Quantized Gravity in ℝ⁴

• Quantized Gravity in ℝ⁴ is a framework that elevates classical curvature actions to quantum operator algebras, establishing a rigorous basis for quantum spacetime.
• It employs methods such as linearized gauged quantization, emergent matrix models, and canonical Wheeler-DeWitt approaches to address gauge invariance and renormalization challenges.
• The theory constructs well-defined symplectic structures, BRST-invariant gauge systems, and nonlocal QFT frameworks to ensure unitary, finite, and predictive quantum gravity models.

Quantized gravity in four-dimensional flat spacetime $\mathbb{R}^4$ refers to rigorous frameworks constructed to realize quantum field theories, algebraic QFTs, gauge theories, and canonical/bundle quantization of the gravitational field and its linearizations on the flat Minkowski manifold or its Euclidean counterpart. In these approaches, the Einstein-Hilbert or quadratic curvature actions are elevated to the status of quantum field operator algebras, with precise constructions for the symplectic structure, quantum states, commutators, and representation theory in both classical and gauge-fixed settings. The resulting theories range from algebraic QFTs for linearized gravitons, through canonical quantization with full Wheeler-DeWitt equations, to renormalizable and even finite quantum gauge theories for gravity, each distinguished by its mathematical infrastructure and physical predictions regarding the nature of quantum spacetime, gauge degrees of freedom, and the spectrum of quantum gravity excitations in $\mathbb{R}^4$.

1. Linearized Gravity and Algebraic Quantum Field Theory on $\mathbb{R}^4$

Linearized quantization starts from a background metric $g_{ab}$ satisfying $R_{ab} = \Lambda g_{ab}$ ($\Lambda$ the cosmological constant) and linearizes around small fluctuations $h_{ab}$: $$g_{ab}(\epsilon) = g_{ab} + \epsilon h_{ab} + O(\epsilon^2)$$ (Fewster et al., 2012). The linearized Einstein tensor operator is

$$E_g(h)_{ab} \equiv L_{ab}(h) = -\tfrac{1}{2} g_{ab} (\nabla^c \nabla^d h_{cd} - \nabla^2 h - \Lambda h) - \tfrac{1}{2} (\nabla_a \nabla_b h + g_{ab} \nabla^c \nabla^d h_{cd}) - \Lambda h_{ab} + \nabla_{(a} \nabla^c h_{b)c}$$

with $h = g^{cd} h_{cd}$.

Gauge freedom is induced by diffeomorphisms, $h_{ab} \mapsto h_{ab} + 2 \nabla_{(a} w_{b)}$, under which $E_g(\nabla_{(a} w_{b)}) = 0$. The physical solution space of linearized gravity is the quotient

$$\text{Sol} = \{ h_{ab} \mid E_g(h) = 0, \text{ spacelike-compact} \}/\sim$$

with equivalence under gauge transformations. The symplectic form on $\text{Sol}$ is

$$\sigma(h^{(1)}, h^{(2)}) = \int_\Sigma [ \nabla^a h^{(1) bc} h^{(2)}_{bc} - h^{(1) bc} \nabla^a h^{(2)}_{bc} ] d\Sigma_a$$

for any smooth Cauchy surface $\Sigma$. This symplectic product is antisymmetric, bilinear, gauge-invariant, and independent of $\Sigma$ when solutions are spacelike-compact.

Classical observables are "smeared fields" $$F_f([h]) = \int_M h^{ab}(x) f_{ab}(x) d\text{vol}_g(x)$$ for test tensors $f_{ab}$ satisfying $\nabla^a f_{(ab)} = 0$ for gauge invariance. Algebraic quantization proceeds by constructing a *-algebra $\mathcal{A}(M, g)$ generated by field operators $\Phi(f)$, subject to linearity, adjointness, field equations, and commutation relations

$[ \Phi(f), \Phi(g) ] = i \sigma(f, g) \mathbf{1}$

with symplectic form $\sigma$ as above.

Specializing to Minkowski space $\mathbb{R}^4$ ($\Lambda = 0$), the transverse-traceless (TT) gauge $\partial^i h_{ij} = 0, h_{0\mu} = 0, h^i{}_i = 0$ yields the graviton mode expansion

$$h_{ij}(x) = \int \frac{d^3k}{(2\pi)^3 2|\mathbf{k}|} \sum_{s=1}^2 \left[ \epsilon_{ij}^s(\mathbf{k})\,a_s(\mathbf{k})\,e^{-i|\mathbf{k}| t + i\mathbf{k}\cdot\mathbf{x}} + \epsilon_{ij}^{s*}(\mathbf{k})\,a_s^\dagger(\mathbf{k})\,e^{i|\mathbf{k}| t - i\mathbf{k}\cdot\mathbf{x}} \right]$$

with commutation relations

$$[a_s(\mathbf{k}), a_{s'}^\dagger(\mathbf{k'})] = (2\pi)^3 2|\mathbf{k}| \delta_{ss'} \delta^3(\mathbf{k} - \mathbf{k}')$$

representing canonical quantization of spin-2 massless quanta.

Hadamard states (quasi-free) are characterized by two-point functions with the microlocal spectrum condition, and the vacuum state yields a manifestly gauge-invariant, positive-energy spectrum for the graviton field (Fewster et al., 2012).

2. Gauge, Matrix, and Emergent Quantum Gravity

Matrix-model and emergent approaches demonstrate quantizing gravity in $\mathbb{R}^4$ as a quantum effect of underlying matrix variables. The IKKT (IIB) matrix model, governed by

$$S_{\rm IKKT} = -\text{Tr}([Y^a,Y^b][Y_a,Y_b]) + (\text{fermions})$$

where $Y^a$ are matrices and $[Y^a, Y^b]$ encodes noncommutative geometry, induces gravity by integrating over backgrounds of the form $\bar{Y}^a = (X^\mu, Z^i)$, where $X^\mu$ generate a noncommutative $3+1$-brane in $\mathbb{R}^4$, and $Z^i$ provide compact fuzzy extra dimensions (Steinacker, 2021).

The emergent metric reads

$$G_{\mu\nu}(x) = \rho^{-2}(x) g_{ab} E^a{}_\mu(x) E^b{}_\nu(x)$$

with $\rho^4 = \det (\theta^{-1}_{\mu\nu})$, and matrix model one-loop expansions yield a finite induced Einstein-Hilbert term: $$\Gamma_{1\text{-loop}} \approx \int_{M_4} d^4x\,\sqrt{|G|} \left\{ \frac{1}{16\pi G_N} R[G] + \cdots \right\}$$ where the induced Newton constant is $1/G_N \sim m_{\rm KK}^2$ with $m_{\rm KK}$ set by the Kaluza-Klein scale of the compact space. All power-law divergences are canceled by maximal supersymmetry.

This matrix model is fully quantized and sidesteps the cosmological constant problem, as the vacuum energy term depends only on the symplectic volume, not the metric, decoupling it from gravitational dynamics (Steinacker, 2021).

3. Canonical, Bundle, and Wheeler-DeWitt Quantization

Canonical quantization in Minkowski space employs the ADM $3+1$ decomposition, eliminating shift and introducing the Hamiltonian constraint

$$H = g^{-1/2} (\pi_{ij} \pi^{ij} - \tfrac{1}{2} \pi^2 ) - g^{1/2} ( R - 2\Lambda )$$

while restricting to block-diagonal metrics and using a fixed Euclidean metric as a reference for volume densities (Gerhardt, 2012). Each spatial point $x\in\Sigma$ is associated with a fiber $F(x)$ of positive-definite symmetric matrices, and the quantized field solves the Wheeler-DeWitt functional equation in the bundle $E = \Sigma \times F(x)$,

$H u = 0$

where $H$ is a normally hyperbolic operator acting only in the fiber. The bundle construction admits a well-posed Cauchy problem for quantum gravity, and quantization proceeds via symplectic forms from Green functions and bosonic Fock space representations.

This structure generalizes to include matter: unified quantization of gravity with Yang–Mills, Higgs, and spinor fields yields second quantized CCR algebras for bosons and CAR algebras for spinors, with the physical solution space determined by invariance under gauge and diffeomorphism group actions, and the net of Haag–Kastler local algebras encoding quantum locality (Gerhardt, 2013, Gerhardt, 2022).

4. Gauge-Theoretic and Nonlocal QFT Approaches

Gravity admits formulation as a renormalizable gauge theory for a Lorentz (SO(1,3)) connection $B_{a\mu}$, where the effective vierbein $e^a_\mu$ is constructed from $B_{a\mu}$ (Wiesendanger, 2019). Gauge fixing (Lorentz gauge), Faddeev–Popov ghosts, physical Fock space with appropriate positivity constraints, BRST symmetry and Zinn–Justin identities enforce quantum consistency and control renormalization, so that all divergences are absorbed into field and coupling redefinitions.

Nonlocal gravity proposes a quantization of general relativity as a gauge theory of local translations (infinite-dimensional nonlocal group) plus local O(4) spin connection with no dynamical metric or tetrad (Diego, 2010). The action is a Yang–Mills-type trace over the curvature of a nonlocal covariant derivative. Path integral, gauge fixing, and Faddeev–Popov quantization proceed analogously to standard Yang–Mills. Propagators and Feynman rules are derived, and power-counting shows the theory is renormalizable in $\mathbb{R}^4$, with all divergences removable by counterterms already present in the action.

5. Higher-Derivative, Finite, and Ghost-Free Quantum Gravity

Quadratic curvature extensions of the Einstein-Hilbert action introduce higher-derivative operators, with the path integral or canonical quantization analyzed using Ostrogradsky formalism. Canonical quantization in either Dirac–Pauli negative-norm representation or BFV extended phase space resolves the classical instability. The spectrum comprises massless spin-2 gravitons, massive spin-2 ghosts, and massive spin-0 modes (Bellorin et al., 19 Nov 2025, Salvio et al., 2015). For appropriately chosen form factors (entire functions), all higher-derivative quantum gravity models can be rendered super-renormalizable, and with further construction, finite (Modesto, 2013). Gauge fixing, Fock–space quantization, and sensitivity to negative-norm states are controlled so that the quantum evolution remains unitary in an indefinite-metric Kreĭn space.

6. Algebraic and Conformal Field Representations

Manifestly covariant, non-perturbative algebraic QFT for gravity in four dimensions is formulated by generating $C^*$-algebras from curvature tensor operators and vector fields, acting on indefinite-unitary representations of the diffeomorphism group $\text{Diff}^+(M)$ (Etesi, 2014). The net of local subalgebras obeys Haag–Kastler isotony, covariance, and locality, and several physically significant representations are constructed: the tautological representation (defining a von Neumann algebra), positive-mass (quasilocal energy–momentum) GNS representations, and classical Einstein metric representations (recovering general relativity in the semiclassical sector). The space of positive-mass representations forms a two-dimensional conformal field theory in the sense of Segal.

7. Loop, Spin-Network, and Gauge Group Quantization

Loop quantum gravity on $\mathbb{R}^4$ and related spaces can be reformulated as quantization of the Holst action with internal gauge group reductions: by partial gauge fixing, $\mathfrak{su}(2)$ is traded for $\mathfrak{su}(1,1)$, yielding new canonical pairs and $\mathfrak{su}(1,1)$-valued connections (Liu et al., 2017). Spin network states on fixed graphs are labeled by unitary irreducible representations of $SU(1,1)$, leading to discrete spectra for space-like area operators and continuous spectra for time-like areas. This construction displays the non-anomalous matching of spectra to the usual $SU(2)$-based LQG, accommodates quantization on Lorentzian hypersurfaces, and provides avenues for black-hole holography and covariant canonical quantization on timelike slices.

Table: Comparison of Major Quantization Frameworks for Gravity in $\mathbb{R}^4$

Approach	Algebra/State Space	Key Technical Feature
Algebraic QFT	Symplectic quotient, *-algebra	Rigorous solution space, Hadamard states
Matrix/Emergent	Matrix variables, induced metric	1-loop E-H generation, fuzzy dimensions
Canonical/Bundled	Fiber bundle, Wheeler-DeWitt	Globally hyperbolic fibers, well-posed QFT
Gauge (SO(1,3))	Gauge field, Fock space	BRST, Zinn–Justin, renormalizability
Non-local Gauge	Infinitesimal translation group	Non-metric, power-counting renorm.
Higher-Derivative	Dirac–Pauli/Krein space	Ghost resolution, super-renorm.
Algebraic C*-QFT	C*-algebra, GNS	Diffeo symmetry, CFT sector
Loop/Spin-Network	SU(2)/SU(1,1) spin networks	Area spectrum, hypersurface quantization

Each framework rigorously quantizes aspects or full sectors of the gravitational field in flat $\mathbb{R}^4$, maintaining gauge invariance, covariance, and carefully analyzing the spectrum, renormalizability, and unitarity structure that governs quantum spacetime.