Global invariant

We have seen that the quark mass dependence of ferromagnetism should be important, while we have treated it as an input parameter. When we consider the realization of chiral symmetry in QCD, the quark mass should be dynamically generated as a result of the vacuum superconductivity qq pairs are condensed in the vacuum. We consider here SU(2)l x SU(2)r symmetry. Then Lagrangian should be globally invariant under the operation of any group element with constant parameters, except the symmetry-breaking term... 

In order to demonstrate that spontaneous symmetry breaking can affect the energy inherent in the vacuum, consider the globally invariant Higgs Lagran-gian ... 

Equation (348) is the globally invariant wave equation defining a, and Eq. (349) is its locally invariant equivalent. Using the locally invariant Lagrangian (345) in Eq. (347) gives the inhomogeneous held equation (SI units)... 

There is therefore a balance between globally invariant Lagrangians ... 

The starting point of our derivation is the globally invariant 0(3) Lagrangian of the Higgs mechanism... 

The non-simply connected U(l) vacuum is considered first to illustrate the method as simply as possible. This is defined as earlier in this review by the globally invariant Lagrangian density... 

The collapse of the global invariant set at d = 2 is caused by a period 4 resonance. The global invariant set rapidly recovers from this collapse as d increases past the resonance at d = 2. From Ref. 44. (continued)... 

In two-dimensional mappings, a global invariant set with a positive area is preserved by the escape dynamics. However, in four-dimensional mappings Arnold diffusion precludes the existence of complete barriers formed by invariant tori. Accordingly, no invariant set of positive Legesgue measure is expected to exist. Nevertheless, numerical integration shows that a quasi-invariant set persists for a very long time. This quasi-invariant set shows a property similar to the invariant... 

It is amazing that variational solutions obey the same general theorems as the exact solutions. This was first pointed out in a remarkable article by Fock [48]. Let us give a few examples. We shall assume that the set of trial wave functions is globally invariant under rescaling r, ->Ar,. 

The observed scaling of p has an important physical interpretation Once p approaches the characteristic invariant and statistical distributions generated by the global invariant manifold, it then evolves everywhere at the same rate as the mean density. In other words, if the mean intermaterial area density is doubled, the local density is doubled everywhere. This is important, because it means that the time evolution of time evolution of p at aU locations of the chaotic flow (i.e., intimacy of mixing improves everywhere by the same factor). Similarly, the striation thickness both locally and globally ... 

Giona, M., A. Adrover, F. J. Muzzio, S. CabeUi, and M. M. Alvarez (1999). The geometry of mixing in time-periodic chaotic flows I. Asymptotic directionality in physically realizable flows and global invariant propaties, Physica D, 132, 298-324. 

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