Marios Christodoulou, Andrea Di Biagio, Pierre Martin-Dussaud

Time at the Planck scale ($\sim 10^{-44}~\mathrm{s}$) is an unexplored physical regime. It is widely believed that probing Planck time will remain for long an impossible task. Yet, we propose an experiment to test the discreteness of time at the Planck scale and show that it is not far removed from current technological capabilities.

Henriette Elvang

This article is an introduction to two currently very active research programs, the Conformal Bootstrap and Scattering Amplitudes. Rather than attempting full surveys, the emphasis is on common ideas and methods shared by these two seemingly very different programs. In both fields, mathematical and physical constraints are placed directly on the physical observables in order to explore the landscape of possible consistent quantum field theories. We give explicit examples from both programs: the reader can expect to encounter boiling water, ferromagnets, pion scattering, and emergent symmetries on this journey into the landscape of local relativistic quantum field theories. The first part is written for a general physics audience. The second part includes further details, including a new on-shell bottom-up reconstruction of the $\mathbb{CP}^1$ model with the Fubini-Study metric arising from re-summation of the $n$-point interaction terms derived from amplitudes.

Damon J. Binder, Daniel Z. Freedman, Silviu S. Pufu

We develop a new embedding-space formalism for AdS$_4$ and CFT$_3$ that is useful for evaluating Witten diagrams for operators with spin. The basic variables are Killing spinors for the bulk AdS$_4$ and conformal Killing spinors for the boundary CFT$_3$. The more conventional embedding space coordinates $X^I$ for the bulk and $P^I$ for the boundary are bilinears in these new variables. We write a simple compact form for the general bulk-boundary propagator, and, for boundary operators of spin $\ell \geq 1$, we determine its conservation properties at the unitarity bound. In our CFT$_3$ formalism, we identify an $\mathfrak{so}(5,5)$ Lie algebra of differential operators that includes the basic weight-shifting operators. These operators, together with a set of differential operators in AdS$_4$, can be used to relate Witten diagrams with spinning external legs to Witten diagrams with only scalar external legs. We provide several applications that include Compton scattering and the evaluation of an $R^4$ contact interaction in AdS$_4$. Finally, we derive bispinor formulas for the bulk-to-bulk propagators of massive spinor and vector gauge fields and evaluate a diagram with spinor exchange.

Fabio D'Ambrosio, Mudit Garg, Lavinia Heisenberg, Stefan Zentarra

We consider a simpler geometrical formulation of General Relativity based on non-metricity, known as Coincident General Relativity. We study the ADM formulation of the theory and perform a detailed Hamiltonian analysis. We explicitly show the propagation of two physical degrees of freedom, as it should, even though the role of boundary terms and gauge conditions is significantly altered. This might represent an alternative promising new route for numerical relativity and canonical quantum gravity. We also give an outlook on the number of propagating degrees of freedom in non-linear extension of non-metricity scalar.

Thiago M. Mergulhão, Carlos Batista

It is well known that in general relativity theory two spacetimes whose metrics are related by a coordinate transformation are physically equivalent. However, given two line elements, it is virtually impossible to implement the most general coordinate transformation in order to check the equivalence of the spacetimes. In this paper we present the so-called Cartan-Karlhede algorithm, which provides a finite sequence of steps to decide whether or not two metrics are equivalent. The point of this note is to illustrate the method through several simple examples, so that the reader can learn the fundamentals and details of the algorithm in practice.