ABSTRACT
In both statistical mechanics and quantum field theory, the Euler-Lagrange equations for interacting systems, such as (11.26) in the Ginzburg-Landau treatment of a ferromagnet or (8.43) for a non-Abelian gauge theory, are nonlinear equations governing the behaviour of fields such as the magnetization density M(x) or the gauge field Aµ(x). Until now, we have dealt with the nonlinearities perturbatively (apart from our qualitative discussion of QCD in §12.5, where we saw that perturbation theory works only at high energies). That is to say, we have identified constant values of the fields that represent the most stable state of the system by minimizing an appropriate potential or free energy, and treated fluctuations about these constant values as excitations that interact only weakly. In quantum field theory (other than confining theories such as QCD), the quantized ‘excitations’ of the vacuum state are, of course, the particles observed by experimenters.
