A note on the foundation of relativistic mechanics. II

I illustrate a simple hamiltonian formulation of general relativity, derived from the work of Esposito, Gionti and Stornaiolo, which is manifestly 4d generally covariant and is defined over a finite dimensional space. The spacetime coordinates drop out of the formalism, reflecting the fact that they are not related to observability. The formulation can be interpreted in terms of Toller's reference system transformations, and provides a physical interpretation for the spinnetwork to spinnetwork transition amplitudes computable in principle in loop quantum gravity and in the spin foam models.

2018

The study of geometrodynamics was introduced by Wheeler in the 50’s decade in order to describe particle as geometrical topological defects in a relativistic framework[1], and, in the last years has becoming a very intensive subject of research[2]. In the last decades Loop Quantum Gravity (LQG) have provided a picture of the quantum geometry of space, thanks in part to the theory of spin networks[3]. The concept of spin foam is intended to serve as a similar picture for the quantum geometry of spacetime. LQG is a theory that attempts to describe the quantum properties of the universe and gravity. In LQG the space can be viewed as an extremely fine favric of finite loops. These networks of loops are called spin networks. The evolution of a spin network over time is called a spin foam. The more traditional approach to LQG is the canonical LQG, and there is a newer approach called covariant LQG, more commonly called spin foam theory. However, at the present time, it is not possible to ...

1993

In this paper, a candidate for pregeometry, Ponzano-Regge spin networks, will be examined in the context of the pregeometric philosophy of Wheeler. Ponzano and Regge were able to construct a theory for 3-dimensional quantum gravity based on 3nj-symbols, obtaining the path integral over the metric in the semiclassical limit. However, extension of this model to 4-dimensions has proven to be difficult. It will be shown that the building blocks for 4-dimensional spacetime are already present in the Ponzano-Regge formalism using a reinterpretation of the theory based on the pregeometric hypotheses of Wheeler.

Cornell University - arXiv, 2016

We propose (1) that the flat space-time metric that defines the traditional covariant Heisenberg algebra commutation rules of quantum theory between the four-vector position and momentum, be generalized to be the space-time dependent Riemann metric following Einstein's equations for general relativity, which determine the metric from the energy-momentum tensor. The metric is then a function of the four-vector position operators which are to be expressed in the position representation. This then allows one (2) to recast the Christoffel, Riemann, and Ricci tensors and Einstein's GR differential equations for the metric as an algebra of commutation relations among the four-vector position and momentum operators (a generalized Lie algebra). This then allows one (3) to generalize the structure constants of the rest of the Poincare algebra with the space-time dependent metric of general relativity tightly integrating it with quantum theory. Finally, (4) we propose that the four-mometumoperator be generalized (to be gauge covariant) to include the intermediate vector bosons of the standard model further generalizing this algebra of observables to include gauge observables. Then the generalized Poincare algebra, extended with a four-vector position operator, and the phenomenological operators of the non-Abelian gauge transformations of the standard model form a larger algebra of observables thus tightly integrating all three domains. Ways in which this may lead to observable effects are discussed.

to appear, 1994

We present a general relativistic approach to quantum mechanics of a spinless charged particle, subject to external classical gravitational and electromagnetic fields in a curved space-time with absolute time. The scheme is also extended in order to treat the n-body quantum mechanics.

We introduce a new basis on the state space of non-perturbative quantum gravity. The states of this basis are linearly independent, are well de ned in both the loop representation and the connection representation, and are labeled by a generalization of Penrose's spin networks. The new basis fully reduces the spinor identities (SU(2) Mandelstam identities) and simpli es calculations in non-perturbative quantum gravity. In particular, it allows a simple expression for the exact solutions of the Hamiltonian constraint (Wheeler-DeWitt equation) that have been discovered in the loop representation. Since the states in this basis diagonalize operators that represent the three geometry of space, such as the area and volumes of arbitrary surfaces and regions, these states provide a discrete picture of quantum geometry at the Planck scale.

Quantum Gravity , 2024

Neste artigo selecionamos uma possível abordagem de relação entre a teoria quântica de campos, e a relatividade geral de Einstein.

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

arXiv: General Relativity and Quantum Cosmology, 2006

We review the canonical analysis of the Palatini action without going to the time gauge as in the standard derivation of Loop Quantum Gravity. This allows to keep track of the Lorentz gauge symmetry and leads to a theory of Covariant Loop Quantum Gravity. This new formulation does not suffer from the Immirzi ambiguity, it has a continuous area spectrum and uses spin networks for the Lorentz group. Finally, its dynamics can easily be related to Barrett-Crane like spin foam models.

2018

We propose (1) that the flat space-time metric that defines the traditional covariant Heisenberg algebra commutation rules of quantum theory between the four-vector position and momentum, be generalized to be the space-time dependent Riemann metric satisfying Einstein’s equations for general relativity (GR), which determine the metric from the energy-momentum tensor. The metric is then a function of the four-vector position operators which are to be expressed in the position representation. This then allows one (2) to recast the Christoffel symbols, and the Riemann and Ricci tensors in Einstein’s GR differential equations for the metric, as an algebra of commutation relations among the four-vector position and momentum operators (a generalized Lie algebra). This then (3) defines the structure constants of the rest of the Poincare algebra with the space-time dependent metric of general relativity tightly integrating it with quantum theory. (4) We propose that the four momentumoperat...