Complementarity in classical dynamical systems

The concept of complementarity, originally defined for non-commuting observables of quantum systems with states of non-vanishing dispersion, is extended to classical dynamical systems with a partitioned phase space. Interpreting partitions in terms of ensembles of epistemic states (symbols) with corresponding classical observables, it is shown that such observables are complementary to each other with respect to particular partitions unless those partitions are generating. This explains why symbolic descriptions based on an \emph{ad hoc} partition of an underlying phase space description should generally be expected to be incompatible. Related approaches with different background and different objectives are discussed.Comment: 18 pages, no figure

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

Galilean Conformal and Superconformal Symmetries

Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: Schroedinger, Galilean conformal and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c equal to infinity from relativistic conformal algebra O(d+1,2) (d - number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D=4 superconformal algebra the Galilei conformal superalgebra (GCSA) is obtained, in the formulation using complex Weyl supercharges.Comment: 16 pages, LateX; talk presented at XIV International Conference "Symmetry Methods in Physics", Tsakhkadzor, Armenia, August 16-22, 201

Black Hole Thermodynamics and Statistical Mechanics

We have known for more than thirty years that black holes behave as thermodynamic systems, radiating as black bodies with characteristic temperatures and entropies. This behavior is not only interesting in its own right; it could also, through a statistical mechanical description, cast light on some of the deep problems of quantizing gravity. In these lectures, I review what we currently know about black hole thermodynamics and statistical mechanics, suggest a rather speculative "universal" characterization of the underlying states, and describe some key open questions.Comment: 35 pages, Springer macros; for the Proceedings of the 4th Aegean Summer School on Black Hole

Mode of origin differentially influences the fitness of parthenogenetic freshwater snails

How parthenogenetic lineages arise from sexual ancestors may strongly influence their persistence over evolutionary time. Hybrid parthenogens often have elevated heterozygosity and ploidy, thus making it difficult to disentangle the influence of reproductive mode, hybridity and ploidy on their relative fitness. By comparing the relative fitness of both hybrid and non-hybrid parthenogens to their sexual ancestors, further insight may be gained into how these three factors influence the maintenance of sexual and parthenogenetic reproduction. In the present study, hybrid and non-hybrid parthenogenetic and sexual snails (Campeloma sp.) were compared for the following characteristics: female size-fecundity curves, offspring size, survivorship, and growth. Compared to nearby sexual populations, triploid hybrid parthenogens from the Florida Gulf coast have similar fecundity and offspring size, five-times higher survivorship, and 60% higher growth. Relative to nearby sexual populations, non-hybrid parthenogenetic C. limum from the Atlantic coast have significantly higher fecundity, smaller offspring size, similar survivorship and slightly lower growth. Given the considerable fitness advantages of parthenogens, especially hybrid parthenogens, it is enigmatic as to why these parthenogens occupy marginal natural habitats