Explicit and spontaneous breaking of SU(3) into its finite subgroups

We investigate the breaking of SU(3) into its subgroups from the viewpoints of explicit and spontaneous breaking. A one-to-one link between these two approaches is given by the complex spherical harmonics, which form a complete set of SU(3)-representation functions. An invariant of degrees p and q in complex conjugate variables corresponds to a singlet, or vacuum expectation value, in a (p,q)-representation of SU(3). We review the formalism of the Molien function, which contains information on primary and secondary invariants. Generalizations of the Molien function to the tensor generating functions are discussed. The latter allows all branching rules to be deduced. We have computed all primary and secondary invariants for all proper finite subgroups of order smaller than 512, for the entire series of groups \Delta(3n^2), \Delta(6n^2), and for all crystallographic groups. Examples of sufficient conditions for breaking into a subgroup are worked out for the entire T_{n[a]}-, \Delta(3n^2)-, \Delta(6n^2)-series and for all crystallographic groups \Sigma(X). The corresponding invariants provide an alternative definition of these groups. A Mathematica package, SUtree, is provided which allows the extraction of the invariants, Molien and generating functions, syzygies, VEVs, branching rules, character tables, matrix (p,q)_{SU(3)}-representations, Kronecker products, etc. for the groups discussed above.Comment: 62 pages, 5 figures; the corresponding software package SUtree can be downloaded from http://theophys.kth.se/~amerle/SUtree/SUtree.html New in v2: Nice figure added, references added, explicit transformation matrices between different embeddings calculated, software package update

Detection of gravitational waves from black holes: Is there a window for alternative theories?

Recently LIGO and VIRGO collaborations reported about observation of gravitational-wave signal corresponding to the inspiral and merger of two black holes, resulting into formation of the final black hole. It was shown that the observations are consistent with the Einstein theory of gravity with high accuracy limited mainly by the statistical error. Angular momentum and mass of the final black hole were determined with rather large allowance of tens of percents. Here we shall show that this indeterminacy in the range of the black-hole parameters allows for some not negligible deformations of the Kerr spacetime leading to the same frequencies of black-hole ringing. This means that at the current precision of the experiment there remain some possibilities for alternative theories of gravity.Comment: 5 pages, 1 figur

Trade Costs, Conflicts, and Defense Spending

This paper develops a quantitative model of trade, military conflicts, and defense spending. Trade liberalization between two countries reduces probability of an armed conflict between them, causing both to cut defense spending. This in turn causes a domino effect on defense spending by other countries. As a result, both countries and the rest of the world are better off. We estimate the model using data on trade, conflicts, and military spending. We find that, after reduction of costs of trade between a pair of hostile countries, the welfare effect of worldwide defense spending cuts is comparable in magnitude to the direct welfare gains from trade