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Quantum mechanics
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In quantum gravity, quantum geometry is the set of mathematical concepts that generalize geometry to describe physical phenomena at distance scales comparable to the Planck length. Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion.

String theory uses quantum geometry to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing transitions, minimal possible distance scale, and other effects that challenge intuition. Generally, string theory is initially explored on a compact six-dimensional manifold to restrict the algebraic data needed for computation. By utilizes compactifications, string theory describes geometric states, where a compactification is a spacetime that looks four-dimensional macroscopically even if its actual dimension is higher. One goal in exploring string compactifications is to find vacuum solutions where the space is maximally symmetric.^[1]

When computing these vacuum solutions, preserving supersymmetry gives a first-order system of equations which can partially give the second-order equations of motion. This supersymmetry enables the use of differential geometry methods by using transition functions^{[disambiguation needed]}. Because the six-dimensional manifold cannot be covered with a single coordinate system, transition functions are grouped together into different G-structures. To define the G-structure, infinitesimal parameters for supersymmetry called spinors are introduced to enable stability during the transition.^[1] More technically, quantum geometry refers to the shape of a spacetime manifold as experienced by D-branes, which includes quantum corrections to the metric tensor, such as the worldsheet instantons. For example, the quantum volume of a cycle is computed from the mass of a brane wrapped on this cycle.^{[citation needed]}

In an alternative approach to quantum gravity called loop quantum gravity (LQG), the phrase "quantum geometry" usually refers to the formalism within LQG where the observables that capture the information about the geometry are well-defined operators on a Hilbert space. In particular, certain physical observables, such as the area, have a discrete spectrum. LQG is non-commutative.^[2] It is possible (but considered unlikely) that this strictly quantized understanding of geometry is consistent with the quantum picture of geometry arising from string theory.^{[citation needed]}

Another approach, which tries to reconstruct the geometry of space-time from "first principles" is Discrete Lorentzian quantum gravity.

References

[edit]

^ ^a ^b Tomasiello, Alessandro (2022). Geometry of String Theory Compactifications. University Printing House, Cambridge, United Kingdom: Cambridge University Press. ISBN 978-1-108-47373-6.
^ Ashtekar, Abhay; Corichi, Alejandro; Zapata, José A. (1998), "Quantum theory of geometry. III. Non-commutativity of Riemannian structures", Classical and Quantum Gravity, 15 (10): 2955–2972, arXiv:gr-qc/9806041, Bibcode:1998CQGra..15.2955A, doi:10.1088/0264-9381/15/10/006, MR 1662415, S2CID 250895945.

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Quantum geometry is a branch of theoretical physics that generalizes classical geometric concepts to incorporate quantum mechanical principles, addressing phenomena where spatial structures and distances are influenced by quantum effects at fundamental scales.^[1] In this framework, geometry is not continuous but quantized, leading to discrete spectra for quantities like area and volume, and it manifests in diverse areas including quantum gravity theories and the band structure of quantum materials.^[2] The core idea emerges from the need to reconcile general relativity's description of curved spacetime with quantum mechanics' probabilistic nature, resulting in non-commutative or discrete spatial frameworks.^[1] In the context of quantum gravity, quantum geometry plays a central role in loop quantum gravity (LQG), a non-perturbative approach to quantizing general relativity.^[1] Here, spacetime is represented by spin networks, where nodes and links encode quantized geometric operators; for instance, the area of a surface pierced by a link with spin quantum number

j

has eigenvalues

8\pi \gamma \ell_P^2 \sqrt{j(j+1)}

8πγℓP2j(j+1)

, with $\gamma$ the Barbero-Immirzi parameter and $\ell_P$ the Planck length.^[1] This discreteness implies that at the Planck scale ( $\approx 10^{-35}$ meters), space consists of indivisible "atoms" of geometry, resolving singularities in black holes and the Big Bang by replacing them with finite, quantum bounces.^[1] Foundational work in LQG, developed since the 1980s by researchers like Abhay Ashtekar and Carlo Rovelli, treats gravity as a gauge theory of SU(2) connections, yielding a background-independent quantization.^[1] In condensed matter physics, quantum geometry describes the intrinsic geometry of the Hilbert space of quantum states parameterized by continuous variables, such as momentum in the Brillouin zone of crystals.^[2] This is quantified by the quantum geometric tensor $Q_{\mu\nu}(\mathbf{k}) = \langle \partial_{\mu} u_{\mathbf{k}} | (1 - |u_{\mathbf{k}}\rangle \langle u_{\mathbf{k}} |) | \partial_{\nu} u_{\mathbf{k}} \rangle$ , where $|u_{\mathbf{k}}\rangle$ are Bloch wavefunctions; its imaginary part yields the Berry curvature $F_{\mu\nu} = 2 \Im Q_{\mu\nu}$ , linked to topological invariants like the Chern number, while the real part defines the quantum metric $g_{\mu\nu} = \Re Q_{\mu\nu}$ , measuring state overlap distances.^[3] Originating from Berry's 1984 phase concept and Thouless's 1982 explanation of the quantum Hall effect, it governs phenomena such as anomalous Hall conductivity, superfluid stiffness in superconductors (e.g., $D_s \propto \int g_{xx}$ ), and fractional Chern insulators in flat-band systems like magic-angle graphene.^[3] Advances as of 2025 include direct measurements of quantum geometry in solids via nonlinear optics, observations of hidden quantum geometries distorting electron paths, and mappings of its structure in materials, highlighting its role in engineering topological phases and quantum technologies.^[4]^[5]^[6] Beyond these domains, quantum geometry influences broader quantum field theories and even abstract constructs like positive geometries in scattering amplitudes, which encode particle interactions without reference to spacetime.^[7] Its interdisciplinary nature bridges mathematics and physics, with ongoing research exploring connections to non-commutative geometry and matrix models for emergent gravity.^[8]

Overview

Definition and scope

Quantum geometry refers to the quantization of spacetime geometry, where the smooth manifolds of classical general relativity are replaced by discrete or noncommutative structures emerging from quantum operators, particularly at the Planck scale of approximately

10^{-35}

meters.^[9] In this framework, geometry is not a fixed background but arises dynamically from the eigenvalues and eigenstates of operators representing physical quantities like area and volume, which exhibit discrete spectra rather than continuous ones.^[1] This approach stems from the broader quest in quantum gravity to reconcile general relativity's geometric description of gravity with quantum mechanics' principles of uncertainty and superposition.^[10] The scope of quantum geometry encompasses deformations of classical differential geometry to incorporate quantum effects, such as uncertainty principles that prevent simultaneous precise measurements of conjugate variables like holonomies and fluxes.^[9] Central to this is the holonomy-flux algebra, which provides the algebraic foundation for quantizing the phase space of general relativity without relying on a background metric, leading to operator-valued metrics and noncommutative spatial relations.^[10] These structures ensure background independence, allowing geometry to fluctuate quantum mechanically while maintaining diffeomorphism invariance.^[1] A core idea is that macroscopic geometry emerges from underlying quantum operators acting on Hilbert spaces of spin networks or similar discrete excitations, where area and volume operators yield quantized eigenvalues that scale with the Planck area and volume.^[9] In the classical limit, as the scale increases far beyond the Planck regime, these quantum geometries recover the smooth Riemannian metrics of general relativity through a process of coarse-graining, where discrete levels densely approximate continuous values.^[10] However, at quantum scales, inherent fluctuations and discreteness introduce deviations, such as a minimal nonzero area gap, fundamentally altering the notion of spacetime continuity.^[1]

Motivations from quantum gravity

Quantum geometry emerges as a central concept in efforts to reconcile general relativity with quantum mechanics, addressing the fundamental incompatibility between the two theories at the Planck scale. General relativity describes spacetime as a smooth, continuous manifold, but quantum mechanics implies that at distances around the Planck length (

\ell_P \approx 1.6 \times 10^{-35}

m), spacetime should exhibit quantum fluctuations and discreteness, leading to a breakdown of classical geometry.^[11] This tension motivates the development of quantum geometry, where geometric quantities like area and volume become operators with discrete spectra, providing a framework for a non-perturbative quantization of gravity.^[1] A primary driver in loop quantum gravity (LQG) is the resolution of singularities in classical general relativity, such as those in black holes and the Big Bang, where curvature diverges and predictability fails. In LQG, quantum geometry quantizes the phase space of general relativity using Ashtekar variables, which reformulate the theory in terms of SU(2) connections and triads, akin to Yang-Mills gauge theories. This leads to holonomy-flux algebras, where geometric operators, like the area operator, have eigenvalues

A = 8\pi \gamma \ell_P^2 \sqrt{j(j+1)}

A=8πγℓP2j(j+1)

for spin quantum number $j$ , indicating a granular structure at the Planck scale that smears out singularities. Seminal work by Ashtekar introduced these variables to simplify constraints and enable canonical quantization.^[12] Rovelli and Smolin's loop representations provided the kinematical arena for quantum states of geometry.^[13] In noncommutative geometry (NCG), the motivation stems from incorporating quantum effects directly into the structure of spacetime, treating it as a noncommutative algebra rather than a classical manifold to couple gravity with quantum fields consistently. Alain Connes' framework uses spectral triples to define geometry via Dirac operators, allowing quantum gravity models where the metric is deformed at short distances, avoiding ultraviolet divergences in quantum field theory on curved spaces. This approach is particularly motivated by the need to unify gravity with the Standard Model, where noncommutativity resolves issues in reconciling diffeomorphism invariance with quantum symmetries, as explored in Connes' foundational papers on gravity coupled to matter.^[14]

Historical development

Precursors in the 20th century

In the mid-20th century, physicist John Archibald Wheeler pioneered early efforts to quantize gravity through geometrodynamics, a framework developed in the 1950s and 1960s that treats gravitational phenomena as the pure dynamics of three-dimensional spatial geometries, or "shapes," without a fixed background metric. This approach emphasized the relational nature of spacetime, where physical laws emerge from the evolution of geometric configurations rather than absolute coordinates.^[15] Central to Wheeler's vision was the 1955 concept of "quantum foam," which posits that at the Planck scale—approximately

10^{-35}

meters—spacetime undergoes turbulent quantum fluctuations, rendering it a frothy, fluctuating structure rather than smooth geometry. These ideas highlighted the need to reconcile quantum uncertainty with gravitational curvature, foreshadowing challenges in quantum geometry.^[16] Influences from quantum field theory further shaped precursors to quantum geometry. In the 1970s, Stephen Hawking employed path integrals to study quantum fields propagating in curved spacetime, enabling calculations of phenomena like black hole evaporation and demonstrating how quantum effects could alter gravitational backgrounds.^[17] Concurrently, the discovery of conformal anomalies in the early 1970s revealed that quantum fields in curved spaces violate classical conformal symmetries, producing trace anomalies in the energy-momentum tensor that depend on spacetime curvature invariants.^[18] On the mathematical front, Paul Dirac's 1950 formulation of generalized Hamiltonian dynamics provided a systematic method for quantizing constrained systems, addressing the first-class constraints arising from gauge symmetries like diffeomorphisms in general relativity.^[19] Symplectic geometry, integral to Hamiltonian mechanics since its classical roots, supplied the phase space structure needed for such quantizations, enabling the preservation of Poisson brackets under quantization.^[20] Early hints of noncommutativity also emerged from Werner Heisenberg's 1927 uncertainty principle, which imposes fundamental limits on simultaneous measurements of position and momentum, naturally extending to suggest that spacetime coordinates themselves might not commute at quantum scales, as later explored in noncommutative frameworks.^[21] General relativity's portrayal of gravity as spacetime curvature thus motivated these diverse threads toward a quantum theory of geometry.

Key advancements since 1980

In the 1980s, a pivotal advancement came with Abhay Ashtekar's introduction of new canonical variables in 1986, reformulating general relativity as an SU(2) gauge theory akin to Yang-Mills theories, which simplified the constraints and paved the way for consistent quantization of gravity.^[22] The 1990s saw the emergence of loop quantum gravity, pioneered by Carlo Rovelli and Lee Smolin, who in 1990 established a rigorous loop space representation for the quantum states of general relativity, deriving exact solutions to the Wheeler-DeWitt equation without relying on a fixed background.^[23] Building on this, Rovelli and Smolin quantumized spin networks—combinatorial structures originally proposed by Roger Penrose in 1971 for discrete quantum geometry—into a complete, diffeomorphism-invariant basis for the Hilbert space of quantum gravity in 1995, resolving issues with overcompleteness in earlier loop representations. Independently, Alain Connes formalized noncommutative geometry in his 1990 monograph (French edition; English 1994), replacing classical manifolds with spectral triples from operator algebras to model quantum spaces where coordinates fail to commute, influencing approaches to quantum spacetime. During the 2000s, spin foam models provided a path-integral formulation for the dynamical evolution of quantum geometry, extending loop quantum gravity to a sum over histories of spin networks; key progress included Alejandro Perez's 2003 review of covariant models and the 2008 EPRL model by Engle, Pereira, Rovelli, Livine, and others, which imposed simplicial geometry constraints while preserving anomaly-free quantization.^[24] The AdS/CFT correspondence, conjectured by Juan Maldacena in 1997, gained traction in quantum geometry post-2000 through holographic dualities, notably the 2006 Ryu-Takayanagi formula linking entanglement entropy in boundary conformal field theories to minimal surfaces in bulk anti-de Sitter spacetime, illuminating emergent geometric structures from quantum information. From the 2010s to 2025, quantum geometry increasingly intertwined with holography and tensor networks, modeling spacetime emergence via entanglement; for instance, 2022 tensor network constructions of AdS/qCFT dualities demonstrated fractional central charges and renormalization group flows approximating quantum critical points in holographic settings.^[25] In 2025, physicists achieved the first direct measurement of quantum geometry in solid-state materials using nonlinear optical techniques, providing experimental validation of theoretical predictions in quantum materials.^[4]

Mathematical foundations

Classical geometric prerequisites

Riemannian geometry provides the foundational framework for understanding curved spaces, essential for quantum geometry's deformations of classical structures. A Riemannian manifold is a smooth manifold equipped with a metric tensor

g_{\mu\nu}

, which defines an inner product on tangent spaces, allowing measurements of lengths, angles, and volumes.^[26] The metric induces a Levi-Civita connection, enabling parallel transport and the definition of geodesics as shortest paths satisfying the equation

\frac{d^2 x^\rho}{d\tau^2} + \Gamma^\rho_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = 0

, where

\Gamma^\rho_{\mu\nu}

are Christoffel symbols derived from the metric.^[27] Curvature is quantified by the Riemann tensor

R^\rho_{\sigma\mu\nu}

, which measures the deviation of geodesics from straight lines and encodes the intrinsic geometry, with contractions yielding the Ricci tensor and scalar curvature.^[26] Differential forms offer a coordinate-independent way to describe geometric objects on manifolds, underpinning integration and cohomology in bundle theory. On principal bundles, an Ehresmann connection specifies horizontal subspaces complementary to vertical fibers, facilitating parallel transport of sections along curves.^[28] Holonomy arises from the path dependence of this transport, represented by the holonomy group acting on fibers, which captures global topological features through the curvature form.^[29] These structures generalize affine connections to fiber bundles, where the connection form

\omega

is a Lie algebra-valued 1-form satisfying equivariance under the structure group.^[28] Symplectic geometry formalizes classical phase spaces, serving as a prerequisite for quantization procedures that replace Poisson structures with commutators. A symplectic manifold is equipped with a closed, non-degenerate 2-form

\omega

, such as the canonical form

\omega = dq \wedge dp

\mathbb{R}^{2n}

, preserving volume under Hamiltonian flows.^[30] The Poisson bracket

\{f, g\} = \omega(X_f, X_g)

, where

X_f

is the Hamiltonian vector field satisfying

df = \iota_{X_f} \omega

, governs the time evolution via Hamilton's equations

\dot{f} = \{f, H\}

.^[30] This bracket structure anticipates Dirac's quantization rule, mapping classical observables to quantum operators. In general relativity, these geometric tools culminate in the Einstein field equations

G_{\mu\nu} = 8\pi T_{\mu\nu}

, where

G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}

relates spacetime curvature to matter-energy content, formulated on a pseudo-Riemannian manifold.^[31] Tetrads, or vielbeins

e^a_\mu

, provide local orthonormal frames bridging Lorentzian geometry with spinors, while spin connections

\omega^a_{b\mu}

ensure covariance under local Lorentz transformations, incorporating torsion-free conditions in standard GR.^[32] These fail at Planck scales, where quantum effects demand non-commutative or discrete alternatives.^[28]

Quantum algebraic structures

In quantum geometry, the algebraic structures underpinning the theory replace classical commutative geometries with noncommutative operator algebras that encode geometric observables as self-adjoint elements. C*-algebras serve as the primary framework for quantizing bounded observables, providing a norm-closed -algebra of operators on a Hilbert space that satisfies the C-identity

\|a^* a\| = \|a\|^2

for all elements

a

. This structure ensures algebraic consistency in the representation of physical quantities, such as distances and volumes, as operators whose spectra yield quantized eigenvalues. Von Neumann algebras extend this to include weakly closed subalgebras of bounded operators, enabling the treatment of unbounded observables through the bicommutant theorem, which characterizes them as the double commutant of a *-algebra acting on the Hilbert space. The spectral theorem for self-adjoint operators in von Neumann algebras is central, asserting that any self-adjoint operator

A

admits a unique spectral resolution

A = \int_{-\infty}^{\infty} \lambda \, dE(\lambda)

, where

E

is a projection-valued measure supported on the spectrum of

A

.^[33] This decomposition allows geometric operators, like those for area or curvature, to have discrete spectra in quantum settings, reflecting the granular nature of spacetime at the Planck scale. In the commutative limit, these algebras reduce to functions on classical manifolds, recovering standard differential geometry. Noncommutative algebras form the core of quantum geometric deformations, generalizing the Heisenberg-Weyl relations

[x, p] = i \hbar

—which underlie position-momentum uncertainty—to spacetime coordinates via

[x^\mu, x^\nu] = i \theta^{\mu\nu}

, where

\theta^{\mu\nu}

is an antisymmetric constant tensor with dimensions of length squared. This relation, first systematically explored in the context of string theory and quantum field theory on noncommutative spaces, implies that coordinates fail to commute at short distances, leading to a fuzzy spacetime structure parameterized by

\theta

. The algebra is realized through star-product deformations of the Moyal plane, where functions multiply via

f \star g = f e^{i \theta^{\mu\nu} \overleftarrow{\partial}_\mu \overrightarrow{\partial}_\nu / 2} g

f⋆g=feiθμν∂

μ∂

ν/2g, ensuring covariance under deformed Poincaré transformations.^[34] In specific quantum gravity frameworks like loop quantum gravity (LQG), the kinematical Hilbert space provides the arena for these algebraic structures, comprising square-integrable cylindrical functions over the space of Ashtekar-Barbero connections $\overline{\mathcal{A}}$ . Cylindrical functions are wave functions $\psi_\gamma[\overline{A}]$ depending on holonomies $h_e(\overline{A})$ along finitely many edges $e$ of a graph $\gamma$ embedded in the spatial manifold, forming a dense subspace completed with respect to the Ashtekar-Lewandowski measure $d\mu_{\mathrm{AL}}$ , which is diffeomorphism-invariant. Formally, the space is $\mathcal{H}_{\mathrm{kin}} = L^2(\overline{\mathcal{A}}, d\mu_{\mathrm{AL}}) = \overline{\bigcup_\gamma \mathrm{Cyl}_\gamma}^{\|\cdot\|_{\mathrm{AL}}}$ , where $\mathrm{Cyl}_\gamma = \mathrm{Span}\{ \sqrt{\dim} \chi_{j_e, m_e}(h_e) \mid j_e \in \frac{1}{2}\mathbb{N}_0, m_e = -j_e, \dots, j_e \}$

χje,me(he)∣je∈21N0,me=−je,…,je} uses SU(2) Wigner matrices contracted over intertwiners.^[35] This construction ensures background independence, as states are gauge-invariant under SU(2) transformations and transform covariantly under diffeomorphisms. A key example of quantized geometry in this framework is the spectrum of the area operator $A(S)$ for a surface $S$ , derived from the holonomy-flux algebra. Classically, the area is $A(S) = \int_S \sqrt{\det q_{ab}} \, d^2\sigma$

d2σ, but in LQG variables, it involves the densitized triad fluxes $P^i(S,f) = \int_S f_i E^i$ , with $E^i = \det(e) e^{i a} \frac{\delta}{\delta \overline{A}_a^i}$ as momentum operators conjugate to connections. The operator is constructed as $A(S) = 8\pi \gamma \ell_P^2 \sum_I \sqrt{j_I(j_I + 1)}$

, where the sum is over intersection points (punctures) of spin-network edges with $S$ , labeled by half-integer spins $j_I$ , and $\gamma$ is the Immirzi parameter regulating the self-dual decomposition of the spin connection.^[36] The derivation proceeds by considering the action on cylindrical states. For a single puncture with holonomy $h_l$ along an edge $l$ piercing $S$ at point $p$ , the flux $P^i(S)$ acts via the Lie algebra su(2), generating infinitesimal rotations. The squared flux operator $|\vec{P}(S)|^2 = \sum_i P^i(S) P^i(S)$

(S)∣2=∑iPi(S)Pi(S) diagonalizes in the spin representation, yielding the Casimir eigenvalue $j(j+1)$ from the right-invariant Maurer-Cartan form $\tau_i = h_l^{-1} (\tau_i / 2) dh_l$ , where $\tau_i$ are Pauli matrices.^[36] Specifically, on a state $|j, m\rangle$ transformed to the tangent frame at $p$ , $P^i(S) |j, m\rangle = \pm \sqrt{j(j+1) - m(m \pm 1)} |j, m \pm 1\rangle$

∣j,m±1⟩ for the raising/lowering components, but the invariant norm $\sqrt{|\vec{P}|^2}$

∣2

gives $\sqrt{j(j+1)}$

as the eigenvalue, independent of $m$ . For multiple punctures, the total area is the sum over independent contributions, assuming orthogonality of edges, with the spectrum consisting of discrete multiples of the Planck area scaled by $\gamma$ . The Immirzi parameter $\gamma$ enters via the Poisson bracket $\{A_a^i, E^b_j\} = 8\pi \gamma G \delta^b_a \delta^i_j \delta^{(3)}$ , fixing the quantization scale and later constrained by black hole entropy matching. This discreteness, with minimal non-zero eigenvalue $A_{\min} = 8\pi \gamma \ell_P^2 \frac{\sqrt{3}}{2}$

for $j=1/2$ , underscores the atomicity of quantum geometry.^[36]

Major approaches

Noncommutative geometry

Noncommutative geometry offers a mathematical framework for describing quantum spaces where classical notions of points and coordinates are replaced by noncommuting algebraic structures, enabling a synthesis of geometry with quantum theory. Developed primarily by Alain Connes, this approach treats spacetime as a noncommutative algebra, capturing quantum effects through spectral properties that generalize Riemannian geometry to operator algebras.^[37] At the heart of this framework lies the spectral triple

( \mathcal{A}, \mathcal{H}, D )

, consisting of a unital involutive

C^*

-algebra

\mathcal{A}

representing coordinates, a Hilbert space

\mathcal{H}

of fermionic degrees of freedom such as spinors, and a self-adjoint unbounded operator

D

(the Dirac operator) encoding the metric structure. The triple satisfies axioms ensuring compatibility: the representation of

\mathcal{A}

\mathcal{H}

is faithful and nondegenerate, the commutators

[D, a]

for

a \in \mathcal{A}

are bounded operators, and an antilinear isometry

J

provides a real structure with appropriate commutation relations to incorporate orientation and charge conjugation. This setup allows the Dirac operator

D

to dictate both the geometry and the quantum dynamics, with the spectrum of

|D|

determining distances and volumes in the noncommutative setting.^[37] The reconstruction theorem asserts that, under suitable conditions on the spectral triple, the underlying smooth manifold

M

and its metric can be uniquely recovered from the algebraic data. Specifically, the algebra

\mathcal{A}

arises as smooth functions on

M

, while the geometry emerges from the commutators: the metric on the cotangent bundle is defined via the principal symbol of

D

, and infinitesimal distances satisfy

ds^2 = [[D, f], [D, f]^\dagger]

for functions

f

, with the Connes distance between states

\phi, \psi

given by

d(\phi, \psi) = \sup \{ |\phi(f) - \psi(f)| : \|[D, f]\| \leq 1, f \in \mathcal{A} \}

, generalizing the geodesic distance in the classical limit. This theorem bridges commutative and noncommutative regimes, showing how classical geometry is a special case where

\mathcal{A} = C^\infty(M)

.^[37]^[38] In applications to physics, noncommutative geometry facilitates models of quantized spaces such as the noncommutative torus and the fuzzy sphere, where finite-dimensional approximations replace continuous manifolds to regulate ultraviolet divergences. The framework unifies gravity with Yang-Mills gauge fields through bimodule connections on projective modules over

\mathcal{A}

, which extend the Levi-Civita connection to incorporate curvature and torsion in a spectral action principle, yielding actions like

\mathrm{Tr} \chi(D/\Lambda)

that couple all interactions at high energies.^[37] A key example is the noncommutative torus, defined by the

C^*

-algebra generated by unitary operators

u, v

satisfying

uv = e^{i\theta} vu

for irrational

\theta

, which deforms the classical 2-torus and quantizes its area in units of

\theta

, leading to a spectral triple with

D

incorporating twisted derivations and enabling gauge theories with quantized magnetic fluxes.^[37]

Loop quantum gravity

Loop quantum gravity (LQG) is a background-independent approach to quantizing general relativity, reformulating the theory in terms of Ashtekar-Barbero variables to facilitate canonical quantization. The classical formulation begins with the Ashtekar-Barbero connection

A_a^i

, a real SU(2) connection incorporating the Barbero-Immirzi parameter

\gamma

, and the densitized triad

E_i^a

, which serves as the momentum conjugate to the connection. These variables cast general relativity as a gauge theory similar to Yang-Mills theory, with the phase space equipped with a Poisson bracket

\{A_a^i(x), E_j^b(y)\} = \gamma \kappa \delta_a^b \delta_j^i \delta^3(x-y)

, where

\kappa = 8\pi G

. The basic quantum variables are holonomies

h_\gamma(A) = \mathcal{P} \exp \left( \int_\gamma A \right)

along one-dimensional loops

\gamma

in space and flux operators

P_S(E) = \int_S E_i^a \, n_a \, \tau^i

, where

S

is a two-dimensional surface,

n_a

its normal, and

\tau^i

the SU(2) generators.^[22] The quantization proceeds by promoting these variables to operators on a Hilbert space consisting of square-integrable functions on the space of Ashtekar-Barbero connections modulo diffeomorphisms and SU(2) gauge transformations, denoted

L^2(\overline{\mathcal{A}}/G)

. Cylindrical functions on this space, defined by holonomies along finite graphs, form a dense domain, with the holonomy operators acting by multiplication and flux operators by Lie derivatives. The theory is regularized on lattices that refine under diffeomorphisms, leading to a spin network basis where states are labeled by graphs with edges carrying SU(2) representations (intertwiner spaces at nodes ensure gauge invariance). Diffeomorphism invariance is imposed projectively via the holomorphic measurable functions framework, yielding a rigorous kinematic Hilbert space with an orthonormal basis of diffeomorphism-invariant spin networks. The holonomy-flux algebra exhibits noncommutativity, with

[h_\gamma(A), P_S(E)] \neq 0

when the loop pierces the surface, encoding the quantum geometry. Dynamics are introduced by quantizing the Hamiltonian constraint, which generates time evolution in the canonical formalism. The quantized constraint acts on spin networks to produce new spin networks, but its implementation requires regularization; Thiemann's anomaly-free regularization yields an operator that is densely defined and preserves the diffeomorphism-invariant subspace. This leads to a path-integral formulation via spin foams, where transitions between spin network states are summed over two-complexes labeled by representations, providing a covariant dynamics. In cosmology, this resolves singularities, replacing the Big Bang with a Big Bounce due to repulsive quantum gravity effects at Planck densities. A key operator in LQG is the volume operator at a vertex

v

of a spin network, whose discrete spectrum quantifies the discrete quantum geometry. In the Ashtekar-Lewandowski formulation, it is given by

\hat{V}(v) = \ell_P^3 \sqrt{ \left| \sum_{I<J<K} \epsilon(IJK) \hat{q}_{IJK} \right| },

V^(v)=ℓP3

I<J<K∑ϵ(IJK)q^IJK

, where $\hat{q}_{IJK} = [(J_{IJ})^2, (J_{JK})^2]$ with $(J_{IJ})^2 = \sum_k (J_k^I + J_k^J)^2$ the squared angular momentum operators associated to pairs of edges, and $\epsilon(IJK)$ the sign from edge orientations; the constant factor is of order 1. The eigenvalues scale as $\ell_P^3 \sqrt{j_1 j_2 j_3}$

for simple configurations with three edges of spins $j_1, j_2, j_3$ , implying that spatial volume is quantized in units of order $\ell_P^3$ . The operator is self-adjoint, positive semi-definite, and the full theory remains anomaly-free under this quantization, preserving classical symmetries at the quantum level without ad hoc additions.^[39]

String theory and mirror symmetry

In string theory, the extra six spatial dimensions are compactified on Calabi–Yau threefolds to yield a four-dimensional theory while preserving N=2 supersymmetry.^[40] These Ricci-flat Kähler manifolds ensure the absence of a scalar curvature term in the effective action, maintaining supersymmetry at the compactification scale. The geometry of the Calabi–Yau determines key features of the low-energy physics, including the spectrum of massless modes and interaction strengths. The moduli space of these compactifications consists of complex structure deformations, governed by the Kähler potential constructed from the periods of the holomorphic three-form over homology cycles, and Kähler deformations, encoded in the potential derived from the triple intersection numbers and volumes of divisor classes. Stabilizing these moduli against quantum fluctuations is crucial for phenomenological viability; for instance, non-perturbative effects like gaugino condensation or D-brane instantons generate potentials that fix the Kähler moduli, preventing runaway behavior in the effective theory. Mirror symmetry provides a profound duality in this framework, relating two Calabi–Yau threefolds whose compactifications yield physically equivalent N=2 superconformal field theories, despite distinct topologies.^[41] Originating from T-duality in the type II string, it exchanges the roles of Kähler and complex structure moduli: variations in the Kähler form of one manifold map to deformations of the complex structure on its mirror, and vice versa, leading to swapped Hodge numbers

h^{1,1}

and

h^{2,1}

. A concrete example is the two-torus

T^2

, whose mirror under T-duality is another torus with inverted complex structure parameter

\tau \to -1/\tau

. The Strominger–Yau–Zaslow conjecture geometrizes this duality by proposing that both manifolds in a mirror pair admit special Lagrangian torus fibrations over a common base, with the fibers interchanged via T-duality along the three torus directions.^[42] This fibration structure resolves singularities in the moduli space near conifold points and provides a semi-classical understanding of the duality, though full mathematical realization remains an active area of research for generic Calabi–Yau examples. Quantum corrections enrich the geometric structure, with worldsheet instantons wrapping holomorphic curves contributing non-perturbative terms to the effective potential and couplings. In particular, these instantons generate suppressed contributions to Yukawa couplings between chiral matter fields, filling gaps in the perturbative spectrum and enabling realistic hierarchies in model-building.^[43] For instance, in type II orientifolds, Euclidean D-brane instantons distant from matter branes can induce such terms via disk worldsheet diagrams intersecting at intersection points. D-brane configurations further probe quantum geometry, where stacks of identical branes support non-Abelian gauge groups, and non-Abelian Wilson lines—constant adjoint-valued gauge fields along the brane worldvolume—parametrize the fuzzy collective coordinates in the transverse Calabi–Yau directions. These lines deform the brane geometry into non-commutative bundles, stabilizing positions against quantum fluctuations and influencing open-string spectra. A central algebraic structure is the quantum cohomology ring of the Calabi–Yau, where the classical cup product is deformed by three-point Gromov–Witten invariants, which virtually count rational holomorphic curves of given degree joining Poincaré dual cycles.^[44] The multiplication in this ring, defined as

\alpha \star \beta = \sum_d q^d \mathrm{GW}_{A,B,C}^d \gamma,

where

\alpha, \beta \in H^*(X, \mathbb{Q})

d

is the curve degree, and

\mathrm{GW}

are the invariants, encodes enumerative geometry and matches the A-model topological string correlators on the mirror side. This ring structure captures quantum corrections to geodesic flows on the moduli space, linking holomorphic curve counts to physical amplitudes in string compactifications.

Discrete and causal methods

Discrete and causal methods in quantum geometry emphasize the discretization of spacetime through structures that preserve causality, aiming to reconcile quantum principles with gravitational dynamics without relying on continuous backgrounds. These approaches treat spacetime not as a smooth manifold but as a collection of discrete elements ordered by causal relations, enabling path-integral formulations that sum over geometries while enforcing Lorentzian signature. Causal set theory posits that spacetime emerges from a fundamental discrete structure known as a causal set, or causet, which is a locally finite partially ordered set (poset) of elements representing events.^[45] The partial order encodes causality, where one element precedes another if it can influence it, mimicking the light-cone structure of Lorentzian geometry. To generate such sets stochastically, the "sprinkling" procedure Poisson-distributes points in a continuous spacetime volume, with the density determining the discreteness scale, and then imposes the causal order from the underlying manifold. This framework ensures that continuous spacetimes approximate causets in the large-volume limit, with the effective dimension recovered from counting chains or antichains in the poset. A key causality condition, inspired by Bell's notions of local independence, requires that the order relations between elements are determined solely by their spacetime separations, preventing acausal influences. Causal dynamical triangulations (CDT) formulate quantum gravity as a path integral over triangulated spacetime histories that respect causality, using simplicial manifolds with a preferred time foliation to maintain Lorentzian signature. Unlike Euclidean dynamical triangulations, CDT restricts configurations to those where spatial slices evolve forward in time without branching, ensuring a well-defined signature while allowing the metric to fluctuate dynamically.^[46] The path integral sums over all such triangulations of fixed topology, typically spherical, with weights from the Einstein-Hilbert action discretized on the lattice, providing a nonperturbative regularization of gravity. Numerical Monte Carlo simulations in four dimensions reveal a phase with de Sitter-like geometry, characterized by positive cosmological constant and exponential expansion, where the Hausdorff dimension measures 4, consistent with classical spacetime dimensionality.^[47] Spin foams extend discrete methods by providing a covariant, path-integral description of quantum geometry, where transition amplitudes connect spin network states representing spatial geometries at different times. In the EPRL model, face amplitudes arise from SU(2) recoupling coefficients that impose simplicity constraints, projecting onto representations compatible with general relativity's geometry.^[48] The full amplitude for a spin foam complex sums products of vertex, edge, and face weights, with the EPRL vertex incorporating the Immirzi parameter to recover semiclassical Regge gravity in the large-spin limit. These models differ from canonical loop quantum gravity by focusing on summed histories rather than Wheeler-DeWitt constraints, facilitating computations of gravitational propagators on discrete lattices. Recent advances as of 2025 include LQG models of nonsingular rotating charged black holes and their observational signatures in gravitational lensing and shadows,^[49] enhanced CDT simulations confirming de Sitter-like phases with improved algorithms,^[50] applications of noncommutative geometry to chronogeometry and cosmic inflation models,^[51] and progress in functorial homological mirror symmetry linking to perturbative topological string theory.^[52]

Applications

In quantum gravity models

In quantum gravity models, quantum geometry provides a framework for addressing key challenges such as the computation of black hole entropy and the resolution of cosmological singularities. By quantizing the geometry of spacetime using discrete structures like spin networks, these models derive thermodynamic properties from underlying microstates, avoiding infinities inherent in semiclassical approximations. A prominent application is the derivation of black hole entropy in loop quantum gravity (LQG), where the Bekenstein-Hawking formula

S = \frac{A}{4 \ell_P^2}

emerges from counting the microstates associated with quantum geometric excitations at the horizon. In this approach, the horizon is punctured by edges of spin networks carrying quantum numbers labeled by half-integer spins

j

, with each puncture contributing an area eigenvalue proportional to

\ell_P^2 \sqrt{j(j+1)}

ℓP2j(j+1)

. The entropy is then given by $S = \sum_i \log(2j_i + 1)$ , where the sum runs over the punctures, and the Barbero-Immirzi parameter is fixed to match the semiclassical result. This counting yields the area proportionality without invoking string theory or AdS/CFT duality, relying instead on the SU(2) Chern-Simons theory for the isolated horizon boundary. In cosmological settings, loop quantum cosmology (LQC) incorporates quantum geometry to modify the Friedmann equation, leading to effective dynamics that resolve big bang singularities. The effective equation takes the form $\left( \frac{\dot{a}}{a} \right)^2 = \frac{8\pi G}{3} \rho \left(1 - \frac{\rho}{\rho_c} \right)$ , where $\rho$ is the energy density and $\rho_c \approx 0.41 \rho_{Pl}$ is a critical density set by the Planck scale and the Immirzi parameter. This modification predicts a quantum bounce at $\rho = \rho_c$ , transitioning from a contracting to an expanding universe without reaching infinite density or curvature. Such dynamics arise from polymer quantization of the Hamiltonian constraint, replacing classical geometry with holonomy-flux algebra. Singularity resolution extends beyond isotropic models to generic spacetimes in LQC and LQG, supported by theorems demonstrating bounded curvature in quantum regimes. For instance, in Bianchi I and II anisotropic cosmologies, quantum effects ensure that the maximum scalar curvature remains finite, of order the Planck scale, regardless of initial conditions near singularity. These results follow from anomaly-free effective Hamiltonians and the discreteness of quantum geometry, which suppress high-curvature configurations. Similar theorems apply to black hole interiors, where the Raychaudhuri equation receives quantum corrections preventing geodesic incompleteness. Efforts toward unification involve constructing effective actions that couple quantum geometry to matter fields, enabling consistent inclusion of Standard Model particles in LQG dynamics. In LQC, scalar fields or fermions are incorporated via anomaly-free modifications to the Hamiltonian, preserving diffeomorphism covariance while allowing for inflationary scenarios post-bounce. These actions, derived from Thiemann regularization, facilitate semiclassical limits and predict observable effects like modified primordial power spectra. Noncommutative effects near horizons further refine these couplings by altering field propagators.

In condensed matter and other fields

To derive the geometry of matter at the nanoscale from known probability distributions, one follows a systematic procedure grounded in quantum information geometry. First, obtain the probability distributions from ground or excited state wavefunctions via

p(\mathbf{r}) = |\psi(\mathbf{r})|^2

for the density, or from measurement statistics such as those obtained via scanning tunneling microscopy (STM), transport measurements, or interferometry. Second, parameterize the system using geometric or control parameters

\theta

(e.g., gate voltages, magnetic field

B

, strain) to generate a family of states

|\psi(\theta)\rangle

. Third, compute the Fisher-Rao metric

g_{ij}(\theta)

on these distributions, which yields the quantum metric tensor that describes the geometry sensed by quantum matter. Fourth, extract the Berry curvature

F_{ij}

and integrate it over the parameter space to obtain topological invariants, such as the Chern number

C = \frac{1}{2\pi} \int F \, d^2k

, to identify topological phases. For mixed states, employ the quantum Fisher information (QFI) to detect topological transitions through singularities or scaling behaviors.^[53]^[54]^[55] In topological insulators, quantum geometry manifests through the Berry curvature, which serves as the imaginary part of the quantum geometric tensor

Q_{\mu\nu} = \langle \partial_\mu u | (1 - |u\rangle\langle u|) | \partial_\nu u \rangle

, where the Berry connection is

A_\mu = i \langle u | \partial_\mu u \rangle

and the curvature is given by

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu

. This geometric structure encodes the topology of the band structure, leading to quantized Hall conductance

\sigma_{xy} = (e^2/h) C

, where

C

is the Chern number derived from the integral of the Berry curvature over the Brillouin zone. In materials like Bi

_2

_3

, this results in robust edge states protected by time-reversal symmetry, enabling dissipationless transport at room temperature. The quantum Hall effect further illustrates quantum geometry in two-dimensional electron gases under strong magnetic fields. Laughlin's wavefunctions,

\psi_m(z_1, \dots, z_N) = \prod_{i<j} (z_i - z_j)^m \exp\left( -\sum_i |z_i|^2 / 4\ell_B^2 \right)

, describe incompressible ground states at filling fractions

\nu = 1/m

(with

m

odd), incorporating geometric phases from adiabatic transport of quasiparticles. These wavefunctions can be reformulated on noncommutative planes, where coordinates satisfy

[x, y] = i \ell_B^2

, capturing the effective geometry induced by the magnetic field and explaining fractional excitations with charge

e/m

. The Berry phase accumulated during cyclic adiabatic evolution,

\gamma = \oint A \cdot dk

, quantifies the geometric contribution to Hall plateaus, distinguishing it from dynamical phases. In two-dimensional systems exhibiting anyonic statistics, quantum geometry governs the braiding of quasiparticles with fractional statistics, intermediate between bosons and fermions. Anyons emerge in fractional quantum Hall states, where the wavefunction acquires a phase

e^{i\theta}

(

0 < \theta < \pi

) upon particle exchange, described by the topological properties of the underlying Hilbert space. Braiding non-Abelian anyons, such as those in

\nu = 5/2

states, implements unitary operations via the braid group representation, forming the basis for fault-tolerant topological quantum computing that protects against local noise. This geometric encoding of statistics has been realized in platforms like Majorana nanowires, where braiding paths trace non-trivial loops in the anyon configuration space. Recent applications in the 2020s leverage holographic duality, inspired by AdS/CFT correspondence, to model strongly correlated electrons in strange metals and high-temperature superconductors. These approaches map boundary quantum field theories of electrons to bulk gravitational geometries, reproducing linear-in-temperature resistivity and Planckian scattering rates observed in cuprates. For instance, charged black hole horizons in anti-de Sitter space dualize to finite-density states with emergent Fermi surfaces, providing a geometric framework for non-Fermi liquid behavior without perturbative expansions. Such models have predicted transport coefficients matching experiments in twisted bilayer graphene, highlighting quantum geometry's role in emergent spacetime structures for condensed matter. Experimental advances have begun to directly probe quantum geometry in solids. In late 2024, researchers used angle-resolved photoemission spectroscopy (ARPES) to measure the quantum metric—the real part of the quantum geometric tensor—in a kagome metal, marking the first such observation and revealing how geometric properties influence electron behavior in quantum materials.^[56] This technique opens pathways for manipulating quantum properties in devices like quantum computers. Additionally, as of 2025, quantum geometry has been linked to enhanced superfluid stiffness in magic-angle twisted bilayer graphene superconductors and the realization of fractional Chern insulators in twisted bilayer MoTe₂, where transport measurements confirm geometric contributions to topological phases and fractional excitations.^[2]

Challenges and future directions

Open theoretical issues

One of the central unresolved challenges in quantum geometry arises from the "problem of time," particularly evident in the Wheeler-DeWitt equation, which describes the quantum state of the universe in canonical quantum gravity approaches like loop quantum gravity (LQG). This equation, derived from the Hamiltonian constraint, imposes a timeless or "frozen" formalism where the wave function of the universe satisfies a time-independent Schrödinger-like equation, lacking an external time parameter to parameterize evolution. As a result, classical notions of time evolution are obscured, complicating the interpretation of quantum dynamics and the emergence of spacetime from quantum geometric structures. Efforts to recover semiclassical time often rely on relational dynamics, where time is defined relative to the evolution of matter degrees of freedom or geometric observables, such as the expansion of volume in minisuperspace models, but a general, non-perturbative resolution remains elusive.^[57]^[58] Renormalization and ultraviolet (UV) completion pose further theoretical hurdles across quantum geometric frameworks. In noncommutative geometry, which underpins approaches to quantizing spacetime coordinates, perturbative control is absent for many models due to the non-local structure induced by the star product, leading to difficulties in defining a consistent UV limit without ad hoc cutoffs or additional symmetries. Although some noncommutative field theories, like φ⁴ on the Moyal plane, admit renormalization to all orders via adapted Hopf algebra techniques, broader applications to gravity encounter persistent UV/IR mixing, where ultraviolet divergences manifest as infrared effects, undermining predictive power. Similarly, in LQG, while the theory provides a non-perturbative quantization of geometry with a discrete spectrum for area and volume operators, infrared divergences emerge in cosmological settings, such as loop quantum cosmology, where long-wavelength modes lead to inconsistencies in the effective Hamiltonian without a full resolution of the anomaly problem. These issues highlight the lack of a complete UV/IR-safe framework for quantum geometry.^[59]^[60]^[61] Incorporating matter fields into quantum geometric structures introduces additional complications, particularly regarding anomalies and consistency. In noncommutative geometries, coupling fermions and gauge fields requires careful treatment to avoid anomalies; for instance, the standard chiral anomaly in even dimensions persists but is modified by the noncommutativity, leading to UV/IR mixing that can render theories inconsistent unless specific representations or regularization schemes are imposed. Chiral issues are especially pronounced, as the noncommutative Dirac operator fails to anticommute perfectly with the chirality projector in curved or deformed spaces, resulting in spectral asymmetries and potential violations of chiral gauge invariance in models attempting to reconstruct the standard model. In LQG, matter coupling via holonomies and fluxes preserves background independence but struggles with fermions, where the absence of a preferred foliation induces anomalies in the presence of gauge fields, and full anomaly-free quantization remains incomplete. These challenges underscore the need for anomaly-cancellation mechanisms that align with the deformed symmetries of quantum geometry.^[62]^[63] A fundamental compatibility issue persists between discrete approaches like LQG, which predict a granular spacetime at the Planck scale, and continuous frameworks such as string theory, where geometry emerges from extended objects in higher dimensions. LQG's spin-network discretization conflicts with string theory's requirement of smooth, supersymmetric manifolds and extra dimensions, as attempts to embed strings on quantized geometries lead to inconsistencies in the low-energy effective action and loss of Lorentz invariance at high energies. Moreover, mirror symmetry in string theory relies on continuous Calabi-Yau geometries, incompatible with LQG's area quantization without additional mechanisms like emergent dimensions. As of 2025, no unified quantum gravity theory incorporating quantum geometry has been achieved, with ongoing research highlighting these irreconcilable foundational differences.^[64]^[65]^[66]

Experimental prospects and predictions

Quantum geometry frameworks, particularly loop quantum cosmology (LQC), predict modifications to the cosmic microwave background (CMB) power spectrum arising from quantum bounces that replace the big bang singularity. These effects lead to a suppression of power at low multipoles (ℓ ≲ 30), potentially explaining observed CMB anomalies such as the low-ℓ suppression in Planck data. Detailed numerical analyses show that LQC with post-bounce inflation yields a scalar power spectrum consistent with observations while introducing subtle deviations testable by future CMB experiments like the Simons Observatory or CMB-S4. In LQC and related approaches, primordial gravitational waves exhibit altered dispersion relations due to quantum corrections to the tensor modes, introducing a frequency-dependent speed that deviates from the canonical c at high frequencies near the Planck scale. These modifications suppress the tensor-to-scalar ratio r and imprint oscillatory features in the B-mode polarization power spectrum, offering a potential signature distinguishable from standard inflationary predictions. High-energy astrophysical probes provide stringent tests of quantum geometry through searches for Lorentz invariance violation (LIV), a common prediction in noncommutative geometry and loop quantum gravity models. Fermi Large Area Telescope (LAT) observations of gamma-ray bursts (GRBs) from 2008 to 2025 have constrained linear LIV parameters, yielding lower limits on the quantum gravity scale E_QG ≳ 10^{19} GeV for several bursts with known redshifts, using time-of-flight delays between high- and low-energy photons. While no violations have been detected, these bounds rule out significant Planck-scale effects manifesting at TeV energies. In particle colliders like the LHC, potential Planck-suppressed effects such as modified dispersion or extra dimensions remain unconstrained below ~10 TeV, as no deviations from standard model predictions have been observed.^[67]^[68]^[69] Laboratory analogs offer tabletop platforms to simulate quantum geometry effects. Bose-Einstein condensates (BECs) enable analog gravity setups where sonic horizons mimic event horizons, allowing probes of quantum corrections like Hawking radiation analogs and modified dispersion in curved effective metrics. Experiments with ultracold atoms have demonstrated horizon formation and phonon propagation mimicking gravitational phenomena, providing indirect tests of discrete spacetime implications for wave propagation. In quantum optics, noncommutative phases can be probed via interferometric setups with entangled photons, detecting deformations in momentum space that signal underlying noncommutativity, though current sensitivities limit bounds to macroscopic scales. Future space-based missions like LISA, scheduled for the 2030s, are poised to detect quantum-corrected waveforms from binary inspirals, including extreme mass-ratio inspirals around supermassive black holes. These could reveal Planck-scale horizon modifications, such as echoing signals or altered phase evolution, with sensitivity to deviations at the 10^{-20} level in the gravitational wave strain. No direct detection of quantum geometry effects has occurred to date, but recent analyses yield tight bounds on the noncommutative parameter θ^{μν} < 10^{-90} m² from gravitational lensing and Shapiro delay observations.^[70]

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