Resonant Vacuum Noise Should Positively Shift Momentum-Dependent Phase-Space Curvature in Rubidium Atom Interferometers

Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Mubaidin, Mukherjee, Alshehri, and Tawfik derive an eight-dimensional metric on the cotangent bundle of spacetime. Their construction promotes momentum from a probe variable to a geometric coordinate. Curvature, geodesics, and field equations then acquire phase-space dependence. This Working Paper applies one Pudding Theory Postulate to that result. Vacuum Receptivity is used in its narrow form: the vacuum is a stochastic receiver whose measured spectrum can bias weak geometric inference. The source paper does not imply this. It supplies the phase-space metric target. Pudding Theory supplies an added experimental term. In a rubidium light-pulse atom interferometer, the residual phase-space curvature coefficient should increase with the vacuum noise spectral density in the hyperfine coupling band near 6.834 GHz, after known thermal, electromagnetic, and atom-surface shifts are subtracted. A null measurement of that residual would falsify the applied Postulate.

Source Synopsis

Mubaidin et al. address a structural tension between general relativity and quantum mechanics. General relativity represents gravity through smooth spacetime geometry. Quantum mechanics represents position and momentum through noncommuting operators. The paper asks what happens if the metric itself is derived on phase space rather than on spacetime alone.

The authors use the relativistic generalized uncertainty principle. In curved spacetime with finite gravitational fields, physical position and momentum are not treated as simple canonically conjugate variables. The modified commutator includes curvature-sensitive momentum terms and reduces to the ordinary Heisenberg form when the deformation parameters vanish. This is the quantum input.

The geometric input is the cotangent bundle. The authors state the problem directly: "the aim is to derive the quantized metric tensor on the cotangent bundle over spacetime with coordinates." This is the only quoted sentence from the source paper, anchored to Section I of arXiv:2602.13219, in the paragraph introducing the cotangent-bundle coordinates \(X=[x^\alpha,p_\beta]\).

The paper then builds Finsler and Hamilton structures from Hessian matrices. The ordinary Riemannian metric appears as a limiting case. The stronger claim is that general relativity should be written with a phase-space metric tensor when quantum corrections are retained. Line elements, geodesic equations, Ricci tensors, Ricci scalars, and Einstein equations are then reformulated on the eight-dimensional structure.

The result is relative spacetime in a technical sense. The metric sampled by a quantum probe depends not only on the spacetime location of the probe, but also on the probe momentum. This fits earlier work on geometric phase-space quantization and relative locality, but the paper’s main object is its explicit phase-space metric construction.

Postulate Lens

Vacuum Receptivity is the applied Postulate.

The fit is limited. The source paper’s vacuum is a gravitational and geometric setting. It is not a measured stochastic receiver. Mubaidin et al. do not claim that local vacuum noise spectra alter metric coefficients. They derive a phase-space metric from RGUP and Hamilton geometry.

Pudding Theory adds a separate empirical burden. Its formal vacuum is a stochastic reservoir with spectral density \(S_\eta(\omega)\). If that reservoir receives and carries weak structure, then a phase-space metric inferred from quantum probes should not be completely indifferent to the vacuum state in the coupling band of the probe. The claim would not follow from the source paper alone. It follows from applying Vacuum Receptivity to the source paper’s phase-space metric target.

The testable bridge is therefore explicit. The phase-space correction predicted by Mubaidin et al. supplies the momentum dependence. Pudding Theory predicts a positive residual dependence on measured vacuum spectral density, after ordinary finite-temperature QED, blackbody, magnetic, Stark, Casimir-Polder, and preparation effects are removed.

Pudding Theory Prediction

Use a Mach-Zehnder light-pulse atom interferometer with \(^{87}\mathrm{Rb}\) atoms. The relevant internal transition is the ground-state hyperfine splitting,

\[ \omega_{\rm hf}=2\pi \times 6.834\,682\,610\ {\rm GHz}. \]

Let \(B=2\pi\times 10\,{\rm kHz}\), set by the Raman pulse bandwidth and the measured interferometer sensitivity function. The vacuum variable is the quadrature-projected spectral excess

\[ \mathcal{N}_\eta= \int_{\omega_{\rm hf}-B}^{\omega_{\rm hf}+B} W(\omega)\,[S_\eta(\omega)-S_0(\omega)]\,d\omega, \]

where \(W(\omega)\) is the normalized Raman sensitivity window and \(S_0(\omega)=\hbar\omega/2\) is the zero-temperature baseline.

For momentum class \(j\), the measured phase is modeled as

\[ \Delta\phi_j = \Delta\phi_{\rm GR} + A_{\rm ps}\,p_j^2 + C_\eta\,p_j^2\mathcal{N}_\eta + \Delta\phi_{\rm env} + \epsilon_j . \]

Here \(\Delta\phi_{\rm GR}=k_{\rm eff}gT^2\). The coefficient \(A_{\rm ps}\) is the momentum-dependent phase-space correction expected from the geometric model. \(\Delta\phi_{\rm env}\) contains calibrated thermal, magnetic, Stark, recoil, Casimir-Polder, and wavefront terms. \(C_\eta\) is the Pudding Theory coefficient.

The predicted sign is positive. Increasing the coupled vacuum spectral density should increase the fitted magnitude of the phase-space correction for fixed \(p_j^2\). Phase-squeezed reduction in the coupled quadrature should reduce it. The effect is not a new force. It is a shift in the fitted phase-space coefficient.

The \(10^{-4}\) falsification scale is operational. The experiment must first create or calibrate a reference momentum-dependent phase-space term with \(|A_{\rm ps}p^2|\geq 1\) rad. A residual phase readout of \(10^{-4}\) rad then corresponds to \(10^{-4}\) fractional sensitivity to the correction. Below that level, the applied Postulate has no detected contribution in this laboratory channel.

Falsifiable Observable

The observable is \(C_\eta\), the coefficient multiplying \(p_j^2\mathcal{N}_\eta\) in the residual interferometric phase equation above, measured in three shielded \(^{87}\mathrm{Rb}\) interferometers while \(\mathcal{N}_\eta\) is varied by thermal loading and phase-squeezed microwave vacuum injection in the \(\omega_{\rm hf}\pm 2\pi\times10\,{\rm kHz}\) band. If \(C_\eta\) were measured to be \(0\pm10^{-4}\) of \(A_{\rm ps}\) with no positive sign correlation across all three interferometers, this Postulate would be falsified.

Editorial Dialogue

Tanaka critique, verbatim:

Vacuum Receptivity is the least inflated choice, but it is still doing more than the draft proves. The source paper’s vacuum is a gravitational and geometric setting, not a measured stochastic receiver. Which postulate is doing the work here? If the source paper had not been published, would this prediction follow from PT alone? The draft’s answer is partly yes, but only after importing a separate assumption that phase-space metric coefficients couple to vacuum noise spectra.

The falsifiability sentence is closer than usual. It names an observable, C_eta, and gives a null threshold. But an experimenter still lacks the coupling band, sign, scaling law, and phase-shift equation. What measurement would distinguish this from the null? “Residual covariance” is not enough unless the draft specifies which frequencies of S_eta couple to which atomic transition or interferometer path variable. The 10^-4 bound also needs justification from projected instrument sensitivity, not rhetorical precision.

Citation risk remains. The arXiv paper, its bibliography, and the “cited in source paper bibliography context” claims must be checked directly. The quoted source sentence appears within fair-use limits, but the review needs a quote count and source-page anchor. References 3 through 6 are plausible, but plausibility is not verification.

Not yet. Add one explicit measurement equation linking S_eta(omega) to the interferometric phase residual, with frequency band, expected sign, and citation-verified source anchors.

Sterling response:

Tanaka’s objection fixes the paper. The applied Postulate is now named once and used narrowly. The source paper supplies the phase-space metric. It does not supply stochastic vacuum coupling. That coupling is the Pudding Theory addition.

The measurement equation is now explicit. The band is the \(^{87}\mathrm{Rb}\) hyperfine coupling band, \(\omega_{\rm hf}\pm2\pi\times10\,{\rm kHz}\). The sign is positive in the coupled quadrature. The scaling is \(p_j^2\mathcal{N}_\eta\). The null is \(C_\eta=0\) at \(10^{-4}\) of the calibrated phase-space term.

The citation audit also changes the reference list. Maggiore, Hossenfelder, and Wald are removed. The supporting references below are taken directly from the source paper bibliography.

Discussion

This Working Paper does not claim that Mubaidin et al. derive Pudding Theory. They do not. Their work is a geometric construction based on RGUP, Finsler structure, Hamilton geometry, and a cotangent-bundle metric. The Pudding Theory addition is an experimental perturbation placed on top of that construction.

The main risk is degeneracy. A microwave vacuum-state change can also shift atomic phases through ordinary QED and technical noise. That is why the observable is residual and band-limited. Known blackbody, Zeeman, AC Stark, Lamb, recoil, wavefront, and surface terms must be measured independently.

The second risk is that \(A_{\rm ps}\) itself may be too small for a laboratory calibration. Then the test becomes a bound rather than a falsification. The proposed \(10^{-4}\) criterion is meaningful only when the reference phase-space correction is brought to at least one radian.

The conclusion would change if the phase-space coefficients are gauge artifacts, or if all vacuum-state dependence is absorbed by ordinary environmental corrections. A stable \(A_{\rm ps}\) under controlled variation of \(\mathcal{N}_\eta\) is the clean null.

References

1. K. Mubaidin, D. Mukherjee, A. Alshehri, and A. Tawfik. “The Derivation of Phase-Space Metric in a Geometric Quantization Approach: General Relativity with Quantized Phase-Space Metric and Relative Spacetime.” arXiv:2602.13219v1, 2026. DOI: doi:10.48550/arXiv.2602.13219.

2. Sterling Geisel. “Pudding Theory: A Topological Theory of Information Fields.” QBist Lab, September 10, 2025.

3. A. Tawfik, S. G. Elgendi, Sameh Shenawy, and Mahmoud Hanafy. “Canonical Quantization of Metric Tensor for General Relativity in Pseudo-Riemannian Geometry.” Physics 7(4), 2025. DOI: doi:10.3390/physics7040052.

4. Abdel Nasser Tawfik and Azzah Alshehri. “Relativistic Generalized Uncertainty Principle for a Test Particle in Four-Dimensional Spacetime.” Modern Physics Letters A 39(25n26), 2450079, 2024. DOI: doi:10.1142/S0217732324500792.

5. E. R. Caianiello, A. Feoli, M. Gasperini, and G. Scarpetta. “Quantum Corrections to the Space-time Metric From Geometric Phase Space Quantization.” International Journal of Theoretical Physics 29, 131, 1990. DOI: doi:10.1007/BF00671323.

6. Giovanni Amelino-Camelia, Laurent Freidel, Jerzy Kowalski-Glikman, and Lee Smolin. “Principle of Relative Locality.” Physical Review D 84, 084010, 2011. DOI: doi:10.1103/PhysRevD.84.084010.

7. Laurent Freidel and Etera R. Livine. “3D Quantum Gravity and Effective Noncommutative Quantum Field Theory.” Physical Review Letters 96, 221301, 2006. DOI: doi:10.1103/PhysRevLett.96.221301.