arXiv subjects
Publications and source records attributed to Rodolfo Gambini.
This is a summary of the talk presented by JP at ICGC2004. It covered some developments in canonical quantum gravity occurred since ICGC2000, emphasizing the recently introduced consistent discretizations of general relativity.
Causality imposes strong restrictions on the type of operators that may be observables in relativistic quantum theories. In fact, causal violations arise when computing conditional probabilities for certain partial causally connected measurements using the standard non covariant procedure. Here we introduce another way of computing conditional probabilities, based on an intrinsic covariant relational order of the events, which differs from the standard one when these type of measurements are included. This alternative procedure is compatible with a wider and very natural class of operators without breaking causality. If some of these measurements could be implemented in practice as predicted by our formalism, the non covariant, conventional approach should be abandoned. Furthermore, the description we promote here would imply a new physical effect where interference terms are suppressed as a consequence of the covariant order in the measurement process.
Smolin has put forward the proposal that the universe fine tunes the values of its physical constants through a Darwinian selection process. Every time a black hole forms, a new universe is developed inside it that has different values for its physical constants from the ones in its progenitor. The most likely universe is the one which maximizes the number of black holes. Here we present a concrete quantum gravity calculation based on a recently proposed consistent discretization of the Einstein equations that shows that fundamental physical constants change in a random fashion when tunneling through a singularity.
The recently introduced consistent discrete lattice formulation of canonical general relativity produces a discrete theory that is constraint-free. This immediately allows to overcome several of the traditional obstacles posed by the ``problem of time'' in totally constrained systems and quantum gravity and cosmology. In particular, one can implement the Page--Wootters relational quantization. This brief paper discusses this idea in the context of a simple model system --the parameterized particle-- that is usually considered one of the crucial tests for any proposal for solution to the problem of time in quantum gravity.
We consider the application of the consistent lattice quantum gravity approach we introduced recently to the situation of a Friedmann cosmology and also to Bianchi cosmological models. This allows us to work out in detail the computations involved in the determination of the Lagrange multipliers that impose consistency, and the implications of this determination. It also allows us to study the removal of the Big Bang singularity. Different discretizations can be achieved depending on the version of the classical theory chosen as a starting point and their relationships studied. We analyze in some detail how the continuum limit arises in various models. In particular we notice how remnants of the symmetries of the continuum theory are embodied in constants of the motion of the consistent discrete theory. The unconstrained nature of the discrete theory allows the consistent introduction of a relational time in quantum cosmology, free from the usual conceptual problems.
It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. Additionally, the traditional lattice field theory approach consists in fixing the gauge in a Euclidean action, which does not appear appropriate for general relativity. We analyze discrete lattice general relativity and develop a canonical formalism that allows to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a ``dynamical gauge'' fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. Among other issues the problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories like BF theory. We discuss a simple cosmological application that exhibits the quantum elimination of the singularity at the big bang.
We have recently introduced a realistic, covariant, interpretation for the reduction process in relativistic quantum mechanics. The basic problem for a covariant description is the dependence of the states on the frame within which collapse takes place. A suitable use of the causal structure of the devices involved in the measurement process allowed us to introduce a covariant notion for the collapse of quantum states. However, a fully consistent description in the relativistic domain requires the extension of the interpretation to quantum fields. The extension is far from straightforward. Besides the obvious difficulty of dealing with the infinite degrees of freedom of the field theory, one has to analyze the restrictions imposed by causality concerning the allowed operations in a measurement process. In this paper we address these issues. We shall show that, in the case of partial causally connected measurements, our description allows us to include a wider class of causal operations than the one resulting from the standard way for computing conditional probabilities. This alternative description could be experimentally tested. A verification of this proposal would give a stronger support to the realistic interpretations of the states in quantum mechanics.
Starting from Ooguri's construction for $BF$ theory in three (and four) dimensions, we show how to construct a well defined theory with an infinite number of degrees of freedom. The spin network states that are kept invariant by the evolution operators of the theory are exact solutions of the Hamiltonian constraint of quantum gravity proposed by Thiemann. The resulting theory is the first example of a well defined, finite, consistent, spin-foam based theory in a situation with an infinite number of degrees of freedom. Since it solves the quantum constraints of general relativity it is also a candidate for a theory of quantum gravity. It is likely, however, that the solutions constructed correspond to a spurious sector of solutions of the constraints. The richness of the resulting theory makes it an interesting example to be analyzed by forthcoming techniques that construct the semi-classical limit of spin network quantum gravity.
We discuss the canonical quantization of systems formulated on discrete space-times. We start by analyzing the quantization of simple mechanical systems with discrete time. The quantization becomes challenging when the systems have anholonomic constraints. We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. We extend the analysis to gauge field theories on the lattice. We consider a complete canonical formulation, starting from a discrete action, for lattice Yang--Mills theory discretized in space and Maxwell theory discretized in space and time. After completing the treatment, the results can be shown to coincide with the results of the traditional transfer matrix method. We then apply the method to BF theory, yielding the first lattice treatment for such a theory ever. The framework presented deals directly with the Lorentzian signature without requiring an Euclidean rotation. The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper.
Up to now it has been impossible to find a realistic interpretation for the reduction process in relativistic quantum mechanics. The basic problem is the dependence of the states on the frame within which collapse takes place. A suitable use of the causal structure of the devices involved in the measurement process allows us to introduce a covariant notion for the collapse of quantum states.
This is a summary of a talk given at the CP01 meeting on possible Lorentz anomalies in canonical quantum gravity. It briefly reviews some initial explorations on the subject that have taken place recently, and should be only be seen as a short pointer to the literature on the subject, mostly for outsiders.
We consider discretizations of the Einstein action of general relativity such that the resulting discrete equations of motion form a consistent constrained system. Upon ``spin foam'' quantization of the system, consistency allows a natural way of recovering the correct semi-classical theory. A consistent set of approximations to the Einstein equations could also have implications for numerical relativity and for the construction of approximate classical observables for the theory.
We analize the relational quantum evolution of generally covariant systems in terms of Rovelli's evolving constants of motion and the generalized Heisenberg picture. In order to have a well defined evolution, and a consistent quantum theory, evolving constants must be self-adjoint operators. We show that this condition imposes strong restrictions to the choices of the clock variables. We analize four cases. The first one is non- relativistic quantum mechanics in parametrized form. We show that, for the free particle case, the standard choice of time is the only one leading to self-adjoint evolving constants. Secondly, we study the relativistic case. We show that the resulting quantum theory is the free particle representation of the Klein Gordon equation in which the position is a perfectly well defined quantum observable. The admissible choices of clock variables are the ones leading to space-like simultaneity surfaces. In order to mimic the structure of General Relativity we study the SL(2R) model with two Hamiltonian constraints. The evolving constants depend in this case on three independent variables. We show that it is possible to find clock variables and inner products leading to a consistent quantum theory. Finally, we discuss the quantization of a constrained model having a compact constraint surface. All the models considered may be consistently quantized, although some of them do not admit any time choice such that the equal time surfaces are transversal to the orbits.
We analyze the quantization of dynamical systems that do not involve any background notion of space and time. We give a set of conditions for the introduction of an intrinsic time in quantum mechanics. We show that these conditions are a generalization of the usual procedure of deparametrization of relational theories with hamiltonian constraint that allow to include systems with an evolving Hilbert Space. We apply our quantization procedure to the parametrized free particle and to some explicit examples of dynamical system with an evolving Hilbert space. Finally, we conclude with some considerations concerning the quantum gravity case.
We have recently introduced an approach for studying perturbatively classical and quantum canonical general relativity. The perturbative technique appears to preserve many of the attractive features of the non-perturbative quantization approach based on Ashtekar's new variables and spin networks. With this approach one can find perturbatively classical observables (quantities that have vanishing Poisson brackets with the constraints) and quantum states (states that are annihilated by the quantum constraints). The relative ease with which the technique appears to deal with these traditionally hard problems opens several questions about how relevant the results produced can possibly be. Among the questions is the issue of how useful are results for large values of the cosmological constant and how the approach can deal with several pathologies that are expected to be present in the canonical approach to quantum gravity. With the aim of clarifying these points, and to make our construction as explicit as possible, we study its application in several simple models. We consider Bianchi cosmologies, the asymmetric top, the coupled harmonic oscillators with constant energy density and a simple quantum mechanical system with two Hamiltonian constraints. We find that the technique satisfactorily deals with the pathologies of these models and offers promise for finding (at least some) results even for small values of the cosmological constant. Finally, we briefly sketch how the method would operate in the full four dimensional quantum general relativity case.
We consider general relativity with a cosmological constant as a perturbative expansion around a completely solvable diffeomorphism invariant field theory. This theory is the $Λ\to\infty$ limit of general relativity. This allows an explicit perturbative computational setup in which the quantum states of the theory and the classical observables can be explicitly computed. The zeroth order corresponds to highly degenerate space-times with vanishing volume. Perturbations give rise to space-times with non-vanishing volumes in a natural way. The spectrum of area- and volume-related observables constructed by coupling the theory to matter can be directly assessed. An unexpected relationship arises at a quantum level between the discrete spectrum of the volume operator and the allowed values of the cosmological constant.
We generalize the idea of Vassiliev invariants to the spin network context, with the aim of using these invariants as a kinematical arena for a canonical quantization of gravity. This paper presents a detailed construction of these invariants (both ambient and regular isotopic) requiring a significant elaboration based on the use of Chern-Simons perturbation theory which extends the work of Kauffman, Martin and Witten to four-valent networks. We show that this space of knot invariants has the crucial property -from the point of view of the quantization of gravity- of being loop differentiable in the sense of distributions. This allows the definition of diffeomorphism and Hamiltonian constraints. We show that the invariants are annihilated by the diffeomorphism constraint. In a companion paper we elaborate on the definition of a Hamiltonian constraint, discuss the constraint algebra, and show that the construction leads to a consistent theory of canonical quantum gravity.
In a companion paper we introduced a kinematical arena for the discussion of the constraints of canonical quantum gravity in the spin network representation based on Vassiliev invariants. In this paper we introduce the Hamiltonian constraint, extend the space of states to non-diffeomorphism invariant ``habitats'' and check that the off-shell quantum constraint commutator algebra reproduces the classical Poisson algebra of constraints of general relativity without anomalies. One can therefore consider the resulting set of constraints and space of states as a consistent theory of canonical quantum gravity.