Simplicial group field theory

Tuesday, May 2nd
Marco Finocchiaro, Albert Einstein Institute
Title: Simplicial Group Field Theory models for Euclidean quantum gravity: recent developments 
PDF of the talk (2M)
Audio+Slides [.mp4 15MB]

by Jorge Pullin, Louisiana State University

The approach to quantum gravity known as “spin foams” is based on the quantization technique known as the path integral. In this technique probabilities are assigned for a given slice of space to transition to a future slice in a space-time. Since in loop quantum gravity spatial slices are associated to spin networks, as these evolve in time transitioning to slices of the future one gets the “spin foams”. Group field theory is a technique in which an ordinary (but non-local) quantum field theory is constructed in such a way that its Feynman diagrams yield the probabilities of the spin foam approach. There is an analogue en 1+1 dimensions known as “matrix models” that were widely studied in the 1990’s. Group field theories can be viewed as their generalization to four dimensions.

 Formulating spin foams in terms of group field theories has several advantages. Results do not depend on the triangulations picked, as one expects it should be but is not obvious in terms of spin foams. One can import techniques from field theories, in particular to introduce notions of renormalizability and a continuum limit.

 In this talk a particular group field theory model is presented and discussed in some detail. In particular a numerical analysis of the resulting probabilities was made. And results were compared to a popular model of spin foams, the EPRL model. Certain insights on the possible choices in the construction of the model and how it could influence the ultraviolet behavior and possible singularities present were discussed.

Loop quantum gravity with homogeneously curved vacuum

Tuesday, Apr 18th
Bianca Dittrich, Perimeter Institute
Title: (3+1) LQG with homogeneously curved vacuum 
PDF of the talk (8M)
Audio+Slides [.mp4 17MB]

by Jorge Pullin, Louisiana State

The way geometries are studied mathematically one starts with a set of points that has a notion of proximity. One can say when points are close to each other. This is not the same as being able to measure distances in the set. That requires the introduction of an additional mathematical structure, a metric. The sets of points with notion of proximity are known as “manifolds”. General relativity is formulated on a manifold and is a theory about a metric to be imposed on that manifold. Ordinary quantum field theories, like quantum electrodynamics, require the introduction of a metric before they can be formulated, so they are of different nature than general relativity. Theories that do not require a metric in order to be formulated are known as “background independent”. Interestingly, although general relativity is a theory about a metric, it can be formulated without any prior metric. There exist quantum field theories that can be formulated without a metric. They are known as topological field theories and they typically, contrary to ordinary field theories, have only a finite number of degrees of freedom. This means that they are much easier to treat and to quantize.

An example of a topological field theory is general relativity in three space-time dimensions. In one dimension less than four, the Einstein equations just say that the metric is flat, except at a finite number of points. So space-time is flat everywhere with curvature concentrated at just a few points. An example of a space that is flat everywhere except at a point is a cone. The only place that is curved is the tip. One has to remember that the notion of curvature we are talking here is one that can be measured from inside the space-time (typically by going around a circle and seeing if a vector carried around returns parallel to itself). If you do that in a cone on any circle that does not thread the tip, the vector comes back parallel to itself. So space-times in three dimensional general relativity are said to have “conical singularities” at the points where the curvature is non-zero. As other topological field theories, general relativity in three space-time dimensions has a finite number of degrees of freedom. This explains why Witten was able to complete its quantization in the mid 1980’s whereas the quantization of four dimensional general relativity is still a big outstanding problem today.


In this talk a generalization of three dimensional general relativity to four dimensions was presented. The resulting theory in four space-time dimensions has curvature concentrated at edges (strings) –as opposed to points as we had in the three dimensional case- and elsewhere the metric is flat. This makes them much easier to quantize than general relativity. Among the results was the construction of four dimensional quantum geometries similar to those in a previous model by Crane and Yetter. Also a role for quantum groups, that had been conjectured to arise when one considers a cosmological constant was found providing more evidence to this assertion. The space of quantum states (Hilbert space) was rigorously constructed and leads to insights about how the continuum limit of the theory could emerge. The hope is that one could build on these theories to construct new representations for loop quantum gravity in four space-time dimensions and hopefully to implement on them the (quantum) dynamics of general relativity.


Also a notion of duality emerges. In this context, duality means a certain relationship between the metric and the curvature of the space-time at a classical level. Here it can be implemented at a quantum level and quantum space of states associated with the metric (areas) and curvatures can be introduced and are dual to each other. Similar spaces had been proposed for general relativity, but here there is much more mathematical control over them, so this provides a controlled arena to test ideas that are being put forward in the context of loop quantum gravity in four space-time dimensions.

Transition times through the black hole bounce

Tuesday, Apr 4th
Parampreet Singh, LSU
Title: Transition times through the black hole bounce 
PDF of the talk (2M)
Audio+Slides [.mp4 18MB]

by Gaurav Khanna, University of Massachusetts Dartmouth


Loop quantum cosmology (LQC) is an application of loop quantum gravity theory in the context of spacetimes with a high degree of symmetry (e.g. homogeneity, isotropy). One of the main successes of LQC is the resolution of "singularities" that generically appear in the classical theory. An example of this is the "big bang" singularity that causes a complete breakdown of general relativity (GR) in the very early universe. Models studied within the framework of LQC replace this "big bang" with a "big bounce" and do not suffer a singular breakdown like in the classical theory.


It is, therefore, natural to consider applying similar techniques to study black holes; after all, these solutions of GR are also plagued with a central singularity. In addition, it is plausible that a LQC model may shed some light on long-standing issues in black hole physics, i.e., information loss, Hawking evaporation, firewalls, etc.


Now, if one restricts the model only to the Schwarzschild black hole interior region, the spacetime can actually be considered as a homogeneous, anisotropic cosmology (the Kantowski-Sachs spacetime). This allows techniques of LQC to be readily applied to the black hole case. In fact, a good deal of study has been performed in this direction by Ashtekar, Bojowald, Modesto and many others for over a decade. While these models are able to resolve the central black hole singularity and include important improvements over previous versions, they still have a number of issues.


Recently, Singh and Corichi (2016), proposed a new LQC model for the black hole interior that attempts to address these issues. In this talk, Singh describes some of the resulting phenomenology that emerges from that improved model.

The main emphasis of this talk is on the following questions:
 

(1) Is the "bounce" in the context of a black hole LQC model, i.e., transition from a black hole to a white hole, symmetric? Isotropic and homogeneous models in LQC have generally exhibited symmetric bounces. But, that is not expected to hold in the context of more general models.
(2) Does quantum gravity play a role only once during the bounce process?
(3) What quantitative statements can be made about the time-scales of this process; and what are the full implications of those details?
 (4) Do all black holes, independent of size, exhibit very similar characteristics?


Based on detailed numerical calculations that Singh reviews in his presentation, he uncovers the following features from this model:


(1) The bounce is indeed not symmetric; for example, the sizes of the parent black hole and the offspring white hole are widely different. Other details on this asymmetry appear below.
(2) Two distinct quantum regimes appear in this model, with very different associated time-scales.
(3) In terms of the proper time of an observer, the time spent in the quantum white hole geometry is much larger than in the quantum black hole. And, in particular, the time for the observer to reach the white hole horizon is exceedingly large. This also implies that the formation of the white hole interior geometry happens a lot quicker than the formation of its horizon.
(4) The relation of the bounce time with the black hole mass, does depend on whether the black hole is large or small.


On the potential implications of such details on some of the important open questions in black hole physics, Singh speculates:


(1) For large black holes, the time to develop a white hole (horizon) is much larger than the Hawking evaporation time. This may suggest that for an external observer, a black hole would disappear long before the white hole appears!
(2) For small black holes, the time to form a white hole is smaller than Hawking time, i.e., small black holes explode before they can evaporate!


These could have some interesting implications for the various proposed black hole evaporation paradigms. Given the concreteness of the results Singh presents, they are also likely to be relevant to the many previous phenomenological studies on black hole to white hole transitions including Planck stars.


The two main limitations of Singh's results are: (1) the current model ignores the black hole exterior entirely; and (2) the conclusions rely on effective dynamics, and not the full quantum evolution. These may be addressed in future work.

Holographic signatures of resolved cosmological singularities

Tuesday, March 21st
Norbert Bodendorfer, LMU Munich
Title: Holographic signatures of resolved cosmological singularities 
PDF of the talk (2M)
Audio+Slides [.mp4 10MB]

By Jorge Pullin, Louisiana State University

One of the most important results in string theory is the so called “Maldacena conjecture” or “AdS/CFT correspondence” proposed by Juan Maldacena. This conjecture states that given a space-time with cosmological constant (known as anti De Sitter space-time or AdS) the behavior of gravity in it is equivalent to the behavior of a field theory living on the boundary of the space-time. These field theories are of a special type known as “conformal field theories”. Hence the AdS/CFT name. Conformal field theories are considerably better understood than quantum gravity so to make the latter equivalent to them opens several new possibilities. The discussion of AdS/CFT has mostly taken place in the context of string theory which has general relativity as a classical limit. This opens the question of what kind of imprint the singularities that are known to exist in general relativity leave in the conformal field theory.

On the other hand, loop quantum gravity is known for eliminating the singularities that arise in general relativity. They get replaced by regions of high curvature and fluctuations of it that are not well described by a semiclassical geometry. However, nothing is singular, physical variables may take large –but finite-values. If AdS/CFT were to hold in the context of loop quantum gravity the question arises of what imprint would the elimination of the singularity leave on the conformal field theory. The seminar dealt with this point by considering certain functions known as correlation functions in the conformal field theory that characterize its behavior. In particular how the singularities of general relativity get encoded in these correlation functions and how their elimination in loop quantum gravity changes them. The work is at the moment only a model in five dimensions of a particular space-time known as the Kasner space-time.

Future work will consist in expanding the results to other space-times. Of particular interest would be the extension to black hole spacetimes, which loop quantum gravity also rids of singularities. As is well known, black hole space-times have the problem of the “information paradox” stemming from the fact that black holes evaporate through the radiation that Hawking predicted leaving in their wake only thermal radiation no matter what process led to the formation of the black hole. It is expected that when the evaporation is viewed in terms of the conformal field theory, this loss of information about what formed the black hole will be better understood.

In addition to the specific results, the fact that this work suggests points of contact between loop quantum gravity and string theory makes it uniquely exciting since both fields have developed separately over the years and could potentially benefit from cross pollination of ideas.

Gravity as the dimensional reduction of a theory of forms in six or seven dimensions

Tuesday, February 21st
Kirill Krasnov, University of Nottingham
Title: 3D/4D gravity as the dimensional reduction of a theory of differential forms in 6D/7D 
PDF of the talk (5M)
Audio+Slides [.mp4 16MB]

by Jorge Pullin, Louisiana State University

Ordinary field theories, like Maxwell’s electromagnetism, are physical systems with infinitely many degrees of freedom. Essentially the values of the fields at all the points of space are the degrees of freedom. There exist a class of field theories that are formulated as ordinary ones in terms of fields that take different values at different points in space,  but that whose equations of motion imply that the number of degrees of freedom are finite. This makes some of them particularly easy to quantize. A good example of this is general relativity in two space and one time dimensions (known as 2+1 dimensions). Unlike general relativity in four-dimensional space-time, it only has a finite number of degrees of freedom that depend on the topology of the space-time considered. This type of behavior tends to be generic for these types of theories and as a consequence they are labeled Topological Field Theories (TFT). These types of theories have encountered application in mathematics to explore geometry and topology issues, like the construction of knot invariants, using quantum field theory techniques. These theories have the property of not requiring any background geometric structure to define them unlike, for instance, Maxwell theory, that requires a given metric of space-time in order to formulate it.

Remarkably, it was shown some time ago by Plebanski, in 1977 and later further studied by Capovilla-Dell-Jacobson and Mason in 1991 that certain four dimensional TFTs, if supplemented by additional constraints among their variables, were equivalent to general relativity. The additional constraints had the counterintuitive effect of adding degrees of freedom to the theory because they modify the fields in terms of which the theory is formulated. Formulating general relativity in this fashion leads to new perspectives on the theory. In particular it suggests certain generalizations of general relativity, which the talk refers to as deformations of GR.

The talk considered a series of field theories in six and seven dimensions. The theories do not require background structures for their definition but unlike the topological theories we mentioned before, they do have infinitely many degrees of freedom. Then the dimensional reduction to four dimensional of these theories was considered. Dimensional reduction is a procedure in which one “takes a lower dimensional slice” of a higher dimensional theory, usually by imposing some symmetry (for instance assuming that the fields do not depend on certain coordinates). One of the first such proposals was considered in 1919 by Kaluza and further considered later by Klein so it is known as Kaluza-Klein theory. They considered general relativity in five dimensions and by assuming the metric does not depend on the fifth coordinate, were able to show that the theory behaved like four-dimensional general relativity coupled to Maxwell’s electromagnetism and a scalar field. In the talk it was shown that the seven dimensional theory considered, when reduced to four dimensions, was equivalent to general relativity coupled to a scalar field. The talk also showed that certain topological theories in four dimensions known as BF theories (because the two variables of the theory are fields named B and F) can be viewed as dimensional reductions from topological theories in seven dimensions and finally that general relativity in 2+1 dimensions can be viewed as a reduction of a six dimensional topological theory.

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At the moment is not clear whether these theories can be considered as describing nature, because it is not clear whether the additional scalar field that is predicted is compatible with the known constraints on scalar-tensor theories. However, these theories are useful in illuminating the structures and dynamics of general relativity and connections to other theories.

Loop Quantum Gravity, Tensor Network, and Holographic Entanglement Entropy

Tuesday, February 7th
Muxin Han, Florida Atlantic University
Loop Quantum Gravity, Tensor Network, and Holographic Entanglement Entropy 
PDF of the talk (2M)
Audio+Slides [.mp4 18MB]

by Jorge Pullin, Louisiana State University


The cosmological constant is an extra term that was introduced into the equations of General Relativity by Einstein himself. At the time he was trying to show that if one applied the equations to the universe as a whole, they had static solutions. People did not know in those days that the universe expanded. Some say that Einstein called the introduction of this extra term his “biggest blunder” since it prevented him from predicting the expansion of the universe which was observed experimentally by Hubble a few years later. In spite of its origin, the term is allowed in the equations and the space-times that arise when one includes the term are known as de Sitter space-times in honor of the Dutch physicist who first found some of these solutions. Depending on the sign of the cosmological constant chosen, one could have de Sitter or anti-de Sitter (AdS) space-times.


It was observed in the context of string theory that if one considered quantum gravity in anti-de Sitter space-times, the theory was equivalent to a certain class of field theories known as conformal field theories (CFT) living on the boundary of the space-time. The result is not a theorem but a conjecture, known as AdS/CFT or Maldacena conjecture. It has been verified in a variety of examples. It is a remarkable result. Gravity and conformal field theories are very different in many aspects and the fact that they could be mapped to each other opens many possibilities for new insights. For instance, an important open problem in gravity is the evaporation of black holes. Although nothing can escape a black hole classically, Hawking showed that if quantum effects are taken into account, black holes radiate particles like a black body at a given temperature. The particles take away energy and the black hole shrinks, eventually evaporating completely. This raises the question of what happened to matter that went into the black hole. Quantum mechanics has a property named unitarity that states that ordinary matter cannot turn into incoherent radiation, so this raises the question of how it could happen in an evaporating black hole. In the AdS/CFT picture, since the evaporating black hole would be mapped to a conformal field theory that is unitary, that would provide a way to study quantum mechanically how matter turns into incoherent radiation.


Several authors have connected the AdS/CFT conjecture to a mathematical construction known as tensor networks that is commonly used in quantum information theory. Tensor networks have several points in common with the spin networks that are the quantum states of gravity in loop quantum gravity. This talk spells out in detail how to make a correspondence between the states of loop quantum gravity and the tensor networks, basically corresponding to a coarse graining or averaging at certain scales of the states of quantum gravity. This opens the possibility of connecting results from AdS/CFT with results in loop quantum gravity. In particular the so-called Ryu-Takahashi formula for the entropy of a region can be arrived from in the context of loop quantum gravity.

Symmetries and representations in Group Field Theory

Tuesday, January 24th
Alexander Kegeles, Albert Einstein Institute
Title: Field theoretical aspects of GFT: symmetries and representations 
PDF of the talk (1M)
Audio+Slides [.mp4 11MB]


by Jorge Pullin, Louisiana State University


In loop quantum gravity the quantum states are labeled by loops, more precisely by graphs formed by lines that intersect at vertices and that are “colored”, meaning each line is associated with an integer. They are known as "spin networks". As the states evolve in time these graphs "sweep" surfaces in four dimensional space-time constituting what is known as a “spin foam”. This is a representation of a quantum space-time in loop quantum gravity. The spin foams connect an initial spin network with a final one and the formalism gives a probability for such “transition” from a given spatial geometry to a future spatial geometry to occur. The picture that emerges has some parallel with ordinary particle physics in which particles transition from initial to final states, but also some differences.

However, it was found that one could construct ordinary quantum field theories such that the transition probabilities of them coincided with those stemming from spin foams connecting initial to final spatial geometries in loop quantum gravity. This talk concerns itself with such quantum field theories, known generically as Group Field Theories (GFTs). The talk covered two main aspects of them: symmetries and representations.

Symmetries are important in that they may provide mathematical tools to solve the equations of the theory and identify conserved quantities in it. There is a lot of experience with symmetries in local field theories, but GFT’s are non-local, which adds challenges. Ordinary quantum field theories are formulated starting by a quantity known as the action, which is an integral on a domain. A symmetry is defined as a map of the points of such domain and of the fields that leaves the integral invariant. In GFTs the action is a sum of integrals on different domains. A symmetry is defined as a collections of maps acting on the domains and fields that leave invariant each integral in the sum. An important theorem of great generality stretching from classical mechanics to quantum field theory is Noether’s theorem, that connects symmetries with conserved quantities. The above notion of symmetry for GFTs allows to introduce a Noether’s theorem for them. The theorem could find applicability in a variety of situations, in particular certain relations that were noted between GFTs and recoupling theory and better understand various models based on GFTs.

In a quantum theory like GFTs the quantum states structure themselves into a mathematical set known as Hilbert space. The observable quantities of the theory are represented as operators acting on such space. Hilbert spaces are generically infinite dimensional and this introduces a series of technicalities both in their own definition and in the definition of observables for quantum theories. In particular one can find different families of inequivalent operators related to the same physical observables. This is what is known as different representations of the algebra of observables. Algebra in this context means that one can compose observables to form either new observables or linear combinations of known observables. An important type of representation in quantum field theory is known as Fock representation. It is the representation on which ordinary particles are based. Another type of representations is the condensate representation which, instead of particles, describes their collective (excitations) behaviour and is very convenient for systems with large (infinite) number of particles. A discussion of Fock and condensate like representations in the context of GFTs was presented and the issue of when representations are equivalent or not was also addressed.

Future work looks at generalizing the notion of symmetries presented to find further non-standard symmetries of GFTs. Also investigating “anomalies”. This is when one has a symmetry in the classical theory that may not survive upon quantization. The notion of symmetry can also be used to define an idea of “ground state” or fundamental state of the theory. In ordinary quantum field theory in flat space-time this is done by seeking the state with lower energy. In the context of GFTs one will invoke more complicated notions of symmetries to define the ground state. Several other results of ordinary field theories, like the spin statistics theorem, may be generalizable to the GFT context using the ideas presented in this talk.