Previous works have shown that bilocal constraint geometry naturally gives rise to twodistinct geometric structures.The radial sector generates a distinguished one-dimensional characteristic flowinterpreted as relational time, whereas the angular sector reduces to a symplecticorbit carrying an internal Hamiltonian symmetry. The present work investigates the next natural step of this construction.Instead of interpreting the reduced orbit merely as a classical internal state space,we propose that it should be regarded as the classical precursor of a quantum statespace. Starting from the reduced symplectic orbit obtained by presymplectic reduction,we show that its geometric quantization naturally leads to a Hilbert space associatedwith every bilocal relation.After localization over spacetime, these Hilbert spaces form a bundle of quantumstates whose sections play the role of wave functions. Within this framework, neither time, gauge fields, nor Hilbert space are introducedas fundamental structures.All three emerge from the geometry of bilocal constraint orbits through differentstages of the same relational construction.
Previous works have shown that bilocal constraint geometry naturally gives rise to twodistinct geometric structures.The radial sector generates a distinguished one-dimensional characteristic flowinterpreted as relational time, whereas the angular sector reduces to a symplecticorbit carrying an internal Hamiltonian symmetry. The present work investigates the next natural step of this construction.Instead of interpreting the reduced orbit merely as a classical internal state space,we propose that it should be regarded as the classical precursor of a quantum statespace. Starting from the reduced symplectic orbit obtained by presymplectic reduction,we show that its geometric quantization naturally leads to a Hilbert space associatedwith every bilocal relation.After localization over spacetime, these Hilbert spaces form a bundle of quantumstates whose sections play the role of wave functions. Within this framework, neither time, gauge fields, nor Hilbert space are introducedas fundamental structures.All three emerge from the geometry of bilocal constraint orbits through differentstages of the same relational construction.
This work presents the construction of a bilocal phase space arising from arelational description of the state Ψ(x1,x2) on an extended configurationspace. We introduce bilocal configuration and gradient variables and analyse thecorresponding symplectic form on T∗R3,3. Angle reduction leads to aneffective phase-space structureωeff=dt∧dE+dr∧dp,described by two canonically conjugate pairs: a relational pair (t,E) and a localpair (r,p). The first pair is associated with the moduli of bilocal configuration andgradient vectors, and its structure is a classical counterpart of the quantum relation[t,E^]=2iℏ. The second pair reproduces the radial part of the standard phasespace. The resulting phase space is entirely relational and can be interpreted as a geometricintertwining of two Minkowski spaces, M1,3(x)×M1,3(∇). This suggeststhat the bilocal construction provides a natural framework for describing structuresin which relations between configuration points, rather than the points themselves,are fundamental. The analysis presented here forms the basis for formulating bilocalextensions of dynamics and quantization, which will be developed in future work.
We consider the bilocal formalism of quantum mechanics, in which the wave functiondefined on the space Q×Q is interpreted as a description of two statesof the same particle.In contrast to earlier approaches, we do not assume the wave function to be scalar-valued,but instead allow it to take values in a finite-dimensional complex vector space. We show that changing the codomain of the wave function does not modify the structureof the configuration space nor its phase-space realization.The entire bilocal geometry, responsible for the emergence of spacetime,mass, and kinematical spin, remains unchanged.New structures appear exclusively at the level of the state-value space. The space of physical states then takes the form of the complex projective spaceCPn−1, which is a natural symplectic manifold and a coadjoint orbitof the group SU(n).For n=2, the resulting geometry duplicates the structural spin spherealready present in the bilocal formalism, whereas the case n=3 leads tothe first non-degenerate representational symmetry. These results suggest that internal symmetries of a representational naturecan be understood as consequences of the geometry of quantum state spaces,rather than as additional postulated degrees of freedom.
We present a three-level organization of a bilocal relational theory inwhich structures usually associated with quantum mechanics arise atlogically distinct stages. The first level is purely geometric. Starting from the bilocalconfiguration space M×M and its canonical cotangent bundle,the relative sector admits a radial--angular decomposition. For N=3,the angular phase space is T∗S2. Its non-singular sector isnaturally organized asR+×SO(3) and therefore possesses the canonical doublecoverR+×SU(2)≃C2∖{0}. A classical two-component SU(2)-spinorial carrier thus appearsbefore any Lorentzian dynamics or quantum postulate; its identificationwith a Lorentz-covariant Weyl spinor is not assumed at this stage.Reduction at fixed internalCasimir yields the Hopf fibration with ∣c1∣=1. For N=3, definingpμ=(pt,P) andμint=κ1+iκ2rewrites the full constraint aspμpμ+∣μint∣2=0.Its Clifford factorization then admits an exact chiral first-order formwith a two-dimensional complex kernel. At this stage the constructionremains classical: the momenta are phase-space variables and the kernelspinor is not yet a quantum wave function.The third level promotes this geometric and dynamical structure toquantum spinor dynamics. Geometry supplies an integer topological classbut not a universal scale of action; the identificationI=nℏ/2 therefore requires the quantum scale ℏ.Together with the operator rulepμ↦−iℏDμ, the classical chiral factorization becomesa quantum Dirac-type equation. If the internal complex contributionadmits a sector or limit in which it reduces to a real constantm, the standard massive Dirac equation(iℏγμDμ−m)ΨD=0is recovered. The construction therefore separates three ingredients that arenormally introduced together: spinorial and topological statestructure, Lorentzian dynamics, and quantum operator dynamics. We donot derive quantum mechanics from classical geometry. Rather, we showthat the spinorial carrier and the algebraic architecture of the Diracequation can arise before quantization, while the quantum scale andoperator realization remain independent inputs. This identifiesprecisely which elements of quantum spinor dynamics are geometric,which depend on the Lorentzian dynamical postulate, and which remaingenuinely quantum assumptions.
Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.
In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of information-theoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function. In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same information-theoretic discussion of the Chapter 3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,with the Fisher information metric replaced by the Jensen-Shannon distance information. Furthermore, we show that the non-linear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of three-qubit pure states with a specific (shifted) spectra of single-particle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations.
Quantum mechanics is traditionally formulated on complex Hilbert spaces, whereas gauge theories are formulated on fiber bundles equipped with connections and curvature. Despite the central role of geometry in modern physics, the Hilbert-space structure of quantum theory is usually postulated rather than derived. In this work, we investigate a differential-geometric formulation in which quantum states are represented as points on a coherence manifold characterized by a radial coherence coordinate and a holonomy-induced phase coordinate. The resulting geometry naturally admits the structure of a fiber bundle whose base space is associated with a coherence lattice and whose fiber corresponds to a phase circle. Within this framework, Hilbert spaces emerge as function spaces defined on the coherence manifold, while Schrödinger dynamics arise from the Laplace–Beltrami operator associated with the underlying geometry. The construction unifies concepts from geometric quantum mechanics, fiber-bundle gauge theory, holonomy, and coherence geometry within a common differential-geometric framework.