Non-perturbative Path Integral Research Articles

Scaling behaviour in random non-commutative geometries

Random non-commutative geometries are a novel approach to taking a non-perturbative path integral over geometries. They were introduced in Barrett and Glaser (2015 arXiv:1510.01377), where a first examination was performed. During this examination we found that some geometries show indications of a phase transition. In this article we explore this phase transition further for geometries of type (1, 1), (2, 0), and (1, 3). We determine the pseudo-critical points of these geometries and explore how some of the observables scale with the system size. We also undertake first steps towards understanding the critical behaviour through correlations and in determining critical exponents of the system.

CDT and cosmology

In the approach of Causal Dynamical Triangulations (CDT), quantum gravity is obtained as a scaling limit of a non-perturbative path integral over space-times whose causal structure plays a crucial role in the construction. After some general considerations about the relation between quantum gravity and cosmology, we examine which aspects of CDT are potentially interesting from a cosmological point of view, focusing on the emergence of a de Sitter universe in CDT quantum gravity. Dans l'approche des triangulations dynamiques causales (CDT), la gravitation quantique est obtenue comme une limite d'échelle d'une intégrale de chemin non perturbative sur des espaces-temps dont la structure causale joue un rôle crucial dans la construction. Après quelques considérations générales sur la relation entre la gravité quantique et la cosmologie, nous examinons quels aspects des triangulations dynamiques causales sont potentiellement intéressants d'un point de vue cosmologique, en mettant l'accent sur l'émergence d'un univers de de Sitter en gravitation quantique dans l'approche CDT.

Causal Dynamical Triangulations without preferred foliation

We introduce a generalized version of the Causal Dynamical Triangulations (CDT) formulation of quantum gravity, in which the regularized, triangulated path integral histories maintain their causal properties, but do not have a preferred proper-time foliation. An extensive numerical study of the associated nonperturbative path integral in 2+1 dimensions shows that it can nevertheless reproduce the emergence of an extended de Sitter universe on large scales, a key feature of CDT quantum gravity. This suggests that the preferred foliation normally used in CDT is not a crucial (albeit convenient) part of its background structure.

Second-Order Phase Transition in Causal Dynamical Triangulations

Causal dynamical triangulations are a concrete attempt to define a nonperturbative path integral for quantum gravity. We present strong evidence that the lattice theory has a second-order phase transition line, which can potentially be used to define a continuum limit in the conventional sense of nongravitational lattice theories.

THE SELF-ORGANIZED DE SITTER UNIVERSE

We propose a theory of quantum gravity which formulates the quantum theory as a nonperturbative path integral, where each space–time history appears with the weight exp (iS EH ), with S EH the Einstein–Hilbert action of the corresponding causal geometry. The path integral is diffeomorphism-invariant (only geometries appear) and background-independent. The theory can be investigated by computer simulations, which show that a de Sitter universe emerges on large scales. This emergence is of an entropic, self-organizing nature, with the weight of the Einstein–Hilbert action playing a minor role. Also, the quantum fluctuations around this de Sitter universe can be studied quantitatively and remain small until one gets close to the Planck scale. The structures found to describe Planck-scale gravity are reminiscent of certain aspects of condensed-matter systems.

Sum over topologies and double-scaling limit in 2D Lorentzian quantum gravity

We construct a combined non-perturbative path integral over geometries and topologies for two-dimensional Lorentzian quantum gravity. The Lorentzian structure is used in an essential way to exclude geometries with unacceptably large causality violations. The remaining sum can be performed analytically and possesses a unique and well-defined double-scaling limit, a property which has eluded similar models of Euclidean quantum gravity in the past.

A Lorentzian cure for Euclidean troubles

There is strong evidence coming from Lorentzian dynamical triangulations that the unboundedness of the gravitational action is no obstacle to the construction of a well-defined non-perturbative path integral. In a continuum approach, a similar suppression of the conformal divergence comes about as the result of a non-trivial path-integral measure.

Making the gravitational path integral more Lorentzian or Life beyond Liouville gravity

In two space-time dimensions, there is a theory of Lorentzian quantum gravity which can be defined by a rigorous, non-perturbative path integral and is inequivalent to the well-known theory of (Euclidean) quantum Liouville gravity. It has a number of appealing features: i) its quantum geometry is non-fractal, ii) it remains consistent when coupled to matter, even beyond the c=1 barrier, iii) it is closer to canonical quantization approaches than previous path-integral formulations, and iv) its construction generalizes to higher dimensions.

Towards a nonperturbative path integral in gauge theories

We propose a modification of the Faddeev–Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato–Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few explicit examples are given to illustrate new features of the formalism developed. The method is applied to the Kogut–Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang–Mills theory. Feynman's conjecture about a relation between the mass gap and the orbit space geometry in gluodynamics is discussed in the framework of the modified path integral.

Non-perturbative path integral of 2d dilaton gravity and two-loop effects from scalar matter

Performing a non-perturbative path integral for the geometric part of a large class of 2d theories without a kinetic term for the dilaton field, the quantum effects from scalar matter fields are treated as a perturbation. When integrated out to two-loops they yield a correction to the Polyakov term which is still exact in the geometric part. Interestingly enough the effective action only experiences a renormalization of the dilaton potential.