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.

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. 

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.

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.