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The 4D Regge action is invariant under 5--1 and 4--2 Pachner moves, which
define a subset of (local) changes of the triangulation. Given this fact one
might hope to find a local path integral measure that makes the quantum theory
invariant under these moves and hence makes the theory partially triangulation
invariant. We show that such a local inva...
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Citations
... This has been successfully implemented for three-dimensional Euclidean gravity in [18], and has allowed interesting applications, e.g. the computation of one-loop partition functions and confirmation of a holographic duality in three-dimensional quantum gravity (without a cosmological constant) [19,20]. On the other hand, it has been shown in [21], that four-dimensional Regge gravity does not admit a local discretization-independent measure. More specifically, there does not even exist a local measure, which is invariant under a specific class of changes of triangulation, which leave the action invariant. ...
... These techniques were adjusted and applied to the computation of the Pachner move Hessians in Euclidean Regge gravity in [18]. The paper [21] streamlined this derivation by introducing a Caley-Menger determinant associated to the Pachner move configurations. We will pick up this technique and apply a further simplification, which will allow us to derive a simple expression for the Lorentzian Regge Hessian for all Pachner moves in general dimensions (larger than two). ...
... See [19] for a similar computation of the partition function for the Euclidean solid torus (proving a holographic relationship to a theory defined on the boundary of the triangulation), which can now also be performed in Lorentzian signature. Regarding the path integral measure, [21] showed that for Euclidean Regge calculus, there does not exist a local invariant measure, even if one restricts only to the 5 − 1 and 4 − 2 Pachner moves. The arguments in [21] are algebraic and we can thus assume that they hold also for Lorentzian signature. ...
A bstract
Lorentzian quantum gravity is believed to cure the pathologies encountered in Euclidean quantum gravity, such as the conformal factor problem. We show that this is the case for the Lorentzian Regge path integral expanded around a flat background. We illustrate how a subset of local changes of the triangulation, so-called Pachner moves, allow to isolate the indefinite nature of the gravitational action at the discrete level. The latter can be accounted for by oppositely chosen deformed contours of integration. Moreover, we construct a discretization-invariant local path integral measure for 3D Lorentzian Regge calculus and point out obstructions in defining such a measure in 4D. We see the work presented here as a first step towards establishing the existence of the non-perturbative Lorentzian path integral for Regge calculus and related frameworks such as spin foams.
An extensive appendix provides an overview of Lorentzian Regge calculus, using the recently established concept of the complexified Regge action, and derives useful geometric formulae and identities needed in the main text.
... is a component of the path integral measure. Various such F [l] have been studied in the context of Euclidean quantum gravity [38][34] [39]. Thus, assuming ergodicity, the partition function for the system undergoing variational dynamics is the Euclidean path integral. ...
A variational phase space is constructed for a compact and piecewise flat Riemannian manifold. An extended action functional is provided such that the variational dynamics generate a symplectic flow on the phase space. This symplectic flow is numerically integrated as it evolves with respect to the variational parameter. Assuming ergodicity, the resulting flow samples the Euclidean path integral.
... Previous work on classical perfect actions [7] considered only scalar fields, vector fields and gravity in three dimensions (which do not feature bulk gravitons), [15][16][17] does discuss scalar and vector fields, [11][12][13] considers in addition fermionic systems. Previous work on the quantum level established a perfect (one-loop) path integral measure for threedimensional gravity [20,21], but also showed, that a perfect measure for four-dimensional gravity, even on one-loop level, is necessarily non-local [20,21]. This has so far precluded explicit constructions of such a measure. ...
... Previous work on classical perfect actions [7] considered only scalar fields, vector fields and gravity in three dimensions (which do not feature bulk gravitons), [15][16][17] does discuss scalar and vector fields, [11][12][13] considers in addition fermionic systems. Previous work on the quantum level established a perfect (one-loop) path integral measure for threedimensional gravity [20,21], but also showed, that a perfect measure for four-dimensional gravity, even on one-loop level, is necessarily non-local [20,21]. This has so far precluded explicit constructions of such a measure. ...
... We can thus easily impose a cut-off, and see this as a further discretization [18,19], or take all modes into account and work with a system that is continuous in three of its dimensions and discrete in the remaining one. The Fourier transform does in particular allow us to deal with the expected non-locality of the one-loop correction [20,21]. ...
A bstract
Lattice actions and amplitudes that perfectly mirror continuum physics are known as perfect discretizations. Such perfect discretizations naturally preserve the symmetries of the continuum. This is a key concern for general relativity, where diffeomorphism symmetry and dynamics are deeply connected, and diffeomorphisms play a crucial role in quantization. In this work we construct for the first time a perfect discretizations for four-dimensional linearized gravity. We show how the perfect discretizations lead to a straightforward construction of the one-loop quantum corrections for manifolds with boundary. This will also illustrate, that for manifolds with boundaries, gauge modes that affect the boundary, need to be taken into account. This work provides therefore an evaluation of the boundary action for the diffeomorphism modes for a general class of backgrounds.
... Then one idea for fixing the measure is to demand discretization independence [62]. This would lead to a non-local measure in 4D [63]. Alternatively, one could adopt simpler local measures and demand that the exact result be obtained only after taking the lattice refinement limit. ...
Evaluating gravitational path integrals in the Lorentzian has been a long-standing challenge due to the numerical sign problem. We show that this challenge can be overcome in simplicial quantum gravity. By deforming the integration contour into the complex, the sign fluctuations can be suppressed, for instance using the holomorphic gradient flow algorithm. Working through simple models, we show that this algorithm enables efficient Monte Carlo simulations for Lorentzian simplicial quantum gravity. In order to allow complex deformations of the integration contour, we provide a manifestly holomorphic formula for Lorentzian simplicial gravity. This leads to a complex version of simplicial gravity that generalizes the Euclidean and Lorentzian cases. Outside the context of numerical computation, complex simplicial gravity is also relevant to studies of singularity resolving processes with complex semi-classical solutions. Along the way, we prove a complex version of the Gauss-Bonnet theorem, which may be of independent interest.
... Then one idea for fixing the measure is to demand discretization independence [62]. This would lead to a non-local measure in 4D [63]. Alternatively, one could adopt simpler local measures and demand that the exact result be obtained only after taking the lattice refinement limit. ...
Evaluating gravitational path integrals in the Lorentzian has been a long-standing challenge due to the numerical sign problem. We show that this challenge can be overcome in simplicial quantum gravity. By deforming the integration contour into the complex, the sign fluctuations can be suppressed, for instance using the holomorphic gradient flow algorithm. Working through simple models, we show that this algorithm enables efficient Monte Carlo simulations for Lorentzian simplicial quantum gravity. In order to allow complex deformations of the integration contour, we provide a new definition of Lorentzian simplicial gravity that is different from Sorkin's non-holomorphic one. The end result is a complex version of simplicial gravity that generalizes both the Euclidean and Lorentzian cases. Outside the context of numerical computation, the complex version of simplicial gravity is also relevant to studies of singularity resolving processes with complex semi-classical solutions. Along the way, we prove a complex version of the Gauss-Bonnet theorem, which may be of independent interest.
... In this task, one key technical and conceptual challenge is to reconcile the regularizations required in quantum field theory with the diffeomorphism symmetry which underlies general relativity. Indeed, a number of approaches employ discretizations as regulators, which is the case for instance of Regge calculus [1], and in this class of theories, where one attempts to represent geometrical data on a triangulation 1 , diffeomorphism symmetry is generically broken [2][3][4]. ...
We construct in this article a new realization of quantum geometry, which is
obtained by quantizing the recently-introduced flux formulation of loop quantum
gravity. In this framework, the vacuum is peaked on flat connections, and
states are built upon it by creating local curvature excitations. The inner
product induces a discrete topology on the gauge group, which turns out to be
an essential ingredient for the construction of a continuum limit Hilbert
space. This leads to a representation of the full holonomy-flux algebra of loop
quantum gravity which is unitarily-inequivalent to the one based on the
Ashtekar-Isham-Lewandowski vacuum. It therefore provides a new notion of
quantum geometry. We discuss how the spectra of geometric operators, including
holonomy and area operators, are affected by this new quantization. In
particular, we find that the area operator is bounded, and that there are two
different ways in which the Barbero-Immirzi parameter can be taken into
account. The methods introduced in this work open up new possibilities for
investigating further realizations of quantum geometry based on different
vacua.
... Concerning diffeomorphisms, strictly speaking (being smooth transformations) they are not defined in a discrete context like that of GFT Feynman amplitudes, i.e. spin foam models and lattice gravity path integrals, and thus the question becomes whether we can identify some analogue of diffeomorphic symmetry that, in a continuum limit, could be then identified with the one characterizing GR. There are several analyses of such question for 3d (topological) models at the level of spin foam amplitudes [29], lattice gravity (see for example [30][31][32]) and corresponding GFT formulation [33], but nothing similar in the 4d gravity case (where the 4d counterpart of the symmetry identified in the 3d case is actually broken, at the discrete level [34]). When attempting a reconstruction of an effective dynamics of geometry in a continuum approximation, as done in the context of GFT cosmology, one has to proceed in terms of observables of the fundamental theory that have a chance to correspond to diffeomorphic invariant observables in GR, since all the structures of continuum GR on which diffeomorphisms act, e.g., manifold points, directions an coordinate functions, but also fields defined on the same manifold, are simply not present in the theory. ...
We discuss motivation and goals of renormalization analyses of group field theory models of simplicial 4d quantum gravity, and review briefly the status of this research area. We present some new computations of perturbative Group field theories amplitudes, concerning in particular their scaling behavior, and the numerical techniques employed to obtain them. Finally, we suggest a number of research directions for further progress.
... However, when integrating out these degrees of freedom, one picks up a non-local factor that cannot be written as a local product. In [55] it is shown that said factor is related to a condition whether the six vertices involved in the Pachner move lie on a 3-sphere. Moreover, these articles reveal that the 4D Regge action itself is not invariant under Pachner moves. ...
In quantum gravity, we envision renormalization as the key tool for bridging the gap between microscopic models and observable scales. For spin foam quantum gravity, which is defined on a discretization akin to lattice gauge theories, the goal is to derive an effective theory on a coarser discretization from the dynamics on the finer one, coarse graining the system in the process and thus relating physics at different scales. In this review I will discuss the motivation for studying renormalization in spin foam quantum gravity, e.g., to restore diffeomorphism symmetry, and explain how to define renormalization in a background independent setting by formulating it in terms of boundary data. I will motivate the importance of the boundary data by studying coarse graining of a concrete example and extending this to the spin foam setting. This will naturally lead me to the methods currently used for renormalizing spin foam quantum gravity, such as tensor network renormalization, and a discussion of recent results. I will conclude with an overview of future prospects and research directions.
... Moreover, again due to the fact that there are no propagating bulk degrees of freedom, the boundary field theories describe so-called flat solution, Asante et al. [32] finds the same type of boundary theory as in 3D. However, due to the fact that 4D Regge calculus does not feature a local discretization independent measure [33], it is hard to extend this result to the (one-loop) quantum theory. 2 To extend these results to more general backgrounds and to tackle the main task, namely including gravitons, we need a framework that is applicable to 4D gravity and for which we can expect to solve the dynamics. Being particularly interested in length observables, we will therefore consider (linearized) metric gravity. ...
... We proceed by inserting the solutions (42) into the action with Lagrange multiplier term (33). Let us first consider the case that we have an outer boundary at r out and an inner boundary at r in . ...
In this work we construct holographic boundary theories for linearized 3D gravity, for a general family of finite or quasi-local boundaries. These boundary theories are directly derived from the dynamics of 3D gravity by computing the effective action for a geometric boundary observable, which measures the geodesic length from a given boundary point to some center in the bulk manifold. We identify the general form for these boundary theories and find that these are Liouville-like with a coupling to the boundary Ricci scalar. This is illustrated with various examples, which each offer interesting insights into the structure of holographic boundary theories.
... Restricting to boundary data which induce a 4D flat solution, [32] finds the same type of boundary theory as in 3D. However, due to the fact that 4D Regge calculus does not feature a local discretization independent measure [33], it is hard to extend this result to the (one-loop) quantum theory. 2 To extend these results to more general backgrounds and to tackle the main task, namely including gravitons, we need a framework that is applicable to 4D gravity and for which we can expect to solve the dynamics. Being particularly interested in length observables, we will therefore consider (linearized) metric gravity. ...
In this work we construct holographic boundary theories for linearized 3D gravity, for a general family of finite or quasi-local boundaries. These boundary theories are directly derived from the dynamics of 3D gravity by computing the effective action for a geometric boundary observable, which measures the geodesic length from a given boundary point to some centre in the bulk manifold. We identify the general form for these boundary theories and find that these are Liouville like with a coupling to the boundary Ricci scalar. This is illustrated with various examples, which each offer interesting insights into the structure of holographic boundary theories.
