Self-dual noncommutative \phi^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory

We study quartic matrix models with partition function Z[E,J]=\int dM \exp(trace(JM-EM^2-(\lambda/4)M^4)). The integral is over the space of Hermitean NxN-matrices, the external matrix E encodes the dynamics, \lambda>0 is a scalar coupling constant and the matrix J is used to generate correlation functions. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula which gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation which must be solved case by case for given E. These results imply that if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing \beta-function. As main application we prove that Euclidean \phi^4-quantum field theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the infinite volume limit, is exactly solvable and non-trivial. This model is a quartic matrix model, where E has for N->\infty the same spectrum as the Laplace operator in 4 dimensions. Using the theory of singular integral equations of Carleman type we compute (for N->\infty and after renormalisation of E,\lambda) the free energy density (1/volume)\log(Z[E,J]/Z[E,0]) exactly in terms of the solution of a non-linear integral equation. Existence of a solution is proved via the Schauder fixed point theorem. The derivation of the non-linear integral equation relies on an assumption which we verified numerically for coupling constants 0<\lambda\leq (1/\pi).Comment: LaTeX, 64 pages, xypic figures. v4: We prove that recursion formulae and vanishing of \beta-function hold for general quartic matrix models. v3: We add the existence proof for a solution of the non-linear integral equation. A rescaling of matrix indices was necessary. v2: We provide Schwinger-Dyson equations for all correlation functions and prove an algebraic recursion formula for their solutio

The effects and consequences of very large explosive volcanic eruptions

Every now and again Earth experiences tremendous explosive volcanic eruptions, considerably bigger than the largest witnessed in historic times. Those yielding more than 450km3 of magma have been called super-eruptions. The record of such eruptions is incomplete; the most recent known example occurred 26000 years ago. It is more likely that the Earth will next experience a super-eruption than an impact from a large meteorite greater than 1km in diameter. Depending on where the volcano is located, the effects will be felt globally or at least by a whole hemisphere. Large areas will be devastated by pyroclastic flow deposits, and the more widely dispersed ash falls will be laid down over continent-sized areas. The most widespread effects will be derived from volcanic gases, sulphur gases being particularly important. This gas is converted into sulphuric acid aerosols in the stratosphere and layers of aerosol can cover the global atmosphere within a few weeks to months. These remain for several years and affect atmospheric circulation causing surface temperature to fall in many regions. Effects include temporary reductions in light levels and severe and unseasonable weather (including cool summers and colder-than-normal winters). Some aspects of the understanding and prediction of super-eruptions are problematic because they are well outside modern experience. Our global society is now very different to that affected by past, modest-sized volcanic activity and is highly vulnerable to catastrophic damage of infrastructure by natural disasters. Major disruption of services that society depends upon can be expected for periods of months to, perhaps, years after the next very large explosive eruption and the cost to global financial markets will be high and sustained