Spherically symmetric geometrodynamics is studied for scalar–tensor theory and Einstein General Relativity minimally coupled to a scalar field. We discussed the importance of boundary terms for non-compact space-like foliations and derived the equations of motion in the Hamiltonian formalism both in the Jordan and Einstein frames. On the reduced phase space obtained by gauge-fixing the lapse and the radial shift functions, the two frames are connected through a Hamiltonian canonical transformation. We discussed the effects of the singularity of the Hamiltonian canonical transformation connecting Jordan and Einstein frames for two static solutions (Fisher, Janis, Newman and Winicour solution in the Einstein frame and Bocharova–Bronnikov–Melnikov–Bekenstein black hole solution in the Jordan frame).

Italy

Vatican

Matteo Galaverni

United States

Gabriele Gionti

Spherically symmetric geometrodynamics in Jordan and Einstein frames

We consider quantum gravity with zero cosmological constant in three dimensions. First, we show that pure quantum gravity can be written as a magnetic Carrollian theory living on null infinity, described by Schwarzian-like degrees of freedom. Next, we couple quantum gravity to massless matter. Transition amplitudes exhibit several features that resemble soft graviton physics in four dimensions, despite the absence of a propagating graviton. As in four dimensions, we find three equivalent results: a soft graviton theorem, an infinite-dimensional BMS asymptotic symmetry, and a gravitational memory effect. We also resolve some extant puzzles concerning the partition function and Hilbert space of pure 3d gravity with zero cosmological constant.

Jordan Cotler

Canada

Kristan Jensen

Austria

Stefan Prohazka

Max Riegler

Belgium

Jakob Salzer

Soft gravitons in three dimensions

It was recently suggested — based on general self-consistency arguments as well as results from the bootstrap [1–3] — that the CFT describing the -state Potts model is logarithmic for generic values of , with rank-two Jordan blocks for and $$ {\overline{L}}_0 $$ in many sectors of the theory. This is despite the well-known fact that the lattice transfer matrix (or Hamiltonian) is diagonalizable in (arbitrary) finite size. While the emergence of Jordan blocks only in the limit → ∞ is perfectly possible conceptually, diagonalizability in finite size makes the measurement of logarithmic couplings (whose values are analytically predicted in [2, 3]) very challenging. This problem is solved in the present paper (which can be considered a companion to [2]), and the conjectured logarithmic structure of the CFT confirmed in detail by the study of the lattice model and associated “emerging Jordan blocks.”

Lawrence Liu

France

Jesper Jacobsen

Hubert Saleur

Emerging Jordan blocks in the two-dimensional Potts and loop models at generic

Collective coordinates are frequently employed in path integrals to manage divergences caused by fluctuations around saddle points that align with classical symmetries. These coordinates parametrize a manifold of zero modes and more broadly provide judicious coordinates on the space of fields. However, changing from local coordinates around a saddle point to more global collective coordinates is remarkably subtle. The main complication is that the mapping from local coordinates to collective coordinates is generically multivalued. Consequently one is forced to either restrict the domain of path integral in a delicate way, or otherwise correct for the multivaluedness by dividing the path integral by certain intersection numbers. We provide a careful treatment of how to fix collective coordinates while accounting for these intersection numbers, and then demonstrate the importance of the fix for free theories. We also provide a detailed study of the fix for interacting theories and show that the contributions of higher intersections to the path integral can be nonperturbatively suppressed. Using a variety of examples ranging from single-particle quantum mechanics to quantum field theory, we explain and resolve various pitfalls in the implementation of collective coordinates.

Arindam Bhattacharya

Aurélien Dersy

Matthew D. Schwartz

Collective coordinate fix in the path integral

With non-perturbative de Sitter gravity and holography in mind, we deduce the genus expansion of de Sitter Jackiw-Teitelboim (dS JT) gravity. We find that this simple model of quantum cosmology has an effective string coupling which is pure imaginary. This imaginary coupling gives rise to alternating signs in the genus expansion of the dS JT S-matrix, which as a result appears to be Borel-Le Roy resummable. Furthermore dS JT gravity is formally an analytic continuation of AdS JT gravity, and behaves like a matrix integral with a negative number of degrees of freedom.

Non-perturbative de Sitter Jackiw-Teitelboim gravity

Carrollian field theories have recently emerged as a candidate dual to flat space quantum gravity. We carefully quantize simple two-derivative Carrollian theories, revealing a strong sensitivity to the ultraviolet. They can be regulated upon being placed on a spatial lattice and working at finite inverse temperature. Unlike in conventional field theories, the details of the lattice-regulated Carrollian theories remain important at long distances even in the limit that the lattice spacing is sent to zero. We use that limit to define interacting continuum models with a tractable perturbative expansion. The ensuing theories are those of generalized free fields, with non-Gaussian correlations suppressed by positive powers of the lattice spacing, and an unbroken supertranslation symmetry.

Amir Raz

Quantizing Carrollian field theories

A longstanding issue is the classical equivalence between the Jordan and the Einstein frames, which is considered just a field redefinition of the metric tensor and the scalar field. In this work, based on the previous result that the Hamiltonian transformations from the Jordan to the Einstein frame are not canonical on the extended phase space, we study the possibility of the existence of canonical transformations. We show that on the reduced phase space – defined by suitable gauge fixing of the lapse and shifts functions – these transformations are Hamiltonian canonical. Poisson brackets are replaced by Dirac’s brackets following the Bergman-Dirac’s procedure. The Hamiltonian canonical transformations map solutions of the equations of motion in the Jordan frame into solutions of the equations of motion in the Einstein frame.

On the canonical equivalence between Jordan and Einstein frames

We study the Hilbert space of the Class 5 model described in <inter-ref xmlns:ns0='http://www.w3.org/1999/xlink' ns0:role='http://www.elsevier.com/xml/linking-roles/preprint' ns0:href='arxiv:1904.12005' id='inf0010'>arXiv:1904.12005</inter-ref>. Despite being integrable, neither its transfer matrix nor its Hamiltonian are diagonalisable, meaning that the usual Algebraic Bethe Ansatz does not provide the full Hilbert space. Instead, we make use of the symmetries of the model to construct the Jordan blocks of the transfer matrix. We also show that the Hamiltonian and the transfer matrix, despite commuting, do not have the same Jordan block structure.

Germany

Juan Miguel Nieto García

Jordan blocks and the Bethe ansatz III: Class 5 model and its symmetries

We study time evolution in a simple model of de Sitter quantum gravity, namely, Jackiw-Teitelboim with a positive cosmological constant. We find that time evolution is isometric rather than unitary. The states that are projected out under time evolution correspond to initial conditions that crunch. Our findings suggest that knowledge of bulk physics, even on arbitrarily large timescales, is insufficient to deduce the de Sitter <math><mrow><mi>S</mi></mrow></math> matrix.

Isometric Evolution in de Sitter Quantum Gravity

The conformal correspondence between FLRW universes in the Einstein and Jordan frames allows for an expansion-collapse duality – an always expanding Einstein frame universe can have a dual Jordan frame description that is contracting forever. The scenario eventually runs into an apparent paradox. When the contracting Jordan frame universe becomes sufficiently small, the classical description becomes inadequate and the universe is expected to develop quantum characteristics. However, at this time, the corresponding Einstein frame universe is expected to behave classically, due to the arbitrarily large size it has grown to. The conformal map here appears to be providing a duality between a quantum effect-dominated universe and a classical universe. We investigate the status of the conformal map at the quantum level in such a scenario, focusing on addressing this paradox. The Einstein and Jordan frame universes are quantized using the Wheeler-DeWitt prescription. We show that the classical conformal map holds at the quantum level when compared through expectation values of scale factors. The relative quantum fluctuation in the scale factor becomes conformally invariant, it increases in both the past and future directions according to the internal clock. Expectedly, the quantum fluctuations in the collapsing Jordan frame increase as it shrinks towards singularity. More surprisingly, the quantum fluctuations in the expanding Einstein frame increase as well, even as its classical scale factor becomes larger. Despite having drastically different cosmological evolutions, the rise in quantum characteristics in a collapsing frame implies the same in its expanding counterpart, thereby resolving the apparent paradox.