Higher-order field corrections

The strength of the field must not be so strong that higher order (m) ) effects come into play and then, should we introduce the basis functions suited for order m to get a correct response of the system up to order , even if we are concerned only with the th-order ... 

This is a relationship between unknown field g and two measured quantities, namely, the distance 5 and time t, provided that we neglect terms proportional to the square of the coefficient a and those of higher order. Besides, this equation contains three unknown parameters, namely, the position of the mass. so at the moment when we start to measure time, the initial velocity, vo, at this moment and, finally, the rate of change of the gravitational field, a, along the vertical. Thus, in order to solve our problem and find the field we have to perform measurements of the distance. s at four instants. If so is known, the number of these measurements is reduced by one. In modern devices the coefficient of the last term on the right hand side of Equation (3.14) has a value around 100 pGal and it is defined by calculations as a correction factor s(vo, g, t, a). In the case when we can let so equal to zero, it is sufficient to make measurements at two instances only. 

Using a similar attack for a fully circulating fluid sphere in a stationary field but using only n = 2 due to inconsistency of the equations for higher order functions, their wall correction factor was... 

In each case, the mean-field model forms only a starting point from which one attempts to build a fully correct theory by effecting systematic corrections (e.g., using perturbation theory) to the mean-field model. The ultimate value of any particular mean-field model is related to its accuracy in describing experimental phenomena. If predictions of the mean-field model are far from the experimental observations, then higher-order corrections (which are usually difficult to implement) must be employed to improve its predictions. In such a case, one is motivated to search for a better model to use as a starting point so that lower-order perturbative (or other) corrections can be used to achieve chemical accuracy (e.g., 1 kcal/mole). 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

Leading recoil corrections in Za (of order (Za) (m/M)") still may be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. 

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