Abstract We revisit quantum field theory in curved spacetime (QFTCS) as a semi-classical framework for quantum matter on classical geometries, emphasizing its limitations, including vacuum ambiguity and background dependence. We briefly review major approaches to quantum gravity (QG), including Loop Quantum Gravity (LQG), string theory, and asymptotic safety, highlighting their conceptual challenges. Motivated by these issues, we outline a teleparallel framework based on coframe and spin-connection variables, where gravity is encoded in torsion rather than curvature. This framework naturally incorporates local Lorentz symmetry and fermionic couplings while displaying a gauge-like structure. We argue that the coframe/spin-connection pair provides an alternative and geometrically refined description of gravitational variables, which may serve as a useful starting point for future investigations of QG. The purpose of this work is not to provide a complete quantization of teleparallel gravity but to identify the geometric and conceptual ingredients that such a formulation would require. 1. Introduction String theory offers a distinct approach, replacing point particles with extended one-dimensional objects and naturally incorporating a massless spin-2 excitation interpreted as the graviton [ 19, 20]. It provides a framework capable of unifying all fundamental interactions and has led to profound insights such as holography and gauge/gravity duality [ 21]. Nevertheless, the theory typically requires extra spatial dimensions and supersymmetry, neither of which has been experimentally confirmed. Furthermore, the vast landscape of possible vacua limits predictive power [ 22], and most formulations remain background-dependent. Despite their diversity, many approaches retain, either directly or through their classical limit, an important relation to metric-based spacetime geometry. This observation suggests that the central difficulty in QG may not lie solely in the quantization procedure, but in the choice of fundamental variables. A deeper issue concerns the role of time. In standard QFT, time is treated as an external parameter, whereas in GR it is dynamical and intertwined with spatial geometry. This mismatch leads to profound conceptual difficulties in defining quantum evolution and observables [ 15, 18]. It is therefore natural to examine whether alternative geometric variables may provide a useful description of spacetime at the quantum level. In this work, we explore the possibility that some conceptual difficulties in QG may motivate the investigation of alternative geometric variables beyond the metric formulation, including formulations based on local frames. From this perspective, a teleparallel formulation based on coframe/spin-connection pairs may provide an alternative geometric setting in which some conceptual issues of QG can be reconsidered. In particular, it offers a framework in which torsion, local Lorentz covariance, and frame variables are treated explicitly. At present, there is no direct experimental evidence that spacetime torsion constitutes an independent propagating degree of freedom in nature. Whether this leads to a complete and internally consistent quantum theory remains an open problem. The contribution of this paper is threefold. First, we clarify the role of coframe and spin-connection variables in teleparallel gravity as candidate variables for a quantum formulation. Second, we compare this framework with curvature-based, connection-based, and metric-affine approaches. Third, we outline a constrained canonical setting in which the coframe is treated as the main gravitational variable while the flat spin connection encodes inertial Lorentz covariance. This paper is therefore programmatic in scope. It does not claim to solve the problem of observables, renormalizability, or the construction of the physical Hilbert space. Rather, it identifies the geometrical structures that must be controlled in any teleparallel QG proposal. Accordingly, the present work should be viewed as a conceptual and geometric analysis rather than as a completed QG construction. Notation Throughout this work we use the metric signature ( − , + , + , + ) and natural units c = ℏ = 1 . Greek indices μ , ν , … denote spacetime coordinate indices, whereas Latin indices a , b , … denote Lorentz-frame indices. The coframe is denoted by h μ a , its inverse by h a μ , and its determinant by h = det ( h μ a ) = − g . The inverse coframe satisfies h μ a h a ν = δ μ ν , h μ a h b μ = δ b a . (1) The gravitational coupling is κ = 8 π G . 2.1. Klein–Gordon Field The dynamics of a scalar field in curved spacetime are governed by the action S ϕ = − 1 2 ∫ d 4 x h g μ ν ∇ μ ϕ ∇ ν ϕ + m 2 ϕ 2 + ξ R ϕ 2 , (2) leading to the Klein–Gordon equation ( □ − m 2 − ξ R ) ϕ = 0 , □ = g μ ν ∇ μ ∇ ν . (3) Here, h = − g = det ( h a μ ) according to the convention introduced above. Physically admissible states are commonly restricted to Hadamard states in order to ensure a well-defined renormalized stress-energy tensor. 2.2. Dirac Field The coframe formalism provides the standard and geometrically transparent framework for coupling spinor fields to curved spacetime, although alternative non-tetrad formulations have been explored. The Dirac action is given by S = ∫ d 4 x h ψ ପ୍ତ ( i γ a h a μ D μ − m ) ψ , (4) where D μ is the spinor covariant derivative containing the spin-connection [ 1, 44]. The corresponding Dirac equation is ( i γ a h a μ D μ − m ) ψ = 0 . (5) Quantization proceeds via a mode expansion and canonical anti-commutation relations. The usefulness of coframes in the standard spinorial formulation highlights the central role of local Lorentz symmetry and motivates the use of frame-based variables in teleparallel gravity. Nonlinear geometric formulations of spinors without introducing an independent orthonormal tetrad have also been discussed in the literature, notably by Ogievetsky and Polubarinov and in later analyses by Pitts [ 45, 46]. Thus, the use of coframes should not be interpreted as the only possible description of spinors in curved spacetime, but rather as the natural language for teleparallel and Lorentz-covariant formulations. 2.3. Proca Field The Proca field describes a massive spin-1 field with action S A = ∫ d 4 x h − 1 4 F μ ν F μ ν − 1 2 m 2 A μ A μ , F μ ν = ∇ μ A ν − ∇ ν A μ , (6) where F μ ν is the antisymmetric field strength. The field satisfies ∇ μ F μ ν − m 2 A ν = 0 . (7) For m ≠ 0 , taking the divergence of the field equation (FEs) yields the Proca constraint ∇ μ A μ = 0 [ 47]. The theory propagates three physical polarization states, reflecting the degrees of freedom of a massive vector field. 2.4. Spin-2 Field and Gravitational Waves Linearized gravity describes perturbations γ μ ν around a background metric. Expanding the Einstein–Hilbert action to second order in the metric perturbation yields the quadratic action for the spin-2 field, together with curvature-dependent interaction terms when the background is not flat. We decompose the metric as g μ ν = g ପ୍ତ μ ν + γ μ ν , (8) where g ପ୍ତ μ ν is the background metric. The trace-reversed perturbation is γ ପ୍ତ μ ν = γ μ ν − 1 2 g ପ୍ତ μ ν γ , γ = g ପ୍ତ μ ν γ μ ν . (9) In curved spacetime, the FEs take the form □ γ ପ୍ତ μ ν + 2 R μ α ν β γ ପ୍ତ α β = 0 , (10) where □ = g ପ୍ତ μ ν ∇ ପ୍ତ μ ∇ ପ୍ତ ν is the d’Alembertian associated with g ପ୍ତ μ ν . 2.5. Critical Assessment Quantum field theory in curved spacetime (QFTCS) provides a consistent and mathematically well-defined framework for describing quantum matter propagating on classical gravitational backgrounds. It successfully accounts for key physical effects such as particle creation in expanding universes [ 49] and black hole evaporation [ 11] and offers a rigorous treatment of renormalized observables through the use of Hadamard states and local covariance [ 1, 43]. However, the theory is intrinsically semi-classical: matter fields are quantized while spacetime geometry remains classical, leading to an inherent inconsistency when backreaction effects become significant [ 1]. This limitation is particularly evident in regimes where quantum fluctuations of the gravitational field cannot be neglected. The problem of observables is not unique to metric formulations and may arise in any generally covariant theory. The teleparallel framework does not by itself solve this problem; rather, it provides a different set of variables in which the issue may be reconsidered. 3.1. Loop Quantum Gravity The fundamental phase space variables are the SU ( 2 ) connection A A i and the densitized triad E i A , satisfying { A A i , E j B } = 8 π G γ δ A B δ j i δ ( x , y ) , (11) where γ is the Barbero–Immirzi parameter. Here, A , B , … denote spatial indices and i , j , … internal S U ( 2 ) indices. Quantum states are represented by spin networks, which form an orthonormal basis of the kinematical Hilbert space [ 17, 51]. These states diagonalize geometric operators such as area and volume, which possess discrete spectra. The dynamics are governed by the Hamiltonian constraint, leading to the Wheeler–deWitt equation defined by [ 52]: H ^ Ψ = 0 , (12) which encodes the dynamics of quantum geometry. However, explicit solutions remain difficult to construct, and the implementation of the Hamiltonian constraint remains technically nontrivial and not uniquely defined. A covariant formulation is provided by spin foam models, which define transition amplitudes between spin network states [ 53]. In symmetry-reduced settings, loop quantum cosmology predicts a resolution of the Big Bang singularity via a quantum bounce [ 54]. Although both LQG and teleparallel gravity use frame-related variables, their geometric interpretations differ. LQG is based on the Ashtekar–Barbero S U ( 2 ) connection and densitized triad, with curvature holonomies playing a central role. Thus, the distinction is not merely between metric and frame variables, but between curvature-based and torsion-based descriptions of gravitational field strength. Teleparallel gravity instead imposes vanishing curvature for the Weitzenböck connection and attributes gravity to torsion. This does not mean that the Levi–Civita curvature associated with the metric vanishes; rather, the curvature of the teleparallel spin connection is set to zero, while the Levi–Civita curvature is encoded equivalently through torsion and a boundary term. Despite its conceptual strengths, LQG faces challenges, including ambiguities in the dynamics, the problem of time, and the recovery of classical spacetime geometry. 3.2. String Theory String theory provides a distinct route to quantum gravity by replacing point particles with one-dimensional extended objects whose perturbative dynamics are described by the Polyakov action [ 19, 20]. Upon quantization, the closed-string spectrum contains a massless spin-2 excitation, naturally interpreted as the graviton. String theory therefore serves here as a contrast with the teleparallel approach: string theory modifies the microscopic objects, whereas teleparallel gravity modifies the geometrical variables used to describe the gravitational field. 3.3. Asymptotic Safety The asymptotic safety program proposes that QG may be defined as a non-perturbatively renormalizable QFT through the existence of a non-trivial ultraviolet (UV) fixed point of the renormalization group flow [ 12, 13, 23, 24, 25]. In this framework, the effective gravitational action is described by a scale-dependent functional whose couplings approach finite values at high energies, ensuring predictivity despite the perturbative non-renormalizability of Einstein gravity. Functional renormalization group (FRG) techniques have provided evidence for such fixed points in truncated theory spaces, typically involving higher-curvature operators such as R 2 and R μ ν R μ ν terms [ 13]. Despite these encouraging results, the asymptotic safety scenario remains subject to important limitations. In particular, the existence and properties of the UV fixed point depend on truncation schemes, raising questions about robustness and convergence. Moreover, many formulations are expressed in terms of the metric field, sharing some conceptual issues of perturbative approaches, including background dependence and difficulties in defining physical observables. The extent to which asymptotic safety can be formulated in a fully background-independent manner remains an active area of research. For the present comparison, its relevance lies in the fact that many standard formulations are metric-based, although tetrad and connection versions have also been studied. In this sense, coframe- and torsion-based formulations provide a useful comparison point for assessing the role of geometric variables. 3.4. Einstein–Cartan and Metric-Affine Approaches The teleparallel framework should also be situated within the broader class of non-Riemannian geometrical formulations of gravity. In Einstein–Cartan theory, the affine connection may have torsion in addition to curvature, and torsion is algebraically related to the spin density of matter [ 44]. In this setting, torsion does not replace curvature as the carrier of gravity; rather, curvature and torsion coexist as geometrical structures associated with the connection. Metric-affine gravity provides a still more general framework in which the metric and affine connection are treated as independent variables. The corresponding geometry may possess curvature, torsion, and nonmetricity as independent field strengths [ 44, 56]. Teleparallel gravity corresponds to the sector in which the curvature of the teleparallel connection and the nonmetricity vanish, while torsion remains nonzero and encodes the gravitational interaction. The Levi–Civita curvature associated with the reconstructed metric need not vanish. 3.5. Alternative and Emergent Approaches The relevance of these approaches for the present work is mainly conceptual. They show that the metric formulation of GR need not be the only starting point for thinking about quantum spacetime. However, the teleparallel proposal considered here is different from emergent-spacetime scenarios: it retains a differentiable spacetime manifold but reorganizes the gravitational variables in terms of a coframe and a flat spin-connection, with torsion playing the role of gravitational field strength. 3.6. Comparative Summary and Critical Analysis The comparison as presented in Table 1 suggests that different quantum-gravity programs are distinguished not only by their quantization methods but also by their choice of geometrical variables. Metric, connection, coframe, torsion, nonmetricity, and extended objects emphasize different aspects of the gravitational field. This observation should be interpreted cautiously. The use of coframes and torsion does not by itself solve the problems of observables, dynamics, renormalizability, or the recovery of the classical limit. In particular, the conceptual issues associated with diffeomorphism invariance and constrained Hamiltonian systems may reappear in any generally covariant theory, including a teleparallel one. The motivation for the teleparallel approach is therefore specific and deliberately limited: it provides a geometrical setting in which the metric is reconstructed from the coframe, the spin-connection is flat and inertial, and torsion carries the gravitational information. Whether this variable choice leads to a consistent quantum theory remains an open question. The next section therefore turns to the teleparallel case in more detail, emphasizing the precise geometrical meaning of the coframe, flat spin-connection, and torsion scalar. 4. Teleparallel Gravity Within the Landscape of Non-Riemannian Geometries Teleparallel gravity provides an alternative geometric formulation of gravitation in which torsion, rather than curvature, encodes the gravitational interaction [ 32, 57]. The fundamental variables are the coframe h μ a and spin-connection ω b μ a , which define a local orthonormal basis on spacetime and reconstruct the metric via: g μ ν = η a b h μ a h ν b and ω a b μ = Λ a c ∂ μ ( Λ − 1 ) c b , (13) where Λ a b ( x ) is a local Lorentz transformation specifying the inertial spin-connection. 4.1. Geometric Foundations The theory is based on the Weitzenböck connection: Γ λ ν μ = h a λ ∂ μ h a ν + ω a b μ h b ν , (14) which is curvature-free but torsionful: T a μ ν = ∂ μ h a ν − ∂ ν h a μ + ω a b μ h b ν − ω a b ν h b μ , (15) T μ ν λ = Γ ν μ λ − Γ μ ν λ , (16) S λ μ ν = 1 2 K μ ν λ + δ λ μ T α ν α − δ λ ν T α μ α . (17) Here, K λ μ ν is the contorsion tensor, relating the Weitzenböck and Levi–Civita connections, and is defined by K λ μ ν = 1 2 T μ λ ν + T ν λ μ − T λ μ ν . (18) The torsion scalar is constructed as: T = 1 4 T μ ν λ T λ μ ν + 1 2 T μ ν λ T λ ν μ − T μ λ λ T ν ν μ . (19) The TEGR action reads: S TEGR = 1 2 κ ∫ d 4 x h T , (20) and is dynamically equivalent to GR provided appropriate boundary conditions are imposed [ 32, 57]. The equivalence with GR follows from the identity R ∘ = − T + B , where B is a boundary term. 4.2. Extension to F ( T ) Gravity A natural modification consists of promoting the torsion scalar to a function: S = 1 2 κ ∫ d 4 x h F ( T ) , (21) leading to modified gravitational dynamics [ 31, 58, 59, 60]. Variation of the action gives the symmetric and antisymmetric parts of the FEs: κ Θ a b = F T T G ∘ a b + F T T T S a b μ ∂ μ T + g a b 2 F T − T F T T , (22) 0 = F T T T S a b μ ∂ μ T , (23) Here, F T = d F d T , F T T = d 2 F d T 2 , Θ a b is the matter energy-momentum tensor, G ∘ a b is the Einstein tensor computed with the Levi–Civita connection, and S a μ ν is the teleparallel superpotential. Adding the GR conservation laws ∇ ∘ ν Θ μ ν = 0 , the antisymmetric equation imposes compatibility conditions on admissible coframe/spin-connection pairs in F ( T ) models. In the covariant formulation, the inclusion of a flat spin connection restores local Lorentz covariance and separates inertial effects from genuine gravitational torsion. In contrast to F ( R ) theories, the FEs remain second order, which avoids Ostrogradsky instabilities and represents a key structural advantage. The role of the spin connection is central in the covariant formulation of F ( T ) gravity. A major conceptual issue in early F ( T ) formulations was the apparent violation of local Lorentz invariance. This problem is resolved in the covariant approach developed by Krššák and Pereira [ 61], where the coframe h μ a and a flat spin-connection ω b μ a are treated as independent variables. This formulation ensures proper separation between inertial and gravitational effects and restores local Lorentz covariance at the level of the action and FEs. However, modified teleparallel theories may introduce additional propagating degrees of freedom whose physical interpretation remains under investigation. 4.3. Recent Developments and Exact Solutions A significant body of recent work has focused on constructing exact solutions and exploring the phenomenology of F ( T ) gravity. Recent studies have constructed several classes of exact solutions in teleparallel F ( T ) gravity, including static spherically symmetric perfect-fluid configurations and anisotropic cosmological models [ 62, 63]. These studies demonstrate that torsion-based dynamics can reproduce a wide range of physically relevant spacetimes while introducing novel structural features absent in curvature-based theories. These constructions illustrate how torsion-based variables can encode nontrivial gravitational configurations in modified teleparallel models. 4.4. New GR and Extended Teleparallel Theories Beyond TEGR and its F ( T ) extensions, a broader class of torsion-based theories has been developed under the framework of New General Relativity (NGR). In this approach, the gravitational Lagrangian is constructed from the three irreducible quadratic invariants of the torsion tensor: L NGR = a 1 T μ ν λ T λ μ ν + a 2 T μ ν λ T λ ν μ + a 3 T μ λ λ T ν ν μ , (24) where a 1 , a 2 , and a 3 are free parameters [ 66, 67]. TEGR corresponds to a specific choice of these coefficients, while NGR allows for a wider parameter space of torsion-based dynamics. This generalization provides a useful testing ground for exploring deviations from GR within a purely torsional framework. NGR may be viewed as a torsion-quadratic sector within the broader landscape of gravitational gauge theories. It remains distinct from Einstein–Cartan and metric-affine models because curvature and nonmetricity are not treated as independent gravitational field strengths. However, NGR models typically introduce additional propagating degrees of freedom, and their physical viability depends sensitively on parameter choices. Stability, ghost freedom, and consistency with observations impose strong constraints, limiting the range of acceptable models. 4.5. Boundary-Term Extensions: F ( T , B ) Gravity An important extension of teleparallel gravity involves the boundary term B, defined through the relation: R = − T + B , B = 2 h ∂ μ ( h T μ ) , T μ = T ν ν μ , (25) which connects the Ricci scalar R to the torsion scalar T. This leads to F ( T , B ) gravity, with action: S = 1 2 κ ∫ d 4 x h F ( T , B ) , (26) which interpolates between F ( T ) and F ( R ) theories [ 31, 34]. Unlike pure F ( T ) models, F ( T , B ) theories generally yield fourth-order FEs due to the presence of higher derivatives through B. As a result, they reintroduce some of the complexities associated with curvature-based modified gravity, including potential instabilities. Nevertheless, F ( T , B ) gravity provides a unifying framework that clarifies the relationship between torsion and curvature formulations and highlights the role of boundary terms in gravitational dynamics. 4.6. Nonmetricity and Hybrid Extensions: F ( Q ) and F ( T , Q ) Theories A further generalization arises in the context of symmetric teleparallel gravity, where gravity is attributed to nonmetricity rather than torsion or curvature [ 68]. In this framework, the fundamental scalar is the nonmetricity scalar Q, leading to F ( Q ) theories. The nonmetricity tensor is Q α μ ν = ∇ α g μ ν , (27) and Q denotes the corresponding quadratic nonmetricity scalar, up to sign conventions. Thus, F ( Q ) models belong to the symmetric teleparallel sector, whereas F ( T ) models belong to the torsional teleparallel sector. More recently, hybrid extensions combining torsion and nonmetricity, such as F ( T , Q ) models, have been proposed as part of a broader unification of geometric formulations [ 69]. These theories suggest that curvature, torsion, and nonmetricity may represent different aspects of a deeper underlying geometric structure. However, they also significantly enlarge the space of possible models, raising concerns about predictability and physical interpretability. 4.7. Conceptual and Critical Implications for Quantum Gravity The teleparallel and coframe-based formulation of gravity offers a conceptually distinct perspective in which the fundamental variables are the coframe fields and spin–connection, rather than the metric tensor. In this framework, gravitation is attributed to torsion rather than curvature, and the dynamical structure resembles that of a gauge theory for translations [ 44]. This shift in geometric interpretation is relevant for QG. In particular, the coframe formalism naturally accommodates fermionic fields and local Lorentz symmetry, addressing limitations already encountered in QFTCS [ 1]. Moreover, the analogy between torsion and field strength suggests that gravitational interactions may potentially be reformulated in a way partly analogous to quantization procedures used in gauge theories. From a conceptual standpoint, the teleparallel approach motivates reconsidering the role of the metric as the fundamental descriptor of spacetime geometry. As highlighted in various QG programs, including LQG and effective field theory approaches [ 13, 51], the identification of the true dynamical degrees of freedom remains an open problem. In this context, the coframe and spin-connection variables provide a different geometric structure, potentially capturing both translational and rotational aspects of spacetime symmetries. This viewpoint is further supported by the natural role of coframes in the standard spinorial formulation and by the gauge-like structure underlying teleparallel gravity, which may provide a useful alternative setting for quantization relative to purely metric-based formulations. Despite these promising features, several challenges remain. The role of torsion at the quantum level, the construction of a consistent quantum theory based on coframe variables, and the recovery of classical GR in an appropriate limit all require further investigation. Additionally, the relation between teleparallel gravity and other approaches, such as string theory or asymptotic safety, is not yet fully understood [ 13, 19]. Nevertheless, the teleparallel framework provides a complementary geometric setting in which some conceptual issues can be reformulated. It thus represents a possible direction for future investigations of torsion-based quantum geometry. 5. Toward a Constrained Quantum Teleparallel Framework The construction below is programmatic, formal, and heuristic in nature. It is not intended as a complete Dirac quantization of teleparallel gravity, but as an identification of the canonical variables, constraints, and consistency conditions that such a quantization would have to address. In the covariant formulation of teleparallel gravity, the gravitational degrees of freedom are encoded in the coframe field h μ a , while the flat spin–connection ω b μ a accounts for inertial effects and ensures local Lorentz invariance [ 32, 57, 61]. Any candidate quantum formulation must specify how this constrained pair is treated. 5.1. Canonical Variables and Extended Phase Space We perform a 3 + 1 decomposition of spacetime, defining spatial coframes h i a and their conjugate momenta π a i . The spin–connection is included as a constrained variable with conjugate momentum Π a b i : ( h i a , π a i ) , ( ω b i a , Π a b i ≈ 0 ) , (28) where i , j , … denote spatial indices on the chosen hypersurface. The pair ( ω a b i , Π b i a ) is introduced in an extended phase-space sense. Since the spin connection is flat and inertial, its conjugate momentum defines a primary constraint rather than an independent propagating degree of freedom. The vanishing of Π a b i reflects the fact that the spin–connection is non-dynamical, constrained by the flatness condition: R b i j a ( ω ) = ∂ i ω b j a − ∂ j ω b i a + ω c i a ω b j c − ω c j a ω b i c = 0 . (29) Classically, these constraints ensure that ω b μ a encodes only inertial effects, preserving the covariant structure of the theory [ 61, 70]. This canonical decomposition introduces a foliation of spacetime and is therefore not manifestly covariant, although it is useful for identifying the constraint structure. 5.2. Canonical Quantization Promoting the classical fields to operators acting on a Hilbert space H , we impose canonical commutation relations: [ h ^ i a ( x ) , π ^ b j ( y ) ] = i ℏ δ b a δ i j δ ( 3 ) ( x − y ) , (30) [ ω ^ b i a ( x ) , Π ^ c d j ( y ) ] = i ℏ δ c a δ b d δ i j δ ( 3 ) ( x − y ) , (31) with all other commutators vanishing. The physical states | Ψ ⟩ satisfy the operator constraints: Π ^ a b i | Ψ ⟩ = 0 , R ^ b i j a ( ω ) | Ψ ⟩ = 0 , (32) which enforce the flatness of the spin connection and restrict the quantum states to the physically admissible sector. These commutation relations are formal and should be understood prior to solving the constraints. A full quantum theory would require the construction of a physical Hilbert space after imposing all first-class constraints. Issues related to operator ordering, regularization, and the definition of an appropriate inner product are left open. 5.3. Torsion Operator and Gauge Structure The torsion operator is defined as T ^ μ ν a = ∂ μ h ^ ν a − ∂ ν h ^ μ a + ω ^ b μ a h ^ ν b − ω ^ b ν a h ^ μ b . (33) It acts as a quantum field strength associated with translational gauge symmetry, with the coframe playing the role of a gauge potential. This analogy with Yang–Mills theory should be interpreted geometrically rather than as a literal dynamical equivalence with an internal gauge theory. In teleparallel gravity, the gauge-like structure is tied to spacetime translations and local Lorentz covariance. The spin–connection ensures that torsion transforms covariantly under local Lorentz transformations: δ h ^ μ a = ϵ b a ( x ) h ^ μ b , (34) δ ω ^ b μ a = − D μ ϵ b a ( x ) , (35) where D μ is the covariant derivative with respect to ω and ϵ a b = − ϵ b a is the infinitesimal Lorentz parameter. The generator of local Lorentz transformations is denoted by J ^ a b . Physical states satisfy J ^ a b | Ψ ⟩ = 0 , (36) implementing Lorentz invariance at the quantum level [ 61, 70]. 5.4. Constraint Algebra and Dynamics The quantum Hamiltonian H ^ , diffeomorphism generators D ^ i , and Lorentz generators J ^ a b act on H as: H ^ | Ψ ⟩ = 0 , D ^ i | Ψ ⟩ = 0 , J ^ a b | Ψ ⟩ = 0 . (37) 5.5. Comparison with Ashtekar–Barbero Variables Although both approaches use frame-related variables, the teleparallel coframe/spin-connection pair differs from the Ashtekar–Barbero variables used in LQG. In LQG, the connection is an S U ( 2 ) connection whose holonomies encode curvature. In the teleparallel setting, the spin-connection is flat and inertial, while the torsion of the coframe encodes the gravitational field. Thus, the distinction is not merely one of notation but one of geometrical field strength: curvature in LQG versus torsion in teleparallel gravity. This comparison clarifies that the teleparallel proposal is not merely a rewriting of LQG variables, since the two frameworks organize gravitational geometry through distinct field strengths. 5.6. Physical Interpretation and Outlook The discussion above suggests that coframe/spin-connection variables provide a coherent geometrical language for formulating a teleparallel quantum-gravity program. This does not establish a complete quantum theory but identifies the variables and constraints that would have to be controlled. In contrast to metric-based approaches, the coframe formalism naturally incorporates local Lorentz symmetry and allows for a direct coupling to fermionic matter fields, a feature already essential in QFTCS [ 1, 44]. Moreover, the teleparallel interpretation of gravity, in which torsion replaces curvature as the mediator of gravitational interactions, offers a gauge-like structure that is structurally reminiscent of gauge-theoretic descriptions of other interactions. This perspective supports the idea that the gravitational degrees of freedom may be organized in terms of local frame variables rather than only through the metric tensor. From a broader viewpoint, these findings resonate with key insights from existing approaches to QG. LQG emphasizes connection variables and discrete geometric structures [ 51, 52], while effective field theory approaches highlight the limitations of perturbative quantization based on the metric [ 13]. Similarly, string theory incorporates gravity through extended objects but typically relies on a fixed background geometry [ 19]. In comparison, the coframe/teleparallel framework offers a different geometrical organization of the gravitational variables. However, the precise relation between torsion-based dynamics and these established frameworks remains an open question, and further work is required to clarify whether these approaches are complementary or fundamentally distinct. Looking forward, several important directions emerge. A key challenge is the construction of a consistent quantum theory based on coframe and spin-connection variables, including the identification of appropriate observables and the treatment of quantum fluctuations of torsion. Additionally, the recovery of classical GR and standard cosmology in the appropriate limit must be established rigorously. From a phenomenological perspective, it is essential to explore potential observational signatures, such as deviations from GR or novel coupling effects involving spin and torsion. Ultimately, the viability of this program will depend on whether it can produce a well-defined physical Hilbert space, a closed anomaly free constraint algebra, a controlled semiclassical limit, and empirically relevant predictions. 6. Discussion and Outlook The comparison of different QG programs in Section 3 further underscores the absence of a universally satisfactory framework. LQG provides a background-independent quantization of geometry in terms of connection variables, leading to discrete spectra for geometric operators [ 51, 52], yet faces challenges in defining dynamics and recovering the classical limit. String theory achieves a perturbatively consistent unification of interactions and naturally incorporates gravity via a spin-2 excitation [ 19, 21], but relies on higher-dimensional backgrounds and exhibits limited predictive power due to the landscape of vacua. Asymptotic safety suggests a non-perturbative ultraviolet completion of gravity [ 13], while alternative and emergent approaches propose that spacetime itself may not be fundamental [ 43]. Despite their diversity, these frameworks leave open the question of which geometrical variables provide the most useful description of gravitational degrees of freedom in the quantum regime. Looking forward, several open problems remain. A key challenge is the construction of a consistent quantum theory based on coframe and spin-connection variables, including the identification of physical observables, the treatment of quantum torsion, and the control of the constraint algebra. Establishing the precise relation between teleparallel gravity and other QG approaches, such as LQG, string theory, and asymptotic safety, remains an important direction for future research. From a phenomenological standpoint, it is also essential to investigate whether torsion-based dynamics could lead to observable deviations from GR or novel spin–torsion effects. Overall, the teleparallel/coframe framework should be understood as a well-motivated geometrical setting for future investigations, whose viability will depend on mathematical consistency, a controlled semiclassical limit, and empirical relevance. Table 1. Comparison of geometrical variables and field strengths in selected approaches to quantum gravity. Approach Basic Variables Field Strength Main Open Issue Metric GR/QFT g μ ν Levi–Civita curvature Perturb. nonrenormalizability LQG A A i , E i A Curvature holonomies Semiclassical limit String theory Worldsheet fields, background metric Spin-2 excitation Vacuum/background selection Einstein–Cartan Coframe, Lorentz connection Curvature and torsion Spin–torsion sector Metric-affine gravity Metric, affine connection Curvature, torsion, nonmetricity Many degrees of freedom Teleparallel gravity h a μ , ω a b μ Torsion Quantum formulation open Share and Cite MDPI and ACS Style Landry, A. Conceptual and Geometric Foundations for a Teleparallel Approach to Quantum Gravity. Axioms 2026, 15, 427. https://doi.org/10.3390/axioms15060427 AMA Style Landry A. Conceptual and Geometric Foundations for a Teleparallel Approach to Quantum Gravity. Axioms. 2026; 15(6):427. https://doi.org/10.3390/axioms15060427 Chicago/Turabian Style Landry, Alexandre. 2026. "Conceptual and Geometric Foundations for a Teleparallel Approach to Quantum Gravity" Axioms 15, no. 6: 427. https://doi.org/10.3390/axioms15060427 APA Style Landry, A. (2026). Conceptual and Geometric Foundations for a Teleparallel Approach to Quantum Gravity. Axioms, 15(6), 427. https://doi.org/10.3390/axioms15060427 Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here. Article Metrics Article metric data becomes available approximately 24 hours after publication online.
