Stability of the Vacuum

The stability of the vacuum is maybe the most important problem in physics. It is related to the van der Waals polarization effects, the cosmological constant, the fine structure constant, the problem of using a reference aether, the Casimir effect, the use of renormalization or regularization schemes to tackle infinities in quantum field theories. As mathematicians, we can look at related problems which make mathematical sense. One of them is to study non-linear sine-Gordon problems Lu=c sin(u) for small c. In the case when L is the discrete Laplacian on the integers, this is the Frenkel-Kontorova model named after Jakov Frenkel and Tatiana Kontorova who studied this in 1938. The stability problem there is already hard. It leads to the study of the Chirikov Standard map. Stable almost periodic solutions are obtained using KAM methods. Hard implicit function theorems are always involved if L is not invertible. I just remind again here that if the Dirac matrix is changed to a connection matrix (essentially replace incidence by intersection and disregard signs), we get a Laplacian L=D^2 that is invertible. By the soft implicit function theorem we know from standard calculus, it follows that Lu = c sin(u) now has only u=0 as a solution for small c. The vacuum is stable. This is also very general in that it works for any abstract simplicial complex that is locally finite. Any graph with a countable set of vertices and a global bound on the vertex degrees would work. The gap size below which we have only vacuum depends on the maximal size of the unit spheres.