II. Institut für Theoretische Physik

Universität Hamburg

Luruper Chaussee 149

2000 Hamburg 50

Germany

Effective field theories encode the predictions of a quantum field theory at low energy. The effective theory has a fairly low ultraviolet cutoff. As a result, loop corrections are small, at least if the effective action contains a term which is quadratic in the fields, and physical predictions can be read straight from the effective Lagrangean.

Methods will be discussed how to compute an effective low energy action from a given fundamental action, either analytically or numerically, or by a combination of both methods. Basically, the idea is to integrate out the high frequency components of fields. This requires the choice of a ``blockspin'', i.e. the specification of a low frequency field as a function of the fundamental fields. These blockspins will be the fields of the effective field theory. The blockspin need not be a field of the same type as one of the fundamental fields, and it may be composite. Special features of blockspins in nonabelian gauge theories will be discussed in some detail.

In analytical work and in multigrid updating schemes one needs interpolation kernels $\A$ from coarse to fine grid in addition to the averaging kernels $C$ which determines the blockspin. A neural net strategy for finding optimal kernels is presented.

Numerical methods are applicable to obtain actions of effective theories on lattices of finite volume. The special case of a ``lattice'' with a single site (the constraint effective potential) is of particular interest. In a Higgs model, the effective action reduces in this case to the free energy, considered as a function of a gauge covariant magnetization. Its shape determines the phase structure of the theory. Its loop expansion with and without gauge fields can be used to determine finite size corrections to numerical data.

Keywords: Effective action, lattice gauge theory, blockspin methods, multigrid updating schemes, neural multilevel algorithm.

(with 7 PS figures included, 2 figures missing)

Contributed May 15, 1992.