Loop quantization from a lattice gauge theory perspective

We present an interpretation of loop quantization in the framework of lattice gauge theory. Within this context the lack of appropriate notions of effective theories and renormalization group flow exhibit loop quantization as an incomplete framework. This interpretation includes a construction of embedded spin foam models which does not rely on the choice of any auxiliary structure (e.g. triangulation) and has the following straightforward consequences: (1) The values of the coupling constants need to be those of an UV-attractive fixed point (2) The kinematics of canonical loop quantization and embedded spin foam models are compatible (3) The weights assigned to embedded spin foams are independent of the 2-polyhedron used to regularize the path integral, $|J|_x = |J|_{x'}$ (4) An area spectrum with edge contributions proportional to $l_{\rm PL}^2 (j+1 / 2)$ is not compatible with embedded spin foam models and/or canonical loop quantizationComment: 11 pages, no figures; completely rewritte

Gauge from holography and holographic gravitational observables

In a spacetime divided into two regions $U_1$ and $U_2$ by a hypersurface $\Sigma$, a perturbation of the field in $U_1$ is coupled to perturbations in $U_2$ by means of the holographic imprint that it leaves on $\Sigma$. The linearized gluing field equation constrains perturbations on the two sides of a dividing hypersurface, and this linear operator may have a nontrivial null space. A nontrivial perturbation of the field leaving a holographic imprint on a dividing hypersurface which does not affect perturbations on the other side should be considered physically irrelevant. This consideration, together with a locality requirement, leads to the notion of gauge equivalence in Lagrangian field theory over confined spacetime domains. Physical observables in a spacetime domain $U$ can be calculated integrating (possibly non local) gauge invariant conserved currents on hypersurfaces such that $\partial \Sigma \subset \partial U$. The set of observables of this type is sufficient to distinguish gauge inequivalent solutions. The integral of a conserved current on a hypersurface is sensitive only to its homology class $[\Sigma]$, and if $U$ is homeomorphic to a four ball the homology class is determined by its boundary $S = \partial \Sigma$. We will see that a result of Anderson and Torre implies that for a class of theories including vacuum General Relativity all local observables are holographic in the sense that they can be written as integrals of over the two dimensional surface $S$. However, non holographic observables are needed to distinguish between gauge inequivalent solutions

Category and Topological Complexity of the configuration space F(G×Rn,2)F(G\times \mathbb{R}^n,2)

The Lusternik-Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik-Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product $G\times\mathbb{R}^n$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively.Comment: 10 pages, 1 figure. Final version. To appear in Bulletin of the Australian Mathematical Societ

Quantum Structure of Geometry: Loopy and fuzzy?

In any attempt to build a quantum theory of gravity, a central issue is to unravel the structure of space-time at the smallest scale. Of particular relevance is the possible definition of coordinate functions within the theory and the study of their algebraic properties, such as non-commutativity. Here we approach this issue from the perspective of loop quantum gravity and the picture of quantum geometry that the formalism offers. In particular, as we argue here, this emerging picture has two main elements: i) The nature of the quantum geometry at Planck scale is one-dimensional, polymeric with quantized geometrical quantities and; ii) Appropriately defined operators corresponding to coordinates by means of intrinsic, relational, constructions become non-commuting. This particular feature of the operators, that operationally localize points on space, gives rise to an emerging geometry that is also, in a precise sense, fuzzy.Comment: 9 pages, no figure

A contracting circumbinary molecular ring with an inner cavity of about 140 AU around Ori 139-409

Sensitive and subarcsecond resolution ($\sim$ 0.7\arcsec) CH$_3$OH(7$_{-2,6}$ $\to$ 6$_{-2,5}$) line and 890 $\mu$m continuum observations made with the Submillimeter Array (SMA) towards the hot molecular circumbinary ring associated with the young multiple star Ori 139-409 are presented. The CH$_3$OH(7$_{-2,6}$ - 6$_{-2,5}$) emission from the ring is well resolved at this angular resolution revealing an inner cavity with a size of about 140 AU. A LTE model of a Keplerian disk with an inner cavity of the same size confirms the presence of this cavity. Additionally, this model suggests that the circumbinary ring is contracting with a velocity of V$_{inf}$ $\sim$ 1.5 km s$^{-1}$ toward the binary central compact circumstellar disks reported at a wavelength of 7 mm. {\bf The inner central cavity seems to be formed by the tidal effects of the young stars in the middle of the ring.} The ring appears to be not a stationary object. Furthermore, the infall velocity we determine is about a factor of 3 slower than the free-fall velocity corresponding to the dynamical mass. This would correspond to a mass accretion rate of about 10$^{-5}$ M$_\odot$/yr. We found that the dust emission associated with Ori 139-409 appears to be arising from the circumstellar disks with no strong contribution from the molecular gas ring. A simple comparison with other classical molecular dusty rings (e.g. GG Tau, UZ Tau, and UY Aur) suggests that Ori 139-409 could be one of the youngest circumbinary rings reported up to date. Finally, our results confirm that the circumbinary rings are actively funneling fresh gas material to the central compact binary circumstellar disks, i.e. to the protostars in the very early phases of their evolution.Comment: Accepted by MNRA