Quantization of the volume of the simplest grain of space

Tuesday, October 19th

Hal Haggard, Bard College

Quantization of the Volume of the Simplest Grain of Space
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By Jorge Pullin, LSU

There exists a concept in mathematics called "asymptotic resurgence" in which an infinite sum of terms exhibits a surprising behavior. The terms that appear late in the summation around one physically relevant point are exactly the same as the terms that appear early in the sum around another physically relevant point. The result is a rich connection between the physics at the two different extremes and has applications in many areas of physics, for instance in the calculations of intensities of rainbows.

This talk applies these ideas to the calculation of the volume in loop quantum gravity. The volume of a region of space is discrete in that theory and has a complicated expression depending on the details of the quantum state one is considering. The expressions are known but are difficult to interpret. Previous studies have dealt with them primarily through numerical methods. In this talk an approximate expression is derived that can be much more easily interpreted and studied. It applies it to the simplest "grain" of space, a tetrahedron. A neat illustration of the power of the insights that can be learned from having an easier to interpret expression is the following movie following tetrahedra of different shape but equal volume:

To quote the concluding slide: "Quantization of geometry provides a remarkable laboratory for understanding resurgent perturbative/non-perturbative relations and, due to the richness of its underlying quantum structure, may even require extensions of this formalism."

Complex critical points and curved geometries in Lorentzian EPRL spinfoam amplitude

 Tuesday, October 5th

Dongxue Qu, Florida Atlantic University

Complex critical points and curved geometries in Lorentzian EPRL spinfoam amplitude
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By Jorge Pullin, LSU

The states of quantum gravity in the loop representation are given by spin networks. These are multivalent graphs with a number associated with each line. Spin foams represent the transition from an initial spin network state to a final one, as show in the figure (credit: Alejandro Perez). The expanded picture on the right is what is known as a "vertex", where new lines in the spin network are created as it transitions forward in time (time is the vertical axis). These diagrammatics correspond to precise mathematical 

equations that embody the dynamics of general relativity at a quantum level. One of the proposals for the vertex is the "EPRL" one (after Engle, Pereira, Rovelli and Livine). There has been controversy over the years about if the vertex correctly encoded the dynamics of general relativity. That requires studying how it behaves in the classical limit, as one expects departures from classical general relativity in situations where quantum effects are important. Previous calculations, done within certain approximations, seemed to suggest that curved geometries were not properly captured by this construction. The talk was about recent numerical results that imply that indeed it does capture the dynamics of classical general relativity in appropriate situations. Connections were made with a discretization of classical general relativity proposed by Tullio Regge known as Regge calculus. This is a very encouraging result indicating that the dynamics of classical general relativity is properly being captured by the "EPRL vertex".

How the cosmological constant q-deforms symmetries in Loop Quantum Gravity

Tuesday, September 21st

Qiaoyin Pan, Perimeter Institute

How the cosmological constant q-deforms symmetries in LQG
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By Jorge Pullin, LSU

Ashtekar showed in 1986 how to rewrite general relativity in terms of variables that make it look like the Yang-Mills theories that describe particle physics. This was the cornerstone that allowed the introduction of loop variables by Rovelli and Smolin and led to the development of loop quantum gravity.

Yang-Mills theories are generalizations of Maxwell's electromagnetism which have several electric and magnetic fields. These theories have a symmetry through which multiple field values correspond to the same physical situations. The transformations among the fields that keep the physical situation invariant form a mathematical structure called a group. The particular group that arises in loop quantum gravity is called SU(2) and is similar to the one that appears in the theory of the weak interactions.

When a cosmological constant is present, things can be reconfigured in such a way that the group that arises is a mathematical structure called q-deformed group, with q a parameter related to the value of the cosmological constant. Observations indicate that our current universe indeed has a cosmological constant present that makes its expansion accelerate, therefore this is a situation of physical interest.

The talk described the mathematical structures that arise when one formulates loop quantum gravity in terms of the q-deformed structures, including the dynamics of the theory. It also points to connections with another mathematical structure known as quantum groups.

Vacuum state for LQC perturbations

 Tuesday, March 23rd

Rita Neves, Universidad Complutense

Vacuum state for LQC perturbations
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by Jorge Pullin

One of the main achievements of the model of cosmology known as Inflation, in which the universe after the Big Bang expands exponentially, is that it predicts the spectrum of cosmic microwave background radiation. This is radiation that comes to us directly from the earliest moments when the universe stopped being a primordial soup and became transparent to light. If one looks in different directions, the frequency of the radiation is not exactly the same. It differs in parts in a million. And the differences are not random. If one looks in a given direction and then looks at a circle in the sky centered in that direction and averages out the frequencies, if things were random, the result would be the same no matter what size the circle. It turns out it does, mathematically one says the signals are correlated. 

If one assumes one starts inflation with a quantum field present, and one assumes the simplest possible state for the quantum field (the vacuum) and evolves the quantum state through inflation, the state develops correlations that correspond precisely to the ones observed in the cosmic microwave radiation. This model is remarkable in its simplicity and efficacy. 

In traditional cosmology, where things start with a Big Bang where the whole universe is concentrated at a point, it appears natural to place the quantum state of the field in the vacuum at the beginning of inflation, as it is impossible to place it at the Big Bang as the theory breaks down there (densities and curvatures are infinite). But in loop quantum cosmology, the Big Bang is replaced by a Big Bounce where everything is finite and there is dynamics of the universe prior to it. Why would then place the state in a vacuum at the beginning of Inflation, as that instant in time does not have any privileged meaning. Perhaps one should place it at the Bounce (now this is possible as the space-time is regular there). Or somewhere else. All that means that at the beginning of Inflation, the quantum field will not be in the vacuum state anymore. This talk addressed these issues.

Supergravity in loop quantum gravity

Tuesday, March 9th

Konstantin Eder, FAU Erlangen

Supergravity in LQG
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By Jorge Pullin, LSU

Supersymmetry is a conjectured symmetry of nature in which to each particle corresponds a "superpartner" particle. The partners of bosons are fermions and vice-versa. So for instance, the electron (a fermion) has a boson superpartner known as the "selectron" and so on. No superpartner has ever been observed in reality so it is conjectured that this symmetry is broken in nature and can only be present at very high energies. Unfortunately, experiments at large accelerators like the Large Hadron Collider at CERN in Switzerland, are putting tighter and tighter bounds on supersymmetry. Supersymmetry is usually incorporated into string theory, hence the name superstrings. 

If one incorporates this symmetry into gravity, one obtains supergravity. This theory potentially has interesting properties. It could avoid the infinities one faces in usual perturbative quantum gravity, although this is not entirely clear yet.

This talk was about applying loop quantum gravity techniques to supergravity. It updated the treatment with modern techniques (older results referred to techniques that are not used anymore, in particular the use of complex variables) and further insights on how supersymmetry can manifest itself in the form of a gauge symmetry, the kind of symmetry that is the basis for the description of the other forces in nature. Among other aspects it explored the behavior of the theory in cosmology and how one could use supersymmetric matter as a "clock" to study the evolution in time. It also considered calculations of black hole entropy and how it could lead to connections with similar calculations in string theory.