A variational method is studied based on the minimum of energy variance. The method is tested on exactly soluble problems in quantum mechanics, and is shown to be a useful tool whenever the properties of states are more relevant than the eigenvalues. In quantum field theory the method provides a consistent second-order extension to the Gaussian effective potential.

We present a general interpolation theory for the phenomenological effects of thermal fluctuations in superconductors. Fluctuations are described by a simple gauge invariant extension of the gaussian effective potential for the Ginzburg-Landau static model. The approach is shown to be a genuine variational method, and to be stationary for infinitesimal gauge variations around the Landau gauge. Correlation and penetration lengths are shown to depart from the mean field behaviour in a more or less wide range of temperature below the critical regime, depending on the class of material considered. The method is quite general and yields a very good interpolation of the experimental data for very different materials.

The Gaussian Effective Potential (GEP) is derived for the non-Abelian SU(2)×U(1) gauge theory of electroweak interactions. First the problem of gauge invariance is addressed in the Abelian U(1) theory, where an optimized GEP is shown to be gauge invariant. The method is then extended to the full non-Abelian gauge theory where, at variance with naive derivations, the GEP is proven to be a genuine variational tool in any gauge. The role of ghosts is discussed and the unitarity gauge is shown to be the only choice which allows calculability without insertion of further approximations. The GEP for the standard model is derived and its predictions are compared to the known phenomenology, thus showing that the GEP provides an alternative non-perturbative description of the known experimental data. By a consistent renormalization of masses the full non-Abelian calculation confirms the existence of a light Higgs boson in the non-perturbative strong coupling regime of the Higgs sector.

The problem of mass generation is addressed by a Gaussian variational method for the minimal left-right symmetric model of electroweak interactions. Without any scalar bidoublet, the Gaussian effective potential is shown to have a minimum for a broken symmetry vacuum with a finite expectation value for both the scalar Higgs doublets. The symmetry is broken by the fermionic coupling that destabilizes the symmetric vacuum, yielding a self-consistent fermionic mass. In this framework a light Higgs is only compatible with the existence of a new high energy mass scale below 2 TeV

A non-perturbative effective model is derived for the Higgs sector of the standard model, described by a simple scalar theory. The renormalized couplings are determined by the derivatives of the Gaussian Effective Potential that are known to be the sum of infinite bubble graphs contributing to the vertex functions. A good agreement has been found with strong coupling lattice simulations when a comparison can be made.