Quantum geometry from phase space reduction

Florian Conrady; Laurent Freidel (Perimeter Inst. Theor. Phys.)

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Quantum geometry from phase space reduction

Florian Conrady, Laurent Freidel (Perimeter Inst. Theor. Phys.)

General Relativity and Quantum Cosmology · Submitted 2 Feb 2009

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

The authors' main claim The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.78 · mentions code or data availability 0.02.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

In this work we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin--Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this work we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra.	contribution 0.90	0.95	0.15	asserted 1.00
s1	The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra.	result 0.93	0.93	0.97	asserted 1.00
s2	This provides an explicit realization of theorems by Guillemin--Sternberg and Hall that describe the commutation of quantization and reduction.	interpretation 0.94	0.52	0.08	asserted 1.00
s3	In the final part of the paper, we use our result to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.	result 0.57	0.94	0.07	asserted 1.00

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The paper

Original abstract

In this work we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin--Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.

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Cited by 11 Pith papers

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

Polyhedra in loop quantum gravity
gr-qc 2010-09 accept novelty 7.0 of 10

An SU(2) intertwiner of arbitrary valence is the quantum state of a convex polyhedron whose faces carry the intertwiner's areas and normals, generalizing the quantum tetrahedron to any number of faces.
U(N) Coherent States for Loop Quantum Gravity
gr-qc 2010-05 accept novelty 7.0 of 10

The authors construct U(N)-covariant coherent states for N-valent SU(2) intertwiners, labeled by Grassmannian elements, with explicit norm, scalar product, and a proposed interpretation as framed polyhedra.
Towards Spinfoam Cosmology
gr-qc 2010-03 conditional novelty 7.0 of 10

The loop-quantum-gravity spinfoam transition amplitude between homogeneous isotropic coherent states is annihilated by an operator whose classical limit is the Friedmann Hamiltonian constraint, at first order in a sin...
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.
Effective Spin Foam Models for Four-Dimensional Quantum Gravity
gr-qc 2020-04 conditional novelty 6.0 of 10

A family of simplified spin foam models keeps the discrete area spectrum while imposing gluing constraints only as strongly as quantum uncertainty permits, and curvature is not suppressed provided the combination gamm...

Show 6 more citing papers

From twistors to twisted geometries
gr-qc 2010-06 accept novelty 6.0 of 10

The twisted-geometry phase space of loop quantum gravity on a fixed graph is the space of null twistors modulo a global phase, obtained by reducing a pair of spinors with an area-matching constraint.
Twisted geometries: A geometric parametrisation of SU(2) phase space
gr-qc 2010-01 accept novelty 6.0 of 10

The loop gravity phase space on a fixed graph is symplectomorphic to a space of twisted geometries labeled by triangle areas, two unit normals, and an extrinsic angle per edge.
The Fine Structure of SU(2) Intertwiners from U(N) Representations
gr-qc 2009-11 conditional novelty 6.0 of 10

The fixed-total-spin space of N-valent SU(2) intertwiners is exactly the U(N) irreducible representation with highest weight [l,l,0,...,0], which yields a new derivation of the LQG black-hole entropy.
Holomorphic Simplicity Constraints for 4d Spinfoam Models
gr-qc 2011-04 conditional novelty 5.0 of 10

A holomorphic splitting of the simplicity constraints yields coherent intertwiners that solve them exactly, and a new spinfoam model is built from those states.
Spin-Foams for All Loop Quantum Gravity
gr-qc 2009-09 conditional novelty 5.0 of 10

Spin-foam histories are generalized from triangulations to arbitrary 2-cell complexes, admitting all loop-quantum-gravity spin-network states, and the EPRL intertwiner map is shown injective for |γ|≥1.
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 later papers used it

55 later papers cite it

Evidence: substantial. 45 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 13 of the 121 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 paper by the same authors is listed below: 1 built on it.

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By the same authors

Aimed to further develop the holomorphic viewpoint on H introduced in this paper, both for n=4 and in more generality.

Holomorphic Factorization for a Quantum Tetrahedron (2009) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

51 arXiv papers cite this in 116 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 · 86 sentences without a confident label

Builds on it 2 papers

The main aim of this paper is to further develop the holomorphic viewpoint on $H_$ introduced in [ref] , both for $n=4$ and in more generality.

Holomorphic Factorization for a Quantum Tetrahedron · Introduction · 0.99
We generalize the semiclassical states used in [ref] , which is obtained via a superposition over spins of spin-network states having nodes labeled by Livine-Speziale coherent intertwiners [ref] , with coefficients proposed by Rovelli in [ref] .

Lorentzian spinfoam propagator · Introduction · 0.97

Uses its method or tool 4 papers

The construction is new: it uses the coherent intertwiners of Livine and Speziale [ref] (see also [ref] ) together with a superposition over spins as done in [ref] .

LQG propagator from the new spin foams · Semiclassical boundary state and the metric operator · 0.95
Indeed, as described in [ref] , there is a gluing map $g: V_k ⊗V_k →$ that can be represented in terms of coherent states as g =_k∫^2 N(z)/ (1+|z|^2)^2k z̅^2k | -1/ |⊗⟩| z|.⟩ In terms of this gluing map the normalized intertwiner is given by C^k_(z_i) = _k ∫C_j_1,j_2,k(z_1,z_2, -1/ ) C_k, j_3, j_4(z,z_3,z_4)/(1+|z|^2)^2k z̅^2k ^2N(z) .

Holomorphic Factorization for a Quantum Tetrahedron · 3-Point Function and Normalization of the 4-Point Intertwiner · 0.82
At fixed spins $j_ab$, the gauge invariant projector at nodes can be written as [ref] [equation] where $|Φ_n(n_na)⟩$ is the coherent intertwiner peaked on the polyhedron with normals $j_nan_na$.

Gluing polyhedra with entanglement in loop quantum gravity · Bell-network state on $Γ_5$ · 0.95

Compares with it 3 papers

This is similar to the analysis performed in [ref] but the present setting is slightly more general (and actually simpler) since the authors were looking at the Kapovich-Milson phase spaces (at fixed individual face areas).

Deformations of Polyhedra and Polygons by the Unitary Group · Invariant Parametrization and Cross-Ratios · 0.80
The situation is analogous to the one of the labels of the Livine-Speziale coherent intertwiners (four unit-normals) [ref] as opposed to the constrained labels of the same states obtained by Conrady and Freidel via geometric quantization of a classical tetrahedron [ref] .

Coherent spin-networks · Conclusions · 0.91
Similarly, the semiclassical behavior considered here needs to be compared with the results on the semiclassical behavior of these new models [ref] .

Second-order amplitudes in loop quantum gravity · Introduction · 0.90

Cites it as support 3 papers

Now, it turns out [ref] , remarkably, that A_v(q) ∼e^iS_R(q) where the equality holds in the limit of large quantum numbers.[For the moment, this considerable result has been proven rigorously only in the case of 5-valent vertices.] It follows that the integral [ref] is a Regge-like discretization of the formal “functional integral over 4-geometries" Z(q)=∫_∂g=q dg e^iS[g]. where $S[g]$ is the Einstein Hilbert action.

Loop quantum gravity: the first twenty five years · Transition amplitudes · 0.99
The reduced states are still an overcomplete basis of $_F$, as a consequence of the Guillemin-Sternberg theorem [ref] .

Polyhedra in loop quantum gravity · Coherent intertwiners and semiclassical polyhedra · 0.97
This is compatible with the argument that the closure constrain in spinfoam models is imposed strongly [ref] .

Analytic Continuation of Spin foam Models · Space action · 0.99

Cites it as background 11 papers

It is now well-known and explained in great detail in [ref] that the space of $4$-valent intertwiners can be uniquely labelled by oriented tetrahedra.

A Discrete and Coherent Basis of Intertwiners · Semi-Classical Limit · 0.89
This representation is associated to spin-networks with nodes labeled by Livine-Speziale coherent intertwiners [ref] .

Spinfoams in the holomorphic representation · I. Spin Foams in various representations · 0.86
In LQG, polyhedron geometries are quantized by intertwiners $||i⟩_N∈Inv_(j_1,⋯,j_N)$ which are invariant in the tensor product of $N$ $$ unitary irreps $_j_1⊗⋯⊗_j_N$ (spins $j≥1/2$ label the irreps) [ref] .

Semiclassical Behavior of Spinfoam Amplitude with Small Spins and Entanglement Entropy · Quantum Polyhedron and Parallel Restriction · 1.00

What it cites · 39 sentences without a confident label

Builds on 1 paper

This follows from a straightforward extension of proposition VII.1, [ref] to a complex with boundary.

On the semiclassical limit of 4d spin foam models · Asymptotics of vertex amplitude · 0.98

Uses the method or tool of 2 papers

One route to the quantisation of this space is the route proposed by Barbieri [ref] in his seminal work and studied since then in the quantum gravity literature: according to this scheme one first quantises the phase space of four vectors $X_i∈S^2_j_i$, promotes these vectors to operators associated to 4 copies of $𝔰𝔲(2)$, where in each case $X_i^2= j_i(j_i+1)$, and then one imposes the constraints $∑_i X_i =0$ at the operatorial level.

Quantum tetrahedra and simplicial spin networks · Quantisation commutes with reduction · 0.88
Also, in order to use the results of [ref] we need to assume that the integral is only over nondegenerate configurations, i.e. we impose in the integral the restriction (g_1 Û,⋯, g_4 Û) ≠0 ,

On the semiclassical limit of 4d spin foam models · Asymptotics of vertex amplitude · 0.80

Compares with 3 papers

Therefore, eq. asymptoticsAv is consistent with the bulk asymptotics derived in our previous paper [ref] .

On the semiclassical limit of 4d spin foam models · Asymptotics of vertex amplitude · 0.97
Up to numerical factors, our result coincides with the nondegenerate contribution in the asymptotic formula by Barrett et al. [ref] .

Asymptotic analysis of the EPRL four-simplex amplitude · Summary and discussion · 1.00
One can show that this definition of the boundary amplitude agrees with the one given in [ref] [For this one uses the coherence property of coherent states, that is, the identity |j^+, N|⊗⟩|j^-, N|=⟩ |k, N| ⟩for k= j^++ j^-. ].

Path integral representation of spin foam models of 4d gravity · FK$γ$ model · 0.84

Cites as support 1 paper

These results follow from proposition VII.1 and proposition B.2 of [ref] .

On the semiclassical limit of 4d spin foam models · Asymptotics of vertex amplitude · 0.97

Cites as background 8 papers

Recently a wealth of important developments has occurred following the proposal of new spin foam models for gravity [ref] .

Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity · Introduction · 1.00
Recently a wealth of important developments has occurred following the proposal of new spin foam models for gravity [ref] .

LQG vertex with finite Immirzi parameter · Introduction · 1.00
Recently a wealth of important developments has occurred following the proposal of new spin foam models for gravity [ref] .

A New Spin Foam Model for 4d Gravity · Introduction · 1.00

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