Deterministic Semiclassical Gravity

Updated 24 December 2025

Deterministic semiclassical gravity is a framework that couples a classical metric with uniquely determined quantum matter field functionals, ensuring fixed evolution via Einstein’s equations.
It relies on precise initial data and Hadamard quantum states to solve a coupled system of PDEs, enabling unique, deterministic spacetime evolution in symmetric settings.
Model implementations, including ultradense star analyses and Bohmian gravity approaches, illustrate both its classical recovery capabilities and limitations such as post-Newtonian inconsistencies.

Deterministic semiclassical gravity is a framework in which the gravitational interaction is described by a classical metric field sourced by specific, uniquely determined functionals of quantum matter fields, without intrinsic randomness in the metric evolution. Unlike fully quantum or stochastic gravity approaches, deterministic semiclassical gravity insists that, given suitable sets of initial data, the ensuing classical geometry evolves according to Einstein’s equations (or some generalization thereof) where the source term is constructed in a state-dependent, but deterministic, manner. While quantum matter fields may possess intrinsic probabilistic or nonlocal structure, the coupling to gravity is implemented via regularized, expectation-valued, or explicitly selected features of the quantum matter sector that allow for well-posed evolution of the classical geometry. This article surveys the principles, main realizations, mathematical structures, and limits of deterministic semiclassical gravity.

1. Foundational Equations of Deterministic Semiclassical Gravity

The central object of deterministic semiclassical gravity is the semiclassical Einstein equation: $G_{\mu\nu}[g]+\Lambda\,g_{\mu\nu} = 8\pi G_N\,\langle T_{\mu\nu}(x)\rangle_{\omega} + \mathcal{C}_{\mu\nu}$ where $g_{\mu\nu}$ is the classical Lorentzian metric, $G_{\mu\nu}$ the Einstein tensor, $\Lambda$ the cosmological constant, and $\mathcal{C}_{\mu\nu}$ possible finite local curvature counterterms arising from the renormalization ambiguity (e.g., multiples of the conserved quadratic tensors $I_{\mu\nu}$ , $J_{\mu\nu}$ ) (Juárez-Aubry, 2 Sep 2025). The right-hand side is the renormalized expectation value of the quantum stress–energy tensor in a (typically Hadamard) quantum state $\omega$ . This equation is intrinsically deterministic: once the quantum state and classical geometric initial data are specified, all subsequent evolution of the geometry is fixed (modulo diffeomorphism freedom).

Physically, this framework describes quantum matter fields (usually QFT in curved spacetime) evolving in a dynamically determined but not quantized metric background, with the backreaction encoded deterministically via $\langle T_{\mu\nu}(x)\rangle$ (Juárez-Aubry et al., 2022, Juárez-Aubry, 2 Sep 2025).

In alternative deterministic semiclassical schemes, such as Bohmian semiclassical gravity (Struyve, 2019), the metric is sourced by the actual (“beable”) field configuration $\phi_B(x)$ rather than any expectation value: $G_{\mu\nu}[g] = 8\pi G\,T_{\mu\nu}[\phi_B]$ where $\phi_B$ follows the guidance equations associated with the quantum wave functional, maintaining deterministic evolution on both the matter and gravitational side.

2. Mathematical Structure and Initial Value Problem

The semiclassical Einstein equation is a system of coupled PDEs where classical geometry and quantum matter influence each other. The mathematical difficulty arises because the renormalized $\langle T_{\mu\nu}(x) \rangle$ is a nonlocal, distributional quantity whose definition typically involves the Hadamard subtraction scheme (to regulate divergences) and the underlying state must be specified on a Cauchy surface (Juárez-Aubry, 2 Sep 2025, Juárez-Aubry et al., 2022).

The initial value formulation, crucial for determinism, requires:

Specification of classical Cauchy data $(h_{ij}, K_{ij})$ (induced 3-metric and extrinsic curvature) on a spacelike hypersurface, along with higher time derivatives if the theory is of higher order due to renormalization ambiguities.
Specification (at the same Cauchy surface) of a Hadamard quantum state for the matter fields that encodes the short-distance structure ensuring finiteness of $\langle T_{\mu\nu}\rangle$ .
For the full (fourth-order) system, higher derivatives may be required, but a central result is that imposing smoothness in Planck’s constant $\hbar$ at $\hbar\rightarrow 0$ (the “physical solution” criterion) eliminates runaway or unphysical solutions: only the two lowest derivatives remain as genuine free data, as in classical GR(Juárez-Aubry et al., 2022).

Once these data are given, deterministic evolution obtains: there is a unique solution to the coupled system in suitable function spaces, subject to current existence and uniqueness theorems for highly symmetric cases (static, FLRW, or spherically symmetric spacetimes) (Juárez-Aubry, 2 Sep 2025, Juárez-Aubry et al., 2022).

3. Examples and Model Implementations

3.1. Regularized Quantum Backreaction and Ultradense Stars

A particularly clear illustration of deterministic semiclassical gravity occurs in the analysis of constant-density ultracompact stars, where the vacuum polarization of a quantized scalar field (modeled via a regulated Polyakov RSET) is included as a source for the static Einstein equations (Arrechea et al., 2021). This produces coupled ODEs for the metric functions and matter profiles, admitting deterministic numerical construction of spacetimes that violate the classical Buchdahl bound (compactness $C(R)>8/9$ ), resulting in singular, ultracompact objects with regular exteriors and nontrivial internal structure.

3.2. Self-Gravitating Quantum Fields in Spherical Symmetry

Quantized Dirac fields in a static, spherically symmetric spacetime, coupled to the metric via the expectation value of the normal-ordered stress–energy operator in specific Fock states, yield self-consistent ODE systems whose solutions are static, stationary “Dirac stars.” This construction is strictly deterministic once the quantum Fock state is chosen; no ensemble averaging or collapse prescription is involved (Kain, 2023).

3.3. Bohmian and de Broglie–Bohm-Based Models

Alternative deterministic realizations invoke the Bohmian paradigm, in which the classical metric is sourced by the energy–momentum tensor computed from the actual configuration of a real scalar or field, evolving under the guidance of a quantum wave functional. In double-scale theory, both “external” (pilot-wave) and “internal” (localized density) wavefunctions play a role; the actual internal profile sources gravity, while the external guides center-of-mass evolution (Struyve, 2019, Gondran et al., 2023).

Scheme	Matter Source for $G_{\mu\nu}$	Description
Standard semiclassical	$\langle \hat{T}_{\mu\nu} \rangle$	Expectation in state $\omega$
Bohmian	$T_{\mu\nu}[\phi_B]$	Actual field (beable)
de Broglie–Bohm	$T_{\mu\nu}[\|\psi_\text{int}\|^2]$	Internal density (localized)

4. State Space, Cat States, and Validity Range

Traditionally, the semiclassical framework was believed valid only for quantum states close to classical configurations, such as coherent or squeezed states. Recent results demonstrate that more general “cat” states (nontrivial superpositions of almost-orthogonal coherent states) can also yield sharply peaked stress–energy expectation values and thus enable deterministic semiclassical gravity, provided certain overlap or phase relations are satisfied (Ahmed et al., 2023). Explicitly,

results in expectation values and higher moments for $\hat{T}_{\mu\nu}$ that approximate classical values to $O(|\langle \alpha | \beta \rangle |)$ corrections. Thus, even macroscopically distinct superpositions can be treated deterministically under certain conditions without invoking stochastic or fully quantum-gravitational corrections.

5. Physical Implications, Limitations, and Controversies

5.1. Post-Newtonian Consistency Bounds

Despite its deterministic character, semiclassical gravity with expectation-value sourcing encounters severe limitations. At first post-Newtonian (1PN) order, state-dependent corrections arise that do not have Planck-mass suppression and dramatically violate known experimental constraints from laboratory and astrophysical systems (Williams, 21 Dec 2025). The scaling of these anomalous terms,

$V_{1\rm PN}^{(\rm SC)} \sim \frac{G^2\,m_t}{r^2} \frac{\hbar}{m_s\,\sigma^2}$

leads to enhancements by many orders of magnitude over classical GR, signaling that deterministic semiclassical gravity, as usually formulated, cannot be a complete relativistic theory for arbitrary quantum superpositions.

5.2. Interpretational Developments and Decoherece-Conditioned Determinism

Recent proposals introduce “Stable Determination Chains” (SDCs): only quantum matter that interacts with its environment (typically via decoherence processes satisfying specific localization, stability, and causal criteria) actually sources a gravitational field (Pipa, 7 Jul 2025). In this framework, the classical metric arises only after outcome selection by the SDC; before that, quantum superpositions remain invisible to gravity, which effectively restores determinism for the classical field once branch selection has occurred. This undermines naive universal coupling, introduces a form of “conditional determinism,” and connects gravity’s emergence to the quantum measurement process.

5.3. Collapse and Energy-Driven Selection

A related approach investigates gravitational self-energy penalties for superposed mass configurations, resulting in finite decay times for quantum superpositions and reproducing Born’s rule for reduction probabilities (Quandt-Wiese, 2017). Collapse occurs with a lifetime $\tau \sim \hbar/\Delta E$ , where $\Delta E$ is the Diósi–Penrose gravitational energy difference—again, deterministic evolution within each sector but collapse at a definable timescale.

5.4. Mathematical and Physical Open Problems

Full mathematical well-posedness in generic (non-symmetric, non-analytic) settings remains conjectural, though “ $\hbar$ -smooth” (physical) solutions appear to recover determinism by eliminating runaway solutions (Juárez-Aubry, 2 Sep 2025, Juárez-Aubry et al., 2022).
Definition and prescription of Hadamard initial data for the matter sector is nontrivial and typically only tractable in high-symmetry or truncated-moment frameworks.
The behavior of the coupled system under gravitational collapse (e.g., black hole evaporation) and information loss remains an open area of investigation, with deterministic semiclassical gravity suggesting singularity development rather than Cauchy horizon extension (Juárez-Aubry, 2 Sep 2025).

Several extensions and alternatives illustrate the spectrum of deterministic semiclassical gravity:

Weyl (Conformal) Gravity: Deterministic semiclassical limits can be obtained for higher-derivative gravity theories, such as Weyl gravity, where the leading WKB expansion yields a Hamilton–Jacobi equation for conformally invariant shape variables, with matter fields governed deterministically by Schrödinger-type equations (Kiefer et al., 2016).
Quantum Measurement-Based Theories: SDC and event-based frameworks tie the determinism of gravity to the actualization of measurement outcomes or decohered branches, decoupling the gravitational field from non-decohered quantum superpositions (Pipa, 7 Jul 2025).
Bohmian and dBB Gravity: Explicit deterministic prescriptions replace the metric coupling to quantum expectation values with coupling to the actual configuration trajectories (“beables” or internal densities) (Struyve, 2019, Gondran et al., 2023).

7. Summary Table: Key Deterministic Semiclassical Gravity Approaches

Approach	Metric Source	Principle of Determinism	Key Limitation
Expectation-value (Hadamard)	$\langle \hat{T}_{\mu\nu} \rangle$	Fixed by initial state and data	State-dependent 1PN terms; Hadamard data
Bohmian (beable-sourced)	$T_{\mu\nu}[\phi_B]$	Guided by wave functional, unique	Empirical interpretational challenge
SDC/event-based	Decohered branch projection	Outcome selected by SDC, then fixed	Non-universality, measurement dependence
dBB/double-scale	Internal wave density	Coupled dynamics for $(\psi_\text{ext}, \psi_\text{int})$	Physical meaning of “internal” wave

Each instantiation represents a deterministic program, but subject to distinct physical, mathematical, and interpretational caveats and domains of validity.

Deterministic semiclassical gravity plays a pivotal role in bridging quantum field theory and general relativity in regimes where full quantum gravity is unnecessary or unavailable. Its mathematical formalism is well-developed in high-symmetry cases and well-posed under analytic or physically smooth solutions. However, fundamental inconsistencies at post-Newtonian order for arbitrary quantum states, the requirement for precise initial data (Hadamard, SDC, or Bohmian configuration), and the dependence on the mode of branch selection or decoherence set key boundaries on its applicability. Current research focuses on sharpening these boundaries and on experimental procedures that might distinguish deterministic semiclassical scenarios from stochastic or fully quantum gravitational alternatives (Juárez-Aubry, 2 Sep 2025, Williams, 21 Dec 2025, Pipa, 7 Jul 2025).