The non-linear model Gel60,Wei67 is built to describe pseudoscalar mesons of which we know that: 1 they exist, 2 their scalar partners don't, and 3 they obey the chiral symmetry of SU(2)SU(2). The first two facts are experimental ones, and the third one comes from the lower level (quark) theory.
The SU(2)SU(2) group is isomorphic to the O(4) group - the
orthogonal group in four dimensions Gil74. Therefore,
the meson fields in question can be described by four real fields
, =1,2,3,4, and all we need is a model for the Lagrangian density.
The non-linear model makes the following postulate:
Let us now consider the classical ground state corresponding to Lagrangian density (32). The lowest energy corresponds to particles at rest, =0, and resting at a lowest point of the potential energy (34). Now we have a problem - which one of the lowest points to choose, because any one such that = is as good as any other one. However, the classical fields at space-time point must have definite values, i.e., they spontaneously pick one of the solutions out of the infinitely-many existing ones. Once one of the solutions is picked, the O(4) symmetry is broken, because the ground-state field is not any more invariant with respect to all O(4) transformations. Using the graphical representation of the ``Mexican hat'', Fig. 4, one can say that the system rolls down from the top of the hat, and picks one of the points within the brim.
It is now clear that fields do not constitute the best
variables to look at the problem, because the physics in the radial and
transversal directions is different. Before proceeding any
further, let us introduce variables and that
separately describe these two directions, namely,
We disregard now the part of the Lagrangian density depending on . Indeed, the initial potential (33) has been postulated without any deep reason, and a detailed form of it is, in fact, totally unknown - it comes from the quark level that we did not at all solved. Any potential that confines the field to values close to is good enough. This field must remain in its ground state, because any excitations of it would bring too much energy into a meson, and again, meson's internal structure remains unresolved.