Covariant evolution operator

Covariant Evolution Operator and the Green s Operator 4.1 Definitions... 

Fig. 6 Comparison between the standard evolution operator, the Green s function and the covariant evolution operator for single-photon exchange in the equal-time approximation. The solid lines between heavy dots represent electron propagators and xh free lines electron creation and absorption operators...

Keywords Perturbation theory Quantum electrodynamics Electron correlation Electron self-energy Green s operator Covariant evolution operator... 

We have during the past decade developed an energy-dependent perturbation procedure, based on a ""covariant evolution operator method [2-5], that will make it possible to handle energy-dependent QED perturbations very much in the same way as the energy-independent ones in MBPT. 

Fig. 1 Comparison between the standard evolution operator and the covariant evolution operator for single-photon exchange in the equaltime approximation...

The covariant evolution operator for a ladder of retarded interactions between the electrons is given by... 

The non-relativistic evolution operator is in first order represented by the first Feynman diagram in Fig. 1. It is non-covariant, since time flows only in the positive direction. If we insert electron propagators in the in- and outgoing orbital lines, time can flow in both directions, and we get a covariant form of the evolution operator, represented by the second diagram. 

ABSTRACT We consider coherent systems subject to common-cause failures described by multiple failure rates. After giving the generalized expression of the failure frequency v as a function of the failure rates and the derivatives of the system reliability TZ, we expand ft, and v in terms of the covariances of the components probabilities of correct operation. Assuming multiple repair rates also, we show that the evolution of the coupled populations of system states may be solved quite generally in successive steps. When all the failure and repair rate are constant, we give the analytical expressions of the steady-state populations for an arbitrary number of dependent components. 

The linear equation 12 can then be easily included in the state-space formulation. After operating upon the state-space equation of the extended system, i.e. that including the linearized equation, and assuming Gaussian behavior of all the responses, one obtains the following differential equation for the evolution of the covariance matrix of the responses ... 

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