On the quantum Geroch group

Classical and Quantum Gravity, 2020

The Geroch group is an infinite dimensional transitive group of symmetries of classical cylindrically symmetric gravitational waves which acts by non-canonical transformations on the phase space of these waves. Here this symmetry is rederived and the unique Poisson bracket on the Geroch group which makes its action on the gravitational phase space Lie-Poisson is obtained. Two possible notions of asymptotic flatness are proposed that are compatible with the Poisson bracket on the phase space, and corresponding asymptotic flatness preserving subgroups of the Geroch group are defined which turn out to be compatible with the Poisson bracket on the group. A quantization of the Geroch group is proposed that is similar to, but distinct from, the sl 2 Yangian, and a certain action of this quantum Geroch group on gravitational observables is shown to preserve the commutation relations of Korotkin and Samtleben's quantization of asymptotically flat cylindrically symmetric gravitational waves. The action also preserves three of the additional conditions that define their quantization. It is conjectured that the action preserves the remaining two conditions (asymptotic flatness and a unit determinant condition on a certain basic field) as well and is, in fact, a symmetry of their model. Our results on the quantum theory are formal, but a possible rigorous formulation based on algebraic quantum theory is outlined.

arXiv: General Relativity and Quantum Cosmology, 2016

A new set of fundamental commutation relations for quantum gravity is presented. The basic variables are the eight components of the unimodular part of the spatial dreibein and eight SU(3) generators which correspond to Klauder's momentric variables. The commutation relations are not canonical, but they have well defined group theoretical meanings. All fundamental entities are dimensionless; and quantum wave functionals are preferentially selected to be in the dreibein representation.

2014

В работе рассмотрена теория гравитации в многомерных пространствах. Сформулирована модель метрики, удовлетворяющая основным требованиям квантовой теории. Показано, что в такой метрике гравитационные волны описываются уравнением Лиувилля. Доказана гипотеза Шредингера о связи волновой функции с гравитационными волнами In this article we consider gravitation theory in multidimensional space. The model of the metric satisfying the basic requirements of quantum theory is proposed. It is shown that gravitational waves are described by the Liouville equation. Schrödinger conjecture about the Schrödinger wave function and gravitational waves has been proved

Cornell University - arXiv, 2021

Il Nuovo Cimento, 1962

In any quantum theory, in which the metric tensor of Einstein's gravitational theory is also quantized, it becomes meaningless to ask for an initial space-like surface on which to specify the conventional field commutators. The covariant quantum formalism, in which all fields either commute or fail to do so only when the field's points coincide, is proposed as being suitable to quantize gravity. The extension of the covariant quantum formalism to general boson fields that interact in an intrisically nonlinear way with external fields is analysed in some detail. This formalism is applied to the case of the free gravitational field. In a functional representation, the measure en metrics is found to be that proposed by Misner. A basic state of the quantized gravitational theory is proposed, which involves a summation over all permissible metrics in the entire space-time manifohl.

2020

It is well-known that quantum groups are relevant to describe the quantum regime of 3d gravity. They encode a deformation of the gauge symmetries parametrized by the value of the cosmological constant. They appear as a form of regularization either through the quantization of the Chern-Simons formulation or the state sum approach of Turaev-Viro. Such deformations are perplexing from a continuum and classical picture since the action is defined in terms of undeformed gauge invariance. We present here a novel way to derive from first principle and from the classical action such quantum group deformation. The argument relies on two main steps. First we perform a canonical transformation, which deformed the gauge invariance and the boundary symmetries, and makes them depend on the cosmological constant. Second we implement a discretization procedure relying on a truncation of the degrees of freedom from the continuum.

Academia Letters, 2022

The search of a theory of quantum gravity (QG) which is consistent both with the principles of quantum mechanics as well as with the postulates of the classical Einstein theory of General Relativity (GR) has represented until recently one of the most challenging, long-standing debated and hard-to-solve conceptual problems of mathematical and theoretical physics alike. In fact, a basic crucial issue is about the possibility of achieving in the context of either classical or quantum relativistic theories, and in particular for a quantum theory of gravity, a truly coordinate-(i.e., frame-) independent representation, realized by 4-tensor notation of physical laws. This means that the latter theory must satisfy both the principles of general covariance and of manifest covariance with respect to the group of local point transformations (LPT-group), i.e., coordinate diffeomorphisms mutually mapping in each other different GR frames. These principles lie at the foundation of all relativistic theories and of the related physical laws. In fact, although the choice of special coordinate systems is always legitimate for all physical systems either discrete or continuous, including in particular classical and quantum gravity, the intrinsic objective nature of physical laws makes them frame-independent. For the same reason, since LPTs preserve the differential-manifold structure of space-time, these principles represent also a cornerstone of the standard formulation of GR, namely the Einstein field equations and the corresponding classical treatment of the gravitational field. The same principles should apply as well to the very foundations of quantum field theory

2012

Physical Review D - PHYS REV D, 2010

The spectral triple approach to noncommutative geometry allows one to develop the entire standard model (and supersymmetric extensions) of particle physics from a purely geometry stand point and thus treats both gravity and particle physics on the same footing. The bosonic sector of the theory contains a modification to Einstein-Hilbert gravity, involving a nonconformal coupling of curvature to the Higgs field and conformal Weyl term (in addition to a nondynamical topological term). In this paper we derive the weak field limit of this gravitational theory and show that the production and dynamics of gravitational waves are significantly altered. In particular, we show that the graviton contains a massive mode that alters the energy lost to gravitational radiation, in systems with evolving quadrupole moment. We explicitly calculate the general solution and apply it to systems with periodically varying quadrupole moments, focusing in particular on the the well know energy loss formula for circular binaries.

Physics Letters B, 2001