Gravitational Quantum Field Theory (GQFT)

• Gravitational Quantum Field Theory is an umbrella term for diverse QFT approaches that reformulate gravity using fields such as the gravigauge, fractional operators, or higher-derivative metrics.
• It employs methodologies like biframe spacetime frameworks and multiscale analyses to explicitly define propagators, vertices, and power-counting, aiming for a perturbatively unitary and renormalizable theory.
• The framework predicts non-GR gravitational-wave polarizations and spin-related forces, linking theoretical advances with weak-field tests and potential dark-sector signatures.

Gravitational Quantum Field Theory (GQFT) denotes a family of attempts to formulate gravity in explicitly quantum-field-theoretic terms rather than taking the spacetime metric alone as the fundamental starting point. The available literature suggests that the label functions as an umbrella term rather than a single canonical theory: it includes spin- and scaling-gauge constructions on a biframe spacetime with a bicovariant gravigauge field $\chi^a{}_\mu$, multiscale or fractal theories in which ultraviolet behavior is controlled by fractional operators $F(\Box)$, and local higher-derivative or Gupta–Feynman quantizations of Einstein or curvature-squared gravity (Wu, 2015, Anselmi, 2017, Briscese et al., 25 Mar 2026, Plastino et al., 2021). Across these strands, the shared objective is to place gravitational dynamics within a QFT framework, specify propagating degrees of freedom and interactions, and address renormalizability, unitarity, classical limits, and observational tests.

1. Scope of the term and recurrent objectives

In one major usage, GQFT treats gravity “on the same footing as the other gauge interactions” in a flat base spacetime while introducing an internal gravigauge spacetime and local spin or scaling gauge symmetries. In this line of work, the central dynamical field is a bicovariant gravigauge vector, the effective spacetime metric is composite, and general relativity appears as a low-energy or symmetry-reduced limit (Chen et al., 2024). A closely related formulation uses the inhomogeneous spin gauge symmetry $WS(1,3)=SP(1,3)\rtimes W^{1,3}$, a biframe spacetime, and a gauge–geometry duality with emergent $GL(1,3,\mathbb{R})$ covariance (Wu, 2022).

A second usage identifies GQFT with a UV-oriented quantum gravity program on multiscale or fractal-like spacetimes. Here the theory ultimately keeps the Lebesgue measure trivial, $v(x)=1$, and encodes multiscaling in covariant fractional operators $K(\Box)$ or $F(\Box)$. The stated aim is a perturbatively unitary, super-renormalizable field theory of gravity with explicit propagators, vertices, and power counting (Briscese et al., 25 Mar 2026).

A third usage applies the label to local higher-derivative quantum gravity, or to alternative quantizations of Einstein gravity, that seek a unitary and renormalizable—or at least mathematically controlled—QFT of the gravitational interaction. The best-defined example in this group is a local curvature-squared theory quantized with Lee–Wick-type prescriptions so that only the physical graviton propagates in cuts when the cosmological constant vanishes (Anselmi, 2017).

This diversity is not merely terminological. It indicates that “GQFT” names a research ambition shared across distinct frameworks: gravity should admit a field-theoretic description with identifiable dynamical variables, quantum amplitudes, and testable consequences, even if the underlying ontology—metric, gravifield, fractional operator, or higher-derivative field—is not the same.

2. Spin-gauge and biframe formulations

The spin-gauge family of GQFT is built on a biframe structure: a globally flat Minkowski coordinate spacetime and a locally flat non-coordinate gravigauge spacetime. The basic field is the gravigauge or gravifield $\chi^a{}_\mu(x)$, with inverse $\hat{\chi}^\mu{}_a$, and the effective metric seen by matter is the composite field

$\chi_{\mu\nu}=\chi_\mu^a\chi_\nu^b\,\eta_{ab}.$

This construction appears already in the spin- and scaling-gauge theory of gravity, where the gravifield is described as the fundamental gauge-type field and the bosonic sector can be rewritten in a hidden geometric form with a symmetric gravimetric tensor and a hidden $F(\Box)$0 or $F(\Box)$1 symmetry (Wu, 2015, Wu, 2022).

Within this program, local gauge symmetry is not limited to Lorentz rotations. Different papers use $F(\Box)$2, or its conformal extension $F(\Box)$3, together with a scaling symmetry $F(\Box)$4 and the Standard Model gauge group. The gravitational effect is attributed to the non-commutative structure of the fiber derivative operators, while the gravigauge field mediates gravity and is identified as the massless graviton in the projected Minkowski-space theory (Wu, 26 Feb 2025).

The effective low-energy field equation used for post-Newtonian analyses has the form

$F(\Box)$5

where $F(\Box)$6 is the correction tensor induced by the spin-gauge sector and only the combination

$F(\Box)$7

controls the weak-field phenomenology considered in that paper (Chen et al., 2024). In the broader 2025 extension to a “general theory of the standard model,” the same line of work introduces a “gravitization equation” for the gravigauge field strength and derives both gauge-type and geometric-type gravitational equations. The symmetric projected equation extends Einstein’s equation, while the antisymmetric part is sourced by spin-related structures and has no analogue in ordinary general relativity (Wu, 26 Feb 2025).

A distinctive theoretical claim of this family is gauge–geometry duality. In the 2024 “zero energy–momentum tensor theorem” paper, translational invariance plus the full equations of motion are said to imply $F(\Box)$8, interpreted as a cancellation law between matter/gauge contributions and gravigauge contributions, and as equivalent to the general gravitational equation in that formulation (Wu, 2024). In the 2015 and 2022 formulations, the same duality is expressed through the emergence of a geometric connection and curvature from the gravifield and spin gauge field, so that Einstein-like structures arise from a theory originally written as a gauge theory on a flat base spacetime (Wu, 2015, Wu, 2022).

3. Linear dynamics, weak-field tests, and gravitational waves

The linearized dynamics of the spin-gauge GQFT program differ sharply from those of general relativity. In vacuum, the linearized theory predicts five physical massless gravitational-wave polarizations—two tensor, two vector, and one scalar—with luminal dispersion $F(\Box)$9 (Gao et al., 2024). In the source-based gravitational-wave treatment, the same mode content is written as $WS(1,3)=SP(1,3)\rtimes W^{1,3}$0, with the symmetric part of the energy–momentum tensor sourcing scalar and tensor modes and the antisymmetric part sourcing scalar and vector modes (Gao et al., 26 Jun 2025).

In the Newtonian and post-Newtonian sector, the weak-field limit gives an exact relation between the fundamental gravitational coupling and the measured Newton constant,

$WS(1,3)=SP(1,3)\rtimes W^{1,3}$1

and yields $WS(1,3)=SP(1,3)\rtimes W^{1,3}$2 for the two scalar metric potentials in the static limit (Gao et al., 2024). The first post-Newtonian analysis maps the theory to PPN parameters,

$WS(1,3)=SP(1,3)\rtimes W^{1,3}$3

$WS(1,3)=SP(1,3)\rtimes W^{1,3}$4

and reports the bound

$WS(1,3)=SP(1,3)\rtimes W^{1,3}$5

obtained from Solar-System constraints, with the Mercury perihelion bound on $WS(1,3)=SP(1,3)\rtimes W^{1,3}$6 currently dominant (Chen et al., 2024).

The gravitational-wave phenomenology is more differentiated. In the linearized source analysis, tensor modes remain GR-like up to a prefactor $WS(1,3)=SP(1,3)\rtimes W^{1,3}$7, scalar breathing radiation can be emitted by eccentric binaries through the trace of the quadrupole, and vector radiation is tied to antisymmetric source structures such as time-varying net spin (Gao et al., 26 Jun 2025). The same paper argues that current interferometers can detect tensor and scalar modes, whereas vector modes require a detector sensitive to $WS(1,3)=SP(1,3)\rtimes W^{1,3}$8 or to spin-induced forces, because for free vector waves the geodesic-deviation observable satisfies $WS(1,3)=SP(1,3)\rtimes W^{1,3}$9 (Gao et al., 26 Jun 2025).

A detector-response study for LISA- and Taiji-like interferometers develops this point operationally. It identifies three observational features: characteristic interference patterns between polarization modes, distinctive null-point signatures enabling mode discrimination, and sky-position-dependent optimal detection windows. The emphasis is on polarization mapping over the sky while avoiding direct breathing-mode measurements that the paper describes as experimentally challenging (Xu et al., 2 Apr 2025). Taken together, these results place the spin-gauge GQFT program in a regime where its strongest near-term discriminants are not the Newtonian limit—already tightly bounded—but non-GR polarizations and spin-related source couplings.

4. Fractal and multiscale GQFT

A conceptually distinct theory appears in “Fractal universe and quantum gravity made simple,” where GQFT means quantum field theory on fractal or multiscale spacetimes. The explicit claim is that this program reaches a field theory of quantum gravity that is “super-renormalizable and unitary at all perturbative orders” (Briscese et al., 25 Mar 2026). The final theory keeps the measure trivial, $GL(1,3,\mathbb{R})$0, and moves the multiscale structure from the measure into covariant fractional operators built from $GL(1,3,\mathbb{R})$1.

The gravitational action in four dimensions is reconstructed as

$GL(1,3,\mathbb{R})$2

with

$GL(1,3,\mathbb{R})$3

and fractional form factor

$GL(1,3,\mathbb{R})$4

The same paper defines the Hausdorff and spectral dimensions in the usual multiscale way, and for $GL(1,3,\mathbb{R})$5 obtains $GL(1,3,\mathbb{R})$6. Dimensional flow is therefore encoded in the momentum-space kinetic operator rather than in a nontrivial measure (Briscese et al., 25 Mar 2026).

The propagator is given in spectral form,

$GL(1,3,\mathbb{R})$7

with a branch point at $GL(1,3,\mathbb{R})$8, no poles in the physical half-plane for $GL(1,3,\mathbb{R})$9, and $v(x)=1$0 for $v(x)=1$1. The paper interprets the continuum part as quartic Green functions associated with complex-conjugate modes and adopts the fakeon prescription so that only the physical graviton polarizations propagate on shell. Power counting gives

$v(x)=1$2

hence super-renormalizability for $v(x)=1$3 and one-loop super-renormalizability for $v(x)=1$4 (Briscese et al., 25 Mar 2026).

This framework is positioned explicitly against several established quantum-gravity programs. Compared with Stelle gravity, it seeks improved UV behavior without real ghost poles; compared with asymptotic safety, it does not rely on a nontrivial UV fixed point; compared with Hořava–Lifshitz gravity, it keeps Lorentz invariance at the level of the effective dynamics; compared with entire-function nonlocal gravity, it uses non-entire fractional powers and relies on fakeons for unitarity; and compared with CDT or LQG, it remains a continuum QFT with explicit propagators and vertices rather than a discrete microscopic model (Briscese et al., 25 Mar 2026).

Its phenomenology is correspondingly specific. The graviton remains massless with standard dispersion $v(x)=1$5, so current constraints on massive gravitons or Lorentz-violating propagation do not apply. The Newtonian potential behaves as $v(x)=1$6 in the infrared and as $v(x)=1$7 in the ultraviolet, exact Ricci-flat solutions such as Schwarzschild remain solutions of the minimal action, and the paper suggests the possible existence of regular microscopic black holes once the UV behavior is taken seriously or finite extensions are added (Briscese et al., 25 Mar 2026).

5. Other quantum-gravity programs also labeled GQFT

The term GQFT is also used for several additional proposals whose assumptions differ substantially from both the spin-gauge and the fractal/multiscale frameworks.

Program	Core construction	Principal claim or status
Local higher-derivative gravity (Anselmi, 2017)	$v(x)=1$8 with Lee–Wick doubling of unphysical poles	Local, renormalizable by power counting, and unitary in the $v(x)=1$9-matrix sense when $K(\Box)$0
Gupta–Feynman Einstein gravity (Plastino et al., 2021)	Schwinger–Feynman variational principle with a new physical-state constraint	No ghosts are introduced; the theory is explicitly non-renormalizable but treated with ultradistributions
Tetrad canonical GQFT (Matwi, 2019)	Canonical states for $K(\Box)$1 and conjugate $K(\Box)$2, path integral depending on $K(\Box)$3	Weak-field propagator and Newtonian potential are derived
Massive spin-2 SSB proposal (Pashitskii et al., 2017)	“Supermassive” tensor bosons acquire mass from spontaneous symmetry breaking	$K(\Box)$4 is proposed as the analogue of $K(\Box)$5, but consistency issues are left unresolved
Finsler gravitational quantum dynamics (Tavernelli, 2018)	Many-body Dirac-field quantum potential induces Finsler curvature	Quantum matter modifies geodesics through a Lorentz-covariant quantum potential

The local higher-derivative construction is the most conventional QFT among these alternatives. Its basic renormalizable Lagrangian is a linear combination of the Einstein–Hilbert term, $K(\Box)$6, $K(\Box)$7, and a cosmological term, and its distinctive move is the Lee–Wick or “fake degrees of freedom” prescription that doubles the unphysical poles and turns them into non-propagating Lee–Wick poles. The paper states that only the graviton propagates in cuts and that the model is unique, within its class, in having a dimensionless gauge coupling under high-energy power counting (Anselmi, 2017).

The Gupta–Feynman approach instead quantizes Einstein gravity with a traceless field $K(\Box)$8 using the Schwinger–Feynman variational principle rather than path integrals. It imposes a physical-state condition $K(\Box)$9 and $F(\Box)$0, and claims that this avoids the non-unitary sector without introducing Faddeev–Popov ghosts. The same paper explicitly acknowledges that the resulting theory is non-renormalizable and proposes ultradistributions as the device that makes perturbative calculations finite order by order (Plastino et al., 2021).

The remaining proposals are more heuristic or alternative in spirit. The tetrad-based 2019 paper treats $F(\Box)$1 as the quantum gravitational field, introduces a conjugate momentum field, and derives a weak-field propagator together with a Newtonian potential between scalar or spinor matter (Matwi, 2019). The 2017 spontaneous-symmetry-breaking model proposes a flat-spacetime theory of massive spin-2 exchange with $F(\Box)$2 and $F(\Box)$3, explicitly by analogy with weak-interaction boson exchange; the summary itself notes unresolved issues concerning the long-range force and massive spin-2 consistency (Pashitskii et al., 2017). The Finsler-space program does not quantize the metric at all, but geometrizes a many-body quantum potential derived from Dirac fields into a non-Riemannian Finsler structure that modifies geodesic motion (Tavernelli, 2018).

6. Conceptual tensions, misconceptions, and open questions

A recurrent source of confusion is to treat all uses of “GQFT” as if they referred to a single developing theory. The literature surveyed here suggests the opposite: the term is applied to incompatible starting points, different fundamental fields, and distinct criteria of success. In some papers the primitive variable is the gravigauge field $F(\Box)$4; in others it is a higher-derivative metric fluctuation, a fractional operator $F(\Box)$5, a tetrad $F(\Box)$6, or even a Finsler structure induced by a quantum potential. This suggests that GQFT is presently a classificatory label for quantum-field-theoretic approaches to gravity rather than a settled framework.

Within the spin-gauge line, open problems remain tightly coupled to phenomenology. Solar-System tests force the key weak-field combination $F(\Box)$7 to be very small, yet the same framework predicts five propagating polarizations, spin-related forces, and additional dark-sector gauge bosons such as the “dark graviton” of the chirality boost-spin sector. That combination of strong weak-field bounds and non-GR dynamical content makes gravitational-wave polarizations, spin-sensitive detection concepts, and dark-sector searches the natural next probes (Chen et al., 2024, Gao et al., 26 Jun 2025, Wu, 26 Feb 2025).

Within the multiscale/fractal program, the perturbative claims are unusually strong—super-renormalizability, perturbative unitarity, and possible finiteness after adding suitable nonlocal $F(\Box)$8 terms—but the paper itself lists nonperturbative definition, detailed cosmology, explicit beta functions for the finite extension, and exact regular black-hole solutions among the open problems (Briscese et al., 25 Mar 2026). In the local higher-derivative Lee–Wick program, the sharpest limitation is equally explicit: the perturbative unitarity proof is tied to $F(\Box)$9, and the status of the theory with nonzero cosmological constant remains unresolved (Anselmi, 2017).

Some of the more speculative usages of GQFT remain limited by unresolved consistency conditions. The massive spin-2 spontaneous-symmetry-breaking proposal states a bosonic-exchange relation for $\chi^a{}_\mu(x)$0, but the summary notes that the paper does not address the standard massive-spin-2 consistency problems or derive the observed long-range $\chi^a{}_\mu(x)$1 potential from its heavy mediator (Pashitskii et al., 2017). The Gupta–Feynman program explicitly accepts non-renormalizability and shifts the mathematical burden to ultradistribution theory (Plastino et al., 2021). Even within the spin-gauge school, the zero-energy–momentum cancellation law $\chi^a{}_\mu(x)$2 is so unlike the standard language of GR or semiclassical gravity that its interpretation is likely to remain a focal point of debate (Wu, 2024).

The present status of GQFT is therefore best understood as plural rather than singular. It names a continuing effort to reformulate gravity in QFT language, but the field is split between gauge-theoretic, higher-derivative, multiscale, and geometrized-quantum-dynamics programs whose agreements lie mainly at the level of ambition: quantization of gravity, control of ultraviolet behavior, preservation of unitarity, and recovery—or extension—of general relativity in the appropriate limit.