Bound high-order corrections

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. 

Unlike for an atom, a shell correction expansion up to high orders is possible for an electron bound in a harmonic-oscillator potential [16]. However, this system is characterized by only one parameter and, hence, does not readily allow to separate kinetic from other contributions. 

The lipid part of the membrane is essentially a two-dimensional liquid in which the other materials are immersed and to which the cytoskeleton is anchored. This last statement is not totally correct, as some membrane bound enzymes require the proximity of particular lipids to function properly and are thus closely bound to them. Simple bilayers formed from lipids in which both hydrocarbon chains are fully saturated can have a highly ordered structure, but for this reason tend to be rigid rather than fluid at physiological temperatures. Natural selection has produced membranes which consist of a mixture of different lipids together with other amphiphilic molecules such as cholesterol and some carboxylic acids. Furthermore, in many naturally occurring lipids, one hydrocarbon chain contains a double bond and is thus kinked. Membranes formed from a mixture of such materials can retain a fluid structure. The temperature at which such membranes operate determines a suitable mixture of lipids so that a fluid but stable structure results at this temperature. It will be seen that the lipid part of a membrane must, apart from its two-dimensional character, be disordered to do its job. However, the membrane bound proteins have a degree of order, as will be discussed below. 

Concluding our consideration we would like to underline, that the study of the g factor of a bound electron [1] offers a new opportunity for us to precisely test bound state QED theory and to determine two important fundamental constants the fine structure constant a and the electron-to-proton mass ratio m/mp. The experiment can be performed at any Z with about the same accuracy [1] and one can expect new data at medium Z which will allow to verify the present ability to estimate unknown higher-order corrections (i. e. theoretical uncertainty) in both low-Z and high-Z calculations. 

Typically, because of mechanistic requirements the lyases are highly specific for the nucleophilic donor component. This includes the necessity for a reasonably high substrate affinity as well as the general difficulty of binding and anchoring a rather small molecule in a fashion that restricts solvent access to the carb-anionic site after deprotonation and shields one enantiotopical face of the nucleophile in order to secure correct diastereofacial discrimination (Figs. 2 and 3) [51]. Usually, approach of the aldol acceptor to the enzyme-bound nucleophile occurs stereospecifically following an overall retention mechanism,... 

Each particular value found outside the plausibility bounds was manually inspected in its measurement context in order to decide between alternative decisions to be taken discard the value as wrong, correct the value for obvious reasons, or expand the confidence interval. The percentage of discarded data was surprisingly high with regard to the fact that the data were already quality-checked before. Frequent typical cases were data on land, obviously invalid, or erratic data, temperatures significantly below the freezing point and extremely low salinities likely measured inside fjords or near the shore. This detailed and careful procedure took most of the efforts in the BALTIC project development from 2000 to 2007. 

Another important application of all-orders in aZ atomic QED is the theory of the multicharged ions. Nowadays all elements of the Periodic Table up to Uranium (Z=92) can be observed in the laboratory as H-like, He-like etc ions. The recent achievements of the QED theory of the highly charged ions (HCI) are summarized in [11], [12]. In principle, the QED theory of atoms includes the evaluation of the QED corrections to the energy levels and corrections to the hyperfine structure intervals, as well as the QED corrections to the transition probabilities and cross-sections of the different atomic processes photon and electron scattering, photoionization, electron capture etc. QED corrections can be evaluated also to the different atomic properties in the external fields bound electron -factors and polarizabilities. In this review we will concentrate mainly on the corrections to the energy levels which are usually called the Lamb Shift (here the Lamb Shift should be understood in a more broad sense than the 2s, 2p level shift in a hydrogen). 

---