A Discrete History of the Lorentzian Path Integral

Abstract

In these lecture notes, I describe the motivation behind a recent formulation of a non-perturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) space-times, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a welldefined Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to convergent sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d = 2 and d = 3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry.

On recovering continuum topology from a causal set

“…In approaches using simplicial methods like Dynamical Triangulations or Spin Foams, the discrete structure can be taken to be a triangulation of the given manifold, which though non-diffeomorphic to the continuum, by construction, carries all continuum homological information [5]. Conversely, an abstract simplicial complex is associated with a given manifold only if it can be be mapped bijectively to a triangulation of that space.…”

Two-dimensional quantum gravity - a laboratory for fluctuating graphs and quenched connectivity disorder

“…Over the past decades, this has prompted active research into several constructive approaches to non-perturbative quantization prescriptions [1]. Among them, Regge calculus [2] and the dynamical triangulations model in its Euclidean [3] and more recently Lorentzian [4] versions have been extensively studied over the last 20 years. The basic idea of both approaches is the same as in Feynman's formulation of quantum mechanics in terms of path integrals [5].…”

That Strange Procedure Called Quantisation

“…This helps to sharpen ones expectations on the quantisation concept in general, which is particularly important for Quantum Gravity since here sources for direct physical input are rather scarce. Expectations on what Quantum Gravity will finally turn out to be are still diverse, though more precise pictures now definitely emerge within the individual approaches, as you will hopefully be convinced in the other lectures [14,15,17], so that reliable statements about similarities and differences on various points can now be made. The present contribution deliberately takes focus on a very particular and seemingly formal point, in order to exemplify in a controllable setting the care needed in formulating 'rules' for 'quantisation'.…”