Toward a Teleparallel Quantum Field Theory of Gravity

1. Introduction

The construction of a consistent theory of quantum gravity (QG) remains one of the most profound and persistent challenges in modern theoretical physics. Despite the extraordinary empirical success of quantum field theory (QFT) in describing the Standard Model of particle physics, and the equally remarkable success of general relativity (GR) in explaining gravitational phenomena across astrophysical and cosmological scales, these two frameworks remain fundamentally incompatible at both conceptual and mathematical levels [1,2,3,4,5,6,7,8].

This incompatibility is not merely technical, but reflects a deep structural tension in our understanding of spacetime. In QFT, spacetime is treated as a fixed, non-dynamical background, typically Minkowskian or weakly curved, on which quantum fields evolve [1,9]. In contrast, GR describes spacetime as a dynamical entity whose geometry is determined by the distribution of matter and energy. This asymmetry becomes unavoidable at the Planck scale, where quantum fluctuations of spacetime itself are expected to be significant [10,11].

Attempts to quantize gravity perturbatively lead to a non-renormalizable theory, requiring an infinite number of counterterms [3,12]. This failure suggests that GR, when treated as a QFT of the metric, is at best an effective field theory valid below the Planck scale [13,14]. Moreover, conceptual problems such as the breakdown of locality, the absence of a preferred time parameter, and ambiguities in defining observables further complicate the formulation of a consistent theory [2,3,4,5,6,7,15].

Over the past decades, several major research programs have attempted to resolve these issues. Loop Quantum Gravity (LQG) provides a non-perturbative and background-independent quantization of geometry, predicting a fundamentally discrete structure of spacetime [16,17]. However, its semiclassical limit and connection to low-energy physics remain under active investigation. In addition, the canonical formulation leads to the so-called ”problem of time”, embodied in the Wheeler–DeWitt equation, which lacks an explicit temporal parameter [10,18].

String theory offers a radically different 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.

The asymptotic safety program proposes that gravity may be non-perturbatively renormalizable due to the existence of an ultraviolet fixed point [23,24,25]. While functional renormalization group techniques provide encouraging evidence, the approach relies on truncations of theory space, and the existence of a fully consistent continuum limit remains an open question.

Beyond these mainstream approaches, alternative frameworks such as causal set theory, causal dynamical triangulations, and emergent gravity models suggest that spacetime itself may not be fundamental [26,27,28,29]. These approaches offer valuable conceptual insights into discreteness and emergence, but face significant challenges in recovering classical spacetime and connecting with observational data.

Despite their diversity, most existing approaches share a common feature: they rely, either explicitly or implicitly, on a metric-based description of 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 question whether the metric tensor is the appropriate fundamental object for describing spacetime at the quantum level.

In this work, we advance the hypothesis that the persistent failure of QG programs may be rooted in the metric-based formulation itself. We propose instead that a consistent theory of QG may require a shift toward a formulation based on local frames.

Teleparallel gravity provides a compelling alternative in this direction. In this framework, gravitation is attributed to torsion rather than curvature, and the fundamental variables are coframe fields instead of the metric tensor [30,31,32,33,34]. This formulation is dynamically equivalent to GR at the classical level, while naturally accommodating spinor fields and admitting a gauge-theoretic interpretation.

From this perspective, a teleparallel QFT based on coframe/spin–connection pairs may offer a more natural and internally consistent framework for QG. Such an approach has the potential to resolve the asymmetry between space and time, provide a unified description of geometry and matter, and circumvent some of the conceptual limitations inherent in metric-based approaches.

The aim of this work is therefore twofold. In Section 2Section 3, we provide a critical analysis of existing approaches to QG, emphasizing their structural limitations. In Section 4Section 5, we explore the possibility that a teleparallel, frame-based formulation may offer a viable path toward a consistent microscopic theory of spacetime.

2. Quantum Field Theory in Curved Spacetime

2.1. Klein–Gordon Field

The dynamics of a scalar field in curved spacetime are governed by the action

$$S=-{\displaystyle \frac{1}{2}}\int d^{4}xh\left(g^{\mu \nu }{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi +m^2{\varphi }^2+\xi R{\varphi }^2\right),$$

(1)

leading to the Klein–Gordon equation

$$(\square -m^2-\xi R)\varphi =0,$$

(2)

where $h=\sqrt{-g}$ is the coframe determinant.

On globally hyperbolic spacetimes, one constructs a complete set of mode solutions and defines a Hilbert space using the Klein–Gordon inner product [1,35]. However, the absence of a preferred timelike Killing vector (KV) leads to ambiguities in the definition of vacuum states [36,37].

2.2. Dirac Field

Fermionic fields require the introduction of a coframe (tetrad for by-default orthonormal frames) and spin–connection. The Dirac action is given by

$$S=\int d^{4}xh\overline{\psi }(i{\gamma }^a{h}_a{}^{\mu }{D}_{\mu }-m)\psi ,$$

(3)

where $D_{\mu }$ includes the spin–connection [1,38]. The Dirac field satisfies:

$$(i{\gamma }^a{h}_a{}^{\mu }{D}_{\mu }-m)\psi =0.$$

(4)

Quantization proceeds via a mode expansion and canonical anti-commutation relations. The necessity of coframes highlights the fundamental role of local Lorentz symmetry and suggests that frame-based formulations may be more fundamental than metric-based ones.

2.3. Proca Field

The Proca field describes a massive spin-1 field with action

$$S=-{\displaystyle \frac{1}{4}}\int d^{4}x\sqrt{-g}(F_{\mu \nu }{F}^{\mu \nu }+m^2{A}_{\mu }{A}^{\mu }).$$

(5)

The field satisfies

$${\nabla }_{\mu }{F}^{\mu \nu }+m^2{A}^{\nu }=0,$$

(6)

together with the constraint ${\nabla }_{\mu }{A}^{\mu }=0$ [39]. 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 $h_{\mu \nu }$ around a background metric. The second order Einstein-Hilbert action integral is described under the form $S=S_{EH}\left(h_{\mu \nu }\right)+S_{Inter}(h_{\mu \nu },R)$ with an interation term between $h_{\mu \nu }$ and R. In curved spacetime, the field equations (FEs) take the form

$$\square {\overline{h}}_{\mu \nu }+2R_{\mu \alpha \nu \beta }{\overline{h}}^{\alpha \beta }=0.$$

(7)

Although a graviton interpretation can be introduced, perturbative quantization leads to non-renormalizable divergences [40,41]. This indicates that the metric-based spin-2 field theory is only an effective description valid at low energies [13].

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 [42] and black hole evaporation [11], and offers a rigorous treatment of renormalized observables through the use of Hadamard states and local covariance [1,37]. 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.

A further conceptual difficulty arises from the absence of a preferred vacuum state in generic curved spacetimes. In the absence of a global timelike KV, the notion of particles becomes observer-dependent, as illustrated by inequivalent quantizations associated with different mode decompositions [36,37]. Moreover, the formalism relies on a fixed background geometry, which stands in tension with the dynamical nature of spacetime in GR. While QFTCS can be interpreted as an effective field theory valid at energies below the Planck scale [13], it does not provide a fundamental description of quantum gravity. These limitations suggest that a consistent quantum theory of spacetime may require a reformulation in terms of more fundamental variables, such as coframes or connections, that treat geometry and matter on an equal footing.

3. Quantum Gravity Approaches

A wide range of approaches have been developed to reconcile quantum mechanics (QM) and GR. While each provides important insights, none has yet achieved a complete and experimentally verified theory.

3.1. Loop Quantum Gravity

LQG is a non-perturbative and background-independent approach to QG based on the canonical quantization of GR in terms of connection variables [43,44,45].

The fundamental phase space variables are the $\mathrm{SU}\left(2\right)$ connection $A_a^i$ and the densitized triad $E_i^a$, satisfying

$$\{A_a^i,E_j^b\}=8\pi G\gamma {\delta }_a^b{\delta }_j^i\delta (x,y),$$

(8)

where $\gamma$ is the Barbero–Immirzi parameter.

Quantum states are represented by spin networks, which form an orthonormal basis of the kinematical Hilbert space [17,44]. 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 [45]:

$$\widehat{H}\Psi =0,$$

(9)

which encodes the dynamics of quantum geometry. However, explicit solutions $\Psi$ remain difficult to construct, and the definition of Hamiltonian operator $\widehat{H}$ is not unique.

A covariant formulation is provided by spin foam models, which define transition amplitudes between spin network states [46]. In symmetry-reduced settings, loop quantum cosmology predicts a resolution of the Big Bang singularity via a quantum bounce [47].

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 and M-Theory

String theory proposes that fundamental interactions arise from one-dimensional extended objects whose dynamics are described by the Polyakov action [19,48].

Quantization of the string leads to a spectrum of excitations, including a massless spin-2 particle identified as the graviton [19]. Consistency requires critical spacetime dimensions and anomaly cancellation.

Supersymmetric extensions give rise to five consistent superstring theories in ten dimensions, which are related by dualities [49]. These theories incorporate gauge interactions and gravity within a unified framework.

To recover four-dimensional physics, extra dimensions must be compactified, typically on Calabi–Yau manifolds, leading to effective field theories with rich phenomenology [50]. String theory admits a wide range of classical solutions, including branes, flux compactifications, and anti-de Sitter backgrounds.

Dualities between string theories led to the conjecture of M-theory, an eleven-dimensional framework whose low-energy limit is supergravity [51]. The AdS/CFT correspondence provides a non-perturbative definition of certain QG theories [21].

Despite its theoretical successes, string theory faces several challenges, including background dependence, the landscape of vacua, and the lack of experimental verification.

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 [13,23,24,25,40]. 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 $R_{\mu \nu }{R}^{\mu \nu }$ 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, the formalism is generally formulated in terms of the metric field, inheriting some of the conceptual issues of perturbative approaches, including background dependence and difficulties in defining physical observables. While asymptotic safety provides a compelling effective-field-theory extension of gravity at high energies, it does not yet offer a fully background-independent or geometrically fundamental description. In this sense, alternative formulations based on coframes and torsion may provide complementary insights into the underlying structure of quantum spacetime.

3.4. Bi-vector Approach

An alternative perspective on the structure of spacetime and quantum fields is provided by the bi-vector approach, in which the fundamental variables are not the metric or connection, but antisymmetric tensorial objects encoding oriented area elements. In this framework, the geometry is described in terms of bivectors that naturally incorporate both metric and gauge-like degrees of freedom, offering a unified description of gravitational and electromagnetic interactions [52,53,54,55,56,57]. The bi-vector formalism emphasizes the role of oriented two-surfaces and suggests that spacetime structure may be fundamentally encoded in extended geometric quantities rather than pointwise fields. Solutions in this approach are typically constructed by decomposing fields into bivector bases, allowing for an alternative representation of curvature, field strengths, and conserved quantities.

Despite its conceptual appeal, the bi-vector approach remains less developed than other QG frameworks. In particular, the precise relation between bivector variables and standard geometric observables, such as curvature invariants or metric components, is not always transparent. Furthermore, the quantization procedure and the construction of a consistent Hilbert space structure are still under active investigation [52,53]. While the formalism offers intriguing connections to gauge theories and may provide a more geometric interpretation of fundamental interactions, its predictive power and physical implications remain to be fully established. Nevertheless, the emphasis on extended geometric structures aligns with other non-metric approaches and supports the broader idea that alternative variables, such as coframes or torsion, may be more fundamental in a quantum theory of gravity.

3.5. Alternative and Emergent Approaches

Beyond the main frameworks discussed above, a variety of alternative and emergent approaches to QG have been proposed, often motivated by the limitations of both perturbative and canonical quantization schemes. A common theme among these approaches is the idea that spacetime geometry may not be fundamental, but rather an emergent structure arising from more primitive degrees of freedom. Examples include induced gravity, entropic gravity, and holographic scenarios, where gravitational dynamics are derived from thermodynamic or information-theoretic principles [21,37]. In particular, the holographic paradigm suggests that gravitational phenomena in a bulk spacetime can be encoded in lower-dimensional quantum field theories, providing a radically different perspective on locality and geometry.

Another class of approaches focuses on reformulating gravity in terms of alternative geometric variables, such as connections, coframes, or generalized gauge structures. These formulations aim to recast gravity in a language closer to that of standard gauge theories, thereby facilitating quantization and unification with other interactions [43,44]. In this context, the role of local frames and spin–connections becomes central, especially in the presence of fermionic matter, reinforcing the idea that the metric tensor may not be the most fundamental variable. Such perspectives are also compatible with extensions of GR involving torsion or non-Riemannian geometry [38].

Despite their diversity, these alternative and emergent approaches share common challenges, including the identification of fundamental degrees of freedom, the recovery of classical spacetime in an appropriate limit, and the formulation of testable predictions. While none of these frameworks currently provides a complete and experimentally verified theory of QG, they offer valuable conceptual insights and highlight possible directions beyond conventional metric-based quantization schemes. In particular, they support the broader viewpoint that a consistent theory of QG may require a reformulation of spacetime geometry in terms of more fundamental structures, such as coframes, connections, or extended geometric objects.

3.6. Critical Analysis

The various approaches to QG reviewed above—LQG, string theory, asymptotic safety, and alternative geometric or emergent frameworks—provide complementary insights into the structure of spacetime at the quantum level. LQG offers a background-independent quantization of geometry with discrete spectra for geometric operators [44,45], while string theory achieves a perturbatively consistent unification of gravity with other interactions through extended objects in higher dimensions [19,21]. Asymptotic safety, in turn, suggests that gravity may be non-perturbatively renormalizable via a UV fixed point [13], and alternative approaches emphasize emergent or non-metric descriptions of spacetime [37,38]. Despite these advances, each framework faces significant conceptual and technical challenges, including the recovery of classical spacetime, the definition of observables, and the connection to experimentally accessible regimes.

A common limitation shared by many of these approaches is their reliance—either explicit or implicit—on the metric tensor as the primary geometric variable, or on background structures that obscure the underlying degrees of freedom. Even in formulations that depart from the metric, such as LQG, the emergence of smooth spacetime geometry from discrete or combinatorial structures remains only partially understood [44]. Similarly, string theory typically presupposes a fixed background geometry, raising questions about true background independence [19]. These considerations suggest that a reformulation of gravity in terms of more fundamental variables—such as coframes, spin–connections, or torsion—may provide a more natural starting point for quantization. In this light, frame-based and gauge-theoretic approaches offer a promising avenue for addressing the conceptual limitations of existing frameworks and for developing a more unified description of quantum spacetime.

4. Teleparallel $f\left(T\right)$ Gravity and Coframe/Spin-Connection Pair Formulation

Teleparallel gravity provides an alternative geometric formulation of gravitation in which torsion, rather than curvature, encodes the gravitational interaction [32,58]. The fundamental variables are the coframe $h_{\mu }^a$ and spin-connection ${\omega }_{b\mu }^a$, which define a local orthonormal basis on spacetime and reconstructs the metric via:

$$g_{\mu \nu }={\eta }_{ab}{h}_{\mu }^a{h}_{\nu }^b\mathrm{and}{\omega }_{b\mu }^a={\Lambda }_c^a{\partial }_{\mu }{\Lambda }_b^c,$$

(10)

where ${\Lambda }_c^a$ is an undetermined Lorentz transformation.

4.1. Geometric Foundations

The theory is based on the Weitzenböck connection:

$${\mathrm{\Gamma }}_{\mu \nu }^{\lambda }=h_a^{\lambda }{\partial }_{\mu }{h}_{\nu }^a,$$

(11)

which is curvature-free but torsionful:

$$T_{\mu \nu }^{\lambda }={\mathrm{\Gamma }}_{\nu \mu }^{\lambda }-{\mathrm{\Gamma }}_{\mu \nu }^{\lambda }.$$

(12)

The torsion scalar is constructed as:

$$T={\displaystyle \frac{1}{4}}{T}_{\mu \nu }^{\lambda }{T}_{\lambda }^{\mu \nu }+{\displaystyle \frac{1}{2}}{T}_{\mu \nu }^{\lambda }{T}_{\lambda }^{\nu \mu }-T_{\mu \lambda }^{\lambda }{T}_{\nu }^{\nu \mu }.$$

(13)

The TEGR action reads:

$$S_{\mathrm{TEGR}}={\displaystyle \frac{1}{2\kappa }}\int d^{4}xhT,$$

(14)

and is dynamically equivalent to GR up to a boundary term [32,58].

4.2. Extension to $f\left(T\right)$ Gravity

A natural modification consists in promoting the torsion scalar to a function:

$$S={\displaystyle \frac{1}{2\kappa }}\int d^{4}xhf\left(T\right),$$

(15)

leading to modified gravitational dynamics [31,59,60]. Applying the least-action principle, we find the symmetric and antisymmetric parts of FEs [61]:

$$\begin{array}{ccc}\hfill \kappa {\mathrm{\Theta }}_{\left(ab\right)}& =& F_T\left(T\right){\stackrel{\circ }{G}}_{ab}+F_{TT}\left(T\right)S_{\left(ab\right)}^{\mu }{\partial }_{\mu }T+{\displaystyle \frac{g_{ab}}{2}}\left[F\left(T\right)-T{F}_T\left(T\right)\right],\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill 0& =& F_{TT}\left(T\right)S_{\left[ab\right]}^{\mu }{\partial }_{\mu }T,\hfill \end{array}$$

(17)

Adding the GR conservation laws ${\stackrel{\circ }{\nabla }}_{\nu }\left({\mathrm{\Theta }}^{\mu \nu }\right)=0$, we can solve the FEs for any coframe/spin-connection pair ($h_{\mu }^a/{\omega }_{b\mu }^a$–pair) in any gauge $g_{ab}$. Teleparallel gravity theory and all its extensions are gauge-invariant as any frame-based approaches. In contrast to $f\left(R\right)$ theories, the FEs remain second order, which avoids Ostrogradsky instabilities and represents a key structural advantage.

However, there is a related problem to the relationship between spin-connection and covariant formulation. A major conceptual issue in early $f\left(T\right)$ formulations was the apparent violation of local Lorentz invariance. This problem is resolved in the covariant approach developed by Krššák and Pereira [62], where the coframe $h_{\mu }^a$ and a flat spin-connection ${\omega }_{b\mu }^a$ 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.

4.3. Recent Developments and Exact Solutions

A significant body of recent work has focused on constructing exact solutions and exploring the phenomenology of $f\left(T\right)$ gravity.

In particular, Landry has developed several classes of exact solutions in teleparallel $F\left(T\right)$ gravity, including static spherically symmetric perfect fluid configurations and anisotropic cosmological models [63,64]. 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.

Such constructions highlight the flexibility of the coframe formalism in encoding gravitational degrees of freedom and suggest that torsion may play a more fundamental role in gravitational dynamics than traditionally assumed.

Complementary studies, including those by Sahoo and collaborators, have investigated cosmological constraints and late-time acceleration scenarios within $f\left(T\right)$ models [65], while dynamical systems analyses by Coley and van den Hoogen emphasize the role of geometric variables in determining the global structure of cosmological phase space [66].

4.4. New GR and Extended Teleparallel Theories

Beyond TEGR and its $f\left(T\right)$ 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:

$${\mathcal{L}}_{\mathrm{NGR}}=a_1{T}_{\mu \nu }^{\lambda }{T}_{\lambda }^{\mu \nu }+a_2{T}_{\mu \nu }^{\lambda }{T}_{\lambda }^{\nu \mu }+a_3{T}_{\mu \lambda }^{\lambda }{T}_{\nu }^{\nu \mu },$$

(18)

where $a_1$, $a_2$, and $a_3$ are free parameters [67,68].

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.

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:

$$R=-T+B,$$

(19)

which connects the Ricci scalar R to the torsion scalar T.

This leads to $f(T,B)$ gravity, with action:

$$S={\displaystyle \frac{1}{2\kappa }}\int d^{4}xhf(T,B),$$

(20)

which interpolates between $f\left(T\right)$ and $f\left(R\right)$ theories [69].

Unlike pure $f\left(T\right)$ 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: $f\left(Q\right)$ 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 [70]. In this framework, the fundamental scalar is the nonmetricity scalar Q, leading to $f\left(Q\right)$ theories.

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 [71].

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 [38]. This shift in geometric interpretation has important implications 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 be formulated in a way more closely aligned with the quantization procedures successfully applied to gauge theories.

From a conceptual standpoint, the teleparallel approach challenges the primacy of the metric as the fundamental descriptor of spacetime geometry. As highlighted in various QG programs, including LQG and effective field theory approaches [13,44], the identification of the true dynamical degrees of freedom remains an open problem. In this context, the coframe and spin-connection variables provide a more refined geometric structure, potentially capturing both translational and rotational aspects of spacetime symmetries. This viewpoint is further supported by the necessity of coframes in coupling spinor fields and by the gauge-like structure underlying teleparallel gravity, which may offer a more natural starting point for quantization than 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 compelling alternative that addresses some of the conceptual limitations of existing theories, particularly regarding background dependence and the identification of fundamental variables. It thus represents a promising avenue for the development of a more unified and geometrically grounded theory of QG.

5. Toward a Quantum Teleparallel Formulation: Coframe and Spin–Connection Formulation

In the covariant formulation of teleparallel gravity, the gravitational degrees of freedom are encoded in the coframe field $h_{\mu }^a$, while the flat spin–connection ${\omega }_{b\mu }^a$ accounts for inertial effects and ensures local Lorentz invariance [32,58,62]. A consistent quantum formulation must therefore treat the coframe/spin-connection pair appropriately.

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 ${\pi }_a^i$. The spin–connection is included as a constrained variable with conjugate momentum ${\mathrm{\Pi }}_a^{bi}$:

$$(h_i^a,{\pi }_a^i),({\omega }_{bi}^a,{\mathrm{\Pi }}_a^{bi}\approx 0).$$

(21)

The vanishing of ${\mathrm{\Pi }}_a^{bi}$ reflects the fact that the spin–connection is non-dynamical, constrained by the flatness condition:

$$R_{bij}^a\left(\omega \right)={\partial }_i{\omega }_{bj}^a-{\partial }_j{\omega }_{bi}^a+{\omega }_{ci}^a{\omega }_{bj}^c-{\omega }_{cj}^a{\omega }_{bi}^c=0.$$

(22)

These constraints guarantee that ${\omega }_{b\mu }^a$ encodes only inertial effects, preserving the covariant structure of the theory [62,72].

5.2. Canonical Quantization

Promoting the classical fields to operators acting on a Hilbert space $\mathcal{H}$, we impose canonical commutation relations:

$$\begin{array}{cc}\hfill [{\widehat{h}}_i^a\left(x\right),{\widehat{\pi }}_b^j\left(y\right)]& =i\hslash {\delta }_b^a{\delta }_i^j{\delta }^{\left(3\right)}(x-y),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\widehat{\omega }}_{bi}^a\left(x\right),{\widehat{\mathrm{\Pi }}}_c^{dj}\left(y\right)]& =i\hslash {\delta }_c^a{\delta }_b^d{\delta }_i^j{\delta }^{\left(3\right)}(x-y),\hfill \end{array}$$

(24)

with all other commutators vanishing. The physical states $|\Psi \rangle$ satisfy the operator constraints:

$${\widehat{\mathrm{\Pi }}}_a^{bi}|\Psi \rangle =0,{\widehat{R}}_{bij}^a\left(\omega \right)|\Psi \rangle =0,$$

(25)

which enforce the flatness of the spin–connection and restrict the quantum states to the physically admissible sector.

5.3. Torsion Operator and Gauge Structure

The torsion operator is defined as

$${\widehat{T}}_{\mu \nu }^a={\partial }_{\mu }{\widehat{h}}_{\nu }^a-{\partial }_{\nu }{\widehat{h}}_{\mu }^a+{\widehat{\omega }}_{b\mu }^a{\widehat{h}}_{\nu }^b-{\widehat{\omega }}_{b\nu }^a{\widehat{h}}_{\mu }^b.$$

(26)

It acts as a quantum field strength associated with translational gauge symmetry, with the coframe playing the role of a gauge potential. The spin–connection ensures that torsion transforms covariantly under local Lorentz transformations:

$$\begin{array}{cc}\hfill \delta {\widehat{h}}_{\mu }^a& ={ϵ}_b^a\left(x\right){\widehat{h}}_{\mu }^b,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \delta {\widehat{\omega }}_{b\mu }^a& =-D_{\mu }{ϵ}_b^a\left(x\right),\hfill \end{array}$$

(28)

where $D_{\mu }$ is the covariant derivative with respect to $\omega$. Physical states satisfy

$${\widehat{\mathcal{J}}}_{ab}|\Psi \rangle =0,$$

(29)

implementing Lorentz invariance at the quantum level [62,72].

5.4. Constraint Algebra and Dynamics

The quantum Hamiltonian $\widehat{\mathcal{H}}$, diffeomorphism generators ${\widehat{\mathcal{D}}}_i$, and Lorentz generators ${\widehat{\mathcal{J}}}_{ab}$ act on $\mathcal{H}$ as:

$$\widehat{\mathcal{H}}|\Psi \rangle =0,{\widehat{\mathcal{D}}}_i|\Psi \rangle =0,{\widehat{\mathcal{J}}}_{ab}|\Psi \rangle =0.$$

(30)

Closure of the constraint algebra is expected to follow the structure of gauge theories with first-class constraints, similar to Dirac’s canonical formulation of constrained systems [32,58,72]. This ensures consistency of quantum evolution within the physical subspace.

5.5. Physical Interpretation and Outlook

The results obtained in this work suggest that a formulation of gravity based on coframe/spin-connection pairs provides a conceptually coherent and physically meaningful framework for addressing the problem of QG. 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 [1,38]. Moreover, the teleparallel interpretation of gravity, in which torsion replaces curvature as the mediator of gravitational interactions, offers a gauge-like structure that parallels the successful description of other fundamental interactions. This perspective supports the idea that the true dynamical variables of gravity may be more closely related to local frame fields than to the metric tensor itself.

From a broader viewpoint, these findings resonate with key insights from existing approaches to QG. LQG emphasizes connection variables and discrete geometric structures [44,45], 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 potentially more geometrically transparent and background-flexible description. 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 coframe-based approach may provide a promising pathway toward a more unified and fundamental description of spacetime, but its viability will depend on its ability to connect with both established theory and empirical data.

6. Discussion and Outlook

The analysis presented in this work highlights the conceptual and structural limitations of QFTCS and of the principal approaches to QG. As discussed in Section 2, QFTCS provides a robust semi-classical framework for describing quantum matter on classical geometries, successfully accounting for phenomena such as particle creation and Hawking radiation [1,11]. However, its intrinsic background dependence and the ambiguity in the definition of vacuum states [36,37] prevent it from offering a fully consistent description of quantum spacetime. These shortcomings motivate the exploration of more fundamental formulations in which geometry itself is dynamical and subject to quantization.

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 [44,45], 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 [37]. Despite their diversity, these frameworks often retain a reliance—explicit or implicit—on metric-based structures or fixed backgrounds, leaving open the question of the true fundamental degrees of freedom.

In contrast, the teleparallel and coframe-based formulation developed in Section 4 offers a conceptually distinct perspective in which gravity is described by torsion rather than curvature, and the fundamental variables are the coframe and spin-connection pair [38]. This approach naturally accommodates fermionic fields and local Lorentz symmetry, and recasts gravity in a gauge-theoretic language closer to that of the Standard Model interactions. The coframe variables encode local geometric information in a manner that may be more directly amenable to quantization, while the spin–connection provides a natural structure for incorporating gauge symmetries. As emphasized in Section 5, this framework suggests that the metric tensor may be viewed as a derived quantity, rather than a fundamental one, pointing toward a reformulation of spacetime geometry in terms of more primitive variables.

Looking forward, several important challenges and opportunities arise. A key open problem is the construction of a consistent quantum theory based on coframe and spin-connection variables, including the identification of physical observables and the treatment of quantum torsion degrees of freedom. Establishing the precise relation between teleparallel gravity and other QG approaches—such as LQG or string theory—remains an important direction for future research. From a phenomenological standpoint, it is essential to investigate potential observational signatures of torsion-based dynamics, as well as their implications for cosmology and black hole physics. Overall, the teleparallel/coframe framework provides a promising and conceptually coherent pathway toward QG, but its ultimate viability will depend on its ability to bridge the gap between mathematical consistency and empirical relevance.