To a certain extent, a way out from the explosion of dimensionality,
discussed in Sec. 4.1, may consist in using a better
single-particle space. Instead of parametrizing fields by
space-spin-isospin points , one can use a parametrization by the
shell-model orbitals that are active near the
Fermi surface of a given nucleus, i.e., by fields
The reduction is now not a mere question of discretizing continuous
fields, but involves a serious limitation of the Hilbert space.
In quantum mechanics one can always split the Hilbert space into
two subspaces,
,
where and are projection operators such that .
Then, the Schrödinger equation =
is strictly equivalent to the following 22 matrix of equations,
The main questions is, of course, whether the Bloch-Horowitz effective interaction, = , can be replaced by a simple phenomenological interaction, and used to describe real systems. In particular, when a two-body, energy-independent interaction is postulated in a very small phase space, one obtains the shell model, which is successfully used since many years in nuclear structure physics.
In order to illustrate the dimensions of the shell-model Hilbert space, in Fig. 9 we show the numbers of many-fermion states that are obtained when states in = medium heavy nuclei are described within the space (20 s.p. states for protons and 20 for neutrons). Currently, complete solutions for the space become available, i.e., dimensions of the order of 10 can effectively be treated. Progress in this domain closely follows the progress in size and speed of computers, i.e., one order of magnitude is gained in about every two-three years. We shell not discuss these methods in any more detail, because dedicated lectures have been presented on this subject during the Summer School.