Topological Quantization of Complex Velocity in Stochastic Spacetimes

T op ological Quan tization of Complex V elo cit y in Sto c hastic Spacetimes Jorge Meza-Dom ´ ınguez ∗ 1 and T onatiuh Matos † 2 1,2 Departamen to de F ´ ısica, Centro de In v estigaci´ on y de Estudios Av anzado s del Instituto P olit´ ecnico Nacional, Av. Instituto P olit ´ ecnico Nacional 2508, San P edro Zacatenco, M´ exico 07360, CDMX. Abstract The h ydro dynamic form ulation of quantum mec hanics features tw o velocity fields: a geo desic (classical) velocity π µ and a sto c hastic (quantum) v elo cit y u µ . W e sho w that a veraging o ver a sto c hastic gra vitational w av e background unifies these in to a single complex v elo cit y η µ = π µ − iu µ , deriv ed from the logarithmic deriv ativ e of a matter amplitude K . This ob ject lives as a section of the pullbac k bundle π ∗ 2 ( T ∗ M ) o ver configuration space and defines a flat U (1) connection, satis- fying D µ K = 0. Crucially , η µ acts as a fundamen tal information-geometric carrier, where u µ maps the v ariance of metric fluctuations ⟨ h µν h αβ ⟩ to the Fisher metric and v on Neumann entrop y . The resulting geometric structure collapses in to an ele- gan t complex geo desic equation η ν ∇ ν η µ = ∇ µ ( 1 2 η ν η ν ), while non-trivial spacetime top ology imp oses a holonom y quantization condition. This top ological phase sug- gests observ able signatures in atom in terferometry and cosmological correlations, pro viding an exp erimental windo w into the sto chastic nature of spacetime at the Planc k scale. 1 In tro duction Quan tum mechanics (QM) and general relativit y (GR) are curren tly the t wo pillars of ph ysics. Ho wev er, while GR has a v ery clear and understandable interpretation, QM lac ks a robust one. In [6] a new interpretation has b een initiated based on the follo wing idea, which we hav e called Sto chastic Quantum Gravit y (SQG). W e know that spacetime is filled with gravitational wa ves of all sizes and frequencies. If this is true, particles on the order of magnitude of gravitational w av es would not mo v e along geo desics, but rather along a geo desic plus a sto chastic term, just as in Brownian motion. F or example, w e observ e this phenomenon in pulsars measured b y Pulsar Timing Arra ys (PT As) (see, for example, [17]), for gra vitational wa v es on the order of parsecs. The idea b ehind SQG is that small particles, suc h as electrons, protons, photons, etc., also perceive the presence of gra vitational wa v es; therefore, these quan tum particles do not follo w geo desics, but rather ∗ E-mail: jorge.meza@cinv estav.mx † E-mail: tonatiuh.matos@cinv estav.mx 1 a geo desic plus a sto c hastic tra jectory , similar to ho w a pulsar mo ves in the galaxy . In fact, in [6] we sho wed that if we add a sto chastic term to any geodesic tra jectory , the field equation for this particle is simply the Klein-Gordon equation for the complex function Φ = √ ne θ , where π µ ∼ ∇ µ θ is the geo desic v ector and u µ ∼ ∇ µ ln( n ) is the sto chastic v elo cit y . This leads to a new in terpretation of QM, since, as we know, the Klein-Gordon equation reduces to the Sc hr¨ od inger equation in its Newtonian limit. Therefore, in this new interpretation, the QM equation remains in tact, but now the phase of the wa v e function determines the particle’s tra jectory . In other words, in this interpretation of QM, particles do ha ve tra jectories, but sto c hastic tra jectories determined b y the sto c hastic nature of QM, without affecting the QM equation itself. The Madelung–Bohm form ulation of quantum mec hanics [18, 3, 19] decomp oses the w av e function Ψ = √ ρe iS/ ℏ in to tw o velocity fields: π µ = 1 m ∇ µ S, u µ = ℏ 2 m ∇ µ ln ρ. (1) While π µ go verns classical tra jectories along geo desics, the origin of the sto c hastic v elo c- it y u µ has remained mysterious [27, 19]. In this framework, particles follo w sto c hastic tra jectories determined not b y hidden lo cal parameters, but by the fundamen tal geomet- ric fluctuations of the spacetime manifold, preserving the non-lo cal structure of quantum mec hanics through the complex v elo cit y η µ . Recent prop osals suggest that spacetime ma y p ossess intrinsic sto chastic fluctuations at the Planck scale [23, 24, 15], arising from quan tum gra vity effects [13, 28]. If gravit y is sto c hastic, quantum fields on fluctuating bac kgrounds inherit effective sto cha stic dynamics, suggesting a deep link: u µ ma y arise from av eraging ov er gravitational fluctuations. W e demonstrate that starting from a master partition function that a v erages o v er a sto chastic gravitational wa v e bac kground, a natural complex velocity η µ = π µ − iu µ emerges. Its geometric structure revea ls a flat U (1) connection with quantized holonomy , directly linking the v ariance of metric fluctuations to observ able top ological phases. This framew ork bridges quantum mechanics, information geometry , and sto chastic gravit y [2, 10, 12], offering new insigh ts into the quan tum-classical transition [14, 34]. 2 Complex V elo cit y from Sto c hastic Gra vit y Consider a matter action S [Φ , A ; g (0) + h ] where h µν are sto c hastic metric fluctuations with distribution P [ h ] satisfying ⟨ h µν ⟩ = 0 and ⟨ h µν h αβ ⟩ = C µν αβ [30]. The master partition function is Z = Z D [ h ] P [ h ] Z D [Φ , A ] e i ℏ S [Φ ,A ; g (0) + h ] . (2) Define the matter amplitude K [Φ , A ] = R D [ h ] P [ h ] e i ℏ S [Φ ,A ; g (0) + h ] . This ob ject av erages o ver gra vitational fluctuations b efore matter in tegration, enco ding the interpla y b etw een matter and sto c hastic gra vity [7, 33]. Its p olar decomp osition K = √ P e i S / ℏ yields an effectiv e probability density P and phase S . Definition 1. The c omplex velo city is η µ := − i ℏ m ∇ µ K K = − i ℏ m ∇ µ ln K . (3) Using the p olar de c omp osition, η µ = 1 m ∇ µ S − i ℏ 2 m ∇ µ ln P = π µ − iu µ . (4) 2 The cumulan t expansion [16, 29] giv es ln P = − 1 ℏ 2 ⟨ S 2 1 ⟩ h + · · · , where S 1 is the linear term in h . Thus u µ = ℏ 2 m ∇ µ ln P = − 1 2 m ℏ ∇ µ ⟨ S 2 1 ⟩ h , (5) directly linking u µ to the v ariance of gra vitational fluctuations. The Fisher metric on configuration space [2, 10] b ecomes H µν ( x ) = Z D [Φ , A ] P ∇ µ ln P ∇ ν ln P = 4 m 2 ℏ 2 ⟨ u µ u ν ⟩ P , (6) establishing u µ as an information-geometric to ol linking spacetime sto chasticit y to the distinguishabilit y of quantum states via the Fisher metric. [9, 4]. 3 Geometric Structure: Pullbac k Bundle and U (1) Connection Let M be spacetime and C the infinite-dimensional configuration space of matter fields [25]. The pro duct C × M carries the pullback bundle E = π ∗ 2 ( T ∗ M ), with fibre T ∗ x M at ( ϕ, x ) [20, 8]. Prop osition 1. η µ is a se ction of E : η ∈ Γ( π ∗ 2 ( T ∗ M ) → C × M ) . Equivalently, η : C → Γ( T ∗ M ) . Smo othness is understo o d in the sense of F r´ echet manifolds, ensuring wel l-define d variational c alculus on C . W riting η µ = ∇ µ ϕ with ϕ = − i ℏ m ln K , w e define a cov arian t deriv ativ e on a U (1) bundle ov er M D µ := ∇ µ − i m ℏ η µ . (7) A direct calculation yields D µ K = 0, so K is a horizon tal section. Since η µ is exact, the curv ature v anishes: [ D µ , D ν ] = − i m ℏ ( ∇ µ η ν − ∇ ν η µ ) = 0 . (8) Th us η µ defines a flat connection. This geometric structure parallels the Berry connection in quan tum mec hanics [26] and the gauge fields in the Aharono v-Bohm effect, but here the connection emerges from stochastic gra vitational fluctuations rather than electromagnetic p oten tials. 4 Holonom y Quan tization and T op ological Phases Despite flatness, non-trivial holonomy can arise if M is not simply connected [20, 8]. Our univ erse is consisten t with trivial top ology at cosmological scales, but lo cal structures suc h as blac k holes introduce non-contractible lo ops. Additionally , Planc k-scale quan tum gra vity suggests a foam-like structure with microscopic top ological defects [13, 15]. F or a closed lo op γ that cannot b e contracted to a p oin t, Hol γ = exp  i m ℏ I γ η µ dx µ  = exp  i m ℏ I γ ∇ µ ϕ dx µ  . (9) 3 P arallel transp ort of K around γ m ust return the same v alue, imp osing the quan tization condition m ℏ I γ η µ dx µ = 2 π n, n ∈ Z . (10) This mirrors the Dirac quan tization condition and the Aharonov–Bohm effect, where phase accum ulations around closed lo ops b ecome ph ysically observ able despite lo cal flat- ness. The multi-v aluedness of ϕ arises from three sources: top ological terms in the classical action S 0 , gravitational corrections ⟨ S 2 ⟩ h , and zeros of P where ln P develops branc h cuts [13, 28, 15]. 5 Unified Geometrical Equation F rom η µ = ∇ µ ϕ , symmetry ∇ ν η µ = ∇ µ η ν follo ws. Computing η ν ∇ ν η µ η ν ∇ ν η µ = η ν ∇ µ η ν (11) = ∇ µ  1 2 η ν η ν  − 1 2 ( ∇ µ η ν ) η ν + 1 2 η ν ( ∇ µ η ν ) . (12) The last t wo terms cancel by symmetry , yielding the complex geo desic equation η ν ∇ ν η µ = ∇ µ  1 2 η ν η ν  . (13) This is equiv alen tly expressed via the Lie deriv ativ e. Using ∇ ν η µ = ∇ µ η ν , ( L η η ) µ = η ν ∇ ν η µ + η ν ∇ µ η ν = 2 η ν ∇ ν η µ , (13) so (13) b ecomes the compact geometric condition L η η = d ( | η | 2 ) , | η | 2 ≡ η ν η ν . (14) Th us the Lie deriv ativ e of the complex velocity along itself is exact, reflecting the flatness of the U (1) connection D µ = ∇ µ − i ( m/ ℏ ) η µ . The con tinuit y equation follows from the mo dified Klein-Gordon equation satisfied by K  ∇ µ ∇ µ − m 2 ℏ 2 + V  K = 0 , (15) with V real. Combining with its complex conjugate gives ∇ µ ( P π µ ) = 0, or ∇ µ π µ = − 2 m ℏ u µ π µ . Hence orthogonalit y u µ π µ = 0 is not imp osed a priori but emerges under additional conditions on spacetime geometry or the flow [14, 19]. 6 Observ able Signatures and Implications The quan tization condition implies that sto chastic gra vitational fluctuations induce top o- logical phases. F or an atom interferometer with paths enclosing area A , the phase shift is ∆ ϕ top = m ℏ I γ η µ dx µ = 2 π n. (14) Using η µ = π µ − iu µ and u µ ∝ ∇ µ ⟨ S 2 1 ⟩ h , the sto c hastic contribution dep ends on the t wo-point correlation C µν αβ ( x, x ′ ) of metric fluctuations. Precision interferometry can 4 directly constrain the v ariance of quantum gravit y fluctuations, providing an exp erimen tal signature of spacetime sto chasticit y . In cosmology , suc h top ological phases ma y leav e imprints in the cosmic micro wa ve bac kground through non-Gaussian correlations or p olarization patterns. The effective Einstein equations deriv ed from Z [ g ] giv e G µν = 8 π G ⟨ T µν ⟩ eff = 8 π G 1 Z Z D [Φ , A ] Z D [ h ] P [ h ] T (clas) µν e iS/ ℏ , (15) where ⟨ T µν ⟩ eff couples to the Fisher metric via u µ , pro viding a backreaction mec hanism [23, 32]. This framework also connects to the von Neumann en trop y [21, 31]: S vN ≈ 1 2 T r ln  ℏ 2 4 m 2 ⟨ η η ∗ + η ∗ η ⟩  + const. , (16) sho wing that η µ as the carrier of v on Neumann en tropy , mapping gra vitational v ariance to quantum statistical uncertaint y . [11, 5, 22]. 7 Discussion and Conclusions W e hav e demonstrated that the deterministic and sto chastic velocities of quan tum me- c hanics, π µ and u µ , are not indep endent entities but comp onents of a single complex v elo cit y field η µ . This unification is a direct consequence of av eraging o ver a sto chastic gra vitational bac kground, where the quantum p otential emerges naturally from metric fluctuations. The most striking result is the collapse of the coupled h ydro dynamic equations in to the compact complex geo desic form η ν ∇ ν η µ = ∇ µ ( 1 2 η ν η ν ). This elegance suggests that the “quan tumness” of a particle is a manifestation of its motion along complex geo desics in a non-trivial fib er bundle E = π ∗ 2 ( T ∗ M ). F urthermore, the identification of u µ with the Fisher metric H µν establishes a robust link b et ween spacetime sto c hasticity and the distin- guishabilit y of quan tum states, mapping gravitational v ariance directly to v on Neumann en tropy . Bey ond the theoretical framework, the holonomy quan tization condition m ℏ H γ η µ dx µ = 2 π n pro vides a clear exp erimental target. Ev en in cosmologically trivial top ologies, the presence of black holes or Planc k-scale defects introduces non-contractible lo ops that should manifest as top ological phases. W e prop ose that precision atom in terferometry could detect these phases as a signature of the sto chastic nature of spacetime, offering a p oten tial windo w in to quan tum gra vit y effects at energy scales curren tly inaccessible to accelerators. 1 References [1] M Ab e et al. MA GIS-100: Next-generation in termediate-baseline detector for dark matter and gra vitational wa v es. Quantum Scienc e and T e chnolo gy , 6(4): 044003, sep 2021. doi: 10.1088/2633- 4356/abf79a. URL https://doi.org/10.1088% 2F2633- 4356%2Fabf79a . 1 While direct detection of Planck-scale stochasticit y remains b eyond the ∼ 10 16 GeV reach of foresee- able colliders, the accum ulated phase shift ∆ ϕ top in long-baseline atom in terferometers like MA GIS-100 offers a high-precision alternative to probe the quan tum-classical transition through phase coherence [1]. 5 [2] Sh un-ic hi Amari. Information Ge ometry and Its Applic ations , v olume 194 of Applie d Mathematic al Scienc es . Springer, 2016. [3] Da vid Bohm. A suggested in terpretation of the quan tum theory in terms of ”hidden” v ariables. i. Physic al R eview , 85(2):166–179, 1952. [4] Ariel Catic ha. Relativ e en tropy and inductive inference. AIP Confer enc e Pr o c e e dings , 707:75–96, 2004. [5] Goffredo Chirco et al. Fisher metric, geometric entanglemen t and spin netw orks. Physic al R eview D , 96:126015, 2017. [6] Eric S. Escobar-Aguilar, T onatiuh Matos, and J. I. Jim ´ enez-Aquino. F undamental Klein-Gordon Equation from Sto chastic Mechanics in Curved Spacetime. F ound. Phys. , 55(4):60, 2025. doi: 10.1007/s10701- 025- 00873- y. [7] L. D. F addeev and V. N. P op o v. F eynman diagrams for the yang-mills field. Physics L etters B , 25(1):29–30, 1967. [8] Theo dore F rank el. The Ge ometry of Physics: An Intr o duction . Cambridge Univ ersity Press, 3rd edition, 2011. [9] B. Roy F rieden. Physics fr om Fisher Information: A Unific ation . Cambridge Uni- v ersity Press, 1998. [10] Akio F ujiwara. F oundations of Information Ge ometry . Asakura Publishing, 2015. [11] Akio F ujiwara and Hiroshi Imai. Quantum parameter estimation of a generalized pauli c hannel. Journal of Physics A: Mathematic al and Gener al , 36(29):8093, 2003. doi: 10.1088/0305- 4470/36/29/314. [12] Akio F ujiwara and Hiroshi Imai. A fibre bundle ov er manifolds of quantum channels and its application to quantum statistics. Journal of Physics A: Mathematic al and The or etic al , 41(25):255304, 2008. doi: 10.1088/1751- 8113/41/25/255304. [13] S. W. Hawking. The dev elopment of irregularities in a single bubble inflationary univ erse. Physics L etters B , 115(4):295–297, 1982. [14] P eter R. Holland. The Quantum The ory of Motion: An A c c ount of the de Br o glie- Bohm Causal Interpr etation of Quantum Me chanics . Cam bridge Universit y Press, 1993. [15] Benjamin Ko ch, Ali Riahinia, and Angel Rinc´ on. Geo desics in quantum gravit y . Physic al R eview D , 2025. doi: 10.1103/PhysRevD.112.0840XX. [16] Ry ogo Kub o. Statistic al Me chanics: An A dvanc e d Course with Pr oblems and Solu- tions . North-Holland, 1962. [17] Kuo Liu and Siyuan Chen. Pulsar Timing Arra y in the past decade. 2 2026. [18] Erwin Madelung. Quan tentheorie in h ydro dynamisc her form. Zeitschrift f¨ ur Physik , 40(3-4):322–326, 1927. 6 [19] Jorge Meza-Dom ´ ınguez, T onatiuh Matos, and Pierre-Henri Chav anis. Energy balance of a b oson gas at zero temp erature in curv ed spacetime, 2026. URL https://arxiv. org/abs/2603.23931 . [20] Mikio Nak ahara. Ge ometry, T op olo gy and Physics . Institute of Ph ysics Publishing, 2nd edition, 2003. [21] Mic hael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th A nniversary Edition . Cambridge Univ ersit y Press, 2010. [22] O. Obreg´ on. Generalized information and entanglemen t entrop y , gravitation and holograph y . International Journal of Mo dern Physics A , 30:1550156, 2015. [23] Jonathan Opp enheim. A p ost-quantum theory of classical gra vit y? Physic al R eview X , 2023. [24] Jonathan Opp enheim et al. Exp erimental test of a p ost-quantum theory of classical gra vity . Natur e Communic ations , 2023. [25] James A. Reid and Charles H.-T. W ang. The configuration space of conformal connection-dynamics. Classic al and Quantum Gr avity , 29:125007, 2012. [26] J.J. Sakurai and S.F. T uan. Mo dern Quantum Me chanics . Addison-W esley , revised edition, 1995. [27] V alery I. Sbitnev. Bohmian tra jectories and the path integral paradigm – com- plexified lagrangian mechanics. In Mohammad Reza P ahlav ani, editor, The or etic al Conc epts of Quantum Me chanics . Intec hOp en, 2012. [28] A. A. Starobinsky . Dynamics of phase transition in the new inflationary universe scenario and generation of p erturbations. Physics L etters B , 117(3-4):175–178, 1982. [29] N. G. V an Kamp en. A cum ulant expansion for sto c hastic linear differen tial equations. i. Physic a , 74(2):215–238, 1974. [30] N. G. V an Kamp en. Sto chastic Pr o c esses in Physics and Chemistry . North-Holland, 3rd edition, 2007. [31] Vlatk o V edral. The role of relative entrop y in quantum information theory . R eviews of Mo dern Physics , 74(1):197–234, 2002. [32] Rob ert M. W ald. Gener al R elativity . Universit y of Chicago Press, 1984. [33] Stev en W einberg. The Quantum The ory of Fields: V olume 1, F oundations . Cam- bridge Universit y Press, Cam bridge, 1995. [34] Rob ert E. Wyatt. Quantum wa ve pack et dynamics with tra jectories: W av e function syn thesis along quan tum paths. The Journal of Chemic al Physics , 111(10):4406– 4413, 1999. 7