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Symmetry breaking in classical systems and nonlinear functional analysis. Lecture notes. (English) Zbl 1082.35005

The author explains the phenomenon of spontaneous symmetry breaking (SSB) as related to the fact that for nonlinear dynamical systems with infinite degree of freedom their solutions fall into classes (or “islands”) stable under time evolution and with the property that they can not be related by physically realizable operations, i.e. starting from the configurations of a given island one can not reach the configurations of a different island by physically realizable modifications. The different islands can be interpreted as the realization of different physical systems or different phases of a system. The SSB in a given island or phase is explained as a result of the instability of the given island under symmetry operations. In this case one can not associate to each configuration the one obtained by the symmetry operation. The existence of such structures involves a mathematical control of the nonlinear time evolution of systems with infinite degrees of freedom and the formalization of the concept of physical disjointness of different islands.
At the mathematical formalization of physical disjointness an island can be characterized by some reference bounded configuration, having the meaning of the “ground state” and its \(H^1\) perturbations, i.e. each island is isomorphic to a Hilbert space sector.
The stability under time evolution is guaranteed by the condition that the reference configuration satisfied a generalized stationary conditions (i.e. it solves the elliptic problem) which is satisfied by the time independent solutions and minima \(\overline{\varphi}\) of the potential in the relevant Hilbert space sector \(H_{\overline{\varphi}}\). The existence of minima of the potential unstable under the symmetry gives here rise to islands in which the symmetry is spontaneous broken.
The described structures are closely connected with the classical Noether theorem and its improved version. The local conservation law, \(\partial ^{\mu}j_{\mu}(x)=0\), associated to a given symmetry of the Hamiltonian or Lagrangian gives rise to a global conservation law in a given island only if the symmetry leaves the island stable. This means that the improved version of the Noether theorem yields the local conservation laws corresponding to the generators of the dynamics group symmetry, but in a given phase one has the global conservation law only for the generators of the stability group of a given island. In the case of SSB in a given island the elements of \(H_{\overline{\varphi}}\) can not be classified in terms of irreducible representations (multiplets) of \(G\).
Examples of the developed theory are contained in Ch. 8. They are (1) nonlinear scalar field in one space dimension, defined by the potential \(U=-\frac{1}{2}m^2\varphi^2 +\frac{1}{4}\lambda \varphi^4=\frac{1}{4}\lambda(\varphi^2-\frac{m^2}{\lambda})^2-\frac{1}{4}\frac{m^4}{\lambda},\) as a nonlinear generalization of the wave equation and (2) the sine-Gordon equation \(\square \varphi=-g \sin\varphi,\) where \(\varphi(x,t)\) is a scalar field in one space dimension. In Ch. 9 with the aid of this theory a classical counterpart of the Goldstone theorem, according to which there are massless modes (solutions of the free wave equation) associated to each broken generator.
A part of applications, illustrating and auxiliary materials are carried out in the Appendices.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
37C80 Symmetries, equivariant dynamical systems (MSC2010)
58E09 Group-invariant bifurcation theory in infinite-dimensional spaces
58J70 Invariance and symmetry properties for PDEs on manifolds
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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