Gravitational length stretching: Curvature-induced modulation of quantum probability densities

1 Introduction

The unification of quantum mechanics and general relativity remains one of the most profound challenges in theoretical physics, with far-reaching implications for understanding quantum behavior in curved spacetime [1,2]. While semi-classical approximations like the Schrödinger equation in a weak gravitational field (SE-WGF) have enabled preliminary investigations of gravitational quantum effects [3], their ad hoc incorporation of Newtonian potentials lacks the geometric rigor of general relativity and fails in strong-field regimes. Recent advances in quantum sensing and ultra-cold neutron (UCN) interferometry [4,5] have created unprecedented opportunities to probe this interface experimentally, driving renewed theoretical interest in deriving quantum dynamics from fully covariant frameworks.

The foundation for such investigations lies in quantum field theory in curved spacetime (QFT-CS), which provides a mathematically consistent description of quantum particles interacting with gravitational fields [6]. Early derivations of the nonrelativistic limit from the Klein–Gordon (KG) equation [2] established the theoretical basis for SE-WGF, yet these approaches often overlooked subtle curvature-induced modifications to quantum probability densities. Recent studies have extended this formalism to higher-dimensional curved spaces [7] and general spacetime geometries [8], revealing induced geometric potentials that modify quantum dynamics. Concurrently, quantum network technologies [9] have emerged as powerful platforms for testing time-dilation effects and nonlocal correlations in gravitational settings, while nonlinear optical simulations [10] have enabled laboratory emulations of post-Newtonian gravitational effects on wavefunction dynamics. Despite these advances, the physical interpretation of amplitude modulation in curved-spacetime wavefunctions – particularly its role in spatial probability distributions – remains underdeveloped, creating a gap between theoretical predictions and observable phenomena.

This work bridges this gap by identifying gravitational length stretching (GLS) as a fundamental quantum-gravitational phenomenon arising from the amplitude modulation function B k ( t , x → ) in WKB solutions of the KG equation. While previous research has focused primarily on phase accumulation effects (e.g., gravitational phase shifts and time-dilation decoherence), we demonstrate that B k encodes a geometric distortion of quantum spatial profiles that is equally significant. Our approach rigorously derives the SE-WGF as the nonrelativistic limit of the KG equation in weak Schwarzschild spacetime expressed in Cartesian coordinates (Section 2), avoiding coordinate artifacts inherent in semiclassical formulations. Crucially, we establish that B k satisfies the covariant continuity equation ∇ μ ( B k 2 k μ ) = 0 [11], coupling quantum probability conservation directly to spacetime geometry. This leads to GLS – a frame-independent stretching of quantum objects in regions of stronger curvature where B k 2 diminishes – manifested as a ∼ 20 % increase in proper spatial extent over millimeter scales for UCNs in Earth’s gravity (Sections 3 and 4).

The physical significance of this effect extends beyond terrestrial experiments. In strong-field regimes (e.g., neutron star surfaces where V ∼ m c 2 ), GLS dominates quantum dynamics, necessitating QFT-CS for accurate modeling. Our framework resolves ambiguities in semi-classical treatments, particularly the noncovariant quantization condition ∫ k z d z = n π in SE-WGF versus the geometrically consistent form ∫ k μ d x μ = n π in QFT-CS (Section 5). The latter ensures general covariance, enabling systematic extension to Kerr spacetimes (where frame-dragging modifies k μ [12]) and cosmological settings [13]. Furthermore, we show that position measurements in curved spacetime require Hilbert spaces normalized by γ (Section 3), implying that GLS represents an intrinsic quantum-geometric effect rather than a perturbative correction.

Quantitative predictions for UCN spectroscopy confirm GLS observability: energy level shifts ( Δ E n ∕ E n ∼ 1 % at n = 1,000 ) and probability density gradients ( ∂ z B k 2 ≈ 202 m − 1 ) distinguish it decisively from semi-classical predictions. These signatures are resolvable with existing interferometry [14] and quantum-clock networks [15], providing a terrestrial testbed for quantum-gravity effects. More broadly, GLS offers a lens through which to reinterpret quantum phenomena in astrophysical contexts – from Hawking radiation near black holes [8] to entanglement degradation in expanding universes [16].

This article aims to establish GLS as a fundamental quantum-gravitational phenomenon through the following specific objectives:

1. To rigorously derive the SE-WGF as the nonrelativistic limit of the Klein–Gordon equation in weak Schwarzschild spacetime expressed in Cartesian coordinates, highlighting the emergence of the amplitude modulation function B k ( t , x → ) from first principles.

2. To demonstrate that B k satisfies the covariant continuity equation ∇ μ ( B k 2 k μ ) = 0 and represents physical probability density modulation in curved spacetime, rather than merely a mathematical normalization factor.

3. To provide quantitative predictions for experimental verification using ultracold neutrons (UCNs), including energy level shifts ( Δ E n ∕ E n ∼ 1 % at n = 1,000 ) and probability density gradients ( ∂ z B k 2 ≈ 202 m − 1 ).

4. To establish a clear theoretical distinction between the QFT-CS and SE-WGF frameworks, particularly through their different quantization conditions ( ∫ k μ d x μ = n π versus ∫ k z d z = n π ) and their implications for general covariance.

5. To demonstrate that GLS represents a frame-independent quantum-geometric effect that dominates in strong-field regimes, where V ∼ m c 2 , necessitating the QFT-CS approach for accurate modeling.

This article is structured as follows: Section 2 derives the SE-WGF from the KG equation in weak Schwarzschild spacetime, highlighting coordinate choices and approximations. Section 3 establishes B k as the physical amplitude modulator through QFT-CS mode solutions, and introduces GLS theoretically and connects it to quantum measurement in curved spacetime. Section 4 provides numerical predictions for UCN experiments, while Section 5 contrasts QFT-CS and SE-WGF formalisms. Section 6 discusses the experimental feasibility in testing GLS effect. Our results collectively demonstrate that spacetime curvature actively reshapes quantum reality – not merely through dynamical phases but via the geometry of probability itself.

Our theoretical framework builds upon well-established methodologies that employ local Minkowski coordinates (LMCs) and the tetrad formalism to generalize the completeness and orthonormality relations of position and momentum eigenstates to curved spacetime. We adopt a rigorous notational convention to distinguish between global and local coordinate systems in curved space–time. Greek indices (e.g., μ , ν ) are utilized to represent global coordinates, which are essential for describing the geometry of curved space–time in the framework of general relativity. Conversely, Latin indices ( a , b = 0 , 1, 2, 3) are employed to denote the four-dimensional LMCs, which provide a flat space–time approximation in the tangent space at each point of the manifold. Specifically, the indices a and b encompass both temporal and spatial components, with a , b = 0 corresponding to the time coordinate and a , b = 1 , 2, 3 representing the spatial coordinates. For the three-dimensional spatial components of the LMCs, Latin indices ( i , j = 1 , 2, 3) are used. The Minkowski metric η a b is defined with the signature η a b = diag ( − 1 , 1 , 1 , 1 ) , consistent with the convention adopted throughout this work.

2 Derivation of the SE-WGF from the Klein–Gordon equation in weak Schwarzschild spacetime

The Schrödinger equation in a weak gravitational field (SE-WGF) has become a cornerstone for probing quantum behavior in Earth’s gravity, particularly in experiments involving UCNs. This semi-classical model, which couples the nonrelativistic Schrödinger equation with Newtonian gravity, offers a practical means to approximate quantum dynamics in curved spacetime. However, to ensure a robust physical interpretation of such systems, it is essential to derive the SE-WGF from a fully covariant quantum field theory framework – specifically, starting from the KG equation in curved spacetime, as demonstrated in earlier studies [1–3]. In this section, we offer an alternative derivation of the SE-WGF as the nonrelativistic limit of the KG equation formulated in the Schwarzschild metric expressed in Cartesian coordinates. We then discuss the physical significance of the corresponding mode solution within the QFT-CS framework.

3 Theoretical foundation of gravitational length stretching

In a recent investigation [11], we derived the solutions for free spin-0 (scalar) and spin-1/2 (fermionic) particles in WGF where the perturbations in the metric and the variations in B k ( x ) are negligible compared to the dominant energy scale of the particle. These solutions take the zeroth-order WKB form [18–21]:

where B k ( t , x → ) represents a spacetime-dependent amplitude modulation, and k μ satisfies the curved spacetime dispersion relation g μ ν k μ k ν + m 2 = 0 . Crucially, B k obeys the covariant form of continuity equation:

ensuring conservation of the probability current B k 2 k μ . While the phase factor e − i ∫ k μ d x μ encodes standard geodesic dynamics, the amplitude B k contains novel information about gravitational modifications to quantum probability densities – a feature previously unexplored in QFT-CS.

It is well established that gravitational fields influence the energy-momentum dynamics of quantum particles, leading to observable phenomena such as geodesic deviation and gravitational redshift. These effects are inherently encoded in the phase factor k μ of the wave function, which governs the particle’s propagation in curved space–time. However, the physical implications of the amplitude function B k ( t , x → ) have not been thoroughly explored in the literature. In this work, we propose and investigate the hypothesis that B k ( t , x → ) represents the probability density distribution of a free particle in curved space–time. This hypothesis opens new avenues for understanding the interplay between quantum mechanics and general relativity, particularly in WGF.

To enrich this discussion, we note that the function B k ( t , x → ) can be interpreted as a modulation of the particle’s wave function due to the curvature of space–time. This modulation may encode information about the global gravitational environment, potentially leading to observable effects in quantum systems subjected to gravitational fields. For instance, variations in B k ( t , x → ) could influence the spatial distribution of particles in quantum interference experiments conducted in noninertial frames or near massive objects. Furthermore, the continuity Eq. (3.2) suggests that B k ( t , x → ) is intricately linked to the geometry of space–time through the covariant derivative ∇ μ , highlighting the deep connection between quantum probability densities and the underlying space–time structure. By systematically analyzing the role of B k ( t , x → ) in the context of curved space–time, this study aims to shed light on the quantum-gravitational interplay and its potential observational signatures. Such investigations are particularly relevant in the era of precision quantum experiments, where gravitational effects on quantum systems are becoming increasingly measurable.

In analogy to the formalism employed in Minkowski space–time, the position eigenstate in curved space–time is represented by the Hilbert space vector ∣ x ⟩ , which describes a particle localized at position x . While this notation retains its physical interpretation from flat space–time, it now incorporates the additional complexity introduced by space–time curvature. A crucial insight is that gravity not only affects the energy-momentum dynamics of particles but also distorts their probability density distribution in real space. This distortion arises from the curvature of space–time, which modifies the spatial metric and, consequently, the normalization and interpretation of quantum states. Using tetrads e μ a to connect global coordinates to LMCs, the completeness relation becomes:

where γ is the determinant of γ μ ν = e μ i e ν j δ i j , which is the induced 3-spatial metric. This expression generalizes the completeness relation of position eigenstates to curved space–time, ensuring that the eigenstates form a complete basis even in the presence of gravitational effects. The inner product between two position eigenstates ∣ x → 1 ⟩ and ∣ x → 2 ⟩ is defined as follows:

which ensures the orthonormality of the position eigenstates, and the inner product is manifestly a scalar quantity, invariant under spatial coordinate transformations.

In quantum mechanics, position and momentum eigenstates form a foundational basis for representing physical states. While the physical interpretation of position eigenstates ∣ x → ⟩ remains locally analogous to their Minkowski space–time counterparts (defined via the Dirac delta normalization), the momentum eigenstates in curved space–time exhibit profound geometric distortions. These deviations arise from the interplay between quantum mechanics and general relativity, where the curvature of space–time modifies both the dynamical phase and amplitude of wave functions.

In Minkowski space–time, momentum eigenstates are globally valid plane-wave solutions ⟨ x → ∣ p ( t ) ⟩ = e − i ( ω t − k → ⋅ x → ) with uniform probability density, reflecting the homogeneity and isotropy of flat geometry. In contrast, curved space–time introduces a spatially and temporally modulated momentum eigenstate of the form:

where N k i is a metric-dependent normalization factor, B k i ( t , x → ) is the amplitude function that encodes the spatial and temporal modulation of the wave function due to the gravitational field. The phase factor e − i ∫ k μ d x μ generalizes the plane-wave phase through a path integral over the covariant four-momentum k μ . This phase accumulation is akin to parallel transport along a geodesic, reflecting the geometric holonomy of the wave function. In Minkowski space–time, the momentum eigenstate reduces to a plane wave e − i ( ω t − k → ⋅ x → ) , which implies a uniform probability density distribution in a globally flat space–time. In contrast, the presence of the amplitude function B k i ( t , x → ) in curved space–time indicates that gravity not only influences the energy-momentum of particles but also distorts their probability density distribution in real space. The amplitude function B k i ( t , x → ) arises from the nontrivial spacetime metric g μ ν ( x ) , which distorts the conserved probability current j μ = B k i 2 k μ [11]. Consequently, B k i incorporates tidal gravitational effects, leading to a spatially varying probability density ∣ ϕ k i ∣ 2 = ∣ N k i ∣ 2 ∣ B k i ∣ 2 – a marked departure from the Minkowski case.

The inner product between the two distorted momentum eigenstates ∣ ϕ k i ⟩ and ∣ ϕ k i ′ ⟩ is expressed as follows:

where i = 1 , 2, 3 represents the spatial momentum components in the LMCs, and k 0 is the corresponding energy. The exponential term e − i ∫ ( k μ − k μ ′ ) d x μ oscillates rapidly compared to the variation of B k i ( t , x → ) , as k μ ≫ ∂ μ B k i ( t , x → ) . By applying the method of stationary phase, we argue that the integral is dominated by the oscillatory exponential, while the slowly varying function B k i ( t , x → ) can be treated as approximately constant over the oscillation scale. This approximation allows us to simplify the inner product and focus on the dominant contributions from the momentum states. It is crucial to emphasis that since LMCs in WGF form a continuum of coordinates defined at each space–time point, the value of the momentum k i may vary depending on the LMC in which it is evaluated. Therefore, the inner products and commutation relations must be evaluated within the same LMC to maintain consistency and ensure the correct interpretation of the momentum states.

Notably, the normalization factor N k i can be absorbed into the amplitude function B k i ( t , x → ) , enabling us to omit N k i in subsequent discussions and assume that the wave function ϕ k i ( t , x → ) = B k i ( t , x → ) e − i ∫ k μ d x μ is normalized. While calculating the exact form of B k ( t , x → ) is technically challenging due to the complexity of curved space–time dynamics, we note that only the ratio B k ( t 1 , x → 1 ) ∕ B k ( t 2 , x → 2 ) – representing the relative probability density between two space–time locations – is physically significant. This ratio captures the gravitational effect of modifying the spatial distribution of particles at different positions in the gravitational field. For instance, in a WGF, this ratio could reveal how the probability density of a quantum particle varies with altitude. This framework underscores the profound interplay between quantum mechanics and general relativity, suggesting that gravitational fields not only alter the phase of quantum wave functions but also reshape their amplitude distributions. Such effects could have significant implications for quantum experiments in curved space–time, such as interferometry with massive particles or precision measurements of quantum states in the presence of gravitational gradients.

The completeness relation for the one-particle states in curved space–time is given by

which ensures that the distorted momentum eigenstates ∣ ϕ k i ⟩ form a complete basis for the Hilbert space of one-particle states, generalizing the flat space–time result to include the effects of curvature. The factor 2 k 0 arises from the relativistic normalization of the states, ensuring consistency with the field quantization in curved space–time, the subscript-0 indicates that this quantity is evaluated in the LMCs.

The quantized KG field in WGF can be expressed as follows:

where a k i + and a k i are the creation and annihilation operators, respectively. These operators act on the vacuum state ∣ 0 ⟩ to generate particle states. The state ∣ x → ⟩ t = Φ ( t , x → ) ∣ 0 ⟩ represents a particle localized at position x → at time t . For a KG particle with momentum k i measured in a LMC system, the state is given by ∣ ϕ k i ⟩ = 2 k 0 a k i + ∣ 0 ⟩ , where ∣ 0 ⟩ is the vacuum state. The creation and annihilation operators satisfy the canonical commutation relation:

which ensures the proper quantization of the field.

To demonstrate that the function B k i ( t , x → ) represents the probability density distribution of the wave ϕ k i in real space x → , we calculate the probability of finding the particle within a small volume Δ V 1 around the space point x → 1 :

Here, γ ( x → 1 ) Δ V 1 represents the proper volume around the space point x → 1 , accounting for the spatial metric induced by the curvature of space–time. The probability P x → 1 is proportional to B k i 2 ( t , x → 1 ) , indicating that the particle’s probability density is modulated by the gravitational field. Specifically, a larger value of B k i 2 ( t , x → 1 ) corresponds to a higher probability density, implying that the particle is “compressed” in regions where B k i 2 ( t , x → 1 ) is large, and “stretched” in regions where it is small. This result highlights the role of B k i ( t , x → ) as a gravitational modulation factor that distorts the probability density distribution of the particle in real space.

A free KG particle can be represented by a wave packet constructed from a superposition of the distorted plane waves:

where ϕ k i ( t , x → ) is the distorted plane wave solution for a single mode with momentum k i given by Eq. (3.5), and f ( k i ) is the weight function that determines the contribution of each mode to the wave packet. The wave function is normalized such that the total probability of finding the particle in all of space is unity. This normalization condition

remains valid through the orthonormality relation ⟨ ϕ k ′ ∣ ϕ k ⟩ ≈ 2 k 0 ( 2 π ) 3 δ 3 ( k i − k i ′ ) . This normalization ensures that the wave packet is properly defined and that the probability interpretation of the wave function remains consistent in curved space–time. Now, suppose the particle is well-localized around a space point x → 1 . The probability of finding the particle within a small volume Δ V 1 centered at x → 1 is approximately given by:

By substituting the wave packet expression into the integral, we arrive at the following expression:

In this expression, the exponential term e − i ( k μ − k μ ′ ) Δ x 1 μ describes the interference between different momentum modes. The functions B k i ( t , x → 1 ) and B k i ′ ( t , x → 1 ) account for the gravitational modulation of the wave function at the position x → 1 . If the interference-induced effect on the wave packet size is negligible in the region where the measurement is performed, and the particle subsequently propagates to a different location x → 2 with a stronger gravitational field, the amplitude function B k i ( t , x → 2 ) will generally be smaller than B k i ( t , x → 1 ) . Consequently, the proper size of the particle, denoted by γ ( x → 2 ) Δ V 2 , will typically satisfy the relation:

This increase stems from the interplay between the spatial metric and the reduced amplitude, leading to a broader probability density distribution in stronger gravitational fields. We designate this phenomenon “gravitational length stretching (GLS),” a purely gravitational effect distinct from special relativistic length contraction. Unlike the latter, which depends on relative velocity and reference frames, GLS is frame-independent because B k ( t , x → ) is a scalar function tied to the global geometry. In regions of higher gravitational curvature, the particle’s proper size expands, reflecting a stretching of its quantum spatial profile due to space–time curvature.

4 Example: UCNs in weak-field Schwarzschild geometry

The interplay between quantum mechanics and general relativity manifests uniquely in the phenomenon of GLS, a subtle yet significant effect that alters the spatial structure of quantum probability densities in curved spacetime. In particular, the presence of a gravitational field modifies the wavefunction characteristics of quantum systems, leading to shifts in energy spectra and deformation of probability distributions. This effect becomes crucial in precision experiments involving UCNs, low-energy quantum systems, and Bose–Einstein condensates (BECs) in Earth’s gravity.

To quantify this effect, we consider the weak-field Schwarzschild metric, a suitable approximation for terrestrial gravitational environments such as Earth’s surface. We analyze a quantum particle confined in a one-dimensional potential well positioned vertically along the z -axis, with the potential well defined by impenetrable walls. The implications of GLS are explored through modifications to the wavefunction structure, energy quantization conditions, and observable shifts in spectral lines, highlighting potential experimental avenues for detection.

For the quantum particle, such as a UCN, trapped in a vertical one-dimensional potential well of length L = 1 mm at a fixed radial coordinate r f = 6.371 × 1 0 6 m (Earth’s surface), the Schwarzschild metric in weak-field approximation provides an appropriate framework for analysis. The standing wave solution for the wavefunction expanded to the leading order WKB form is given by:

where ω ≡ c k t is conserved along the Schwarzschild geodesic, and A 1 and A 2 are constants determined by boundary conditions. The factor B k ( z ) is given by [11]:

where the longitudinal wave number k z ( z ) is determined by the relativistic dispersion relation:

where r s denotes the Schwarzschild radius of Earth. The factor B k 2 ∝ 1 ∕ k z suggests that in stronger gravitational fields (smaller z − r s ), the amplitude of the wavefunction is suppressed due to increased k z , thereby stretching the probability density.

Applying the boundary conditions ϕ k ( t , r f ) = 0 and ϕ k ( t , r f + L ) = 0 , the wavefunction assumes the form:

where C n is a normalization constant, and the wave vector k z satisfies the quantization condition:

Solving Eq. (4.5) yields the energy spectrum (see Appendix):

where l n ≡ n π + n 2 π 2 + a s 2 L and a s ≡ r s m 2 c 2 L 3 ℏ 2 r f 2 . The factor a s encapsulates the gravitational correction, shifting l n from its flat-space counterpart n π ∕ L . Neglecting the small correction terms with ε , we arrive at:

The spacing between energy levels is then given by:

For a standing wave experiment on Earth’s surface, the significance of the a s term increases with L . With L = 1 mm , we estimate a s ≈ 5 × 1 0 6 , comparable to n ≈ 1 0 3 . This leads to a deviation in the wave vector:

which contrasts with the flat-space value k z ≈ 3.14 × 1 0 6 m − 1 . Consequently, the energy level spacing at n = 1,000 is:

compared to the Minkowski case, where Δ E n = n π 2 ℏ 2 m L 2 ≈ 4.09 × 1 0 − 13 eV . This corresponds to a relative energy shift of ∼ 1 % , confirming that Earth’s gravity introduces a small yet potentially measurable perturbation to the quantum well spectrum.

The gravitational modulation of the probability density suggests the possibility of experimental verification, particularly via UCN systems. The radial gradient of the probability density is given by

for the UCNs on the surface of earth with wavevector given by Eq. (4.9), indicating a cumulative probability density variation of ∼ 0.2 over L = 1 mm . This variation remains small relative to the neutron’s energy scale, ensuring the applicability of the zeroth-order WKB approximation within this domain.

However, as k z → 0 , the gradient ∂ r B k 2 diverges, marking the breakdown of the zeroth-order WKB approximation. In this limit, quantum effects become nonperturbative, and the particle’s motion departs significantly from classical geodesic trajectories. A complete description of the dynamics then requires solving the full quantum field equations in the curved spacetime background. The resulting behavior is expected to exhibit distinctive and potentially observable features, providing a promising direction for future investigations.

5 Comparison of WKB methods from QFT-CS and SE-WGF formalisms

Using the WKB approximation, the leading-order wavefunction solution for the SE-WGF equation is

where the local momentum is given by

which is equivalent as Eq. (4.3) by recognizing that ℏ ω = m c 2 + E as indicated by Eq. (2.6). Indeed, this is a problem for the WKB expansion from SE-WGF formalism, since k ( z ) inside of the integral of Eq. (5.1) is k z rather than k z as indicated in the quantization condition of Eq. (4.5), which is developed from the QFT-CS formalism. In other words, the quantization condition in SE-WGF formalism employs

which is not covariant, and it is coordinate dependent. By contrast, the quantization condition in Eq. (4.5) is manifestly covariant and invariant under coordinate transformations.

Table 1 summarizes the key differences between the two approaches.

6 Experimental feasibility analysis

This section provides a detailed analysis of the experimental feasibility for detecting GLS effects in UCN systems. Our assessment is based on current experimental capabilities and established physical principles.

7 Discussion and conclusion

This work establishes GLS as a fundamental quantum-gravitational phenomenon arising from amplitude modulation B k ( t , x → ) in curved spacetime. Through rigorous derivation within quantum field theory in curved spacetime (QFT-CS), we demonstrate that GLS represents a geometric distortion of quantum probability densities governed by the covariant continuity equation ∇ μ ( B k 2 k μ ) = 0 , distinct from-phase accumulation effects. Our key findings are as follows:

• Quantum-geometric distortion: GLS causes frame-independent stretching of quantum objects in stronger curvature where B k 2 diminishes. For UCNs in Earth’s gravity, this manifests as a ∼ 20 % increase in proper spatial extent over millimeter scales.

• Experimental signatures: Quantitative predictions confirm GLS observability:

  1. Energy level shifts: Δ E n ∕ E n ∼ 1 % at n = 1,000

  2. Probability gradients: ∂ z B k 2 ≈ 202 m − 1

  3. Spectral deviations from semi-classical models ( δ E ∼ 1 0 − 13 eV)

  These signatures are resolvable with existing UCN interferometry [14].

• Theoretical superiority of QFT-CS: Critical distinctions from semi-classical SE-WGF formalisms include:

  1. Covariant quantization: ∫ k μ d x μ = n π (QFT-CS) vs. coordinate-dependent ∫ k z d z = n π (SE-WGF)

  2. Physical interpretation of B k as probability modulator in curved Hilbert spaces ( ⟨ x → 1 ∣ x → 2 ⟩ = δ 3 ( x → 1 − x → 2 ) ∕ γ )

  3. Robustness in strong fields ( Φ ∕ c 2 ≳ 0.1 ), enabling neutron star and cosmological applications

Connections to recent advances in gravitational quantum phenomena

Our investigation of GLS connects to recent advances in the study of quantum phenomena in extreme gravitational environments. The study by Rayimbaev et al. [23] on test particle dynamics around regular black holes in modified gravity provides important context for understanding how quantum systems might behave in the vicinity of nonsingular compact objects. Their analysis of particle dynamics in these extreme environments complements our QFT-CS approach by offering a classical counterpart against which quantum deviations – including those arising from GLS – might be measured.

The observational framework developed by Rink et al. [24] for testing general relativity using quasi-periodic oscillations from X-ray black holes presents potential applications for detecting quantum gravitational effects in astrophysical settings. Their methodology for extracting information from black hole systems could be adapted to search for signatures of quantum phenomena like GLS in the high-curvature regions near event horizons, where our predicted effects would be substantially amplified.

The investigation of gravitational lensing phenomena by [25] in Ellis–Bronnikov–Morris–Thorne wormhole geometries with topological defects provides an interesting comparison to our work. While their focus is on classical light propagation, the mathematical framework for studying how spacetime geometry affects wave propagation shares conceptual parallels with our analysis of quantum probability distributions in curved spacetime. The inclusion of global monopoles and cosmic strings in their work suggests potential extensions of our GLS formalism to spacetimes with topological defects.

Finally, the study of test particle dynamics and scalar perturbations around Ayón–Beato–García black holes coupled with a cloud of strings by [26] provides additional context for understanding how different types of matter configurations affect gravitational phenomena. Their approach to analyzing perturbations in nonvacuum spacetimes could inform future extensions of our work to more complex matter distributions.

These recent studies collectively highlight the vibrant research activity surrounding quantum and classical phenomena in extreme gravitational environments. Our work on GLS contributes to this growing body of knowledge by providing a specifically quantum-field-theoretic perspective on how curvature affects quantum probability distributions – a effect that may become increasingly important as observational capabilities improve and allow us to probe ever more extreme gravitational regimes.

Broader implications and future directions:

• Extreme gravity: Dominant GLS effects on neutron star surfaces where V ∼ m c 2 .

• Cosmology: Stretching of quantum fluctuations during inflation.

• Quantum measurement: Position measurements require curvature-distorted Hilbert spaces.

• Spin-curvature coupling: Dirac field extensions may amplify GLS through spin-gravity interactions.

Future research directions should include experimental detection of GLS signatures using ultracold neutron interferometry, extension to Dirac fields where spin-curvature coupling may amplify these effects, and investigation of GLS in cosmological contexts where expanding spacetime modifies quantum probability distributions. The integration of advanced mathematical methods from recent works [27–29] may further enhance our understanding of nonlinear quantum phenomena in curved spacetime. In conclusion, spacetime curvature actively reshapes quantum reality – not merely through dynamical phases but via probability geometry itself. While SE-WGF suffices for weak terrestrial fields, QFT-CS emerges as the fundamental framework for gravitational quantum phenomena. Experimental detection of GLS signatures and extensions to Dirac fields represent critical next steps in probing quantum-gravity interfaces.