The Feynman diagrams and rules of a field theory with local four fermion coupling are transformed into a theory of interacting fermions in a finite dimensional Hilbert space constructed on Z3 points of a cubic lattice. The propagator is derived for the lattice space theory and the formal similarity and the qualitative differences are discussed and compared with conventional field theories. The mass produced by the interaction is calculated by compensation of the first order self-energy diagrams (Hartree-Fock-approximation). The solution for coupling constant W > 0 violates the chiral symmetry, the solution for W < 0 the cubic symmetry of the theory. The second order corrections to the mass operator are calculated. They diverge in the limit Z→∞ like if W is proportional to which is necessary for a finite mass in the HFapproximation. Under the same assumption the maximum divergence is like in any order of the perturbation theory. The mass operator is finite including second order contributions if W is proportional to . In this case the higher order contributions decrease like .
Symmetrical hydrogen bonds with a double minimum potential well or a largely flat potential are extremely easily polarisable, which leads to strong mutual interactions of such bonds and to interactions of the bonds with their environment. Particularly in the range of the rearrangement frequencies of the medium, the interactions depend to a marked extent on the fluctuation frequencies of the electromagnetic field and thus on the tunelling frequency of the proton in the hydrogen bond. On the other hand, the tunnelling frequency decreases due to the interactions. The correlation of the proton movements in two adjacent symmetrical hydrogen bonds and the resulting decrease of the tunnelling frequency is studied for the lowest group of levels. In addition, the decrease of the tunnelling frequency due to the effect of ion and dipole fields is calculated. In the tunnelling approximation the result depends in both cases on simple dimensionless parameters and indicates delay of the proton transfer in the bonds, but not complete hindrance. The lingering times in the potential wells are simply connected in both cases with the transition frequencies of the systems.
The displacement of the excess charge of the proton in acid solutions is caused by a structure migration of groupings H5O2+ or H9O4+ . The processes which take place during structure migration are discussed on the basis of results gained in IR-investigations. In an electrical field the structure migration is given a preferred direction. The hydrogen bond with the tunneling proton in H5O2+ and the grouping H3O+ in H5O2+ become polarized. Comparison of both polarizabilities demonstrates that, contrary to previous assumptions, the polarization of the hydrogen bond is the field-dependent mechanism. This conclusion is reached upon calculating the polarizability of the hydrogen bond with a symmetrical double minimum potential well. It is shown that the polarizability is extremely large, being approximately two orders of magnitude greater than that of H3O+ . Despite the large polarizability, the shift of the weights of the proton boundary structures is very small for the external fields usually applied in conductivity measurements. It is demonstrated, however, that this slight shift is large enough for the structure diffusion to obtain a preferred direction consistent with the anomalous high proton conductivity.