Single-particle propagators

The single particle propagator [20] G(x,x E) is an energy-dependent (or time-dependent) function of two compound space spin variables. It satisfies the Dyson-like equation... 

The formal definition (29) of the extended Green s function implies a sum of propagators according to the different components of the extended state s vector. The terms introduced by the product states can be transformed into convolutions of single-particle propagators if we resort to real energies. For... 

The directed lines are single-particle propagators which in general have self-energy renormalization, also contains... 

Fig. 18.15 Advanced methods for single particle analysis in LC ARROW chips, (a) Nanopore added to reservoir for single particle entry into LC ARROW (b) Optical dual beam particle trap based on balancing the scattering force due to counter propagating beams...

The conversion process occurs both on macro- and micro-scale, that is, on single particle level and on bed level. In other words, the conversion process has both a macroscopic and microscopic propagation front. One example of the macroscopic process structure is shown in Figure 10. The conversion front is defined by the process front closest to the preheat zone, whereas the ignition front is synonymous with the char combustion front. 

Figure 3.1 The motion of surfaces of constant S in configuration space. At t = 0 the surfaces S — a and S = b coincide with the surfaces for which W = a and W = b. Surfaces of constant W have fixed locations in space. Surfaces of constant S represent wavefronts propagating in configuration space. The trajectory of a single particle in 3D-space lies along the wave normals.

Due to the fact that the Lagrangian incorporates the creation and destruction of field quanta, not even the time-development of a single particle is a simple matter. The time development can be expressed in terms of the electron (fermion) and photon propagators, which are defined as the vacuum expectation values of the time-ordered product of field operators. For the fermions one has... 

Both P and Q are sums of excitation operators (with weighting coefficients p and 9 )- Thus, P and Q applied to 0> create a polarization of 0> and we call P 6 a polarization propagator. In the special case where P and Q are both single particle-hole excitations, i.e. only one term in Eqs (5) and (6), we talk about the particle-hole propagator. It is important to note that only the residues of the polarization propagator and not of the particle-hole propagator determine transition moments (Oddershede, 1982). We must have the complete summations in Eqs (5) and (6) in order to represent the one-electron operator that induces the transition in question. 

Relation (13) is the main result. It allows us to determine the effective wave number at the propagation of surface waves along the interface with two-dimensional scatterers on the basis of the scattering characteristics on a single particle. As it could be expected, the difference from the three-dimensional case only consists in the form of coefficients f(0) and f(it) and also in the form of the expressions for the scattered amplitudes in themselves. 

We discussed the structure of the repeated ring operator in Section VII.D and pointed out that it contains a variety of dynamic events such as a series of correlated collisions identical to those that appear in the singlet theory via the operators and R. These operators represent the correlated collisions of a single solute molecule with the solvent and serve to renormalize the single-particle motion. Other events in R represent the coupling of the motion of the two solute molecules. In view of this, it is convenient to introduce the propagator for independent motion of the pair... 

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