Quasi local energy from loop quantum gravity boundary modes

Tuesday, Oct 22

Wolfgang Wieland, Perimeter Institute
Title: Quasi-local energy from LQG boundary modes
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By Jorge Pullin

In physical theories situations can be broken up into subsystems that one can follow individually characterized by a finite number of variables.

However, for nonlinear theories like general relativity one does not necessarily know how to do that. This is related to several problems in the theory like the averaging problem in cosmology. In the latter one typically concentrates on a small number of degrees of freedom of the universe, like its scale, and writes equations for them, pretending the universe is homogeneous, it does not change from one point to the next. This is a coarse approximation, but several important results can be derived from it. The matter content of the universe is not homogeneous and therefore to treat it one typically considers an average. But the average of non-linear functions of some variables is not the same as evaluating those quantities with the average values of the variables. This is known as the averaging problem in cosmology.

In this talk it was proposed to build degrees of freedom for subsystems in general relativity by constructing the degrees of freedom of the whole system starting from those of subsystems. The construction was discussed first in the context of classical general relativity. In the last part of the talk a connection with loop quantum gravity was presented. When one considers subregions of space time in loop quantum gravity the loops pierce the boundaries of the subregions and constitute field theories of "punctures" on them. These naturally embody the degrees of freedom of the subregion. A connection with a particular formulation of loop quantum gravity known as the spinor formulation was also suggested.

Effective dynamics from full loop quantum gravity

Tuesday, Oct 8

Muxin Han, Florida Atlantic University
Title: Effective dynamics from full loop quantum gravity
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By Jorge Pullin, LSU


A hot topic of research is how to derive the equations of loop quantum cosmology from loop quantum gravity. Initial investigations started by freezing most degrees of freedom and keeping the ones relevant for cosmology and proceeding to quantize them using loop quantum gravity inspired techniques. In recent years the focus has moved towards trying to derive things directly from full loop quantum gravity. In this talk a proposal along these lines is put forward. The idea is to use the path integral approach to quantization. This is an approach in which the quantum theory is built by considering all possible paths of the dynamics of the system and assigning probabilities to them. The idea is to perform the path integral using a set of states known as coherent states and study the resulting equations of motion. The technique is applied to several proposals for the evolution operator (Hamiltonian) of the theory that have been put out in the literature. The technique is suitable for numerical evolution opening a contact with numerical relativity. It may be applicable in other situations of interest like cosmological perturbations and binary black holes.

New Loop Quantum Cosmology modifications from Symplectic Structures

Tuesday, May 14th

Klaus Liegener, LSU
Title: New Loop Quantum Cosmology modifications from Symplectic Structures
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By Jorge Pullin, LSU

Loop quantum gravity is based on a new set of variables for describing general relativity that were introduced by Abhay Ashtekar. These variables have a certain amount of redundancy, known as a gauge symmetry.

Loop quantum cosmology is an approximation to loop quantum gravity that attempts to model cosmologies by following only a very limited number of degrees of freedom. A current topic of great interest is to understand how and if this approximation captures behaviors of the full theory. This talk concentrated on the role the redundancies in the variables Ashtekar introduced play in the construction of loop quantum cosmologies. It proposes to use certain variables that better behave under the presence of these redundancies and draws implications for the dynamics of the resulting theory. In particular it implies, as is common in loop quantum cosmology, that the Big Bang at the beginning of the universe is replaced by a "bounce" from a previous universe. However, the dynamics of the current model implies the bounce is asymmetrical, the universe before and after it do not look the same. This had been encountered in some previous proposals for loop quantum cosmologies, but not in all of them.

Quantum geometry from higher gauge theory

Tuesday, Sep 10th

Bianca Dittrich, Perimeter Institute
Title: Quantum geometry from higher gauge theory
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By Jorge Pullin, LSU


Physical theories are usually formulated in terms of sets of redundant variables. A simple example would be to consider the position of a pendulum indicated by its x and y coordinates. One knows that x and y are not arbitrary, they are constrained by the length of the pendulum and the position it is hanged from. More complicated field theories, like electromagnetism or Yang-Mills theories are also formulated in terms of redundant gauge variables and there are constraints between them. When one quantizes theories, the constraints have to be promoted to quantum relations between operators, and this is typically problematic.

This talk advocates for an approach to quantum gravity in which one increases the level of redundancy by introducing extra variables with extra constraints. The resulting theories are equivalent to the original ones as classical theories, but their quantization can be more favorable. The idea is not entirely new, it has been carried out more or less implicitly in quantizations of gravity in three dimensions (or two spatial and one time dimension). In three dimensions the Einstein equations imply all space-times are flat, so the resulting theories are relatively simple to handle. The talk proposes following a similar route for gravity in three spatial and one time dimensions, as we think the real universe has. There are numerous technical issues that need to be overcome to complete this approach and this talk addresses some of them. Among them, implementing the new constraints implied by the new formulation. However, the constraints of the traditional formulation of gravity, that have proven to be very problematic, appear geometrically very transparent in the current approach.

Some comments on canonical gauge theories with boundaries

Tuesday, Sep 24th

Alejandro Corichi, UNAM Morelia
Title: Some comments on cannonical gauge theories with boundaries
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By Jorge Pullin, LSU


When field theories are usually formulated, it is assumed that the portion of space they live in is infinite. There are no boundaries to space, or more precisely, they are placed at infinity. Many physical situations, however, concern situations  with a bounded finite domain. The formulation of field theories requires modifications when boundaries are introduced. The equations of field theories are typically derived through a procedure from a function called the action. Extra terms have to be added to the action if one has a situation with finite boundaries.

Ordinary field theories like electromagnetism have infinitely many degrees of freedom. This means its variables are fields that are functions of space and one can view the value at each point in space as a different degree of freedom. In that sense they are generalizations of usual mechanical systems, which have a finite number of degrees of freedom. Topological field theories are models that, although they have variables that are fields that are functions of space, its equations imply that one only has a finite number of degrees of freedom. This implies many simplifications, and they have proved useful as models in which to test quantization techniques, avoiding the many complexities introduced by having an infinite number of degrees of freedom. Both ordinary and topological field theories are typically described in terms of redundant variables. That means that a several mathematical configurations correspond to a physical configuration. This implies there are symmetries in the theory, this is what is usually called "gauge" symmetries. This talk addressed the issue of topological field theories with boundaries and also ordinary field theories coupled to topological theories. A procedure to treat them was given and shown to work in a pair of examples. Some of the techniques would be of interest to explore fields in the vicinity of black holes, where the horizon (the surface beyond which nothing can return) acts as a natural boundary for fields living in the vicinity of the black hole.