How to construct diffeomorphism symmetry on the lattice

Bianca Dittrich

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How to construct diffeomorphism symmetry on the lattice

Bianca Dittrich

General Relativity and Quantum Cosmology · Submitted 18 Jan 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim After discussing the meaning and implications of diffeomorphism symmetry in the discrete, in particular for the continuum limit, we introduce a perturbative framework to construct discretizations with an exact notion of diffeomorphism symmetry.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.70 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.11 · mentions code or data availability 0.02.

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Diffeomorphism symmetry, the fundamental invariance of general relativity, is generically broken under discretization. After discussing the meaning and implications of diffeomorphism symmetry in the discrete, in particular for the continuum limit, we introduce a perturbative framework to construct discretizations with an exact notion of diffeomorphism symmetry. We will see that for such a perturbative framework consistency conditions need to be satisfied which enforce the preservation of the gauge symmetry to the perturbative order under discussion. These consistency conditions will allow structural investigations of diffeomorphism invariant discretizations.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Diffeomorphism symmetry, the fundamental invariance of general relativity, is generically broken under discretization.	background 0.98	0.11	0.06	asserted 0.98
s1	After discussing the meaning and implications of diffeomorphism symmetry in the discrete, in particular for the continuum limit, we introduce a perturbative framework to construct discretizations with an exact notion of diffeomorphism symmetry.	contribution 0.93	0.84	0.37	asserted 0.97
s2	We will see that for such a perturbative framework consistency conditions need to be satisfied which enforce the preservation of the gauge symmetry to the perturbative order under discussion.	result 0.51	0.63	0.19	asserted 0.98
s3	These consistency conditions will allow structural investigations of diffeomorphism invariant discretizations.	interpretation 0.55	0.42	0.17	asserted 0.77

What kind of paper the classifier reports this is: method or tool 0.91 · evidence from proof or derivation 0.88.

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Original abstract

Diffeomorphism symmetry, the fundamental invariance of general relativity, is generically broken under discretization. After discussing the meaning and implications of diffeomorphism symmetry in the discrete, in particular for the continuum limit, we introduce a perturbative framework to construct discretizations with an exact notion of diffeomorphism symmetry. We will see that for such a perturbative framework consistency conditions need to be satisfied which enforce the preservation of the gauge symmetry to the perturbative order under discussion. These consistency conditions will allow structural investigations of diffeomorphism invariant discretizations.

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References

Forward citations

Cited by 8 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Decorated tensor network renormalization for lattice gauge theories and spin foam models
gr-qc 2014-09 conditional novelty 7.0 of 10

A new tensor network coarse graining algorithm preserves gauge symmetry explicitly and locates the 3D Ising gauge model phase transition within 8% of the Monte Carlo value at lowest order.
Effective Spin Foam Models for Lorentzian Quantum Gravity
gr-qc 2021-04 conditional novelty 6.0 of 10

A Lorentzian version of the effective spin foam path integral is constructed and numerically probed, showing a semi-classical regime for small Barbero-Immirzi parameter on simple triangulations with one bulk triangle.
Discrete Renormalization Group for SU(2) Tensorial Group Field Theory
hep-th 2014-07 conditional novelty 6.0 of 10

Within a truncated four-coupling continuous flow, the rank-3 SU(2) tensorial group field theory is not asymptotically free for positive marginal couplings, contrary to prior expectation.
A new vacuum for Loop Quantum Gravity
gr-qc 2014-01 conditional novelty 6.0 of 10

A flat-connection (BF) vacuum dual to the Ashtekar-Lewandowski vacuum is constructed for 2+1 dimensional loop quantum gravity, together with a modified holonomy-flux algebra that is claimed to be consistent under Pach...
Tensorial methods and renormalization in Group Field Theories
hep-th 2013-10 conditional novelty 6.0 of 10

Group Field Theories with tensorial interactions and a closure constraint admit a consistent multiscale renormalization analysis, and a 3d SU(2) model is shown just-renormalizable at the perturbative level.

Show 3 more citing papers

From the discrete to the continuous - towards a cylindrically consistent dynamics
gr-qc 2012-05 conditional novelty 6.0 of 10

A coarse graining scheme with dynamics-selected embedding maps produces fixed point Hamilton's principal functions for a 2D scalar field, but the quantum extension with the standard loop quantum gravity embedding does...
The continuum limit of loop quantum gravity - a framework for solving the theory
gr-qc 2014-09 accept novelty 3.0 of 10

The paper argues that the continuum limit of loop quantum gravity is equivalent to the physical Hilbert space and sketches an iterative coarse-graining program, with diffeomorphism symmetry as the indicator that the l...
The Spin Foam Approach to Quantum Gravity
gr-qc 2012-05 accept novelty 2.0 of 10

The review presents the spin foam quantization of gravity as a constrained BF path integral, focusing on the EPRL and Freidel-Krasnov models and on 2+1 gravity as a solvable guide.

How the literature uses it

37 arXiv papers cite this in 71 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 58 sentences without a confident label

Compares with it 1 paper

In fact, this formalism constitutes a canonical complement to the covariant coarse graining techniques developed in [ref] .

Classification of constraints and degrees of freedom for quadratic discrete actions · Conclusions and outlook · 0.80

Cites it as support 5 papers

There exists strong evidence that the amplitudes of the system can be systematically improved via coarse graining [ref] , such that the symmetry is broken less.

Coarse graining spin foam quantum gravity -- a review · Fate of diffeomorphism symmetry · 1.00
Indeed recent work [ref] has shown that demanding diffeomorphism symmetry is a very strong requirement: there are many examples where it is equivalent to demanding triangulation (or more generally discretization) independence.[This notion of diffeomorphism invariance implies that the positions of vertices in a given solution are gauge parameters and can be chosen freely.

From the discrete to the continuous - towards a cylindrically consistent dynamics · Introduction · 0.97
These findings support the strategy advertised in [ref] to look for triangulation invariant models (in the gravitational context tight to diffeomorphism symmetry [ref] ) by coarse graining.

Topological lattice field theories from intertwiner dynamics · Fixed points and triangulation invariant models · 1.00

Cites it as background 6 papers

While some of this work directly addresses emergent models [ref] , much of it is in the context of lattice regularization of general relativity [ref] .

Challenges for Emergent Gravity · Diffeomorphism invariance I · 0.87
Related is the question whether diffeomorphism symmetry, which is deeply rooted into the dynamics of general relativity, can be restored [ref] .

Coarse graining flow of spin foam intertwiners · Introduction · 0.82
The challenge arises on many levels, for instance in recovering the usual notions of a spacetime manifold from combinatorial structures [ref] ; recovering an effective description in terms of coordinates and restoring the continuum notion of diffeomorphisms or coordinate changes [ref] ; and understanding the intricate interplay between a continuum and semiclassical limit.

Towards anisotropic cosmology in group field theory · Introduction · 0.95

What it cites · 44 sentences without a confident label

Uses the method or tool of 2 papers

The improved or perfect action to zeroth order can be calculated in many ways, one is as fixed points of the coarse graining flow [ref] , the other is to explicitly obtain the inverse matrices $(M^-1)_xy$ and then $M^XY$ using Fourier transform as in [ref] .

Coarse graining free theories with gauge symmetries: the linearized case · Zeroth order · 0.95
The improved or perfect action to zeroth order can be calculated in many ways, one is as fixed points of the coarse graining flow [ref] , the other is to explicitly obtain the inverse matrices $(M^-1)_xy$ and then $M^XY$ using Fourier transform as in [ref] .

Perfect discretization of reparametrization invariant path integrals · Zeroth order · 0.95

Compares with 1 paper

This is analogous to the situation in Regge calculus [ref] , where linearized diffeomorphism symmetry is realized around the flat solutions (based on arbitrary lattices).

Linearized dynamics from the 4-simplex Regge action · Discretized parametrized fields · 0.87

Cites as support 3 papers

Basically, the perturbation assumption, namely that the solution is analytical in a small parameter $ϵ$ is not valid [ref] .

From covariant to canonical formulations of discrete gravity · Introduction and overview · 0.98
The problem is, that linearized Regge gravity displays the linearized form of diffeomorphism symmetry, i.e. one can identify longitudinal lattice modes, that do not propagate as these are null modes of the Hessian of the action [ref] .

Coarse graining free theories with gauge symmetries: the linearized case · Introduction and overview · 0.96
This has been proven for 1D reparametrization invariant (quantum) systems in [ref] .

Perfect discretization of reparametrization invariant path integrals · Introduction and overview · 0.98

Cites as background 5 papers

(This kind of diffeomorphism symmetry in the discrete was termed ditt–invariance in [ref] .) That is this symmetry can change the geometrical distance between the vertices.

Discretizing parametrized systems: the magic of Ditt-invariance · Introduction and overview · 0.81
In [ref] the zeroth order of (free) field theories has been discussed including systems with gauge symmetries such as $U(1)$ gauge theory and gravity.

Coarse graining free theories with gauge symmetries: the linearized case · Introduction and overview · 0.90
Let us consider a simple but popular example [ref] , the discrete parametrized harmonic oscillator.

Breaking and restoring of diffeomorphism symmetry in discrete gravity · Discrete parametrized harmonic oscillator · 0.90

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