The group structure of dynamical transformations between quantum reference frames
Angel Ballesteros, Flaminia Giacomini, Giulia Gubitosi
Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part. While such transformations were shown to be symmetries of the system’s Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.
I am co-organizing the following online meeting, which will be held next week:
It’s now 20 years since the first Doubly Special Relativity papers appeared on ArXiv. Several research groups have worked on DSR-relativistic models, leading to significant progress, but some grey areas remain on the conceptual side and additional phenomenological avenues are much needed. The meeting “DSR20” (an online meeting, using zoom) intends to be an opportunity for an exchange of ideas on these matters. DSR20 will be held from december 14 to december 16 and it will be scheduled so to allow colleagues in different parts of the world to attend at least some of the sessions.
Generalized noncommutative Snyder spaces and projective geometry
Giulia Gubitosi, Angel Ballesteros, Francisco J. Herranz
Given a group of kinematical symmetry generators, one can construct a compatible noncommutative spacetime and deformed phase space by means of projective geometry. This was the main idea behind the very first model of noncommutative spacetime, proposed by H.S. Snyder in 1947. In this framework, spacetime coordinates are the translation generators over a manifold that is symmetric under the required generators, while momenta are projective coordinates on such a manifold. In these proceedings we review the construction of Euclidean and Lorentzian noncommutative Snyder spaces and investigate the freedom left by this construction in the choice of the physical momenta, because of different available choices of projective coordinates. In particular, we derive a quasi-canonical structure for both the Euclidean and Lorentzian Snyder noncommutative models such that their phase space algebra is diagonal although no longer quadratic.
Next week I will attend the virtual “Second Miniworkshop on Quantum Gravity – The Phenomenology of Quantum Gravity“.
I will give a presentation about “Quantum spacetime symmetries, curved momentum spaces and relative locality – the interacting particles picture”.
The κ-Newtonian and κ-Carrollian algebras and their noncommutative spacetimes
Angel Ballesteros, Giulia Gubitosi, Ivan Gutierrez-Sagredo and Francisco J. Herranz
We derive the non-relativistic c→∞ and ultra-relativistic c→0 limits of the κ-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the κ-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the κ-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding κ-Newtonian and κ-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the κ-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter κ, the curvature parameter Λ and the speed of light parameter c.
Next week I will be attending the conference “Quantum gravity phenomenology in the multi-messenger approach” in Granada, Spain.
This is the first annual conference of the COST Action CA18108, for which I co-lead the Working Group 1: Theoretical Approaches.
The first annual conference of the COST Action CA18108 – Quantum gravity phenomenology in the multi-messenger approach – will be held in Granada, Spain, from the 10th of March to the 13th March, 2020.
Registration is open and can be done at this link.
Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry
Angel Ballesteros, Giulia Gubitosi and Francisco J. Herranz
We show that the Lorentzian Snyder models, together with their non-relativistic (?→∞) and ultra-relativistic (?→0) limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of these spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant kinematical symmetries. While the momentum space of the Lorentzian Snyder models is given by certain projective coordinates on (Anti-) de Sitter spaces, we discover that the momentum space of the Galilean (Carrollian) Snyder models is given by certain projective coordinates on curved Carroll (Newton–Hooke) spaces. This exchange between the non-relativistic and ultra-relativistic limits emerging in the transition from the geometric picture to the phase space picture is traced back to an interchange of the role of coordinates and translation operators. As a physically relevant feature, we find that in Galilean Snyder spacetimes the time coordinate does not commute with space coordinates, in contrast with previous proposals for non-relativistic Snyder models, which assume that time and space decouple in the non-relativistic limit. This remnant mixing between space and time in the non-relativistic limit is a quite general Planck-scale effect found in several quantum spacetime models.
Next week in Barcelona there will be the first Network Activity of the COST Action CA18108 QG-MM “Quantum gravity phenomenology in the multi-messenger approach”.
The objective of the meeting is to initiate a discussion among the different communities involved in the project, with the immediate aim of reviewing the present experimental and theoretical status in the search of quantum gravity signatures in the phenomenology of the different cosmic messengers.
I will give a review talk for the WG1 – Theoretical frameworks for quantum gravity effects below the Planck energy.
Next week I will be attending the conference “Corfu2019: Quantum Geometry, Field Theory and Gravity” in Corfu, Greece.
I will give a presentation about “Lorentzian and Galilean Snyder spacetimes from projective geometry”