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In this paper, we propose that at the beginning of the universe gravity existed in a limbo either because it was switched off or because it was only conformally coupled to all particles. This picture can be reverse-engineered from the requirement that the cosmological perturbations be (nearly) scale-invariant without the need for inflation. It also finds support in recent results in quantum gravity suggesting that spacetime becomes two-dimensional at super-Planckian energies. We advocate a novel top-down approach to cosmology based on the idea that gravity and the Big Bang Universe are relics from the mechanism responsible for breaking the fundamental conformal invariance. Such a mechanism should leave clear signatures in departures from scale-invariance in the primordial power spectrum and the level of gravity waves generated. [ABSTRACT FROM AUTHOR]

Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the 2D conformal algebra. In the case without external forces, the symmetry of the conformally-invariant transport equation is first generalized by considering the particle momentum as an independent variable. This new conformal representation can be further extended to include an external force. The construction and possible physical applications are outlined. [ABSTRACT FROM AUTHOR]

The convex-transitivity property can be seen as a convex generalization of the almost transitive (or quasi-isotropic) group action of the isometry group of a Banach space on its unit sphere. We will show that certain Banach algebras, including conformal invariant Douglas algebras, are weak-star convex-transitive. Geometrically speaking, this means that the investigated spaces are highly symmetric. Moreover, it turns out that the symmetry property is satisfied by using only 'inner' isometries, i.e. a subgroup consisting of isometries which are homomorphisms on the algebra. In fact, weighted composition operators arising from function theory on the unit disk will do. Some interesting examples are provided at the end. [ABSTRACT FROM AUTHOR]

The triangular ratio metric is studied in subdomains of the complex plane and Euclidean n-space. Various inequalities are proven for this metric. The main results deal with the behavior of this metric under quasiconformal maps. We also study the smoothness of metric disks with small radii. [ABSTRACT FROM AUTHOR]

Atomic physics and hadronic physics are both governed by the Yang Mills gauge theory Lagrangian; in fact, Abelian quantum electrodynamics can be regarded as the zero-color limit of quantum chromodynamics. I review a number of areas where the techniques of atomic physics can provide important insight into hadronic eigenstates in QCD. For example, the Dirac-Coulomb equation, which predicts the spectroscopy and structure of hydrogenic atoms, has an analog in hadron physics in the form of frame-independent light-front relativistic equations of motion consistent with light-front holography which give a remarkable first approximation to the spectroscopy, dynamics, and structure of light hadrons. The production of antihydrogen in flight can provide important insight into the dynamics of hadron production in QCD at the amplitude level. The renormalization scale for the running coupling is unambiguously set in QED; an analogous procedure sets the renormalization scales in QCD, leading to scheme-independent scale-fixed predictions. Conversely, many techniques which have been developed for hadron physics, such as scaling laws, evolution equations, the quark-interchange process and light-front quantization have important applicants for atomic physics and photon science, especially in the relativistic domain. [ABSTRACT FROM AUTHOR]

In this paper the geometrical features of the fluid propagation in two-dimensional petroleum reservoir described by Darcy equations are studied. The porous media are considered to be tuned by the occupancy parameter p being the probability that a pore is occupied. We analyze the statistical geometrical observables of the Darcy model. To this end we let the water to be injected into random sites of the porous media and solve numerically the Darcy equations to describe the flow motion pattern, using the control volume finite difference (CVFD) method. The fractal dimension of the frontier of the avalanches (defined as the set of the sites through which the fluid passed) and the distribution functions of gyration radius, loop length and cluster mass are numerically obtained revealing that at p = p c (the critical occupancy parameter above which there is definitely a spanning cluster in the system) this model lies within a universality class compatible with the Ising model. We observe that for p > p c , although the model shows critical behaviors, this duality is broken. The mentioned exponents are reported in this paper. [ABSTRACT FROM AUTHOR]

We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local times, entrance laws and excursion laws is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh’s Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh’s Brownian motion, recently obtained by Fitzsimmons and Kuter (2014) using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD). [ABSTRACT FROM AUTHOR]

In this review article, we discuss the distinction and possible equivalence between scale invariance and conformal invariance in relativistic quantum field theories. Under some technical assumptions, we can prove that scale invariant quantum field theories in d = 2 space–time dimensions necessarily possess the enhanced conformal symmetry. The use of the conformal symmetry is well appreciated in the literature, but the fact that all the scale invariant phenomena in d = 2 space–time dimensions enjoy the conformal property relies on the deep structure of the renormalization group. The outstanding question is whether this feature is specific to d = 2 space–time dimensions or it holds in higher dimensions, too. As of January 2014, our consensus is that there is no known example of scale invariant but non-conformal field theories in d = 4 space–time dimensions under the assumptions of (1) unitarity, (2) Poincaré invariance (causality), (3) discrete spectrum in scaling dimensions, (4) existence of scale current and (5) unbroken scale invariance in the vacuum. We have a perturbative proof of the enhancement of conformal invariance from scale invariance based on the higher dimensional analogue of Zamolodchikov’s c -theorem, but the non-perturbative proof is yet to come. As a reference we have tried to collect as many interesting examples of scale invariance in relativistic quantum field theories as possible in this article. We give a complementary holographic argument based on the energy-condition of the gravitational system and the space–time diffeomorphism in order to support the claim of the symmetry enhancement. We believe that the possible enhancement of conformal invariance from scale invariance reveals the sublime nature of the renormalization group and space–time with holography. This review is based on a lecture note on scale invariance vs conformal invariance, on which the author gave lectures at Taiwan Central University for the 5th Taiwan School on Strings and Fields. [ABSTRACT FROM AUTHOR]

If the three dimensional self-avoiding walk (SAW) is conformally invariant, then one can compute the hitting densities for the SAW in a half-space and in a sphere (Kennedy in Phys Rev Lett 111: 165703, ). The ensembles of SAW's used to define these hitting densities involve walks of arbitrary lengths, and so these ensembles cannot be directly studied by the pivot Monte Carlo algorithm for the SAW. We show that these mixed length ensembles should have the same scaling limit as certain weighted ensembles that only involve walks with a single length, thus providing a fast method for simulating these ensembles. Preliminary simulations which found good agreement between the predictions and Monte Carlo simulations for the SAW were reported in Kennedy (Phys Rev Lett 111: 165703, ). In this paper we present more accurate simulations testing the predictions and find even stronger support for the prediction that the SAW is conformally invariant in three dimensions. [ABSTRACT FROM AUTHOR]