Quantum energy conditions in two dimensions

Energy cannot become negative. Energy conditions are mathematical formulations of this apparently obvious physical fact. They play a key role in various branches of physics, e.g. in Hawkings proof that the total black hole area must increase. However, the first sentence is incorrect if quantum effects are taken into account: energy can become negative and quantum effects violate all classical energy conditions. In a sense, one can borrow energy from the vacuum, provided one pays it back with interest. Quantum energy conditions, which are valid universally, allow to quantify how much energy one can borrow and how big is the interest rate. In recent years quantum energy conditions were conjectured and proven for relativistic quantum field theories. The main research goal of this project is to find (and prove) for the first time quantum energy conditions for non-relativistic quantum field theories in two dimensions. Apart from the fascinating research aspects the collaborators hired through this project will have the unique opportunity to contribute to the biggest conference in our field, Strings 2022, which for the first time will be organized in Vienna by the project leader.

Energy conditions, critical in Penrose's Nobel-worthy singularity theorems and Hawking's area theorem, face challenges when confronted with quantum matter. Classical energy conditions are violated by quantum matter, necessitating the development of quantum energy conditions (QECs). These QECs involve the expectation value of the stress-energy tensor and aim to be universally valid for any quantum state in reasonable quantum field theories. In simple words, this research project posed the question "How much energy can we borrow from the vacuum (and if we have to pay an interest rate, how much is it)?". Technically, the project had two primary foci. Firstly, it explored the so-called Quantum Null Energy Condition (QNEC) in two-dimensional conformal field theories with a holographic dual. The team uncovered conditions under which QNEC saturates and examined subleading corrections in a large central charge expansion. Secondly, they proposed and proved QECs for specific non-relativistic quantum field theories in two dimensions, particularly those with a holographic dual, thereby providing new entries in the flat space holographic dictionary. These results contribute to the long-term goal of the PI, namely to find out how generally the holographic principle works, and in particular, if and how it works in our Universe. Several of our investigations revolved around the holographic principle, particularly Celestial and Carrollian holography. Our team was the first to uncover a precise relation between these two complementary approaches to flat space holography, thereby bridging two different communities. Moreover, we discovered the precise structure of Celestial conformal field theories (which turned out to be so-called logarithmic conformal field theories). Finally, we constructed the first models of Carroll gravity, proved that these models have black hole-like solutions, and described mathematically the orbits of test particles placed in the background of Carroll black holes. The research team comprised Principal Investigator Daniel Grumiller, the postdocs Adrien Fiorucci, Max Riegler, and Romain Ruzziconi, and several international collaborators. Together with colleagues from the University of Vienna, they organized the prestigious Strings 2022 conference in Vienna, creating unique opportunities for junior members and outreach activities. The latter included public talks at TU Wien, the University of Vienna, the Austrian Academy of Sciences, the Wiener Volkshochschulen, Pint of Science, Kinderuni, and several high schools. Additionally, they established a new workshop series "Carroll Physics" that commenced at the TU Wien in 2022 and that will culminate in an extensive program at the Erwin Schrödinger Institute for Mathematics and Physics in April 2024. Another spinoff of this project is the textbook "Black Hole Physics" by the PI (together with Sheikh-Jabbari), published in 2022 by Springer. This research project has expanded our understanding of QECs and set the stage for further advancements in the exciting intersection of quantum field theories and holography.