of a well defined state space, which is neither a classical particle, nor a classical field. Its nature is fully and accurately defined by studying how it evolves and interacts and this is the only way that it can be completely and correctly understood. The evolution and interaction is accurately described by the Schrbdinger equation or the Heisenberg equation or the Feynman propagator or any other representation of the dynamical equation. 

We note that the integrand in Eq. 40 does not depend on k , whose integration gives the density of states at the Fermi surface, and the it prescription is consistent with the Feynman propagator. The pole occurs at... 

If there is no explicit external electromagnetic field, the covariant field equations determine a self-interaction energy that can be interpreted as a dynamical electron mass Sm. Since this turns out to be infinite, renormalization is necessary in order to have a viable physical theory. Field quantization is required for quantitative QED. The classical field equation for the electromagnetic field can be solved explicitly using the Green function or Feynman propagator GPV, whose Fourier transform is —gllv/K2, where k = kp — kq is the 4-momentum transfer. The product of y0 and the field-dependent term in the Dirac Hamiltonian, Eq. (10.3), is... 

Hard scale contributions are obtained by Taylor expanding the scattering amplitude in small external momenta. All relevant two-loop integrals (some examples are shown in Fig.l) can be parameterized by nine Feynman propagators ... 

In all the above 2-loop graphs one has the external momentum p, which satisfies the mass-shell constraint p2 = —m2, me being the mass of the electron, and 2 loop momenta, ki and k2 with them one can form the 5 scalar products k2,k2, (p.ki), (p.k2), (ki.k2). Any of the occurring graphs involves a subset of the following Feynman propagators (more exactly inverse Feynman propagators in the momentum representation)... 

To afford a basis for constructing Thomas-Fermi-like approximations to the electronic structure of atoms in intense applied fields, the canonical density matrix (or equivalently the Feynman propagator) for free electrons is first set up. This is done for both intense magnetic and intense electric fields in each case, exact results are available for arbitrary static field strengths. 

A model of confined atoms in an arbitrary static electric field, which can also be solved analytically, will then be discussed in some detail. Contact will be made with results on atomic ions in non-degenerate plasmas, with illustrative examples being presented. A brief treatment follows of the time-dependent uniform electric field Feynman propagator. 

Throughout this section, the canonical density matrix and the Feynman propagator can be used interchangeably, the transformation P = it taking C into the propagator K, with t the time. While most frequently we shall use the coordinate representation r and r, it will be convenient in this section to work in k or momentum representation, by taking a double Fourier transform with respect to r and r. ... 

Let us first then effect the generalization of free electrons in a static uniform electric field F to treat a uniform time-varying electric field. The Feynman propagator for this case has been discussed, for example, by Fallieros and Friar [47]. These workers give the propagator X(k,k, f) in k space as... 

As to future directions, the problem of the canonical density matrix, or equivalently the Feynman propagator, for hydrogen-like atoms in intense external fields remain an unsolved problem of major interest. Not unrelated, differential equations for the diagonal element of the canonical density matrix, the important Slater sum, are going to be worthy of further research, some progress having already been made in (a) intense electric fields and (b) in central field problems. Finally, further analytical work on semiclassical time-dependent theory seems of considerable interest for the future. 

Sp represents the Feynman propagator in the field of the external potential of the nucleus and obeys the equation... 

The Feynman propagator accounts for the motion of an electron or positron from a space-time point Xj = (ctj, xj) to another point 2 = ct2, X2). It is defined by the vacuum expectation value... 

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, Monte Carlo integration of the Feynman propagator in imaginary time, J. Comp. Phys. 3,416-443 (1969). 

The amplitudes will contain various Feynman propagators (which are irrelevant to our discussion) and, as is easy to see, there will be overall factors of... 

Fano interference, 32, 38 Fast electron distribution, 134 Fast electron generation, 123 Fast electron transport, 125 Fast electrons, 176 Fast-ignition, 124 Femtosecond supercontinuum, 94 Feynman s path integral, 73 Feynman s propagator, 76 Field parameter, 172 Filamentation, 82, 84, 112 Floquet ladder, 11 Fluorescence, 85, 125 FROG, 66 FROG-CRAB, 66... 

Band Theory of Metals, Three approaches predict the electronic band structure of metals. The first approach (Kronig-Penney), the periodic potential method, starts with free electrons and then considers nearly bound electrons. The second (Ziman) takes into account Bragg reflection as a strong disturbance in the propagation of electrons. The third approach (Feynman) starts with completely bound electrons to atoms and then considers a linear combination of atomic orbitals (LCAOs). 

Our objective is to design an optical device that will change the polarization from horizontal to vertical linear polarization - a rotation of the Stokes vector 5i,52,53 from // = (1,0,0 to V = (0,1,0 - and to do so independently of the wavelength. For this purpose, we require a propagation equation for the Stokes vector, obtained from Eq. (5.14) and the definitions (5.16) in much the same way that Feynman et al. [9] convert the two-state TDSE into a torque equation for combinations of products of probability amplitudes see Appendix 5.B. The equations... 

Obviously, such interpetation led to disregard advanced solutions as nonphysical. For instance, Ritz [30] and Tetrode [31] considered that the mathematical existence of advanced solutions was a major weakness of Maxwell s equations. An attempt to provide a physical basis for advanced potentials is due to Lewis, who proposed focusing on the process of propagation from an emitter to an absorber far away from the emitter [32]. This concept also appears in the work of Wheeler and Feynman [33]. However, such model constitutes another form of causality violation. Lewis [32, p. 25] himself stated I shall not attempt to conceal the conflict between these views and common sense. ... 

Here for c 1 we have the Feynman gauge, and , = 0 is the Landau gauge. This gauge fixing term will enter into the massive boson propagators for the A 3 field. The propagator will be of the form... 

The results of this analysis show that anomalous dispersion of light in a cesium cell is a consequence of superluminal motion of electrons and superluminal propagation of electromagnetic waves. The Feynman diagram, presented in Fig. 8, is used in the analysis, to explain the phenomena that are taking place in cesium atomic cell and that cause superluminal effects [30]. 

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