Classical aspects and fluctuation-behaviour of two dimensional models in statistical mechanics and many body physics

The author sketches the quasiclassical aspects of the D=2 Lenz-Ising-model which are similar to those of the A4-theory. In particular, the conclusion that there are new coherent states which appear to be Majorana fermions are similar to the A4-theory. It would be tempting to speculate that there is an equivalent description in terms of a Lagrangian which contains a Majorana fermion field in the same way as the sine-Gordon equation and the Thirring field seem to be related, but such an equivalence has not yet been established. The only independent evidence for this speculation comes from the investigation of the critical region. An explicit construction is given for the scale invariant limit (corresponding to the A4-theory where the two minima coalesce for the first time) and it is shown that its most simple field theoretical description can be given in terms of a free D=2 Majorana field. All fields, including the basic field variables, can be written as functions of the Majorana field but for some fields (including all odd powers of the basic field) the relation is somewhat nonlocal (involving line-integrals). (Auth.)