1 Introduction↩︎

Quantum mechanics and general relativity are the two foundational pillars of modern physics. A longstanding challenge is to understand how quantum behavior interacts with classical gravitational backgrounds, and whether gravitational degrees of freedom can play a role in the emergence of quantum phenomena. The framework of Stochastic Quantum Gravity (SQG), initiated in [14], explores one specific aspect of this question: the possibility that a classical stochastic background of gravitational waves induces an effective stochastic dynamics in matter fields. This approach builds on the path integral formulation of quantum field theory [15]–[17] and the functional methods pioneered by Schwinger [18] and Faddeev-Popov [19], adapted to a background with intrinsic stochasticity.

The physical picture is straightforward. Spacetime is filled with gravitational waves across a vast range of scales. This is established observationally at parsec scales by Pulsar Timing Arrays (see, e.g., [20]). If such a stochastic background extends to smaller scales—a hypothesis that remains to be tested, and which has been explored in post-quantum gravity scenarios [21], [22] and quantum gravity models [23]–[25]—then microscopic particles would not follow pure geodesics. Instead, their motion would acquire a stochastic component, analogous to a Brownian particle in a fluctuating medium. In [14] it was shown that a particle following a geodesic plus a stochastic term satisfies the Klein–Gordon equation for the complex function \(\Phi = \sqrt{n}\,e^{i\theta}\), with \(\pi_{\mu} \sim \nabla_{\mu}\theta\) the geodesic velocity and \(u_{\mu} \sim \nabla_{\mu}\ln n\) the stochastic velocity. In the Newtonian limit this reduces to the Schrödinger equation. The present work adopts this semiclassical framework as a working hypothesis and develops its geometric and information-theoretic consequences.

The hydrodynamic formulation of quantum mechanics, introduced by Madelung [26] and further developed by Bohm [12], provides the natural language for this investigation. From the polar decomposition \(\Psi = \sqrt{\rho}\,e^{iS/\hbar}\), two real velocity fields are defined: \[\pi_{\mu} = \frac{1}{m}\nabla_{\mu}S, \qquad u_{\mu} = \frac{\hbar}{2m}\nabla_{\mu}\ln\rho. \label{eq:velocities}\tag{1}\] The field \(\pi_{\mu}\) governs the geodesic component of the flow, while the stochastic (or osmotic) velocity \(u_{\mu}\) encodes density gradients [27], [28]. In Nelson’s stochastic quantization framework, reviewed recently by Chavanis [1], both velocities arise from a Brownian motion model. The SQG approach [14] unifies them into a single complex velocity \(\eta_{\mu} = \pi_{\mu} - i u_{\mu}\), where the stochastic component is traced to the statistical properties of a fluctuating gravitational background rather than postulated as an independent Wiener process.

We emphasize the methodological status of this framework. It is a geometrization of the hydrodynamic description of quantum fields on a stochastic classical background, not a proposal for a new interpretation of quantum mechanics. No claim is made about the fundamental nature of the wave function, the existence of hidden variables, or the ontology of quantum states. The stochastic velocity \(u_{\mu}\) is treated as a derived quantity whose physical origin—whether classical gravitational waves, quantum metric fluctuations, or other sources—is not specified by the formalism itself. The only assumption is that the background fluctuations admit a statistical description and that the Markovian approximation holds. The wave function \(\Phi = \sqrt{n}\,e^{i\theta}\) retains its role as the fundamental descriptor of the boson gas, and the Born rule is preserved.

This paper provides the differential-geometric foundation for this framework and an explicit analytical illustration. Our goals are:

1. To prove that \(\eta_{\mu}\) is a section of the pullback bundle \(\pi_2^*(T^*M)\) and defines a flat \(U(1)\) connection.

2. To establish the information-geometric content via the bundle isomorphism to the symmetric logarithmic derivative operator.

3. To derive the complex geodesic equation and the associated von Neumann entropy [29]–[31].

4. To resolve the holonomy quantization, including the role of multi-valued phases [7].

5. To demonstrate the entire machinery in an analytically solvable toy model: a free scalar field on a conical spacetime [32], [33].

2 The Averaged Matter Amplitude↩︎

Let \(M\) be an \(n\)-dimensional spacetime manifold. The total action for matter fields \(\Phi\) is \(S[\Phi; g]\). We consider a stochastic metric \(g_{\mu\nu} = g_{\mu\nu}^{(0)} + h_{\mu\nu}\), where \(h_{\mu\nu}\) is drawn from a probability distribution \(P[h]\) with \(\langle h_{\mu\nu} \rangle = 0\) and \(\langle h_{\mu\nu}(x) h_{\alpha\beta}(x') \rangle = C_{\mu\nu\alpha\beta}(x,x')\). The rigorous construction of such measures on infinite-dimensional spaces follows from the framework of constructive quantum field theory [34], [35].

The full partition function is \(Z = \int \mathcal{D}[\Phi] \mathcal{K}[\Phi]\). We assume \(\mathcal{K}\) is non-vanishing and admits a smooth polar decomposition: \[\mathcal{K}[\Phi] = \sqrt{\mathcal{P}[\Phi]} \, e^{\frac{i}{\hbar}\mathcal{S}[\Phi]}, \qquad \mathcal{P}[\Phi] > 0,\] where \(\mathcal{P}[\Phi]\) and \(\mathcal{S}[\Phi]\) are real-valued functionals on \(\mathcal{C}\). Expanding the action in powers of \(h\), \(S = S_0 + S_1 + S_2 + \cdots\), and using the cumulant expansion [8]–[10] with \(\langle S_1 \rangle_h = 0\), we obtain: \[\begin{align} \mathcal{S}[\Phi] &= S_0[\Phi] + \langle S_2[\Phi; h] \rangle_h + \cdots, \tag{2} \\ \mathcal{P}[\Phi] &= \exp\left\{ -\frac{1}{\hbar^2}\langle S_1^2[\Phi; h] \rangle_h + \cdots \right\}. \tag{3} \end{align}\] Thus, \(\mathcal{S}[\Phi]\) is the classical action corrected by the averaged second-order gravitational coupling, and \(\mathcal{P}[\Phi]\) is generated by the variance of the linear matter-gravity coupling.

3 The Geometry of the Complex Velocity↩︎

3.1 Definition and Bundle Structure↩︎

Using the polar decomposition, we obtain the unification of the Madelung-Bohm velocities: \[\eta_{\mu}[\Phi] = \frac{1}{m}\nabla_{\mu}\mathcal{S}[\Phi] - i\frac{\hbar}{2m}\nabla_{\mu}\ln\mathcal{P}[\Phi] \equiv \pi_{\mu}[\Phi] - i u_{\mu}[\Phi]. \label{eq:unification}\tag{4}\] The stochastic velocity \(u_{\mu}\) is directly linked to the variance of gravitational fluctuations: \[u_{\mu}[\Phi] = \frac{\hbar}{2m}\nabla_{\mu}\ln\mathcal{P}[\Phi] = -\frac{1}{2m\hbar} \nabla_{\mu}\langle S_1^2[\Phi; h] \rangle_h.\] This unification parallels Nelson’s stochastic mechanics [1], where the complex velocity emerges from a Brownian motion model. In our framework, the stochasticity originates from spacetime fluctuations rather than an ad hoc Wiener process.

3.1.1 The Double Bundle Structure↩︎

The complex velocity \(\eta_{\mu}\) has a dual dependence: it is a functional on the configuration space \(\mathcal{C}\) and a covector field on spacetime \(M\). To formalize this, let \(\mathcal{C}\) be the infinite-dimensional Fréchet manifold of all field configurations [36]. The product manifold is \(\mathcal{E} := \mathcal{C} \times M\), with canonical projections \(\pi_1: \mathcal{E} \to \mathcal{C}\) and \(\pi_2: \mathcal{E} \to M\).

A section of \(E\) is a smooth map \(\eta: \mathcal{E} \to E\) satisfying \(\pi_E \circ \eta = \operatorname{id}_{\mathcal{E}}\), where \(\pi_E: E \to \mathcal{E}\) is the bundle projection.

This bundle structure cleanly separates the two roles of \(\eta_{\mu}\):

1. For fixed \(\Phi_0\): \(\eta_{\mu}[\Phi_0](x)\) is an ordinary covector field on \(M\). Its covariant derivative, divergence, and holonomy are computed using the spacetime connection.

2. Functional dependence on \(\mathcal{C}\): The variation under \(\Phi \to \Phi + \delta\Phi\) is encoded in \(\delta\eta_{\mu}(\Phi, x)/\delta\Phi(y)\), linking local spacetime quantities to global information on configuration space.

The double fibration is visualized in Figure 1.

3.2 Connection to Quantum Estimation Theory↩︎

Following the rigorous construction in [2], we introduce a fixed Gaussian measure \(\nu_0\) on \(\mathcal{C}\) via Minlos’ theorem [40], [41]. The existence and equivalence properties of such Gaussian measures are governed by the Feldman-Hájek theorem [42], [43]. The Hilbert space is \(\mathcal{H}_0 = L^2(\mathcal{C}, \nu_0)\). The state vectors are \(|\Psi_x\rangle = \sqrt{\mathcal{P}} e^{i\mathcal{S}/\hbar} \in \mathcal{H}_0\), parametrized by \(x \in M\), with \(\| \Psi_x \|_0 = 1\) for all \(x\).

The symmetric logarithmic derivative (SLD) \(L_{\mu}(x)\) is the central object in quantum estimation theory [3], [4]. For a smooth family of pure states \(\rho_x = |\Psi_x\rangle\langle\Psi_x|\), the SLD is the self-adjoint operator satisfying the defining equation [5], [6]: \[\partial_{\mu}\rho_x = \frac{1}{2}\{L_{\mu}(x), \rho_x\}.\] For pure states, the general solution with zero expectation value \(\langle \Psi_x | L_{\mu}(x) | \Psi_x \rangle = 0\) is given by the non-local operator [2]: \[L_{\mu}(x) = 2|\partial_{\mu}\Psi_x\rangle\langle\Psi_x| + 2|\Psi_x\rangle\langle\partial_{\mu}\Psi_x| - 2\langle\partial_{\mu}\Psi_x|\Psi_x\rangle_0 \mathbb{I} - 2\langle\Psi_x|\partial_{\mu}\Psi_x\rangle_0 \mathbb{I}. \label{eq:SLD95nonlocal}\tag{5}\]

To express this in terms of the complex velocity, we use the identities derived from the polar decomposition \(\Psi_x = \sqrt{\mathcal{P}} e^{i\mathcal{S}/\hbar}\). A direct calculation gives: \[\partial_{\mu}\Psi_x = \left( \frac{1}{2}\partial_{\mu}\ln\mathcal{P} + \frac{i}{\hbar}\partial_{\mu}\mathcal{S} \right) \Psi_x = \frac{im}{\hbar}\eta_{\mu}\Psi_x,\] where \(\eta_{\mu} = \pi_{\mu} - i u_{\mu}\) and we used \(\partial_{\mu}\ln\mathcal{P} = -\frac{2m}{\hbar}u_{\mu}\) and \(\partial_{\mu}\mathcal{S} = m\pi_{\mu}\). Similarly, for the conjugate: \[\overline{\partial_{\mu}\Psi_x} = -\frac{im}{\hbar}\bar{\eta}_{\mu}\overline{\Psi_x}, \qquad \bar{\eta}_{\mu} = \pi_{\mu} + i u_{\mu}.\]

Introducing multiplication operators \((\hat{\eta}_{\mu}\psi)(\Phi) = \eta_{\mu}(\Phi, x)\psi(\Phi)\) and the orthogonal projector \(\hat{P}_x = |\Psi_x\rangle\langle\Psi_x|\), the SLD 5 takes the compact form: \[\widetilde{\mathcal{T}}(\eta)_{\mu} = \frac{2im}{\hbar}\big(\hat{\eta}_{\mu}\hat{P}_x - \hat{P}_x\hat{\bar{\eta}}_{\mu}\big) + \frac{2im}{\hbar}\big(\langle\hat{\bar{\eta}}_{\mu}\rangle_x - \langle\hat{\eta}_{\mu}\rangle_x\big)\mathbb{I}, \label{eq:isomorphism95nonlocal}\tag{6}\] where \(\langle \cdot \rangle_x = \langle \Psi_x | \cdot | \Psi_x \rangle_0\). This defines the bundle isomorphism \(\widetilde{\mathcal{T}}: \Gamma(E/\sim) \to \Gamma(\mathcal{L})\) established in [2]. The gauge equivalence \(\eta_{\mu} \sim \eta_{\mu} + i c_{\mu}(x)\) with \(c_{\mu} \in \mathbb{R}\) leaves \(\widetilde{\mathcal{T}}(\eta)_{\mu}\) invariant modulo operators that annihilate \(|\Psi_x\rangle\), which are invisible to the quantum Fisher metric.

The quantum Fisher information metric, which coincides with the Fubini-Study metric for pure states, is defined as [5]: \[g_{\mu\nu}^{FS}(x) = \frac{1}{2}\langle \Psi_x | \{L_{\mu}(x), L_{\nu}(x)\} | \Psi_x \rangle_0.\] Using the simplified action ?? , we compute: \[\begin{align} \langle \Psi_x | \widetilde{\mathcal{T}}_{\mu} \widetilde{\mathcal{T}}_{\nu} | \Psi_x \rangle &= \frac{4m^2}{\hbar^2} \langle (\hat{\eta}_{\mu} - \langle\hat{\eta}_{\mu}\rangle_x)(\hat{\eta}_{\nu} - \langle\hat{\eta}_{\nu}\rangle_x) \rangle_x. \end{align}\] Since the Fisher metric is the real part of this expectation, and writing \(\hat{\eta}_{\mu} = \hat{\pi}_{\mu} - i\hat{u}_{\mu}\), \(\hat{\bar{\eta}}_{\mu} = \hat{\pi}_{\mu} + i\hat{u}_{\mu}\), we expand: \[\begin{align} (\hat{\eta}_{\mu} - \langle\hat{\eta}_{\mu}\rangle)(\hat{\bar{\eta}}_{\nu} - \langle\hat{\bar{\eta}}_{\nu}\rangle) &= (\Delta\hat{\pi}_{\mu} - i\Delta\hat{u}_{\mu})(\Delta\hat{\pi}_{\nu} + i\Delta\hat{u}_{\nu}) \nonumber \\ &= \Delta\hat{\pi}_{\mu}\Delta\hat{\pi}_{\nu} + \Delta\hat{u}_{\mu}\Delta\hat{u}_{\nu} + i(\Delta\hat{\pi}_{\mu}\Delta\hat{u}_{\nu} - \Delta\hat{u}_{\mu}\Delta\hat{\pi}_{\nu}), \end{align}\] where \(\Delta\hat{A} = \hat{A} - \langle\hat{A}\rangle_x\). Taking the real part and the expectation value, the antisymmetric imaginary term vanishes, yielding the manifestly positive-definite result: \[g_{\mu\nu}^{FS}(x) = \frac{4m^2}{\hbar^2}\Big[\mathrm{Cov}_x(\pi_{\mu}, \pi_{\nu}) + \mathrm{Cov}_x(u_{\mu}, u_{\nu})\Big], \label{eq:fisher95final}\tag{7}\] where \(\mathrm{Cov}_x(A,B) = \langle AB \rangle_x - \langle A \rangle_x \langle B \rangle_x\). This is a fundamental result: both the geodesic velocity \(\pi_{\mu}\) and the stochastic velocity \(u_{\mu}\) contribute positively and independently to the quantum Fisher metric. The appearance of \(\mathrm{Cov}(\pi_{\mu}, \pi_{\nu})\) is a genuine quantum enhancement absent in classical Fisher information, which would only contain the covariance of the score function \(\partial_{\mu}\ln\mathcal{P} \propto u_{\mu}\) [44], [45]. The non-locality of the SLD 6 is precisely what captures this full quantum information.

By the Feldman-Hájek theorem [42], [43], both the isomorphism \(\widetilde{\mathcal{T}}\) and the Fisher metric 7 are independent of the reference Gaussian measure \(\nu_0\), making the entire construction intrinsic to the physical probability density \(\mathcal{P}\). Projecting the classical Fisher metric on \(\mathcal{C}\) [44] onto spacetime using field gradients [46] yields the induced metric: \[H_{\mu\nu}(x) = \int \mathcal{D}\Phi \, \mathcal{P}[\Phi] \, \nabla_{\mu}\ln\mathcal{P}[\Phi] \, \nabla_{\nu}\ln\mathcal{P}[\Phi] = \frac{4m^2}{\hbar^2}\langle u_{\mu}u_{\nu}\rangle_{\mathcal{P}},\] identifying \(u_{\mu}\) as the information-geometric potential. This connects our framework to broader investigations in quantum information geometry [47]–[50].

3.3 Flat \(U(1)\) Connection and Complex Geodesic Equation↩︎

The flatness is the geometric origin of the unified dynamics. From \([D_{\mu}, D_{\nu}] = 0\), we have \(\nabla_{\nu}\eta_{\mu} = \nabla_{\mu}\eta_{\nu}\). Computing the convective derivative: \[\begin{align} \eta^{\nu}\nabla_{\nu}\eta_{\mu} = \eta^{\nu}\nabla_{\mu}\eta_{\nu} = \nabla_{\mu}\left(\frac{1}{2}\eta^{\nu}\eta_{\nu}\right) - \frac{1}{2}(\nabla_{\mu}\eta^{\nu})\eta_{\nu} + \frac{1}{2}\eta^{\nu}(\nabla_{\mu}\eta_{\nu}). \end{align}\] The last two terms cancel identically, yielding:

Decomposing \(\eta_{\mu} = \pi_{\mu} - i u_{\mu}\), the real part reproduces the modified Hamilton-Jacobi equation, and the imaginary part gives the potential flow equation for \(u_{\mu}\) [14].

4 Physical Consequences↩︎

4.1 Continuity Equation and Born Rule↩︎

The amplitude \(\mathcal{K}\) satisfies a modified Klein-Gordon equation (\(\mathcal{A}=1/2+\mathcal{V}\) in [14]): \[\left(\nabla_{\mu}\nabla^{\mu} - \frac{m^2}{\hbar^2} + \mathcal{V}\right)\mathcal{K} = 0,\] with a real effective potential \(\mathcal{V}\). Computing \(\mathcal{K}^*\nabla^{\mu}\nabla_{\mu}\mathcal{K} - \mathcal{K}\nabla^{\mu}\nabla_{\mu}\mathcal{K}^* = 0\), and using the identity \[\mathcal{K}^* \nabla_{\mu}\nabla^{\mu} \mathcal{K} - \mathcal{K} \nabla_{\mu}\nabla^{\mu} \mathcal{K}^* = \nabla_{\mu}\left(\mathcal{K}^* \nabla^{\mu} \mathcal{K} - \mathcal{K} \nabla^{\mu} \mathcal{K}^*\right),\] we substitute the polar decomposition \(\mathcal{K} = \sqrt{\mathcal{P}}e^{i\mathcal{S}/\hbar}\). The current is: \[\begin{align} \mathcal{K}^* \nabla^{\mu} \mathcal{K} - \mathcal{K} \nabla^{\mu} \mathcal{K}^* &= \mathcal{P}\left[\left(\frac{1}{2}\nabla^{\mu}\ln\mathcal{P} + \frac{i}{\hbar}\nabla^{\mu}\mathcal{S}\right) - \left(\frac{1}{2}\nabla^{\mu}\ln\mathcal{P} - \frac{i}{\hbar}\nabla^{\mu}\mathcal{S}\right)\right] \nonumber \\ &= \frac{2i}{\hbar}\mathcal{P}\nabla^{\mu}\mathcal{S} = \frac{2im}{\hbar}\mathcal{P}\pi^{\mu}. \end{align}\] The divergence-free condition then yields the standard continuity equation: \[\nabla_{\mu}\left(\mathcal{P}\pi^{\mu}\right) = 0. \label{eq:continuity}\tag{8}\] Expanding the derivative using \(\nabla_{\mu}\mathcal{P} = \frac{2m}{\hbar}\mathcal{P}u_{\mu}\), we obtain the fundamental relation coupling the divergence of the classical velocity to the projection of the two velocities: \[\nabla_{\mu}\pi^{\mu} = -\frac{2m}{\hbar}u_{\mu}\pi^{\mu}. \label{eq:div95pi}\tag{9}\] This equation is a direct consequence of the continuity equation 8 and the definition \(u_{\mu} = \frac{\hbar}{2m}\nabla_{\mu}\ln\mathcal{P}\). It couples the deterministic and stochastic sectors without imposing orthogonality \(u_{\mu}\pi^{\mu}=0\) a priori; that condition would follow from additional assumptions such as incompressibility of the flow (\(\nabla_{\mu}\pi^{\mu}=0\)).

The Born rule emerges from the cumulant expansion 3 : \(\mathcal{P} = \exp(-\langle S_1^2\rangle_h/\hbar^2)\) is the effective probability density induced by Gaussian metric fluctuations. The squaring of the amplitude \(|\mathcal{K}|^2 = \mathcal{P}\) follows from the stochastic nature of the background, not as an additional postulate. The product \(u_{\mu}\pi^{\mu}\) appearing in 9 has a direct information-geometric interpretation through the quantum Fisher metric 7 : while \(\mathrm{Cov}(u_{\mu}, u_{\nu})\) measures the distinguishability of nearby probability distributions, the coupling \(\mathrm{Cov}(\pi_{\mu}, \pi_{\nu})\) captures the quantum coherence encoded in the phase.

4.2 Geometrization vs. Hidden Variables↩︎

This framework is a geometrization of quantum mechanics, fundamentally distinct from hidden-variable theories such as the de Broglie-Bohm interpretation [12], [13]. The key differences are:

1. Origin of stochasticity: In hidden-variable theories, \(u_{\mu}\) arises from an assumed initial distribution of particle positions. Here, it derives from the variance of spacetime fluctuations \(\langle S_1^2\rangle_h\), a physical field with a well-defined gravitational origin.

2. Non-locality: The complex velocity \(\eta_{\mu}\) is a functional on the entire configuration space \(\mathcal{C}\), geometrically encoded as a section of the pullback bundle \(\pi_2^*(T^*M)\) over \(\mathcal{C}\times M\). This preserves quantum non-locality without invoking supplementary particle degrees of freedom.

3. Quantum potential: The Bohmian quantum potential \(Q \propto \nabla^2\sqrt{\rho}/\sqrt{\rho}\) is replaced by the geometric structure of the flat \(U(1)\) connection. The flatness \([D_{\mu}, D_{\nu}] = 0\) encodes the absence of a classical force, while the connection itself carries the quantum information through the Fisher metric.

4. Emergent Born rule: The probability density is not postulated as an initial condition but emerges from the cumulant expansion as \(\mathcal{P} = \exp(-\langle S_1^2\rangle_h/\hbar^2)\). The wave function \(\Psi = \sqrt{\mathcal{P}}e^{i\mathcal{S}/\hbar}\) retains its fundamental status.

5. No additional degrees of freedom: There are no hidden particles, trajectories, or supplementary parameters. The quantum behavior is entirely encoded in the geometry of the complex velocity and the stochastic gravitational background.

4.3 Von Neumann Entropy↩︎

The density operator is \(\hat{\rho} = Z^{-1}\int \mathcal{D}[\Phi] \mathcal{K}[\Phi] |\Phi\rangle\langle\Phi|\) [29], [30]. In the Gaussian approximation [31], the von Neumann entropy \(S_{\text{vN}} = -\operatorname{Tr}(\hat{\rho}\ln\hat{\rho})\) becomes: \[S_{\text{vN}} \approx \frac{1}{2}\operatorname{Tr} \ln\left( \frac{\hbar^2}{4m^2} \langle \eta \eta^* + \eta^* \eta \rangle \right) + \text{const.}\] This establishes \(\eta_{\mu}\) as the fundamental carrier of quantum statistical uncertainty, linking the \(U(1)\) geometry to information theory [51]–[53]. The connection between information geometry, entropy, and gravity has been explored in various contexts [51]–[53].

4.4 Effective Einstein Equations and Fisher Metric from the Path Integral↩︎

The full partition function \(Z[g]\) is a functional of the background metric. Varying the effective action \(\Gamma[g] = -i\hbar\ln Z[g]\) yields the effective Einstein equations [54]: \[\frac{\delta\Gamma[g]}{\delta g^{\mu\nu}(x)} = 0 \quad \Longrightarrow \quad G_{\mu\nu}(x) = 8\pi G \, \langle T_{\mu\nu}(x) \rangle_{\text{eff}}.\] Using the definition of the matter amplitude ?? , the effective energy-momentum tensor is: \[\langle T_{\mu\nu}(x) \rangle_{\text{eff}} = \frac{1}{Z} \int \mathcal{D}[\Phi] \int \mathcal{D}[h] P[h] \, T_{\mu\nu}^{\text{(clas)}}(x) \, e^{\frac{i}{\hbar}S[\Phi, A; g+h]}.\] This is the average of the classical energy-momentum tensor over both matter configurations and gravitational fluctuations. Expressing this in terms of the polar decomposition \(\mathcal{K} = \sqrt{\mathcal{P}}e^{i\mathcal{S}/\hbar}\), the conditional average over metric fluctuations for a fixed matter configuration can be written using functional derivatives of \(\mathcal{S}\) and \(\mathcal{P}\) with respect to the metric.

The connection to information geometry arises through the stochastic velocity \(u_{\mu}\). From the cumulant expansion 3 , \(\ln\mathcal{P} = -\frac{1}{\hbar^2}\langle S_1^2\rangle_h + \cdots\), so functional derivatives of \(\ln\mathcal{P}\) probe the variance of the matter-gravity coupling. The induced Fisher metric on spacetime, obtained by projecting the classical Fisher information metric on \(\mathcal{C}\) [44], [46], is: \[H_{\mu\nu}(x) = \int \mathcal{D}\Phi \, \mathcal{P}[\Phi] \, \nabla_{\mu}\ln\mathcal{P}[\Phi] \, \nabla_{\nu}\ln\mathcal{P}[\Phi] = \frac{4m^2}{\hbar^2} \langle u_{\mu} u_{\nu} \rangle_{\mathcal{P}}.\] Thus, the effective energy-momentum tensor couples to the Fisher metric through their mutual dependence on the variance of gravitational fluctuations. The backreaction of quantum matter on spacetime geometry is encoded in the information-geometric structure of the complex velocity [52], [53].

5 Holonomy Quantization and a Toy Model↩︎

5.1 The Logical Tension and Its Resolution↩︎

The flatness condition \([D_{\mu}, D_{\nu}] = 0\) implies the connection is locally pure gauge, but the complex potential \(\phi = -i\frac{\hbar}{m}\ln\mathcal{K}\) can be multi-valued on non-simply connected spacetimes [32], [33]. The multi-valuedness has three independent origins:

1. Classical topological terms: The action \(S_0\) may contain topological invariants (e.g., winding number, Chern-Simons terms) that make \(e^{iS_0/\hbar}\) single-valued but \(S_0/\hbar\) multi-valued modulo \(2\pi\).

2. Effective gravitational corrections: The cumulant expansion gives \(\mathcal{S} = S_0 + \langle S_2 \rangle_h + \cdots\). The term \(\langle S_2 \rangle_h\) can generate effective topological contributions through backreaction of metric fluctuations.

3. Phase singularities of \(\mathcal{P}\): If \(\mathcal{P}[\Phi](x) = 0\) at isolated points, \(\ln\mathcal{P}\) develops branch cuts. While \(\mathcal{P}\) itself remains single-valued, the analytic structure of \(\mathcal{K} = \sqrt{\mathcal{P}}e^{i\mathcal{S}/\hbar}\) is affected [7].

The resolution of the apparent tension comes from the single-valuedness of the full matter amplitude \(\mathcal{K}\). Writing the logarithmic derivative: \[d\ln\mathcal{K} = \frac{1}{2}d\ln\mathcal{P} + \frac{i}{\hbar}d\mathcal{S}.\] Since \(\mathcal{P} > 0\) is strictly positive and single-valued by Assumption 2.2, the integral of \(d\ln\mathcal{P}\) around any closed loop vanishes: \[\oint_{\gamma} d\ln\mathcal{P} = 0. \label{eq:integral95P95zero}\tag{10}\] Therefore, only the phase contributes to the loop integral: \[\oint_{\gamma} d\ln\mathcal{K} = \frac{i}{\hbar}\oint_{\gamma} d\mathcal{S}.\]

Single-valuedness of \(\mathcal{K}\) requires that it return to its original value after parallel transport around a closed loop, which means: \[\oint_{\gamma} d\ln\mathcal{K} = 2\pi i N, \qquad N \in \mathbb{C}.\] The integer \(N\) is complex in general, \(N = n + i m\) with \(n, m \in \mathbb{Z}\). However, the condition 10 forces the real part of the loop integral to vanish, which implies \(\operatorname{Re}(\oint_{\gamma} d\ln\mathcal{K}) = \frac{1}{2}\oint_{\gamma} d\ln\mathcal{P} = 0\). This is automatically satisfied by 10 , so \(m = 0\) and \(N = n \in \mathbb{Z}\).

The action functional \(\mathcal{S}\) may contain contributions beyond the purely classical term. Collecting all contributions, we write: \[\oint_{\gamma} d\mathcal{S} = 2\pi\hbar n + \hbar\Delta\phi_{\text{top}},\] where \(\Delta\phi_{\text{top}} \in \mathbb{R}\) absorbs all non-integer offsets from classical topological terms, gravitational corrections, and phase singularities.

Using the definition \(\eta_{\mu} = -i\frac{\hbar}{m}\nabla_{\mu}\ln\mathcal{K}\), the holonomy is: \[\frac{m}{\hbar}\oint_{\gamma} \eta_{\mu} dx^{\mu} = -i\oint_{\gamma} d\ln\mathcal{K} = -i\left(\frac{i}{\hbar}\oint_{\gamma} d\mathcal{S}\right) = \frac{1}{\hbar}\oint_{\gamma} d\mathcal{S} = 2\pi n + \Delta\phi_{\text{top}}.\]

5.2 Toy Model: Scalar Field on a Conical Spacetime↩︎

We now demonstrate the entire machinery with an explicit analytical calculation.

5.2.1 Geometry and Classical Action↩︎

Consider a conical spacetime metric with a topological defect along the \(z\)-axis: \[ds^2 = -dt^2 + dr^2 + \alpha^2 r^2 d\theta^2 + dz^2, \label{eq:conical}\tag{11}\] where \(0 < \alpha < 1\) is the deficit angle parameter (\(\alpha = 1 - 4G\mu\) for a cosmic string with linear mass density \(\mu\)). The spatial sections are locally flat but globally non-simply connected: circles of constant \(r\) have circumference \(2\pi\alpha r\). This geometry is a classic example in topological physics [32], [33].

Consider a free, minimally coupled, massive scalar field on this background: \[S_0[\Phi] = -\frac{1}{2}\int d^4x \sqrt{-g}\left(g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi + \frac{m^2}{\hbar^2}\Phi^2\right),\] where \(\sqrt{-g} = \alpha r\). We study a single angular mode with fixed radial and longitudinal quantum numbers: \[\Phi_{\ell}(t,r,\theta,z) = \varphi_{\ell}(r) e^{-i\omega t + ik_z z + i\ell\theta/\alpha},\] where \(\ell \in \mathbb{Z}\) ensures \(2\pi\)-periodicity, and \(\varphi_{\ell}(r)\) is a normalized radial profile.

5.2.2 Stochastic Metric Fluctuations↩︎

We introduce stochastic fluctuations that couple to the angular component of the metric: \[\langle h_{\mu\nu}(x) h_{\alpha\beta}(x') \rangle = \sigma^2 r^2 \frac{\delta^{(4)}(x-x')}{\sqrt{-g}} \delta_{\mu}^{\theta}\delta_{\nu}^{\theta}\delta_{\alpha}^{\theta}\delta_{\beta}^{\theta},\] where \(\sigma\) encodes the strength of metric fluctuations. This noise model affects the \(\theta\theta\) component that controls the circumference of circles around the defect.

5.2.3 Matter Amplitude in the Gaussian Approximation↩︎

The first-order coupling is \(S_1 = \int d^4x \sqrt{-g} \, h^{\theta\theta} \, T_{\theta\theta}\), where \(T_{\theta\theta} = \partial_{\theta}\Phi_{\ell}\partial_{\theta}\Phi_{\ell} - \frac{1}{2}\bar{g}_{\theta\theta}\bar{g}^{\mu\nu}\partial_{\mu}\Phi_{\ell}\partial_{\nu}\Phi_{\ell}\). With \(\partial_{\theta}\Phi_{\ell} = i(\ell/\alpha)\Phi_{\ell}\), we obtain: \[\langle S_1^2\rangle_h = \sigma^2 \left(\frac{\ell}{\alpha}\right)^4 D_{\ell},\] where \(D_{\ell}\) is a positive constant from the radial, temporal, and longitudinal integrals. The cumulant expansion gives: \[\mathcal{P}_{\ell} = \exp\left(-\frac{\sigma^2}{\hbar^2}\left(\frac{\ell}{\alpha}\right)^4 D_{\ell}\right), \qquad \mathcal{S}_{\ell} = S_0[\Phi_{\ell}].\] At this order, \(\mathcal{P}_{\ell}\) is constant, so \(u_{\mu} = 0\) and \(\eta_{\mu} = \pi_{\mu}\).

5.2.4 Holonomy Calculation↩︎

The \(\theta\)-component of the geodesic velocity is: \[\pi_{\theta} = \frac{1}{m}\partial_{\theta}\mathcal{S}_{\ell} = \frac{\hbar}{m}\frac{\ell}{\alpha}.\] Since \(\mathcal{P}_{\ell}\) is constant at this order, \(u_{\theta} = 0\), and the complex velocity is purely real: \(\eta_{\theta} = \pi_{\theta} = \frac{\hbar}{m}\frac{\ell}{\alpha}\).

For a circular loop \(\gamma\) at fixed \(t, r, z\) encircling the defect: \[\frac{m}{\hbar}\oint_{\gamma} \eta_{\mu} dx^{\mu} = \frac{m}{\hbar}\int_0^{2\pi} \eta_{\theta} d\theta = \frac{m}{\hbar}\int_0^{2\pi} \frac{\hbar}{m}\frac{\ell}{\alpha} d\theta = \frac{2\pi\ell}{\alpha}. \label{eq:holonomy95toy}\tag{12}\]

Comparing with the quantization condition ?? , \(\frac{m}{\hbar}\oint_{\gamma} \eta_{\mu} dx^{\mu} = 2\pi n + \Delta\phi_{\text{top}}\), we identify: \[\Delta\phi_{\text{top}} = 2\pi\ell\left(\frac{1}{\alpha} - 1\right).\] This is the classical topological offset produced by the deficit angle of the conical spacetime. For a GUT-scale cosmic string with \(G\mu \sim 10^{-6}\), \(\alpha \approx 0.999996\), and \(\Delta\phi_{\text{top}} \approx 8\pi\ell \times 10^{-6}\) rad, which is in principle observable in precision interferometry. The integer \(n\) in ?? is then the nearest integer to \(\ell/\alpha\). This is a direct analog of the Aharonov-Bohm effect [55].

The condition \(\oint_{\gamma} d\ln\mathcal{P} = 0\) is automatically satisfied since \(\mathcal{P}_{\ell}\) is constant. When the noise model includes spatial correlations that generate an angular-dependent \(\mathcal{P}_{\ell}\), the stochastic velocity \(u_{\theta}\) becomes non-zero, and the holonomy receives an additional contribution \(\delta\phi_{\text{stoch}} = -\frac{1}{2}\oint_{\gamma} d\ln\mathcal{P}\). In that case, the full quantization condition becomes: \[\frac{m}{\hbar}\oint_{\gamma} \eta_{\mu} dx^{\mu} = 2\pi n + \Delta\phi_{\text{top}} + \delta\phi_{\text{stoch}},\] where \(\delta\phi_{\text{stoch}}\) is determined by the circulation of \(u_{\mu}\) around the defect.

6 Observable Signatures↩︎

The theoretical framework developed above yields two distinct types of observable phases: a topological holonomy for non-simply connected spacetimes, and a geometric Berry phase for simply connected spacetimes. We discuss both, with emphasis on the latter as the primary target for near-term atom interferometry [11].

6.1 Topological Holonomy for Non-Simply Connected Spacetimes↩︎

When the spacetime \(M\) possesses non-contractible loops—due to cosmic strings, black holes, or Planck-scale topological defects—the holonomy of the complex velocity is quantized according to the theorem of Section 5.1: \[\frac{m}{\hbar}\oint_{\gamma} \eta_{\mu} dx^{\mu} = 2\pi n + \Delta\phi_{\mathrm{top}}.\] This is a direct analog of the Aharonov-Bohm effect [55], where the magnetic flux is replaced by the topological offset \(\Delta\phi_{\mathrm{top}}\) generated by the effective action \(\mathcal{S}\). The phase shift is purely topological: it depends only on the winding number of \(\gamma\) around the defect, not on the detailed geometry of the path.

For a cosmic string with deficit angle \(\alpha = 1 - 4G\mu\), the toy model of Section 5.2 yields \(\Delta\phi_{\mathrm{top}} = 2\pi\ell(1/\alpha - 1)\). If such a defect passes through the interferometer area during a measurement, the phase shift would manifest as a discrete jump as the defect enters and exits the sensitive region. The probability of such an event depends on the unknown density of cosmic strings or Planck-scale defects and is not the primary observable for near-term experiments.

6.2 Berry Phase in Simply Connected Spacetimes↩︎

For experiments conducted in simply connected spacetimes, such as the approximately Minkowski background of MAGIS-100 [11], the loops traversed by the atomic wave packets are contractible. In this case, the holonomy of the classical complex velocity \(\eta_{\mu}\) is trivial because the classical connection \(D_{\mu} = \nabla_{\mu} - i\frac{m}{\hbar}\eta_{\mu}\) is flat (\([D_{\mu}, D_{\nu}] = 0\)). However, the quantum state \(|\Psi_x\rangle\) on the Hilbert space \(\mathcal{H}_0 = L^2(\mathcal{C}, \nu_0)\) defines a non-trivial geometric structure over the parameter manifold \(M\) that gives rise to an observable Berry phase [7].

6.2.1 Berry Connection and Curvature↩︎

The family of states \(|\Psi_x\rangle = \sqrt{\mathcal{P}} e^{i\mathcal{S}/\hbar}\) parametrized by spacetime coordinates \(x \in M\) defines the Berry connection [7]: \[\mathcal{A}_{\mu}^{\mathrm{B}}(x) := i \langle \Psi_x | \partial_{\mu} \Psi_x \rangle_0.\] Using the identity \(\partial_{\mu} \Psi_x = \frac{im}{\hbar} \eta_{\mu} \Psi_x\) derived in Section 3.2, we obtain: \[\label{eq:berry95connection} \mathcal{A}_{\mu}^{\mathrm{B}}(x) = -\frac{m}{\hbar} \langle \eta_{\mu} \rangle_x = -\frac{m}{\hbar} \langle \pi_{\mu} \rangle_x + i \frac{m}{\hbar} \langle u_{\mu} \rangle_x.\tag{13}\] The imaginary part involves the expectation value of the stochastic velocity. Using \(u_{\mu} = \frac{\hbar}{2m}\nabla_{\mu}\ln\mathcal{P}\) and the normalization condition \(\int_{\mathcal{C}} \mathcal{P}[\Phi](x) d\nu_0[\Phi] = 1\) for all \(x \in M\) (Assumption 2.2 of [2]), we compute: \[\langle u_{\mu} \rangle_x = \int_{\mathcal{C}} \frac{\hbar}{2m} (\nabla_{\mu} \ln \mathcal{P}) \, \mathcal{P} \, d\nu_0 = \frac{\hbar}{2m} \int_{\mathcal{C}} \nabla_{\mu} \mathcal{P} \, d\nu_0 = \frac{\hbar}{2m} \nabla_{\mu} \int_{\mathcal{C}} \mathcal{P} \, d\nu_0 = \frac{\hbar}{2m} \nabla_{\mu} 1 = 0.\] Thus, the Berry connection is purely real: \(\mathcal{A}_{\mu}^{\mathrm{B}}(x) = -\frac{m}{\hbar} \langle \pi_{\mu} \rangle_x\), and there is no loss of visibility from the imaginary part at this order.

The Berry curvature is the field strength associated with this connection: \[\label{eq:berry95curvature95def} \Omega_{\mu\nu}^{\mathrm{B}} := \partial_{\mu} \mathcal{A}_{\nu}^{\mathrm{B}} - \partial_{\nu} \mathcal{A}_{\mu}^{\mathrm{B}}.\tag{14}\] A direct calculation using the Leibniz rule and the definition of the expectation value yields: \[\begin{align} \Omega_{\mu\nu}^{\mathrm{B}} &= -\frac{m}{\hbar} \int_{\mathcal{C}} \Big[ (\partial_{\mu} \eta_{\nu} - \partial_{\nu} \eta_{\mu}) \mathcal{P} + \eta_{\nu} \partial_{\mu} \mathcal{P} - \eta_{\mu} \partial_{\nu} \mathcal{P} \Big] d\nu_0 \nonumber \\ &= -\frac{m}{\hbar} \int_{\mathcal{C}} \Big[ \eta_{\nu} \partial_{\mu} \mathcal{P} - \eta_{\mu} \partial_{\nu} \mathcal{P} \Big] d\nu_0, \end{align}\] where we used the flatness condition \(\partial_{\mu} \eta_{\nu} - \partial_{\nu} \eta_{\mu} = 0\) for a fixed field configuration \(\Phi \in \mathcal{C}\). Substituting \(\partial_{\mu} \mathcal{P} = \frac{2m}{\hbar} \mathcal{P} u_{\mu}\) from the definition of the stochastic velocity: \[\label{eq:berry95curvature95final} \Omega_{\mu\nu}^{\mathrm{B}} = -\frac{2m^2}{\hbar^2} \Big( \langle \eta_{\nu} u_{\mu} \rangle_x - \langle \eta_{\mu} u_{\nu} \rangle_x \Big).\tag{15}\] Decomposing \(\eta_{\mu} = \pi_{\mu} - i u_{\mu}\), the real and imaginary parts are: \[\begin{align} \operatorname{Re} \Omega_{\mu\nu}^{\mathrm{B}} &= -\frac{2m^2}{\hbar^2} \Big( \langle \pi_{\nu} u_{\mu} \rangle_x - \langle \pi_{\mu} u_{\nu} \rangle_x \Big), \tag{16} \\ \operatorname{Im} \Omega_{\mu\nu}^{\mathrm{B}} &= \frac{2m^2}{\hbar^2} \Big( \langle u_{\nu} u_{\mu} \rangle_x - \langle u_{\mu} u_{\nu} \rangle_x \Big) = 0. \tag{17} \end{align}\]

6.2.2 Physical Origin of the Berry Curvature↩︎

The crucial point is that \(\Omega_{\mu\nu}^{\mathrm{B}} \neq 0\) even though the classical connection is flat. The non-vanishing curvature arises from the configurational average over \(\mathcal{C}\). The cross-correlator \(\langle \pi_{\nu} u_{\mu} \rangle_x\) in 16 couples the deterministic and stochastic velocities. This is a distinct signature of the SQG framework: in standard quantum mechanics without stochastic gravity, \(u_{\mu}\) and \(\pi_{\mu}\) are both determined by the same wave function \(\Psi_x = \sqrt{\rho} e^{iS/\hbar}\) and are not independent fields. Their correlation is fixed by the wave function alone and does not generate an independent Berry curvature beyond what is already encoded in the Fubini-Study metric.

In SQG, \(u_{\mu}\) encodes the variance of gravitational fluctuations via the cumulant expansion 3 : \[u_{\mu} = -\frac{1}{2m\hbar} \nabla_{\mu} \langle S_1^2 \rangle_h,\] providing an independent physical field whose correlation with \(\pi_{\mu}\) generates the Berry curvature 15 . This cross-correlation is a direct probe of the interplay between the classical phase \(\mathcal{S}\) (through \(\pi_{\mu} = \frac{1}{m}\nabla_{\mu}\mathcal{S}\)) and the variance of the matter-gravity coupling \(\langle S_1^2 \rangle_h\).

6.2.3 Berry Phase in Atom Interferometry↩︎

For a closed loop \(\gamma\) in spacetime traversed by the atomic wave packet, the geometric Berry phase is: \[\label{eq:berry95phase} \gamma_B = \oint_{\gamma} \mathcal{A}_{\mu}^{\mathrm{B}} dx^{\mu} = \iint_S \Omega_{\mu\nu}^{\mathrm{B}} d\Sigma^{\mu\nu},\tag{18}\] where \(S\) is any surface bounded by \(\gamma\), and \(d\Sigma^{\mu\nu}\) is the area element. Using the expression 16 for the Berry curvature: \[\gamma_B = -\frac{2m^2}{\hbar^2} \iint_S \Big( \langle \pi_{\nu} u_{\mu} \rangle_x - \langle \pi_{\mu} u_{\nu} \rangle_x \Big) d\Sigma^{\mu\nu}.\] In the non-relativistic limit relevant for MAGIS-100 [11], the dominant contribution comes from the spatial components. For an interferometer oriented in the \(x\)-\(z\) plane with the sensitive axis along \(z\), the Berry phase simplifies to: \[\gamma_B \approx -\frac{2m^2}{\hbar^2} \iint_S \Big( \langle \pi_z u_x \rangle_x - \langle \pi_x u_z \rangle_x \Big) dx dz.\] This phase is proportional to the enclosed area and to the antisymmetric part of the cross-correlation tensor \(\langle \pi_i u_j \rangle_x\).

6.3 Connection to the Quantum Fisher Metric↩︎

The Berry curvature 15 and the quantum Fisher metric 7 provide complementary probes of the stochastic velocity field. The Fisher metric involves the symmetric covariances: \[g_{\mu\nu}^{FS} = \frac{4m^2}{\hbar^2}\Big[\mathrm{Cov}(\pi_{\mu}, \pi_{\nu}) + \mathrm{Cov}(u_{\mu}, u_{\nu})\Big],\] while the Berry curvature involves the antisymmetric cross-correlation: \[\Omega_{\mu\nu}^{\mathrm{B}} = -\frac{2m^2}{\hbar^2}\Big(\langle \pi_{\nu} u_{\mu} \rangle_x - \langle \pi_{\mu} u_{\nu} \rangle_x\Big).\] Together, they fully characterize the quantum geometry of the state family \(|\Psi_x\rangle\) on the parameter manifold \(M\). The Fisher metric bounds the precision of parameter estimation via the quantum Cramér-Rao bound [5], [6], while the Berry phase provides a direct interferometric signature. The SLD isomorphism \(\widetilde{\mathcal{T}}\) of Section 3.2 links both quantities to the complex velocity \(\eta_{\mu}\), establishing a unified operational framework for testing stochastic gravity with atom interferometry.

7 Conclusion↩︎

We have established a rigorous geometric framework unifying the deterministic and stochastic aspects of quantum mechanics through a complex velocity \(\eta_{\mu} = \pi_{\mu} - i u_{\mu}\) emerging from a stochastic gravitational background. The main results are:

1. Double bundle geometry: \(\eta_{\mu}\) is a section of the pullback bundle \(E = \pi_2^*(T^*M)\) over \(\mathcal{C}\times M\), encoding both functional field dependence and local spacetime structure. The functionals \(\mathcal{S}[\Phi]\) and \(\mathcal{P}[\Phi]\) are explicitly derived from the cumulant expansion of gravitational fluctuations [8]–[10].

2. Flat \(U(1)\) connection and complex geodesic equation: \(\eta_{\mu}\) defines a flat connection \(D_{\mu} = \nabla_{\mu} - i\frac{m}{\hbar}\eta_{\mu}\) whose curvature vanishes identically. The flatness is the geometric origin of the unified complex geodesic equation \(\mathcal{L}_{\eta}\eta = d(|\eta|^2)\), which collapses the coupled Hamilton-Jacobi and potential flow dynamics into a single geometric statement [14].

3. Information geometry: Through the bundle isomorphism \(\widetilde{\mathcal{T}}\) established in [2], \(\eta_{\mu}\) maps to the symmetric logarithmic derivative of quantum estimation theory [3]–[6]. The stochastic velocity \(u_{\mu}\) is the information-geometric potential, with the quantum Fisher metric \(g_{\mu\nu}^{FS} \propto \langle u_{\mu}u_{\nu}\rangle_{\mathcal{P}}\) and the von Neumann entropy \(S_{\text{vN}} \sim \operatorname{Tr}\ln\langle\eta\eta^*\rangle\) [29]–[31]. Both quantities are intrinsic, independent of the reference Gaussian measure [42], [43].

4. Geometrization, not hidden variables: This framework constitutes a genuine geometrization of quantum mechanics [12], [13], [26]. The stochastic velocity \(u_{\mu}\) is the imprint of spacetime fluctuations on matter, not an ad hoc postulate. The Born rule emerges from the cumulant expansion as \(\mathcal{P} = \exp(-\langle S_1^2\rangle_h/\hbar^2)\). Non-locality is preserved through the functional dependence of \(\eta_{\mu}\) on the entire configuration space \(\mathcal{C}\), geometrically encoded in the pullback bundle structure [36]. The Bohmian quantum potential is replaced by the flat \(U(1)\) connection and its associated holonomy. No supplementary hidden variables or particle trajectories are required; the wave function retains its fundamental status.

5. Holonomy quantization and toy model: The tension between flatness and multi-valuedness is resolved by recognizing that single-valuedness of \(\mathcal{K}\) quantizes the total phase: \(\frac{m}{\hbar}\oint_{\gamma}\eta_{\mu}dx^{\mu} = 2\pi n + \Delta\phi_{\text{top}}\). The toy model of a scalar field on a conical spacetime [32], [33] demonstrates this analytically: the classical Aharonov-Bohm phase \(2\pi\ell/\alpha\) [55] receives a stochastic correction \(\delta\phi_{\text{stoch}}\) from metric fluctuations, providing a clear experimental target for atom interferometry [11].

6. Observable signatures: The quantized holonomy predicts discrete topological phase jumps distinguishable from continuous classical noise. Together with the Berry curvature 15 and the Fisher metric 7 , this framework provides a complete geometric characterization of the quantum state manifold, testable with current and near-future atom interferometry [11].

This framework establishes a closed triangle between stochastic gravity, quantum information geometry [44]–[48], and observable topological phases [7], [55], offering a testable geometric foundation for the quantum-classical transition.

References↩︎

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1. E-mail: jorge.meza@cinvestav.mx↩︎

2. E-mail: tonatiuh.matos@cinvestav.mx↩︎