This is the first post in an upcoming series covering some recent papers that deal with quantum cosmology models that are strongly relying on an SU(1,1), or equivalently SL(2,R), structure. In brief, one identifies classical phase space functions whose Poisson algebra is isomorphic to the Lie algebra su(1,1) and then quantises the cosmological model by promoting those functions to the generators of su(1,1) in some representation. The main advantage of this quantisation method is that the representation theory of the group under consideration is well known, so that the crucial "find a representation" step in constructing the quantum theory is essentially trivial.

# relatively quantum

A blog around my research in quantum gravity

## Friday, May 10, 2019

## Friday, May 3, 2019

### Singularity resolution in LQG inspired black holes

It is expected that quantum gravity will somehow resolve the singularities that are generically present in classical gravitational theories. For example, this may be the Big Bang singularity that one encounters when applying Einstein's theory of General Relativity all the way to the beginning of the universe. An example of how this singularity is resolved in the context of loop quantum cosmology was discussed in this post.

Another place where singularities prominently occur is inside black holes. Matter that falls through the horizon of a black hole will eventually hit this singularity and the theory describing its propagation breaks down in this instant. On the other hand, a compete theory of quantum gravity should provide a well-defined description of such a process.

Another place where singularities prominently occur is inside black holes. Matter that falls through the horizon of a black hole will eventually hit this singularity and the theory describing its propagation breaks down in this instant. On the other hand, a compete theory of quantum gravity should provide a well-defined description of such a process.

## Friday, May 18, 2018

### PhD Position available

A PhD Position in my group is available starting this fall. The full advertisement is below:

EDIT: The position has been filled.

EDIT: The position has been filled.

## Thursday, April 12, 2018

### Quantum Gravity meets Lattice QFT

Together with some colleagues, I am organizing a workshop on the intersection of quantum gravity and lattice QFT at ECT* in Trento, Italy, September 3-7.

Click here for the conference website.

The conference abstract goes as follows:

AdS/CFT has been one of the most fruitful approaches to analyse the qualitative aspects of the dynamics of strongly interacting QFTs, most prominently QCD. As an approach to understanding the early stage of high energy heavy ion collisions, but also proton-proton collisions at LHC, it is, in fact, one of very few systematic approaches. However, it is not clear how reliable the description is quantitatively, because QCD is not a N=4, supersymmetric, conformal, SU(N) gauge theory with infinite N and the QCD coupling constant is of limited size. Individual contributions exist on both sides of the duality calculating the size of the relevant corrections (like the perturbative calculation of quantum corrections on the gravity side for finite N and finite coupling strength, the lattice simulation of SU(N) gauge theories with N>3, the calculation of perturbative corrections from non conformality on the QFT side, lattice simulation with partial supersymmetry …) but no systematic effort. In addition, more general scenarios for gauge/gravity dualities have been studied, extending beyond the realms of AdS, CFT, and string theory. The probability is high that quantitative contact can only be made on the basis of non-perturbative calculations on both sides, which is a very tall order. On the QFT side, lattice QFT is the best established tool to do so, while on the quantum gravity side resummed string theory is the main approach. In addition, there is an increased recent interest within loop quantum gravity in holographic computations.

The aim of the workshop is to bring some of the internationally leading experts in these fields together, formulate a more systematic strategy, and realize a few projects in the direction of a quantitative application of quantum gravity techniques to QCD in subsequent months.

Click here for the conference website.

The conference abstract goes as follows:

AdS/CFT has been one of the most fruitful approaches to analyse the qualitative aspects of the dynamics of strongly interacting QFTs, most prominently QCD. As an approach to understanding the early stage of high energy heavy ion collisions, but also proton-proton collisions at LHC, it is, in fact, one of very few systematic approaches. However, it is not clear how reliable the description is quantitatively, because QCD is not a N=4, supersymmetric, conformal, SU(N) gauge theory with infinite N and the QCD coupling constant is of limited size. Individual contributions exist on both sides of the duality calculating the size of the relevant corrections (like the perturbative calculation of quantum corrections on the gravity side for finite N and finite coupling strength, the lattice simulation of SU(N) gauge theories with N>3, the calculation of perturbative corrections from non conformality on the QFT side, lattice simulation with partial supersymmetry …) but no systematic effort. In addition, more general scenarios for gauge/gravity dualities have been studied, extending beyond the realms of AdS, CFT, and string theory. The probability is high that quantitative contact can only be made on the basis of non-perturbative calculations on both sides, which is a very tall order. On the QFT side, lattice QFT is the best established tool to do so, while on the quantum gravity side resummed string theory is the main approach. In addition, there is an increased recent interest within loop quantum gravity in holographic computations.

The aim of the workshop is to bring some of the internationally leading experts in these fields together, formulate a more systematic strategy, and realize a few projects in the direction of a quantitative application of quantum gravity techniques to QCD in subsequent months.

## Monday, July 24, 2017

### New group + PhD positions

The Elite Network of Bavaria recently awarded me with a grant to start my own research group at the University of Regensburg starting this fall. I am currently looking for PhD students to work with me on projects at the intersection of loop quantum gravity and string theory, more precisely in applying ideas about quantum gravity corrected geometries in the context of the gauge / gravity duality. The announcement for the positions can be found here.

EDIT: All currently open positions have been filled.

## Monday, April 3, 2017

### Chamseddine and Mukhanov reinvent loop quantum cosmology

In a recent paper, Chamseddine and Mukhanov have proposed a modification of general relativity that features a limiting curvature. The model is in the framework of their earlier work on "mimetic gravity" and coupled general relativity to a scalar field $\phi$ that is constrained as

\[

g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi = 1

\]

in the mostly minus signature convention. In addition to this constraint, the action contains the term $f(\Box \phi)$. In the homogeneous and isotropic sector, $\Box \phi$ is proportional to the Hubble rate $\dot a / a$.

The main input that is needed to fully define this model is the function $f$, which can a priori be chosen arbitrarily up to a consistency requirement with general relativity at low curvatures. In their paper, Chamseddine and Mukhanov make the proposal

\[

f(\chi) = - \chi_m^2 \, g\left( \sqrt{\frac{2}{3}} \frac{\chi}{\chi_m} \right), ~~~ g(y) = -1 -\frac{y^2}{2} + y \arcsin y + \sqrt{1-y^2}

\]

which leads in the homogeneous and isotropic sector to a very simple correction to the Friedmann equation of the form

\[

\frac{\dot a}{a} \propto \rho \left(1- \frac{\rho}{\rho_\text{crit}}\right)

\]

where $\rho$ is the matter energy density and $\rho_\text{crit}$ a free parameter. This equation leads to a bouncing universe at the critical matter energy density $\rho_\text{crit}$.

In the homogeneous but non-isotropic sector, one can again solve the equations of motion to find singularity resolution, however with somewhat more complicated details. An analogous modification for Schwarzschild black holes is discussed in a second paper.

It is interesting to note that the corrected Friedmann equation above is already known from loop quantum gravity, more precisely from the effective equations of loop quantum cosmology. This already suggests that at least in the homogeneous and isotropic sector, the two theories should agree.

In fact, the choice for $f$ can be read off from the effective action of loop quantum cosmology, as computed first in this paper, equation 7. A more detailed investigation in based on the Hamiltonian formulation of the model proposed by Chamseddine and Mukhanov confirms this. The details have been written up in this paper, see also this paper for an intendedly simultaneous publication of related results.

In addition, a quantum mechanical argument is given there why this choice of $f$ is the simplest one consistent with quantum mechanics: Mukhanov and Chamseddine motivate the structure of their model with non-commutative geometry, in particular the quantisation of three-volume. In a canonical quantisation setting, this leads one directly to the simplest version of loop quantum cosmology. The choice for $f$ then follows by a Legendre-transform.

If one goes beyond the homogeneous and isotropic setting, the effective loop quantum cosmology deviates from the model of Chamseddine and Mukhanov in the details. Still, the qualitative behavior is the same for homogeneous but non-isotropic metrics. Beyond this, a comparison is hard to make because the effective LQG dynamics are in general not known. This on the other hand suggests to simply use the new model as a toy model for loop quantum gravity, i.e. as a toy model effective action that is completely known. The main application of this would be to obtain an understanding for the effective geometries that can emerge from the quantum corrections build into loop quantum gravity beyond the cosmological setting.

\[

g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi = 1

\]

in the mostly minus signature convention. In addition to this constraint, the action contains the term $f(\Box \phi)$. In the homogeneous and isotropic sector, $\Box \phi$ is proportional to the Hubble rate $\dot a / a$.

The main input that is needed to fully define this model is the function $f$, which can a priori be chosen arbitrarily up to a consistency requirement with general relativity at low curvatures. In their paper, Chamseddine and Mukhanov make the proposal

\[

f(\chi) = - \chi_m^2 \, g\left( \sqrt{\frac{2}{3}} \frac{\chi}{\chi_m} \right), ~~~ g(y) = -1 -\frac{y^2}{2} + y \arcsin y + \sqrt{1-y^2}

\]

which leads in the homogeneous and isotropic sector to a very simple correction to the Friedmann equation of the form

\[

\frac{\dot a}{a} \propto \rho \left(1- \frac{\rho}{\rho_\text{crit}}\right)

\]

where $\rho$ is the matter energy density and $\rho_\text{crit}$ a free parameter. This equation leads to a bouncing universe at the critical matter energy density $\rho_\text{crit}$.

In the homogeneous but non-isotropic sector, one can again solve the equations of motion to find singularity resolution, however with somewhat more complicated details. An analogous modification for Schwarzschild black holes is discussed in a second paper.

It is interesting to note that the corrected Friedmann equation above is already known from loop quantum gravity, more precisely from the effective equations of loop quantum cosmology. This already suggests that at least in the homogeneous and isotropic sector, the two theories should agree.

In fact, the choice for $f$ can be read off from the effective action of loop quantum cosmology, as computed first in this paper, equation 7. A more detailed investigation in based on the Hamiltonian formulation of the model proposed by Chamseddine and Mukhanov confirms this. The details have been written up in this paper, see also this paper for an intendedly simultaneous publication of related results.

In addition, a quantum mechanical argument is given there why this choice of $f$ is the simplest one consistent with quantum mechanics: Mukhanov and Chamseddine motivate the structure of their model with non-commutative geometry, in particular the quantisation of three-volume. In a canonical quantisation setting, this leads one directly to the simplest version of loop quantum cosmology. The choice for $f$ then follows by a Legendre-transform.

If one goes beyond the homogeneous and isotropic setting, the effective loop quantum cosmology deviates from the model of Chamseddine and Mukhanov in the details. Still, the qualitative behavior is the same for homogeneous but non-isotropic metrics. Beyond this, a comparison is hard to make because the effective LQG dynamics are in general not known. This on the other hand suggests to simply use the new model as a toy model for loop quantum gravity, i.e. as a toy model effective action that is completely known. The main application of this would be to obtain an understanding for the effective geometries that can emerge from the quantum corrections build into loop quantum gravity beyond the cosmological setting.

## Friday, December 23, 2016

### Holographic signatures of resolved cosmological singularities

After some longer silence partly due to moving to a new location (LMU Munich) and teaching my first regular lecture (Theoretical mechanics for lyceum teachers and computational science at Regensburg University), I hope to write more regularly again in the future.

As a start, a new paper on using loop quantum gravity in the context of AdS/CFT has finally appeared here. Together with Andreas SchÃ¤fer and John Schliemann from Regensburg University, we asked the question of what happens in the dual CFT if you assume that the singularity on the gravity side is resolved in a manner inspired by results from loop quantum gravity.

Building (specifically) on recent work by Engelhardt, Hertog, and Horowitz (as well as many others before them) using classical gravity, we found that a finite distance pole in the two-point-correlator of the dual CFT gets resolved if you resolve the singularity in the gravity theory. Several caveats apply to this computation, which are detailed in the papers. We view this result therefore as a proof of principle that such computations are possible, as opposed to some definite statement of how exactly they should be done.

As a start, a new paper on using loop quantum gravity in the context of AdS/CFT has finally appeared here. Together with Andreas SchÃ¤fer and John Schliemann from Regensburg University, we asked the question of what happens in the dual CFT if you assume that the singularity on the gravity side is resolved in a manner inspired by results from loop quantum gravity.

Building (specifically) on recent work by Engelhardt, Hertog, and Horowitz (as well as many others before them) using classical gravity, we found that a finite distance pole in the two-point-correlator of the dual CFT gets resolved if you resolve the singularity in the gravity theory. Several caveats apply to this computation, which are detailed in the papers. We view this result therefore as a proof of principle that such computations are possible, as opposed to some definite statement of how exactly they should be done.

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