Group Averaging and Refined Algebraic Quantization: Where are we now?

Donald Marolf

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Group Averaging and Refined Algebraic Quantization: Where are we now?

Donald Marolf

General Relativity and Quantum Cosmology · Submitted 30 Nov 2000

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

The authors' main claim The main goal is provide a description of outstanding problems and possible research topics in the field.

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Refined Algebraic Quantization and Group Averaging are powerful methods for quantizing constrained systems. They give constructive algorithms for generating observables and the physical inner product. This work outlines the current status of these ideas with an eye toward quantum gravity. The main goal is provide a description of outstanding problems and possible research topics in the field.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Refined Algebraic Quantization and Group Averaging are powerful methods for quantizing constrained systems.	background 1.00	0.07	0.08	asserted 1.00
s1	They give constructive algorithms for generating observables and the physical inner product.	background 0.91	0.10	0.07	asserted 1.00
s2	This work outlines the current status of these ideas with an eye toward quantum gravity.	contribution 0.93	0.57	0.09	asserted 0.94
s3	The main goal is provide a description of outstanding problems and possible research topics in the field.	future work 0.54	0.41	0.18	asserted 0.97

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Original abstract

Refined Algebraic Quantization and Group Averaging are powerful methods for quantizing constrained systems. They give constructive algorithms for generating observables and the physical inner product. This work outlines the current status of these ideas with an eye toward quantum gravity. The main goal is provide a description of outstanding problems and possible research topics in the field.

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References

Forward citations

Cited by 18 Pith papers

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

Quantum Nature of the Big Bang
gr-qc 2006-02 conditional novelty 8.0 of 10

A semiclassical universe in loop quantum cosmology bounces at Planck-scale density instead of hitting the big bang singularity, connecting expanding and contracting classical branches.
Chaos and the Emergence of the Cosmological Horizon
hep-th 2024-11 conditional novelty 7.0 of 10

Finite-mass observers in dS2 have non-commuting type I algebras, and their OTOC shows a Lyapunov exponent 4π/β_dS, twice the de Sitter bound.
The Trinity of Relational Quantum Dynamics
quant-ph 2019-11 conditional novelty 7.0 of 10

For noninteracting clock-system models with a continuous-spectrum clock, relational Dirac observables, the Page-Wootters formalism, and the symmetry-reduced Heisenberg picture are proven to be equivalent descriptions ...
Inner Product of Closed Universe from Gravitational Path-integral
hep-th 2026-11 accept novelty 6.0 of 10

The gravitational path integral on a cylinder yields an exact physical inner product for closed universes in JT gravity, with co-invariant and invariant Hilbert spaces shown to be equivalent.
Exact DeWitt kernels in Lorentzian quantum cosmology
gr-qc 2026-09 accept novelty 6.0 of 10

DeWitt kernels in minisuperspace quantum cosmology are evaluated exactly for closed, flat, and open de Sitter models using closed-form integrals of Airy functions and elementary functions.

Show 13 more citing papers

A Holographic Map from AdS$_3$ to CFT$_2$
hep-th 2026-07 conditional novelty 6.0 of 10

Semiclassical pure AdS3 gravity states, labelled by fixed-area geodesic networks, are mapped to CFT2 primary states whose wavefunctions are networks of OPE coefficients.
On the relation between perspective-neutral, algebraic, and effective quantum reference frames
quant-ph 2025-07 conditional novelty 6.0 of 10

For ideal quantum reference frames with a single constraint, the perspective-neutral, algebraic, and effective semiclassical approaches describe the same physics and the same frame-switching rules.
Observers and Timekeepers: From the Page-Wootters Mechanism to the Gravitational Path Integral
hep-th 2025-06 conditional novelty 6.0 of 10

Metric summation produces the problem of time, topology summation produces the one-dimensional Hilbert space, and a timekeeper path integral gives observer-dependent holography.
Nonperturbative Quantum Gravity in a Closed Lorentzian Universe
hep-th 2025-05 conditional novelty 6.0 of 10

In a closed universe with a one-dimensional Hilbert space per alpha-microstate, tracing out unobserved environmental degrees of freedom yields robust, einselected probabilities with errors exponentially suppressed by ...
Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy
hep-th 2024-12 accept novelty 6.0 of 10

Gravitational subregion entropy is observer-dependent: different quantum clocks produce different von Neumann algebras and different entropy functionals for the same global state.
Gravitational entropy is observer-dependent
hep-th 2024-04 conditional novelty 6.0 of 10

The entanglement entropy of a spacetime subregion in quantum gravity is observer-dependent: different clock quantum reference frames assign different entropies to the same region and global state.
Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information
hep-th 2020-02 accept novelty 6.0 of 10

Spacetime wormholes make the dimension of the dual Hilbert space a random variable, and the resulting null states enforce a Bekenstein-Hawking entropy bound in every member of the ensemble.
A change of perspective: switching quantum reference frames via a perspective-neutral framework
quant-ph 2018-09 accept novelty 6.0 of 10

Choosing a quantum reference frame is a gauge fixing and switching frames is a symmetry transformation, with Dirac quantization as the frame-neutral bookkeeping and reduced quantization as a single frame's view.
Observables in effective gravity
hep-th 2005-12 accept novelty 6.0 of 10

Diffeomorphism-invariant observables in effective gravity can approximate local field theory observables in suitable states, but quantum and gravitational limits cap the resolution and the number of degrees of freedom...
Switching quantum reference frames in the N-body problem and the absence of global relational perspectives
quant-ph 2018-09 conditional novelty 5.0 of 10

For the translation- and rotation-invariant three-body problem, quantum reference frame changes can be built as 'quantum coordinate maps' through a perspective-neutral Hilbert space, even though no globally valid clas...
Partial observables
gr-qc 2001-10 conditional novelty 5.0 of 10

Splitting 'observable' into measurable-but-unpredictable (partial) and predictable (complete) is claimed to resolve confusions about time in quantum mechanics and observables in general relativity.
Gravitational Hilbert spaces: invariant and co-invariant states, inner products, gauge-fixing, and BRST
hep-th 2025-09 unverdicted novelty 4.0 of 10

Reviews construction of physical inner products in canonical quantum gravity via group averaging and BRST formalism, illustrated in mini-superspace models and connected to path integrals.
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

57 later papers cite it

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2 later papers by other authors are listed below: 2 used its method or tool.

Uses its method or tool 2 of 21 papers with this label

We can find solutions by group averaging [ref] via the projector [equation] where we used that when $[Â,B̂] = 0$ the identity $e^ÂB̂ = e^Â e^B̂$ holds such that we can project on each constraint individually by $δ_Φ_i$.

Quantum reference frames: derivation of perspective-dependent descriptions via a perspective-neutral structure (2021) · Mathematical preliminaries · 1.00
To this end, we employ group averaging (or refined algebraic quantization) [ref] and define an (improper) projector onto solutions of ([ref]): [equation] Projecting an arbitrary state of $^kin$ in momentum representation, |ϕ⟩^kin=∫ dp_A dp_B dp_C ϕ^kin(p_A,p_B,p_C) |p_A⟩|p_B⟩|p_C⟩ ,a general solution becomes, depending on which particle's momentum is solved for, |ϕ⟩^phys = ∫d p_B dp_C ϕ_BC|A(p_B,p_C) |-p_B-p_C⟩_A |p_B⟩_B |p_C⟩_C = ∫d p_A dp_C ϕ_AC|B(p_A,p_C) |p_A⟩_A |-p_A-p_C⟩_B |p_C⟩_C = ∫d p_A dp_B ϕ_AB|C(p_A,p_B) |p_A⟩_A |p_B⟩_B |-p_A-p_B⟩_C . where for later use we have defined ϕ_BC|A(p_B,p_C) := ϕ^kin(-p_B-p_C,p_B,p_C) , ϕ_AC|B(p_A,p_C) := ϕ^kin(p_A,-p_A-p_C,p_C) , ϕ_AB|C(p_A,p_B) := ϕ^kin(p_A,p_B,-p_A-p_B) .All three lines in ([ref]) give different descriptions of the same physical state $|ϕ⟩^phys$ and we shall exploit this below.

A change of perspective: switching quantum reference frames via a perspective-neutral framework (2018) · Dirac quantization – the perspective-neutral quantum theory · 1.00

How the literature uses it

57 arXiv papers cite this in 88 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 · 50 sentences without a confident label

Uses its method or tool 21 papers

It is possible to construct a physical Hilbert space for all dynamical fields, which is based on the Rieffel induced inner product [ref] and on which quantum Dirac observables act as operators and evolve unitarily beyond the semiclassical regime.

Construction of quantum Dirac observables and the emergence of WKB time · Conclusions · 0.89
The method combines refined algebraic [ref] and master constraint [ref] quantization and results in a physical Hilbert space that can be expressed as a $L^2$ Hilbert space over a continuous (momentum) parameter $ρ$ [ref] .

Can chaos be observed in quantum gravity? · Quantization (Refined Topology) · 0.89
Using group averaging [ref] , the post– and pre–physical states can be constructed from kinematical states $ψ^kin_n∈^kin_n$ via an improper projection [ref] ^+ψ^phys_n=^+ℙ_n ψ^kin_n,^-ϕ^phys_n=^-ℙ_n ϕ^kin_n, with post– and pre–projector ^+ℙ_n:=∏_I=1^kδ(^+Ĉ^n_I),^-ℙ_n:=∏_J=1^lδ(^-Ĉ^n_J), where δ(Ĉ)=12πħ∫_ℝds e^isĈ/ħ.

Quantization of systems with temporally varying discretization II: Local evolution moves · Global evolution moves in the quantum theory · 0.98

Cites it as background 13 papers

We shall not worry about this last issue here (for a general discussion of group averaging, see [ref] ).

Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces · Group averaging · 0.81
The reader may consult the by now extensive literature (e.g. [ref] ) for more details.

A global picture of quantum de Sitter space · de Sitter invariant states · 0.99
For instance, in recent years, several methods for implementing the Dirac approach have been developed, such as the refined algebraic quantization (RAQ) [ref] and coherent state quantization [ref] .

On the geometry of quantum constrained systems · Introduction · 0.83

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