Coarse graining free theories with gauge symmetries: the linearized case

Benjamin Bahr; Bianca Dittrich; Song He

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Coarse graining free theories with gauge symmetries: the linearized case

Benjamin Bahr, Bianca Dittrich, Song He

General Relativity and Quantum Cosmology · Submitted 16 Nov 2010

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The authors' main claim Finally we provide a discretization of linearized gravity as well as a coarse graining map and show that with this choice the 3D linearized gravity action is invariant under coarse graining.

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arXiv ID1011.3667 (version 2)

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CitationBenjamin Bahr, Bianca Dittrich, Song He. "Coarse graining free theories with gauge symmetries: the linearized case." arXiv:1011.3667 (2010). https://pith.science/paper/QWP3HNZU

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.90 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.17 · mentions code or data availability 0.02.

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Discretizations of continuum theories often do not preserve the gauge symmetry content. This occurs in particular for diffeomorphism symmetry in general relativity, which leads to severe difficulties both in canonical and covariant quantization approaches. We discuss here the method of perfect actions, which attempts to restore gauge symmetries by mirroring exactly continuum physics on a lattice via a coarse graining process. Analytical results can only be obtained via a perturbative approach, for which we consider the first steps, namely the coarse graining of the linearized theory. The linearized gauge symmetries are exact also in the discretized theory, hence we develop a formalism to deal with gauge systems. Finally we provide a discretization of linearized gravity as well as a coarse graining map and show that with this choice the 3D linearized gravity action is invariant under coarse graining.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Discretizations of continuum theories often do not preserve the gauge symmetry content.	background 1.00	0.07	0.03	asserted 0.94
s1	This occurs in particular for diffeomorphism symmetry in general relativity, which leads to severe difficulties both in canonical and covariant quantization approaches.	background 1.00	0.05	0.02	asserted 0.99
s2	We discuss here the method of perfect actions, which attempts to restore gauge symmetries by mirroring exactly continuum physics on a lattice via a coarse graining process.	method 0.67	0.17	0.07	asserted 0.96
s3	Analytical results can only be obtained via a perturbative approach, for which we consider the first steps, namely the coarse graining of the linearized theory.	limitation 0.69	0.55	0.03	asserted 0.98
s4	The linearized gauge symmetries are exact also in the discretized theory, hence we develop a formalism to deal with gauge systems.	result 0.67	0.62	0.10	asserted 1.00
s5	Finally we provide a discretization of linearized gravity as well as a coarse graining map and show that with this choice the 3D linearized gravity action is invariant under coarse graining.	result 0.73	0.96	0.82	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.59 · evidence from proof or derivation 1.00.

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Original abstract

Discretizations of continuum theories often do not preserve the gauge symmetry content. This occurs in particular for diffeomorphism symmetry in general relativity, which leads to severe difficulties both in canonical and covariant quantization approaches. We discuss here the method of perfect actions, which attempts to restore gauge symmetries by mirroring exactly continuum physics on a lattice via a coarse graining process. Analytical results can only be obtained via a perturbative approach, for which we consider the first steps, namely the coarse graining of the linearized theory. The linearized gauge symmetries are exact also in the discretized theory, hence we develop a formalism to deal with gauge systems. Finally we provide a discretization of linearized gravity as well as a coarse graining map and show that with this choice the 3D linearized gravity action is invariant under coarse graining.

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References

Forward citations

Cited by 7 Pith papers

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

Causal structure and light-conical singularities in Lorentzian simplicial quantum gravity
gr-qc 2026-08 conditional novelty 7.0 of 10

Light-conical singularities dominate random Lorentzian Regge geometries, with trouser-type irregularities entropically preferred for bones shared by many 4-simplices.
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...
Path integral measure and triangulation independence in discrete gravity
gr-qc 2011-10 accept novelty 6.0 of 10

A local path integral measure for linearized 3D Regge calculus makes the amplitude triangulation independent and reproduces the Ponzano-Regge asymptotics; in 4D only partial invariance survives.
Coarse graining methods for spin net and spin foam models
gr-qc 2011-09 conditional novelty 6.0 of 10

Two coarse-graining schemes are applied to finite-group spin net and spin foam models, with a new Gauss-constraint-preserving tensor network algorithm and a demonstration that certain cutoff models are equivalent.
Perfect discretization of reparametrization invariant path integrals
gr-qc 2011-01 conditional novelty 5.0 of 10

For one-dimensional reparametrization invariant systems, requiring the discrete path integral to respect vertex translation symmetry forces it to equal the continuum propagator, and an iterative Wilsonian scheme const...

Show 2 more citing papers

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.
Zakopane lectures on loop gravity
gr-qc 2011-02 accept novelty 2.0 of 10

Loop quantum gravity is presented as a covariant framework in which discrete spin-network quanta of space evolve through spinfoam amplitudes whose semiclassical limit is claimed to reproduce general relativity.

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38 arXiv papers cite this in 83 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 · 66 sentences without a confident label

Uses its method or tool 7 papers

A way out, suggested in [ref] and also followed up here, is to improve the action perturbatively, so that gauge invariance can be obtained order by order.

Perfect discretization of reparametrization invariant path integrals · Reparametrization invariance in the discrete and discretization independence · 0.81
To write this in a compact form we introduce spin zero mode and spin two mode projections $^0Π_L,^2Π_L$ on the lattice [ref] , see Appendix [ref].

From spin foams to area metric dynamics to gravitons · The Hessian block for the length metric fluctuations · 0.94
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] .

How to construct diffeomorphism symmetry on the lattice · Zeroth order · 0.95

Compares with it 2 papers

Hence, this kind of blocking is much closer to the blocking used in the gravitation models in [ref] , which is derived by geometrical arguments.

Coarse graining methods for spin net and spin foam models · Coarse graining methods · 0.82
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 2 papers

Thus Regge calculus is a perfect discretization of 3D gravity [ref] , that is it captures exactly the continuum properties of this (topological) theory.

3D holography: from discretum to continuum · The Regge action and equations of motion · 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] .

How to construct diffeomorphism symmetry on the lattice · Introduction and overview · 0.96

Cites it as background 4 papers

In a series a papers Bahr and Dittrich and collaborators [ref] , and Dittrich and Höhn [ref] , studied the discretization of reparametrization invariant systems, especially the Regge approximation to general relativity.

Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity · Broken symmetries · 0.91
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.

How to construct diffeomorphism symmetry on the lattice · Introduction and overview · 0.90
(See [ref] for the first construction of the Lorentzian Regge action.) The associated dynamics is consistent with Einstein's gravity in the sense that in the limit of an infinitely refined triangulation, it reproduces the continuum dynamics encoded in the Einstein-Hilbert action —for more detailed discussions, see [ref] .

Causal structure and light-conical singularities in Lorentzian simplicial quantum gravity · Regge geometries · 0.88

What it cites · 28 sentences without a confident label

Cites as support 2 papers

The reason is, that gauge breaking at third order is related to the non-invariance of the second order Hamilton–Jacobi functional of the theory under vertex displacements, see [ref] .

From covariant to canonical formulations of discrete gravity · Gauge degrees of freedom under coarse graining · 0.99
However this invariance is broken to higher order [ref] , that is the second order gauge modes do appear in the higher than second order (potential) terms.

(Broken) Gauge Symmetries and Constraints in Regge Calculus · Gauge degrees of freedom under coarse graining · 0.96

Cites as background 16 papers

One popular example is provided by Regge gravity [ref] .

Discrete structures in gravity · The coarse graining · 0.85
For further discussions of these points and related issues see [ref] and references therein.

(Broken) Gauge Symmetries and Constraints in Regge Calculus · Introduction · 0.97
For further discussions of these points and related issues see [ref] and references therein.

A Kirchhoff-like conservation law in Regge calculus · Introduction · 0.97

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