Self–Fibre Bundle Geometry: A Mathematical–Physical Framework for Local Spacetime Superposition and Emergent Quantum Dynamics

Self–Fibre Geometry: Classical Gravity and Quantum–Like Interference from a Unified Geometric Structure Qian Miao Email: [email protected] November 2025 Abstract We present a geometric framework—Self–Fibre geometry—in which classical general relativity and key kinematical features of quantum theory (phase interference and Born–type probability weights) can be encoded within a single, unified structure. The central idea is that local spacetime does not merely form the base of a bundle, but also provides its own fibre: each point of a configuration manifold Blabels an entire Lorentzian spacetime (M, gb), and the relations between these local spacetimes are governed by a connection Ab:TbB→TgbLor(3,1) on the infinite–dimensional manifold of Lorentzian metrics. A natural action functional is constructed from (i) a Yang–Mills–type curvature term 1 2⟨F, F⟩, (ii) a vertical Einstein–Hilbert term κR[g]on each fibre spacetime, and (iii) a fibre vacuum functional ΛV[g]. Variation with respect to gyields a fibrewise Einstein equation whose low–curvature regime reproduces classical general relativity; variation with respect to Ayields a Yang–Mills–type equation for the Self–Fibre curvature. Quantum–like behaviour arises in this framework not from canonical quantisation, but from the geometry of the Self–Fibre connection and the DeWitt supermetric on the space of metrics. Holonomies of Agive geometric phases analogous to quantum interference phases, while formal Gaussian measures induced by the DeWitt supermetric lead to Born–type probability weights on a configuration manifold B. We emphasise that this constitutes an emergent representation of quantum–like structures, rather than a rigorous derivation of quantum theory. In the present paper we concentrate on two families of phenomenological consequences: (i) curvature–dependent modifications of double–slit interference patterns, and (ii) effective variations of an emergent Planck constant ℏeff in strongly curved regimes. These provide, in principle, experimentally testable signatures of the Self–Fibre framework. Contents 1 Introduction 2 2 Mathematical preliminaries 3 2.1 The manifold of Lorentzian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 TheDeWittsupermetric................................ 4 2.3 Fréchet bundles and connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 FormalGaussianmeasures............................... 4 3 The Self–Fibre bundle 5 3.1 Base manifold and its physical interpretation . . . . . . . . . . . . . . . . . . . . 5 3.2 Fibre manifold: local spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Gaugesymmetry .................................... 5 3.4 Self–Fibreprinciple................................... 6 1 4 Connection and curvature on the Self–Fibre bundle 6 4.1 Self–Fibreconnection.................................. 6 4.2 Curvature, holonomy, and geometric phase . . . . . . . . . . . . . . . . . . . . . . 6 5 Action functional and field equations 7 5.1 Self–Fibreaction .................................... 7 5.2 Variation with respect to the connection . . . . . . . . . . . . . . . . . . . . . . . 7 5.3 Variation with respect to the fibre metric . . . . . . . . . . . . . . . . . . . . . . 8 5.4 Classical regime and recovery of general relativity . . . . . . . . . . . . . . . . . . 8 6 Quantum–like regime: phase, multiplicity and Born–type measure 8 6.1 Geometric phase from Self–Fibre holonomy . . . . . . . . . . . . . . . . . . . . . 8 6.2 Internal multiplicity and fibre measure . . . . . . . . . . . . . . . . . . . . . . . . 9 6.3 Emergent Born–type measure on configuration space . . . . . . . . . . . . . . . . 9 7 Toy model and mapping to interference experiments 9 7.1 One–dimensionalbase ................................. 10 7.2 Gaussian weight and emergent coordinate distribution . . . . . . . . . . . . . . . 10 7.3 Two–pathinterference ................................. 10 8 Phenomenological predictions: double–slit interference and effective Planck constant 11 8.1 Double–slit interference as Self–Fibre holonomy . . . . . . . . . . . . . . . . . . . 11 8.2 Curvature–dependent visibility suppression . . . . . . . . . . . . . . . . . . . . . 11 8.3 Emergent effective Planck constant . . . . . . . . . . . . . . . . . . . . . . . . . . 12 9 Discussion and outlook 12 1 Introduction The aim of this work is to explore a geometric framework in which both classical gravitation and quantum–like interference can be encoded within a single structural principle: spacetime provides its own fibre. Instead of beginning from a Hilbert space and postulating quantisation rules, we ask whether a sufficiently rich configuration–space geometry can exhibit: (i) an approximate regime reproducing Einstein’s general relativity, and (ii) a nonclassical regime whose kinematics resembles quantum phase interference and Born– type probability weights. The central object is an infinite–dimensional bundle π:E=B×Lor(3,1) −→ B, whose base Bis a finite–dimensional manifold interpreted as a configuration space of local spacetimes, and whose fibre over b∈Bis the space Lor(3,1) of Lorentzian metrics on a fixed four– manifold M. A point b∈Bthus labels a whole Lorentzian spacetime (M, gb); the family {gb}b∈B is the fundamental dynamical variable, rather than a single metric on a single spacetime. The dynamical relations between different local spacetimes are encoded in a connection Ab:TbB→TgbLor(3,1),(1) whose curvature Fmeasures how families of local spacetimes fail to fit together consistently. When Fis small, the theory reduces to classical general relativity on each fibre. When F is significant, holonomies of Agive rise to geometric phases, and—combined with a suitable measure on the space of metrics—to Born–type probability weights. 2 Scope and limitations It is important to state at the outset what this paper does not claim: •We do not claim a rigorous derivation of the full formalism of quantum mechanics from classical geometry. •We do not construct a mathematically complete measure theory on the Fréchet manifold Lor(3,1); our Gaussian measures are formal in exactly the same sense as Euclidean path integrals. •We do not attempt to quantise gravity in the canonical or path–integral sense. Instead, our more modest goal is: to show that a single geometric structure—a Self–Fibre bundle equipped with a connection and an action functional—naturally admits two limiting regimes: a classical regime corresponding to general relativity, and a quantum–like regime whose kinematics reproduces phase interference and Born–type probability weights on a configuration manifold. In this initial presentation, we further focus our phenomenological discussion on two concrete arenas: (a) modifications of double–slit interference patterns, and (b) an emergent, curvature–dependent effective Planck constant ℏeff. We deliberately set aside broader cosmological or high–energy predictions to keep the analysis technically and conceptually focused. Structure of the paper Section 2 reviews the geometry of the space of Lorentzian metrics, the DeWitt supermetric, infinite–dimensional bundles, and the formal Gaussian measures we shall employ. Section 3 defines the Self–Fibre bundle, emphasising the physical interpretation of the base manifold B. Section 4 introduces the Self–Fibre connection and curvature. Section 5 presents the action functional and derivation of the field equations, including the emergence of classical general relativity in the low–curvature regime. Section 6 discusses the quantum–like regime: geometric phases, multiplicity weights and an emergent Born–type measure on B. Section 7 illustrates these ideas in a solvable one–dimensional toy model, and Section 8 articulates concrete predictions for double–slit interference and effective Planck constants. We conclude in Section 9 with a brief discussion of open problems. 2 Mathematical preliminaries We summarise the basic geometric ingredients needed for the Self–Fibre construction. Our presentation is intentionally concise; standard references on global analysis, canonical gravity and infinite–dimensional geometry provide further background. 2.1 The manifold of Lorentzian metrics Let Mbe a smooth, oriented, time–oriented four–manifold. The configuration space of Lorentzian metrics is Lor(3,1) = g∈Γ(S2T∗M) : gsmooth, nondegenerate, signature (3,1).(2) 3 As is well known, Lor(3,1) is an open subset of the Fréchet space Γ(S2T∗M)of smooth symmetric 2–tensors; hence it inherits a natural structure as a smooth Fréchet manifold. For each g∈ Lor(3,1), TgLor(3,1) ∼ =Γ(S2T∗M),(3) with a tangent vector δg interpreted as an infinitesimal deformation of the metric. 2.2 The DeWitt supermetric Canonical gravity endows Lor(3,1) with the DeWitt supermetric. For g∈Lor(3,1) and δg, δg′∈ TgLor(3,1), ⟨δg, δg′⟩g=ZM Gabcd(g) (δg)ab (δg′)cd d4x, (4) where Gabcd(g)is the (inverse) DeWitt supermetric, schematically Gabcd(g)∝p|g|gacgbd +gadgbc −λgabgcd(5) with a conventional parameter λ. In Lorentzian signature the DeWitt metric is indefinite due to the conformal mode; for measure–theoretic purposes one typically works with a Euclideanised or Wick–rotated version, formally replacing ⟨·,·⟩gby a positive quadratic form. We adopt this standard physics practice without attempting to resolve the underlying functional–analytic subtleties. 2.3 Fréchet bundles and connections Let Bbe a finite–dimensional smooth manifold and Fa Fréchet manifold. The trivial product E=B×F, π(b, f) = b, (6) is a Fréchet bundle over B. A connection on Eis a smooth assignment Ab:TbB→TfF, (7) depending smoothly on (b, f). Equivalently, A∈Ω1(B;TFvert),(8) a one–form on Bwith values in the vertical tangent bundle. In our construction F= Lor(3,1), so that E=B×Lor(3,1), TgF∼ =Γ(S2T∗M),(9) and a connection Aassigns to each v∈TbBan infinitesimal deformation Ab(v)of the metric gb. 2.4 Formal Gaussian measures The nonexistence of translation–invariant Lebesgue measure in infinite dimensions is a classical result; nevertheless, Gaussian measures on Fréchet spaces can be defined in a cylindrical or projective–limit sense. In the present work we adopt the standard physics viewpoint familiar from Euclidean path integrals: we treat Gaussian measures formally, as devices for assigning relative weights to field configurations, without attempting a complete measure–theoretic construction. For a reference configuration g0∈Lor(3,1), the DeWitt norm ∥g−g0∥2 g=ZM Gabcd(g0) (g−g0)ab(g−g0)cd d4x(10) 4 suggests a formal Gaussian measure dµfib(g)∝exp−1 2ℏeff ∥g−g0∥2 gDg, (11) where Dgdenotes a heuristic “flat” measure and ℏeff is an emergent scale parameter. In the Self–Fibre framework we shall interpret such measures as encoding an internal multiplicity of fibre configurations compatible with a given macroscopic outcome. 3 The Self–Fibre bundle We now define the central geometric object of the framework: a bundle whose fibre at each point of a configuration manifold Bis an entire Lorentzian spacetime. 3.1 Base manifold and its physical interpretation Let Bbe a smooth, oriented, finite–dimensional manifold equipped with a Riemannian metric gB. The crucial point is that: Bis not physical spacetime. Instead, it is a configuration manifold parametrising local spacetime structures. Each point b∈Blabels a possible local spacetime, represented by a Lorentzian metric gbon a fixed four–manifold M. One may compare this with nonrelativistic quantum mechanics, where a wavefunction ψ(x)assigns amplitudes to points xof a configuration space. Here Bplays an analogous role, but its points label entire Lorentzian metrics rather than particle positions. The base metric gBinduces a canonical volume form on B, dµbase(b) = dvolgB(b),(12) which in a local inertial chart reduces to Lebesgue measure. This measure will later play the role of the spatial factor in a Born–type probability measure on B. 3.2 Fibre manifold: local spacetimes The fibre over each b∈Bis the Fréchet manifold Lor(3,1). We denote the corresponding metric by gb, so that the pair (b, gb)∈B×Lor(3,1) represents the local spacetime (M, gb)associated with the configuration point b. The total space of the Self–Fibre bundle is thus E=G b∈B Lor(3,1) ∼ =B×Lor(3,1),(13) with projection π:E→B,π(b, g)=b. 3.3 Gauge symmetry The natural gauge group of the Self–Fibre construction is the diffeomorphism group Diff(B). For φ∈Diff(B), φ·(b, g) = φ(b), g,(14) acts fibre–preservingly. Physical observables are required to be invariant under this action, reflecting the fact that Bis a configuration space rather than an observable spacetime. This symmetry plays a central role in the uniqueness of the emergent probability measure on B discussed later. 5 3.4 Self–Fibre principle We can now state the guiding principle precisely. Self–Fibre Principle. The fundamental dynamical object is a smooth assignment b7→ gb∈ Lor(3,1) together with a connection Ab:TbB→TgbLor(3,1) describing how local spacetimes vary across the configuration manifold B. The true degrees of freedom lie not in a single spacetime geometry, but in the geometry of this entire family of local spacetimes and the connection that relates them. This viewpoint replaces the “quantisation of a single spacetime” by a geometry on the space of possible local spacetimes. 4 Connection and curvature on the Self–Fibre bundle Having specified the Self–Fibre bundle E→B, we now introduce the connection that governs how local spacetimes change as one moves in configuration space. 4.1 Self–Fibre connection At each b∈B, the associated metric gbdefines a point in Lor(3,1), whose tangent space is canonically TgbLor(3,1) ∼ =Γ(S2T∗M).(15) A Self–Fibre connection is a smooth bundle map Ab:TbB→TgbLor(3,1),(16) so that for a tangent vector v∈TbB,Ab(v)is a symmetric 2–tensor on M, interpreted as the infinitesimal change of the local spacetime metric gbin the direction v. Equivalently, A∈Ω1B;TLor(3,1)vert,(17) a one–form on Bwith values in the vertical tangent bundle of E. In local coordinates (x1, . . . , xn)on B, a vector v∈TbBdecomposes as v=vi∂i. The connection takes the form A=Ai(gb) dxi, Ab∂i=Ai(gb)∈Γ(S2T∗M),(18) so that an infinitesimal displacement dxiinduces the metric deformation δgb=Ai(gb) dxi. 4.2 Curvature, holonomy, and geometric phase The curvature of the Self–Fibre connection is defined in the usual way, F= dA+1 2[A, A]∈Ω2B;TLor(3,1)vert,(19) where [·,·]is the commutator of vertical vector fields on Lor(3,1). In local coordinates, Fij =∂iAj−∂jAi+ [Ai, Aj], F =1 2Fij dxi∧dxj.(20) Physically, Fmeasures the obstruction to consistently identifying local spacetimes along different routes in configuration space. If F= 0, the family {gb}can be globally trivialised; if F= 0, parallel transport around a loop in Bproduces a nontrivial holonomy. Let γ: [0,1] →Bbe a loop based at b0, and let Uγdenote the parallel–transport operator induced by Aalong γ. Formally one may write Uγ=Pexp Zγ A,(21) 6 where Pdenotes path ordering. If γbounds an oriented surface Σ⊂B, a generalised Stokes theorem gives formally Uγ∼exp ZΣ F.(22) The quantity Φ(Σ) = ZΣ F(23) is a geometric phase associated with the Self–Fibre curvature. When mapped into a complex phase factor, eiΦ, such holonomies provide a natural geometric representation of quantum interference phases. 5 Action functional and field equations We now introduce an action functional whose Euler–Lagrange equations govern both the dynamics of the family {gb}and the Self–Fibre connection A. 5.1 Self–Fibre action Let dµB= dvolgBbe a fixed volume form on B. The Self–Fibre action is S[g, A] = ZB1 2⟨F, F⟩g+κ R[g] + Λ V[g]dµB,(24) where: •⟨F, F⟩gis a Yang–Mills–type curvature energy, defined using the DeWitt supermetric on Lor(3,1); •R[g]is the scalar curvature of the fibre metric gbon M(Einstein–Hilbert term on each fibre); •V[g]is a fibre vacuum functional (e.g. higher–curvature or effective potential terms); •κand Λare coupling constants. More explicitly, using the DeWitt metric (4), the curvature term can be written schematically as ⟨F, F⟩g=Gabcd(gb)Fijab(b)Fij cd(b).(25) The action (24) is invariant under the gauge group Diff(B), since both ⟨F, F ⟩gand R[g]are constructed from geometric invariants on the fibres and the base volume form is geometric. 5.2 Variation with respect to the connection A variation δA of the connection induces a variation of the curvature δF =D(δA), where Dis the covariant derivative on Bassociated with A. The variation of the curvature term is δ1 2⟨F, F⟩g=⟨D(δA), F⟩g.(26) Integrating by parts on Band discarding boundary terms, one obtains ZB ⟨δA, D∗F⟩gdµB= 0,(27) for arbitrary δA. The Euler–Lagrange equation for Ais therefore a Yang–Mills–type equation D∗F= 0,(28) expressing covariant conservation of the Self–Fibre curvature. 7 5.3 Variation with respect to the fibre metric Variations of the fibre metric gbaffect all three terms in the action. The variation of the Einstein–Hilbert term is standard: δR[gb]=Gµν(gb)δgµν b+(total derivative),(29) where Gµν is the Einstein tensor on the fibre spacetime (M, gb). The variation of the vacuum functional V[g]can be written formally as δV [g] = δV δgµν b δgµν b.(30) The curvature term contributes an additional stress–energy–like tensor, which we denote by Πµν(b), defined by Πµν(b) = δ δgµν b1 2⟨F, F⟩g.(31) The Euler–Lagrange equation for gbthen takes the form κ Gµν(gb)+Λ δV δgµν b + Πµν(b)=0.(32) 5.4 Classical regime and recovery of general relativity In a regime where the Self–Fibre curvature is small, F→0,Πµν(b)→0,(33) and where variations of Vare negligible at the scales of interest, the field equation (32) reduces to Gµν(gb)+Λeff gb µν = 0,(34) with Λeff = Λ/κ. Thus in the low–curvature regime, the Self–Fibre framework recovers vacuum general relativity (with cosmological term) on each fibre spacetime M. In this sense classical general relativity appears as the “shadow” of the unified Self–Fibre theory in the limit where fibre curvature and fibre fluctuations are negligible. 6 Quantum–like regime: phase, multiplicity and Born–type measure We now turn to the nonclassical regime where the Self–Fibre curvature Fand fluctuations of the fibre metric become significant. Our aim is not to reconstruct the full formalism of quantum mechanics, but to identify geometric structures that mirror its key kinematical features: phase interference and Born–type probabilities. 6.1 Geometric phase from Self–Fibre holonomy As discussed in Section 4, the curvature Fof the Self–Fibre connection gives rise to holonomies Uγ=PexpZγ A(35) along loops γin configuration space B. Whenever the holonomy takes values in a compact subgroup that can be represented as complex phases (for instance via a suitable projection or effective abelianisation), one may associate to γa real phase Φγ∈R,eiΦγ∈U(1),(36) 8 and interpret eiΦγas a quantum–like interference phase. Different paths between the same endpoints in Bthen contribute with different geometric phases, generating constructive or destructive interference in an amplitude sum. 6.2 Internal multiplicity and fibre measure Let us fix a macroscopic experimental configuration, represented by a subset E⊂Bof configuration space. Within the Self–Fibre picture, a “detection at b∈E” corresponds not to a single fibre metric gb, but to a collection Cb⊂Lor(3,1) of fibre configurations compatible with that outcome. The internal multiplicity of the outcome bis formally w(b) = µfib(Cb),(37) where µfib is a Gaussian–type measure on the fibre, induced by the DeWitt supermetric as described in Section 2. We now define an emergent complex function ψ:B→Cby |ψ(b)|2:= w(b),arg ψ(b) := Φ(b),(38) where Φ(b)encodes an appropriate geometric phase (e.g. from Self–Fibre holonomy along paths ending at b). In this sense ψ(b) = pw(b) eiΦ(b)(39) plays the role of a quantum–like amplitude on configuration space, but its modulus and phase are defined geometrically: the modulus by fibre multiplicities, the phase by connection holonomy. 6.3 Emergent Born–type measure on configuration space The total space E=B×Lor(3,1) carries a formal product measure dµE(b, g)=dµbase(b) dµfib(g|b),(40) where dµbase is the geometric measure on Band dµfib(g|b)is a Gaussian–type measure on the fibre over b. The probability of detecting a macroscopic outcome within a region E⊂Bis obtained by integrating over all fibre configurations compatible with that outcome: P(E) = Zb∈EZg∈Cb dµfib(g|b)dµbase(b) = Zb∈E w(b) dµbase(b).(41) In terms of the emergent amplitude ψthis becomes P(E) = Zb∈E |ψ(b)|2dµbase(b),(42) formally identical to the standard Born rule, but here interpreted as the projection of a product measure on the Self–Fibre bundle onto the configuration manifold B. We emphasise that (42) does not constitute a theorem about quantum mechanics; rather, it shows that the Self–Fibre geometry naturally admits a probability measure on Bof Born type whenever internal multiplicities and Self–Fibre symmetry are taken into account. 7 Toy model and mapping to interference experiments To make the abstract constructions more concrete and to prepare for phenomenological applications, we now examine a simple toy model in which the base manifold Bis one–dimensional. Although highly idealised, this model already captures geometric phase, Gaussian weights and Born–type probabilities. 9