Wednesday, October 24, 2018

A unified geometric framework for boundary charges and dressings

Tuesday, Oct 23rd

Aldo Riello, Perimeter Institute
Title: A unified geometric framework for boundary charges and dressings 
PDF of the talk (2M)
Audio+Slides of the talk (41M)
By Jorge Pullin, LSU (with some help from Aldo)

The electromagnetic force and all the subatomic interactions are described by a class of theories known as “gauge theories”. Even gravitation, in its modern formulation due to Einstein, is a gauge theory of sorts, although a more complicated one. The mathematical formulation of these theories is characterized by peculiar redundancies, as if the simplest way to describe the system is through a plethora of different descriptions rather than through a single “true” one. This is most often seen as a mathematical quirk rather than as a hint of some deep property of nature. This talk explores the latter possibility and build on the idea that the rationale for gauge theories must be found not so much in some property of a single system taken in isolation, but rather in the way systems can come together and talk to each other. The first hint of this can be found in the fact that the natural objects populating a gauge theory (“observables”) are intrinsically nonlocal and therefore can’t be easily localized in a given region, without carefully keeping track of what happens at its boundaries. The simplest example of this phenomenon can be found in the electron, that can never be separated from its electric field, which in turn can be detected even at a distance from the electron. This talk presents a novel mathematical framework that by embracing the relational perspective unifies many seemingly unrelated aspects of gauge theories and might – in its future developments – clarifies the analogous but harder conceptual issues one finds on their way to quantum gravity.

Thursday, October 18, 2018

Quantum extension of black holes

Tuesday, Oct 9th

Javier Olmedo, LSU
Title: Quantum Extension of Kruskal Black Holes 
PDF of the talk (500k)
Audio+Slides of the talk (17M)

By Jorge Pullin, LSU

In the interior of black holes the coordinates t and r swap roles. As one falls "towards the center" one is actually moving forward in time. This makes the interior of a black hole look like a contracting cosmology of a particular type, known as Kantowski-Sachs cosmology. This has allowed the use of loop quantum cosmology techniques to treat the interior of black holes. There have been several discussions of this, but they have some shortcomings. To begin with, they only cover the interior of the black hole. Moreover, some of the proposals have physical quantities with undesirable dependences on fiducial elements introduced in order to quantize or on the mass of the space-time.

This talk discusses overcoming these problems. To begin with, it is shown that the quantum treatment eliminates the singularity inside black holes and replaces with a region of large curvature. The value of the maximum curvature is universal and independent on the mass of the space-time. Moreover, it gives the same mass for the black hole to the past and to the future (unlike other treatments). In addition, the quantum theory is extended to the exterior region of the black hole. In the future it is expected to extend these ideas to other type of black hole space-times, like those with charge, spin and cosmological constant.

Monday, October 8, 2018

Computing volumes in spin foams

Tuesday, Sep 25th

Benjamin Bahr, DESY
Title: 4-volume in spin foam models from knotted boundary graphs 
PDF of the talk (3M)
Audio+Slides of the talk (15M)

by Jorge Pullin, LSU

There is an approach to quantum mechanics known as the path integral approach. In it, one considers all possible classical trajectories, not only the ones satisfying the equations of motion and assigns probabilities to each of them using a formula. The probabilities are summed and that gives the quantum probability to go from an initial state to a final state. In loop quantum gravity the initial and final states are given by spin networks, which are graphs with intersections and "colors" (a number) assigned to each edge. The trajectories connecting initial and final states therefore resemble a "foam" and are given the names of spin foams.

In this talk it was shown how to compute volumes of polytopes (regions of space-time bounded by flat sides, a generalization to higher dimensions of polyhedra of 3d) in spin foam quantum gravity. The calculation has nice connections with knot theory, the branch of math that studies how curves entangle with each other.

One of the central elements of spin foams is the formula that assigns the probabilities, known as a "vertex". The construction in this talk gives ideas for extending the current candidates for vertices, including the possibility of adding a cosmological constant and suggests possible connections with Chern-Simons theories (a special type of field theories) and also with quantum groups.

Tuesday, May 1, 2018

Cosmological perturbations in terms of observables and physical clocks

Tuesday, Apr 17th

Kristina Giesel, FAU Erlangen-Nürnberg
Title: Gauge invariant observables for cosmological perturbations 
PDF of the talk (8M)
Audio+Slides of the talk (15M)

By Jorge Pullin, LSU

When one sets up to quantize general relativity something unusual happens. When one constructs a key quantity for describing the evolution called the Hamiltonian, it turns out it vanishes. What the framework is telling us is that since in general relativity one can choose arbitrary coordinates, the coordinate t that one normally associated with time is arbitrary. That means that the evolution described in terms of it is arbitrary.



Of course this does not mean that the evolution predicted by general relativity is arbitrary. It is just that one is choosing to describe it in terms of a coordinate that is arbitrary. So how can one get to the invariant part of the evolution? Basically one needs to construct a clock out of physical quantities. Then one asks how other variables evolve in terms of the variable of the clock. The relational information between such variables is coordinate independent and therefore characterizes the evolution in an invariant way.

Cosmological perturbation theory is an approximation in which one assumes that the universe at large scales is homogeneous and isotropic plus small perturbations. One can then expand the Einstein equations keeping only the lower order terms in the small perturbations. That makes the equations much more manageable. Up to now most studies of cosmological perturbations were done in coordinate dependent fashion, in particular the evolution was described in terms of a coordinate t. This talk discusses how to formulate cosmological perturbation theory in terms of physical clocks and physically observable quantities. Several choices of clocks are discussed.

Sunday, April 22, 2018

Quantum gravity inside and outside black holes

Tuesday, Apr 3rd

Hal Haggard, Bard College
Title: Quantum Gravity Inside and Outside Black Holes 
PDF of the talk (5M)
Audio+Slides of the talk (19M)
By Jorge Pullin, Louisiana State University

The talk consisted of two distinct parts. The second part discussed black holes exploding into white holes. We have covered the topic in this blog before, and the new results were a bit technical for a new update, mainly a better handle on the time the process takes, so we will not discuss them here.

The first part concerned itself with how the interior of a black hole would look like in a quantum theory. Black holes are regions of space-time from which nothing can escape and are bounded by a spherical surface called the horizon. Anything that ventures beyond the horizon can never escape the black hole. Black holes develop when stars exhaust their nuclear fuel and start to contract under the attraction of gravity. Eventually gravity becomes too intense for anything to escape and a horizon forms.

The interior of the horizon however, is drastically different if a black hole has rotation or not. If the black hole does not rotate, anything that falls into the black hole is crushed in a central singularity where, presumably, all the mass of the initial star concentrated. If the black hole has rotation however, the structure is more complicated and infalling matter can avoid hitting the singularity and move into further regions of space-time inside the black hole.

This raises the question: what happens with all this in a quantum theory of gravity. Presumably a state representing a non-rotating black hole will consist of a superposition of black holes with rotation, peaked around zero rotation, but with contributions from black holes with small amounts of rotation. How does the interior of a non-rotating quantum black hole look when it is formed through a superposition of rotating black holes? This is an interesting question since the interior of rotating black holes are so different from their non-rotating relatives.

The talk concludes that the resulting interior actually does resemble that of a non-rotating black hole. The key observation is that one cannot trust the classical theory all the way to the singularity and that leads to the superposition having large curvatures where one would have expected the singularity of the non-rotating black hole to be.

Sunday, March 25, 2018

Cosmological non Gaussianity from loop quantum cosmology

Tuesday, Mar 6th

Ivan Agullo, LSU
Title: Non-Gaussianity from LQC 
PDF of the talk (22M)
Audio+Slides [.mp4 19MB]
By Jorge Pullin,  LSU

The standard picture of cosmology is that the universe started in the "big bang" and then underwent a period of rapid expansion, called inflation. During those initial moments, densities are very high and matter is fused into a primordial "soup" that is opaque, light cannot travel through it. As the universe expands and cools, eventually electrons and protons form atoms and the universe becomes transparent to light. The afterglow of that initial phase can then travel freely through the universe and eventually reaches us. Due to the expansion of the universe that light "cools" (its frequency is lowered). In the 1960's to Bell Telephone Co. engineers were working on a microwave antenna and discovered a noise they could not get rid of. That noise was the afterglow of the Big Bang, that by then had cooled off into microwaves. That afterglow has been measured with increasing precision using satellites. It is remarkably homogeneous, if one looks into two different directions of the universe, the difference in temperature (frequency) of the microwave radiation is equal to one part in 100,000. The diagram below has those temperature differences magnified 100,000 times to make them visible, different colors correspond to different temperatures. The whole celestial sphere is mapped into the oval.
At first, it appears that the distribution of temperature is sort of random. But it is not, it has a lot of structure. To characterize the structure, one picks a direction and then moves away from it a certain angle and draws a circle of all directions forming the same angle with the original direction one picked. One then averages the temperature along the circle.  Then one averages the result for all possible initial choices of direction. If the distribution were truly random, if one plotted the average computed as a function of the angle, one would get a constant, no angle would be preferred over others. But what one gets is shown in the following diagram,
In the vertical are the averages, in the horizontal, the angles. The dots are experimental measurements. The continuous curve is what one gets if one evolves a quantum field through the inflationary period, starting from the most "quiescent" quantum state possible at the beginning, called "the vacuum state". The incredibly good agreement between theory and experiment is a great triumph of the inflationary model. The quantity plotted above is technically known as the "two point correlation". Loop quantum cosmology slightly changes the predictions of standard inflation, mostly for very large angles. There, the experimental measurements have a lot of uncertainty and are not able to tell us if loop quantum cosmology or traditional inflation give a better result. Perhaps in a few years better measurements will allow us to distinguish between them. If loop quantum cosmology is favored it would be a tremendously important experimental confirmation. But we are not there yet.

One can generalize the construction we made with two directions and an angle between them to three directions and three angles between them, and so on for higher number of directions. These would be known technically as the three point correlation, four point correlation, etc. If the distribution of temperatures were given by a probabilistic distribution known as a Gaussian, all the higher order correlations are determined by the two point correlation, there is no additional information in them. 

In this talk a study of the three point correlations for loop quantum cosmology was presented. It was shown that non-Gaussianities appear. That is, the three point correlation is not entirely determined by the two point one. Satellites are able to measure non-Gaussianities. In the talk it was shown that depending on the values chosen for the quantum fields at the beginning of the universe, the non-Gaussianities predicted by loop quantum gravity can be made compatible with experiment. This is not strictly speaking an experimental confirmation since one had a parameter one could adjust. But the good news is that the values needed to fit the data appear very natural. Again, future measurement should place tighter bounds on all this.

Image credits: Cosmic microwave background Wikipedia page.

Quantum spacetimes on a quantum computer

Tuesday, Mar 20th

Keren Li, Tsinghua University
Title: Quantum spacetime on a quantum simulator 
PDF of the talk (3M)
Audio+Slides [.mp4 11MB]

By Jorge Pullin, LSU


In loop quantum gravity the quantum states are labeled by objects known as "spin networks". These are graphs in space with intersections. If one evolves a spin network in time one gets a "spin foam". If one had a static situation, the various spatial slices of a spin foam would be the same, as shown in the figure,
If one were in a dynamical situation, new vertices are created,
To compute the probability of transitioning from a spin network to another is what calculations in spin foams are about. The details of these computations resemble computations people do in quantum mechanics of systems with spins. This allows to make a parallel between these computations and the ones that are involved in setting up a quantum computer, specifically the qubits that are constructed using nuclear magnetic resonance systems (NMR). In this talk it was described how the evolution of a very simple spin foam known as the tetrahedron can be simulated on an NMR quantum computer of four qubits and how the experimental measurements reproduce very well theoretical calculations of spin foam models.

Tuesday, February 6, 2018

Using symmetries to determine the dynamics

Tuesday, Feb 6th

Ilya Vilensky, Florida Atlantic University
Title: The unique form of dynamics in LQC 
PDF of the talk (0.5M)
Audio+Slides [.mp4 11MB]




By Jorge Pullin, LSU

Loop quantum cosmology is the application of ideas of loop quantum gravity to the context of cosmology, where one freezes most degrees of freedom and studies just a few large scale ones, like the volume of the universe or its anisotropy. Loop quantum cosmology is not "derived" from loop quantum gravity, in the sense of choosing in the full theory quantum states that are very symmetric with only a few degrees of freedom and study their evolution. That is at the moment, too complicated. In loop quantum cosmology one first freezes the degrees of freedom one wishes to ignore and then proceeds to quantize the remaining ones. It is not clear that this coincides with "quantizing and then freezing". It is therefore important to run cross checks to make sure that at least within the approximation considered, things are consistent.

In spite of the enormous simplification one obtains when one first freezes most degrees of freedom and then quantizes, there are still quite a few ambiguities in the quantization process. This talk showed in the example of anisotropic universes, how imposing the residual symmetries and left after freezing most degrees of freedom, and demanding that the correct classical limit follow, allows to cut down on the number of ambiguities present. This increases the confidence in results previously obtained in loop quantum cosmology, some of which may have observable implications for the anisotropies of the cosmic microwave background radiation.

Monday, January 29, 2018

New dynamics for quantum gravity

Tuesday, Jan 23rd

Cong Zhang, Univ. Warsaw/Beijing
Title: Some analytical results about the Hamiltonian operator in LQG 
PDF of the talk (1.7M)
Audio+Slides [.mp4 10MB]



by Jorge Pullin, LSU

One of the central elements when building quantum theories using the approach known as "canonical" is to define a quantity known as the Hamiltonian. This quantity is responsible for the time evolution of the system under study. In general relativity, when one tries to construct such quantity one notices it vanishes. This is because in general relativity one can choose any arbitrary time variable and therefore there is not a uniquely selected evolution. One needs to make a choice. One such choice is to use matter to play the role of a clock. That leads to one having a non-vanishing Hamiltonian. In this work a detailed construction for the quantum operator associated with such Hamiltonian in loop quantum gravity was presented. The implementation presented differs from others done in the past. Among the attractive elements is that it can be shown in certain circumstances that the operator has the desirable mathematical property known as "self-adjointness". This property ensures that physical quantities in the theory are represented by real (as opposed to complex) numbers.

A discussion was also presented of how the operator acts on certain states that behave semi-classically known as "coherent states", in particular in the context of cosmological models. It was observed that it leads to an expanding universe.

Monday, January 15, 2018

Construction of Feynman diagrams for group field theory

Tuesday, Dec 5th

Marco Finocchiaro, Albert Einstein Institute
Title: Recursive graphical construction of GFT Feynman diagrams 
PDF of the talk (1M)
Audio+Slides [.mp4 24MB]

By Jorge Pullin, LSU.
A common technique for computing probability amplitudes in quantum field theory consists in expanding such objects as power series in term of the coupling constant of the theory. Each term in the expansion, usually involving complicated expressions, can be represented in a pictorial way by using diagrams. This graphical technique (known as "Feynman diagrams method") allows to write down and organize the terms in the perturbative series in a much easier way.

Group field theories (GFTs) are ordinary quantum field theories on group manifolds. Their Feynman amplitudes (i.e. amplitudes associated to Feynman graphs) correspond by construction to Quantum Gravity Spinfoam amplitudes. There exists an analogue situation in 1+1 dimensional theories known as matrix models, which are quantum field theories whose Feynman diagrams are related to the path integrals for gravity in 1+1 dimensions. From this point of view group field theories can be seen as a four dimensional generalization of matrix models.

The seminar, articulated in three parts, dealt with several aspects concerning the construction of GFT's Feynman diagrams and the evaluation of the corresponding amplitudes. In the first part a general introduction to group field theory was provided, stressing the importance of studying the divergences appearing in the amplitudes' computations. Indeed they can be used as tools to constraint and test the type of theories that can be built. In the second part the main methods to extract the amplitudes' divergences were briefly reviewed. Moreover a new GFT/Spinfoam model for Euclidean quantum gravity was presented. The last part was devoted to the seminar's main topic, namely the generation of Feynman graphs in group field theory. Beyond the leading order in the power series expansion this is often a difficult task. It was shown how to construct GFT's Feynman diagrams using recursive graphical relations that are suitable for implementations in computers. Future works will deal with making the computations parallelizable.

Entanglement in loop quantum gravity

Tuesday, Nov 7th

Eugenio Bianchi, PennState
Title: Entanglement in loop quantum gravity 
PDF of the talk (9M)
Audio+Slides [.mp4 19MB]


By Jorge Pullin, LSU

Entanglement is one of the most fascinating new concepts introduced in quantum mechanics. When quantum systems interact, the resulting systems properties cannot be described by considering the properties of the individual systems. One needs to consider global properties of the set of systems as a whole. Not only one cannot reconstruct the properties of the whole from the properties of the constituent parts. It turns out that the properties of the constituent parts cannot be determined if one does not know the properties of the whole. Entanglement entropy is a quantity that measures "how much entanglement" there is in a set of quantum systems. This seminar dealt with the application of this concept to the quantum states of loop quantum gravity. Here one tries to understand how different regions of space become entangled with each other in a quantum geometry and how the entanglement entropy measures such entanglement. 

This is not a mere theoretical development. Quantum theory plays an important role in cosmology. We now know that the fluctuations we see in the cosmic microwave background radiation are the product of the evolution of the vacuum state of the inflaton field during inflation. If one assumes that before inflation the field was in a vacuum state and evolves it, the state develops non-trivial correlations that are precisely the ones observed in the cosmic background radiation fluctuations. 
The cosmic microwave background fluctuations. Credit: NASA/WMAP team.


The vacuum state of a quantum field is a highly entangled state. Therefore the correlations one observes in the cosmic microwave background are directly related to entanglement. This seminar raises the mesmerizing possibility that the particular type of entanglement that occurs in the states of loop quantum gravity could leave an observable imprint in the cosmic microwave background radiation. This occurs through their evolution from the big bounce that loop quantum cosmology replaces the big bang with  up to the beginning of inflation influencing the type of vacuum the inflaton starts in. 

Wednesday, January 10, 2018

Black holes exploding into white hole fireworks

Tuesday, Oct 24th

Marios Christodoulou, Aix Marseille U/SUSTec Shenzen
Title: Geometry transition in covariant LQG: black to white 
PDF of the talk (3M)
Audio+Slides [.mp4 11MB]


By Jorge Pullin, LSU

Black holes are regions of space-time where gravity is so intense that nothing, including light, can escape, hence they are black. They are expected to form as stars exhaust their nuclear fuel and start to contract due to gravitational attraction. Eventually they become so dense that a black hole forms. According to classical general relativity, the star matter continues to contract inside the black hole until the density diverges. That is what is known as a "singularity". Obviously nothing can diverge in nature so it is believed that the singularities are an indication that one has pushed general relativity beyond its domain of validity. One expects that at high densities quantum effects should arise and a theory of quantum gravity is needed. There has been some progress in spherically symmetric loop quantum gravity that indicates that the singularity is replaced by a highly quantum region that eventually leads to another classical region of space-time beyond it. 

At the same time Hawking showed in the 70's that if one puts quantum fields to live on the classical background of a black hole, radiation is emitted as if the black hole behaved as a black body with a temperature inversely proportional to the black hole's mass. There is no contradiction with the black hole radiating because the radiation is produced by the quantum field outside the black hole. If the black hole radiates, then it should lose energy. Hawking's calculation cannot study this, because it assumes the quantum field lives in a fixed black hole background. It is expected that more precise calculations including the back-reaction of the field on the background should make the black hole shrink as it emits radiation. As the temperature increases as the black hole loses mass (it is inversely proportional to the mass) the black hole heats up and radiates more. Eventually it should evaporate completely. No detailed analysis of such evaporation is available at present. Such evaporation raises many questions, in particular what happened to the singularity inside the black hole (or the highly  quantum region that apparently replaces it). What happened to all the information of the matter that formed the black hole? Is it lost?

The work described in this seminar posits that the highly quantum region inside the black hole transitions into the future into a "white hole" (the time reverse of a black hole). A great explosion in which all the information that entered the black hole exits. This scenario is known as "fireworks". An important question is: does the explosion happen fast enough for it to make the loss of information through Hawking radiation irrelevant? In this seminar spin foams are used to try to address the question. The calculation at hand is to compute the probability of transition from a black hole to a white hole. There are many assumptions needed to make such calculation, so the results are at the moment tentative. However, the main conclusion seems to be that the explosion takes as long as the process of Hawking evaporation to take place. This may rule out the "fireworks" as candidates for fast radio bursts that have been observed by astronomers, but may keep in play other astrophysical predictions associated with them.

Cosmological dynamics from full loop quantum gravity

Tuesday, Sept 26th
Andrea Dapor and Klaus Liegener, FAU Erlangen
Title: Cosmological Effective Hamiltonian from full Loop Quantum Gravity 
PDF of the talk (2.2M)
Audio+Slides [.mp4 13MB]




By Jorge Pullin, LSU

Due to the complexity of theories like general relativity, a common line of attack to understand the theory is to consider situations with high symmetry. In them, one freezes almost all degrees of freedom but a few and studies them. Examples are the studies of homogeneous cosmologies, where the only degrees of freedom left are the volume of the universe and perhaps variables characterizing its anisotropy. In some cases people choose to freeze most of the degrees of freedom and then quantize the remaining ones. This is called "minisuperspace" quantization. In the case of cosmologies that means that one is left with just a handful of degrees of freedom, turning a field theory with infinitely many degrees of freedom like general relativity into a "mechanical system" in the sense of having only a finite number of degrees of freedom. The resulting quantization is therefore much simplified and a lot of progress can be made. The field of study of these quantizations is known as "loop quantum cosmology". The hope is that the resulting theories resemble what happens when one follows the evolution of a highly symmetric state in the full theory. This is, however, not guaranteed. There are known examples where "reducing then quantizing" does not yield the same result as "quantizing then reducing".

The seminar dealt with an attempt to "quantize then reduce" loop quantum gravity and see if the results of loop quantum cosmology follow for such an approach. This requires choosing quantum states in the full theory whose probabilities are "peaked" around homogeneous geometries and that evolve maintaining the homogeneity. States that are peaked around certain classical solutions and follow their evolution in quantum theory are known as "coherent states". In this talk such states for loop quantum gravity based on cubic lattices were constructed and their evolution was studied. It was noted that the resulting evolution does not coincide with the one usually chosen in loop quantum cosmology. When one quantizes theories there are ambiguities in how one proceeds and choices need to be made in how one write certain classical equations as quantum operators. It turns out that one of the choices usually made in loop quantum cosmology does not match with what one gets in the "quantize then reduce" approach. This suggests novel dynamics to study in the context of loop quantum cosmology that may affect the emerging picture of how our universe's Big Bang got replaced by a Big Bounce. In the traditional loop quantum cosmology approach the bounce is preceded by a large classical universe like ours. In the new dynamics suggested in this talk the bounce is preceded by a large but very quantum universe with a large Planck-scale cosmological constant. In the distant past our universe asymptotes to a very symmetrical universe known as De Sitter space. Further studies need to be done to check the consistency of the approach. 




Intrinsic time geometrodynamics

Tuesday, Sept 12th
Eyo Eyo Ita, University of South Africa
Title: Intrinsic time quantum geometrodynamics: emergence of General Relativity and cosmic time 
PDF of the talk (1.5M)
Audio+Slides [.mp4 8MB]


By Jorge Pullin, LSU

Usual Newtonian mechanics describes the motions of systems with respect to an absolute time variable usually called t. Already special relativity introduces the idea that time is not absolute and that it ticks at different rates for different observers. General relativity goes beyond that: one can pick any variable to play the role of time. The result of that is that if one tries to understand the dynamics of the theory as an "evolution in time" one runs into difficulties. This is important because many of our ideas of how to quantize theories are implemented dynamically. One needs what is known as a "Hamiltonian formulation" of the theory in order to implement what is known as "canonical quantization". In the Hamiltonian formulation there is a quantity known as the Hamiltonian that is responsible for time evolution. If one attempts to construct a Hamiltonian formulation for general relativity one discovers that the Hamiltonian vanishes. This reflects the fact that if one is allowed to pick any time variable one essentially can get any evolution one wants. This was the source of quite a bit of confusion and explains why a suitable Hamiltonian formulation took almost 50 years to emerge, being general relativity from 1915 and the Hamiltonian formulation only finally understood in the early 60's. Today we know that if one wants to have a defined Hamiltonian and evolution one needs to choose a time variable. The intrinsic geometrodynamics essentially chooses the volume of space as time variable. This seminar discussed the details and its implications for quantization in particular in the so-called "path integral quantization". Among the results a natural vacuum for the theory is found that involves the well known mathematical invariant related to the Chern-Simons form, suggesting perhaps that general relativity could be turned into a renormalizable quantum field theory.

Saturday, January 6, 2018

Gravitational path integral and group theory

Pietro Dona, Penn State
Title: SU(2) graph invariants, Regge actions and polytopes 
PDF of the talk (10M)
Audio+Slides [.mp4 16MB]

By Jorge Pullin, LSU


In the loop quantum gravity approach to quantum gravity the quantum states are given by spin networks, graphs with intersections and "colors" associated with each link. The colors are a shorthand to characterize that each link in the graph is associated with a mathematical quantity known as an element of a group. A group is a type of mathematical set with a composition law that is associative, has a neutral element and has an opposite element. For instance, real numbers form a group under addition. Matrices of numbers also form groups under multiplication. When links of a spin network meet at an intersection, the respective group elements associated with them get multiplied into a mathematical entity known as "intertwiner". Such intertwiners are constructed with what are known as invariant tensors in the group. 

One of the approaches to quantizations of field theories is the path integral approach. In it, one assigns probabilities to each physical trajectory and sums over all possible trajectories. When applied in the context of loop quantum gravity one gets trajectories in time of spin networks, which give rise to what are known as "spin foams". The probability of a given trajectory is quantified in terms of a number related to how the spin networks branch out into the future known as a "vertex". There are several proposals for such vertices to represent the dynamics of general relativity, at present it is not clear which one of the proposed ones represents nature more accurately. One of the most studied ones is the EPRL (Engle-Pereira-Rovelli-Livine) vertex. Other vertices that have simpler nature have also been proposed. This seminar deals with the evaluation of these vertices. This requires calculations in group theory. These calculations may have broader applicability than in just quantum gravity as these types of mathematical entities appear in many physical domains. Numerical calculations of the vertices have been carried out and asymptotic analyses performed for some of the more simplified vertices. The objective is to later extend the results to the EPRL vertex.