Reduced Thiemann Coherent States

• Reduced Thiemann coherent states are specialized adaptations of Hall–Thiemann states achieved through graph truncations, symmetry impositions, and finite-dimensional reductions.
• Key methodologies include fixed-graph constructions, symmetry-adapted cosmological models, and an exact SU(1,1) quantization in the FLRW complexifier framework.
• These states exhibit emergent semiclassical behavior with distinct coherent-state operators and asymptotic equivalence to heat-kernel states under large-parameter limits.

Reduced Thiemann coherent states denote a family of constructions derived from, adapted from, or compared with the Hall–Thiemann heat-kernel coherent states of loop quantum gravity in settings where the graph, the phase space, the label set, or the Hilbert space has been reduced. The phrase does not identify a single standardized object. In the literature it covers, at minimum, fixed-graph Hall–Thiemann states and their coherent-state operators, symmetry-adapted cosmological subfamilies on cubic graphs, an exact $SU(1,1)$ coherent-state quantization of the reduced FLRW complexifier–volume–Hamiltonian algebra, superposition-type states in all-dimensional $SO(D+1)$ LQG that are asymptotically equivalent to Thiemann heat-kernel states in a large-$\eta$ regime, and QRLG-projected one-edge heat-kernel packets that emerge numerically in a near-kernel sector of a reduced Hamiltonian (Alesci et al., 2015, Achour et al., 2017, Dapor et al., 2017, Long et al., 2021, Mäkinen et al., 18 May 2026).

1. Terminological scope and principal meanings

The literature uses “reduced” in several technically distinct senses. In one line of work, the reduction is a fixed-graph truncation: one keeps the standard Hall–Thiemann coherent states but only on the Hilbert space of a chosen graph. In another, the reduction is kinematical and symmetry-adapted: one imposes homogeneity and isotropy directly on the labels of coherent states living on a regular cubic graph. A third usage is a genuine minisuperspace reduction: the full-theory complexifier is reduced to a finite-dimensional FLRW observable, and the quantum theory is reorganized exactly as an $SU(1,1)$ system. A fourth usage is asymptotic and label-based: one replaces heat-kernel states by superposition states adapted to generalized twisted-geometry data and shows large-$\eta$ consistency with the Thiemann family. A fifth usage is QRLG-specific: one projects Hall–Thiemann states to the reduced sector and obtains effective $U(1)$-type packets in the reduced spin basis (Alesci et al., 2015, Achour et al., 2017, Dapor et al., 2017, Long et al., 2021, Mäkinen et al., 18 May 2026).

Setting	Sense of reduction	Relation to Thiemann coherent states
Fixed graph LQG	Graph truncation	Same Hall–Thiemann family on $\Gamma$
Flat FLRW with scalar	Reduced phase space	Reduced $SU(1,1)$ analogue
Cubic-graph cosmology	Symmetry-adapted labels	Subfamily of standard gauge coherent states
All-dimensional $SO(D+1)$ LQG	Label reduction, large-$\eta$ asymptotics	Alternative family asymptotically consistent
One-vertex QRLG	Reduced-sector projection	Projected heat-kernel packets

A recurrent misconception is to treat all of these constructions as equivalent. They are not. Some are literal subfamilies of Hall–Thiemann states; some are only asymptotically equivalent; some are reduced analogues rather than the original full-theory construction.

2. Fixed-graph Hall–Thiemann coherent states as the baseline construction

On a single link, the classical phase space is

$SO(D+1)$0

and the paper on coherent-state operators further identifies it with $SO(D+1)$1. The Hilbert space is

$SO(D+1)$2

and the Hall–Thiemann heat-kernel coherent state labeled by $SO(D+1)$3 is

$SO(D+1)$4

Its spin-network components are

$SO(D+1)$5

and the states resolve the identity with

$SO(D+1)$6

where

$SO(D+1)$7

For a graph $SO(D+1)$8 with $SO(D+1)$9 links, the coherent states are tensor products of the single-link states. This is the principal sense in which the construction is reduced in that work: it is graph-fixed and not cylindrically consistent. Gauge-invariant graph states are obtained by averaging over $\eta$0 at the nodes. The same framework yields a coherent-state quantization map for graph-phase-space functions,

$\eta$1

This fixed-graph baseline is important because later reduced constructions either specialize its labels or compare themselves against it. It also establishes a characteristic feature of coherent-state operators: they need not coincide exactly with the canonical operators. The coherent-state holonomy differs from the canonical multiplication operator by a nontrivial factor $\eta$2; the coherent-state flux satisfies

$\eta$3

with $\eta$4 at large $\eta$5. The coherent-state holonomy–flux commutator therefore differs from the canonical holonomy–flux algebra away from the large-spin regime. Area, angle, and volume operators can also be defined by coherent-state quantization. In particular, the coherent-state area operator is diagonal, differs from the canonical $\eta$6 spectrum at low spin, and has $\eta$7 because positivity of the classical function is inherited by the quantized operator (Alesci et al., 2015).

3. Reduced FLRW complexifier and the $\eta$8 coherent-state analogue

In flat, homogeneous, isotropic FLRW gravity coupled to a massless scalar field, the complexifier survives symmetry reduction as an explicit finite-dimensional observable. In Ashtekar–Barbero variables,

$\eta$9

one introduces

$SU(1,1)$0

The Hamiltonian constraint becomes

$SU(1,1)$1

The reduced Thiemann complexifier is

$SU(1,1)$2

equivalently

$SU(1,1)$3

It generates dilatations,

$SU(1,1)$4

and in $SU(1,1)$5 variables its flow implements a rescaling of the Barbero–Immirzi parameter. A key point emphasized in the reduced model is that $SU(1,1)$6 disappears from the classical phase space and Hamiltonian when one works with $SU(1,1)$7, whereas the loop regularization scale $SU(1,1)$8 enters through polymerization; the paper therefore distinguishes sharply between $SU(1,1)$9 and $\eta$0.

The central algebraic structure is the reduced CVH algebra generated by the complexifier, volume, and Hamiltonian. For the gravitational sector,

$\eta$1

and the algebra closes. It can be rewritten as $\eta$2, with the Hamiltonian as a null generator and the complexifier as a boost. In the loop-regularized theory the polymer rule

$\eta$3

leads to

$\eta$4

Because $\eta$5 is not a good operator, the regularized complexifier is defined structurally by

$\eta$6

To close the full regularized CVH algebra with matter, the paper further introduces

$\eta$7

so that $\eta$8 acts as a true dilatation on $\eta$9.

The reduced quantum theory is then an exact $U(1)$0 group quantization. The coherent states are Perelomov $U(1)$1 coherent states. In this precise reduced setting, they function as a cosmological analogue of complexifier coherent states: they are adapted to the reduced CVH algebra rather than to the full holonomy–flux phase space. Their covariance is exact,

$U(1)$2

their expectation values reproduce the classical reduced observables, and they are described as “semi-classical states with minimal spread.” The regularized complexifier becomes a Hermitian operator $U(1)$3, generates unitary transformations

$U(1)$4

and implements unitary Immirzi transformations in the reduced quantum theory. When the scalar field is used as a clock, the deparametrized Hamiltonian is the complexifier itself: $U(1)$5 classically, and

$U(1)$6

in the fully regularized model. The reduced coherent states are therefore organized simultaneously by cosmological evolution and by complexifier-generated canonical transformations (Achour et al., 2017).

4. Symmetry-adapted cosmological gauge coherent states on a cubic graph

A different notion of reduced Thiemann coherent states arises from a fixed regular cubic graph embedded in a compact 3-torus. Here the states are not new coherent states distinct from the Hall–Thiemann family; they are a cosmological subfamily of the standard gauge coherent states, obtained by imposing homogeneity and isotropy directly on the graph labels. The graph has $U(1)$7 vertices and spacing

$U(1)$8

The continuum isotropic data are

$U(1)$9

For an edge $\Gamma$0 of coordinate length $\Gamma$1 and its dual plaquette $\Gamma$2,

$\Gamma$3

Substituting these data into the standard left-polar decomposition of the Hall–Thiemann states yields the $\Gamma$4 labels

$\Gamma$5

Equivalently,

$\Gamma$6

The full cosmological state is the tensor product over all edges of the cubic graph, with labels identical on edges in the same fiducial direction and related by rigid rotations among directions. This is a kinematical, graph-based reduction rather than a continuum symmetry reduction before quantization.

The resulting states are peaked on the isotropic connection and triad data. The expectation value of the fundamental holonomy reproduces the classical holonomy $\Gamma$7 at leading order, and the expectation value of the right-invariant vector field reproduces the classical flux label $\Gamma$8. The volume expectation value also has the correct leading classical FRW behavior for the lattice region. The semiclassicality parameter is

$\Gamma$9

and the normalization of the one-edge state is computed explicitly by Poisson summation. The paper further derives a systematic first-order calculus for expectation values of holonomy–flux polynomials, including a Giesel–Thiemann replacement for the Ashtekar–Lewandowski volume operator,

$SU(1,1)$0

for “good” coherent states.

These states are best described as symmetry-adapted or cosmological Thiemann coherent states. They are not ordinary LQC Gaussian states, and the reduction does not proceed by solving the full symmetry reduction in the continuum. The construction is designed to prepare the evaluation of the AQG-style non-graph-changing Hamiltonian in Thiemann regularization,

$SU(1,1)$1

with explicit Euclidean and Lorentzian parts. The paper itself remains kinematical: it builds the states and the expectation-value technology, while the decisive dynamical comparison with LQC is deferred to its sequel (Dapor et al., 2017).

5. Asymptotically reduced superposition states in all-dimensional $SU(1,1)$2 LQG

In all-dimensional $SU(1,1)$3 LQG, the phrase “reduced Thiemann coherent states” requires particular care. The superposition-type coherent states introduced there are not constructed by truncating Thiemann’s heat-kernel states to a smaller Hilbert space, nor are they a finite-parameter deformation of those states. They are an alternative coherent-state family built from simple $SU(1,1)$4 spin-network basis states, simple coherent intertwiners, and Gaussian superpositions over simple representation labels $SU(1,1)$5. On a graph $SU(1,1)$6,

$SU(1,1)$7

and after imposing edge simplicity strongly, the relevant representations are the simple irreps labeled by $SU(1,1)$8.

The gauge-variant superposition state is

$SU(1,1)$9

with a gauge-invariant analogue obtained by group averaging. The labels

$SO(D+1)$0

are generalized twisted-geometry variables. A central point is that these states are independent of the extra variables $SO(D+1)$1, which parametrize pure-gauge directions associated with the simplicity constraint. This is one precise sense in which they are reduced: the labels retain the twisted-geometry data relevant for discrete gravity and drop the simplicity-gauge directions.

Their relation to Thiemann coherent states is asymptotic. In the simple $SO(D+1)$2 sector, the heat-kernel state on one edge is

$SO(D+1)$3

with polar decomposition

$SO(D+1)$4

For large $SO(D+1)$5, the highest-weight contribution dominates, and the heat-kernel state reduces asymptotically to the superposition form. The identification is

$SO(D+1)$6

This is the precise sense in which the construction is consistent with Thiemann states: it is an alternative family whose large-$SO(D+1)$7 asymptotics reproduces the simple-sector heat-kernel states.

The paper also proves a resolution of identity for the superposition states and analyzes peakedness in a one-loop example. There,

$SO(D+1)$8

and the expectation values show that $SO(D+1)$9 plays the role of the area/flux label while $\eta$0 controls the holonomy phase peak. The states are therefore best described as twisted-geometry-adapted, asymptotically reduced counterparts of Thiemann heat-kernel coherent states, rather than as an exact reduced sector (Long et al., 2021).

6. QRLG projection and emergent near-kernel reduced Thiemann coherent states

A further development appears in the one-vertex model of quantum reduced loop gravity. The reduced Hilbert space is adapted to cuboidal graphs and diagonal triads aligned with three fiducial Cartesian directions. The basis states are

$\eta$1

with positive integer spins, reduced fluxes

$\eta$2

and reduced volume

$\eta$3

The symmetric Hamiltonian contains both Euclidean and Lorentzian contributions,

$\eta$4

and the variational problem minimizes

$\eta$5

in truncated spaces with

$\eta$6

In this setting, reduced Thiemann coherent states are introduced as QRLG-projected Hall–Thiemann states. Their one-edge coefficients are

$\eta$7

The parameter $\eta$8 controls the width distribution between momentum and holonomy representations, $\eta$9 controls localization in spin space together with $SO(D+1)$00, and $SO(D+1)$01 controls localization in holonomy space together with $SO(D+1)$02. These states are not part of the neural ansatz used in the variational calculation; they enter afterward as a diagnostic of the optimized near-kernel states.

The numerical analysis finds three qualitatively distinct classes of low-$SO(D+1)$03 states. At low cutoffs there are correlated, non-factorized states. At larger cutoffs there are two strongly factorized branches. One branch, found by the structured ansatz, is accurately fitted by products of reduced Thiemann coherent states. At representative cutoff $SO(D+1)$04, the one-edge fit fidelities are

$SO(D+1)$05

with fitted parameters

$SO(D+1)$06

$SO(D+1)$07

$SO(D+1)$08

The extracted factors are almost phase-flat and localized at substantially larger spins than the other factorized branch. The agreement extends beyond fidelity to the first two moments and effective phase slopes.

The second high-cutoff factorized branch, found with an MLP ansatz, is not well described by the same coherent-state family. For the same representative cutoff, the best-fit reduced coherent-state fidelities are only

$SO(D+1)$09

with $SO(D+1)$10, which the paper identifies as the lower edge of the admissible fit range. Its one-edge marginals are boundary-dominated near the smallest spins. The low-cutoff correlated branch is also distinct: at $SO(D+1)$11 it shows nontrivial inter-edge conditional structure and only partial factorization, with

$SO(D+1)$12

The significance of this QRLG result is restricted but concrete. The paper does not establish that reduced Thiemann coherent states solve the Hamiltonian constraint exactly, nor that they exhaust the near-kernel sector. What it establishes is that one distinguished large-cutoff factorized branch of low-$SO(D+1)$13 states is fitted with very high accuracy by QRLG-projected Hall–Thiemann packets. This is why the authors describe the coherent-state structure as emergent rather than imposed (Mäkinen et al., 18 May 2026).

Taken together, these constructions show that “reduced Thiemann coherent states” is best understood as a family resemblance term. Depending on context, the reduction may occur at the level of graph truncation, symmetry-adapted labeling, finite-dimensional phase-space reduction, simplicity-adapted label selection, or projection to a reduced Hilbert space. The resulting coherent states are not interchangeable, but all are organized by a common question: how much of the semiclassical content of Hall–Thiemann coherent states survives when one passes from full LQG to a reduced setting.