Lorentzian Quantum Gravity

Updated 26 July 2025

Lorentzian Quantum Gravity is a research program that quantizes general relativity while preserving the causal Lorentzian signature of spacetime, avoiding the pitfalls of Wick rotation.
It integrates both continuum and discrete methods—including causal dynamical triangulations, Regge calculus, and causal sets—to achieve robust ultraviolet fixed points and emergent classical behavior.
Advanced techniques such as Lorentzian path integrals, affine group quantization, and spectral geometry are employed to address challenges in unitarity, regulator choices, and continuum limits.

Lorentzian Quantum Gravity is the broad research program aimed at quantizing general relativity while rigorously maintaining the causal, Lorentzian signature of spacetime at the quantum level. Distinguished from approaches relying on formal Euclideanization (Wick rotation), Lorentzian quantum gravity frameworks insist on causal structure, signature, and dynamical time as irreducible physical data, both nonperturbatively and in the continuum or discrete regularizations. The field encompasses asymptotic safety scenarios, causal dynamical triangulations, Regge calculus, causal sets, spin foam models, and path integral approaches—each of which seeks to respect the fundamental Lorentzian nature of spacetime while formulating mathematically sound and predictive theories of quantum gravity.

1. Covariant Lorentzian Renormalization Group Methods and Asymptotic Safety

The asymptotic safety scenario for quantum gravity, motivated by the search for an ultraviolet (UV) complete and predictive theory, has traditionally been developed in the Euclidean sector. Recent advancements have generalized the Functional Renormalization Group Equation (FRGE) to Lorentzian signature, incorporating causal structure and addressing longstanding ambiguities related to Wick rotation and the conformal factor problem (1102.5012, D'Angelo, 2023, D'Angelo et al., 7 Feb 2025, Ferrero et al., 28 Apr 2024).

A typical Lorentzian FRGE is formulated as

$\partial_k \Gamma_k(\bar{g};\phi) = \frac{i}{2} \int_{\mathcal{M}} \mathrm{Tr}\big[ \partial_k k(x) :G_k:(x,x) \big]$

with $\Gamma_k$ the effective average action on a globally hyperbolic spacetime ( $\bar{g}$ ), $k(x)$ a local, Lorentzian-invariant infrared regulator, and $:G_k:(x,x)$ the Hadamard-normal ordered interacting propagator. Importantly, regulators are chosen to be functions of the spatial Laplacian to avoid the unboundedness and noncompactness of the Lorentzian d'Alembertian, thus preserving causality (1102.5012).

Lorentzian RG analyses in the Einstein-Hilbert truncation consistently find a non-Gaussian UV fixed point:

Euclidean: $g_* \approx 0.19$ , $\lambda_* \approx 0.31$
Lorentzian: $g_* \approx 0.21$ , $\lambda_* \approx 0.30$ with critical exponents nearly identical in both signatures, indicating both universality and the insensitivity of high-energy behavior to spacetime signature (1102.5012, D'Angelo, 2023). More recent studies extend this to cosmological (de Sitter) backgrounds, using graviton propagators constructed with full gauge and mass dependence and explicit Hadamard subtraction, confirming fixed points are robust under gauge and renormalization-scheme changes (D'Angelo et al., 7 Feb 2025).

These advances establish that UV completeness ("asymptotic safety") is achievable within a fully Lorentzian, locally covariant QFT framework, with causal structure preserved and strong background independence via Hadamard state subtraction (D'Angelo, 2023).

2. Discrete Lorentzian Gravity: Causal Triangulations, Regge Calculus, and Causal Sets

Discrete approaches enforce causal structure at the level of spacetime microstructure, distinguishing them categorically from their Euclidean counterparts (1108.4965, Borissova et al., 2023, Eichhorn, 2017, Eichhorn, 2019, Brunekreef et al., 2022, Duin et al., 26 Apr 2024).

Lorentzian Dynamical Triangulations (CDT): Lorentzian DT models build the path integral from simplices that admit a foliation into spatial slices, each respecting light-cone structure. These triangulations avoid "triangle inequalities" inhibiting Lorentzian topologies—unlike in Euclidean DT, time-like edges in Lorentzian simplices are unbounded (1108.4965, Brunekreef et al., 2022).
Quantum Regge Calculus in Lorentzian Signature: The Regge action in Lorentzian signature is built from the signed squared edge lengths. The path integral, after complexifying dihedral angles and volumes, allows for the use of advanced integration-contour (e.g., Picard–Lefschetz) methods, treating "wrong-sign" (indefinite) directions as physically meaningful and avoiding the conformal factor problem (Borissova et al., 2023).
Causal Set Quantum Gravity: Causal sets postulate a locally finite, partially ordered discrete set as the substratum of spacetime, with causality built in at the kinematic level. All physically meaningful actions and observables are written as label-invariant functions of the link matrix. An exact matrix-model RG flow can be constructed even in the absence of background geometry (Eichhorn, 2017). Tools have been developed to recover continuum-like geometric features, such as spatial distances, directly from the partial order and counting structure (Eichhorn, 2019). Coarse-graining and renormalization-group equations have been constructed for these settings, enabling nonperturbative investigations of universality and continuum limits.

In all these approaches, the Lorentzian structure fundamentally shapes the configuration space, the allowed measure, and the quantum dynamics, leading to discretized models that rigorously respect causality and the temporal direction. Analyses of Hausdorff dimensions, entropy exponents, Ricci curvature, and universality classes in 3D CDT suggest strong emergence of expected quantum-geometric behavior in the appropriate limits (Brunekreef et al., 2022, Duin et al., 26 Apr 2024).

3. Lorentzian Path Integral Techniques and Quantum Cosmology

Path integral formulations in Lorentzian signature, particularly relevant to quantum cosmology, face technical challenges due to oscillatory integrands and boundary-condition ambiguities. Modern treatments employ Picard–Lefschetz theory to deform integration contours for lapse and metric variables, converting conditionally convergent oscillatory integrals into absolutely convergent thimble integrals (Feldbrugge et al., 2017, Honda et al., 29 Dec 2024, Rosabal, 2023).

Lorentzian Path Integrals in Minisuperspace: By focusing on homogeneous, isotropic metrics and employing suitable boundary conditions, the path integral is reduced to lower-dimensional integration over lapse functions, with semiclassical saddle points identified in the complex plane. Only those saddles which can be reached from the original Lorentzian contour (as determined by upward flow intersection numbers) contribute (Feldbrugge et al., 2017, Honda et al., 29 Dec 2024).
Unitary and Non-unitary Evolution: Precise treatment of boundary conditions on Lorentzian strips uncovers non-unitary time evolution under generic conditions, while unitarity can recover under specific spatial AdS $_2$ limits (Rosabal, 2023). For Dirichlet boundaries, the amplitude reduces to modified Bessel $K_0$ functions of complex arguments, with saddle points classified via Lefschetz thimble structure (Honda et al., 29 Dec 2024).
Hartle–Hawking No-Boundary and Vilenkin Tunneling: In Lorentzian JT gravity, the no-boundary proposal is closely approximated: for specific parameter sectors, the dominant contributing saddle lies in the complex hemisphere, yielding an exponentially suppressed amplitude that realizes the spirit of the no-boundary prescription (Honda et al., 29 Dec 2024, Feldbrugge et al., 2017).
Perturbations and Quantum Genesis: Lorentzian JT gravity analysis shows that a well-defined quantum genesis in 2D is achieved only if the initial dilaton dominates, ensuring perturbative regularity for the resulting two-dimensional universe (Honda et al., 29 Dec 2024).

These techniques resolve the conformal factor problem—ubiquitous in Euclidean gravity—by making the correct choice of integration thimbles a consequence of the analytic structure of the Lorentzian action.

4. Algebraic and Representation-Theoretic Approaches: Affine Quantization and Loop Variables

Maintaining the causal structure at the quantum level affects not only the dynamics and path integrals, but also the algebraic and representation-theoretic frameworks:

Affine Group Quantization: The reduction of the Hamiltonian constraint in self-dual Ashtekar variables to a commutator between the imaginary part of the Chern–Simons functional ( $Q$ ) and the volume operator ( $V$ ),

$[Q, V(x)] = -i (G\Lambda) V(x)$

shows that gravitational quantum mechanics in 4D Lorentzian signature is naturally formulated as an affine algebra (1207.7263). This algebra requires a nonzero cosmological constant, introduces new uncertainty relations (e.g., $(\Delta V/\langle V\rangle)\Delta Q\geq 2\pi\Lambda L^2_{Pl}$ ), and leads to Hilbert spaces constructed from unitary irreducible representations of the affine group.

Loop Quantum Gravity and Spectral Geometry: Lorentzian loop quantum gravity approaches, particularly in the spinfoam setting, generalize Connes' spectral triple construction. The spinfoam Dirac operator, dressed with time-like Minkowski vectors associated with edges, encodes a notion of Lorentzian spectral distance. The associated Hilbert space is Lorentz-covariant (rather than invariant), reflecting the absence of a universal notion of invariant time in Lorentzian quantum gravity (Rovelli, 2014). The space of Dirac operators is a finite product over future light-cones, providing a geometric moduli space for discrete quantum geometries.

Through these algebraic structures, Lorentzian signature enters not only kinematically through the gauge group ( $SL(2,\mathbb{C})$ ) but also dynamically in the construction of physical Hilbert spaces and coherent states.

5. Physical Observables, Correlators, and Emergent Classicality

Formulating physical observables in a dynamically fluctuating Lorentzian geometry requires diffeomorphism-invariant constructions. Recent work involves several strategies:

Diffeomorphism-Invariant Correlators: Nonperturbative 2D Lorentzian quantum gravity (using CDT) enables manifestly invariant two-point curvature correlators using the quantum Ricci curvature, constructed via lattice-regularized, geodesic-distance-separated averages. Monte Carlo simulations show that the connected correlator vanishes at long distances, consistent with the absence of propagating degrees of freedom (Duin et al., 26 Apr 2024).
Emergent Classicality and Partial Observability: In closed universes with UV wormholes, the gravitational Hilbert space reduces to a one-dimensional, real sector for each set of "α-parameters". Projections onto a basis of states become meaningless; however, upon restricting to subsystems accessible to observers, tracing over the environment yields reduced density matrices that support classical observables with probabilities and pointer bases, and exponentially suppressed uncertainties proportional to environmental entropy (Nomura et al., 26 May 2025). This provides a mechanism whereby quantum gravity in closed Lorentzian universes naturally produces robust, classical predictions without external augmentation.
Cosmological Perturbations from Quantum Gravity: Group field theory condensates, with both spacelike and timelike quanta, allow causal reference frames to be implemented through four massless scalar fields. Entanglement among geometric "atoms" sources small scalar inhomogeneities, which reproduce classical cosmological perturbation theory on super-horizon scales but yield quantum gravitational corrections for sub-horizon (trans-Planckian) modes (Jercher et al., 2023).

These developments solidify the operational significance of Lorentzian quantum gravity, showing that meaningful predictions and observable effects can be extracted even in nonperturbative and background-independent frameworks.

6. Challenges and Future Directions

While significant technical and conceptual progress has been achieved, Lorentzian quantum gravity research continues to face open challenges:

Regulator Choices, Gauge Dependence, and Scheme Ambiguities: Physical predictions—such as the precise location and critical exponents of UV fixed points—may exhibit dependence on gauge fixing and finite renormalization choices. Although the existence and universality of fixed points is robust, quantitative universality classes may require further control, especially in higher truncation or beyond the Einstein-Hilbert approximation (D'Angelo et al., 7 Feb 2025).
Discrete-to-Continuum Transitions and Universality: Rigorous demonstrations of continuum limits, particularly in causal set and triangulated approaches, remain challenging, with universality classes only partially mapped—especially in $3+1$ dimensions (Eichhorn, 2017, Brunekreef et al., 2022).
Incorporation of Matter, Observables, and Relational Quantization: Relational approaches that use physical "clock and rod" fields to reduce the phase space and construct gauge-invariant path integrals constitute a promising bridge between canonical quantization and asymptotic safety (Ferrero et al., 28 Apr 2024). However, their full extension to realistic matter couplings and observables remains under active investigation.
Unitarity, Topology Change, and the Structure of Amplitudes: The identification of correct boundary terms, the precise impact of topology change, and insight into when physical amplitudes preserve or violate unitarity have only recently been systematically addressed in models such as JT gravity (Rosabal, 2023, Honda et al., 29 Dec 2024).
Phenomenology and Observable Effects: The construction of physically meaningful predictions, such as cosmological fluctuation spectra or graviton scattering amplitudes, based on Lorentzian quantum-gravitational computations, is a major frontier, relying on further development of both foundational and computational techniques (Fehre et al., 2021, Jercher et al., 2023).

Continued cross-fertilization between renormalization group theory, canonical/relational quantization, discrete regularizations, and numerical simulations is likely to drive the next advances in Lorentzian quantum gravity research.