In the standard formulation of quantum mechanics, time appears as an externalevolution parameter rather than as an operator. In this work we analyze an alterna-tive perspective: time as an emergent quantity resulting from relations between twoquantum states of the same system. A pair of wave functions defines a point in the six-dimensional bilocal configuration space R3,3 , in which a natural “quantum distance”operator leads to an invariant with Minkowski structure ds2 = dt2 − dr2 .Constructing a basis adapted to this quantity yields a metric equivalent to M4 + S 2 —four-dimensional Minkowski spacetime coupled to an internal sphere of radius equal tothe quantum distance t. On this space we analyze the scalar field equations and showthat the spherical harmonics correspond to Klein–Gordon equations with geometric,time-dependent masses. The lowest modes l = 0, 1 form a four-component multipletwhich can be linearized in the spirit of Dirac, producing a first-order equation withadditional terms arising from the time-varying geometric mass.
In the standard formulation of quantum mechanics, time appears as an externalevolution parameter rather than as an operator. In this work we analyze an alterna-tive perspective: time as an emergent quantity resulting from relations between twoquantum states of the same system. A pair of wave functions defines a point in the six-dimensional bilocal configuration space R3,3 , in which a natural “quantum distance”operator leads to an invariant with Minkowski structure ds2 = dt2 − dr2 .Constructing a basis adapted to this quantity yields a metric equivalent to M4 + S 2 —four-dimensional Minkowski spacetime coupled to an internal sphere of radius equal tothe quantum distance t. On this space we analyze the scalar field equations and showthat the spherical harmonics correspond to Klein–Gordon equations with geometric,time-dependent masses. The lowest modes l = 0, 1 form a four-component multipletwhich can be linearized in the spirit of Dirac, producing a first-order equation withadditional terms arising from the time-varying geometric mass.
Time-Energy Uncertainty Principle in Bilocal Geometry and the Dual Minkowski Structure in R3,3
In the bilocal space R3,3, quantum time can be defined as the distance between two states of the same particle. This construction leads to the Minkowski intervalds2=t2−r2and to the interpretation of t as a time coordinate emerging geometrically. In the present paper we develop the dual side of this construction by introducing the gradient energy operatorE^=−iℏ(∇2−∇1).We show that the time and gradient energy operators form a canonically conjugate pair:[t^,E^]=2iℏ,which leads to a relational uncertainty principle of the time--energy typeΔtΔE≳ℏ.At the same time the intervals (t,r) and (E,p) form two equivalent Minkowski spacetimes arising from a single bilocal geometry. This formalism reveals a hidden geometric structure of quantum theory, in which time, energy and mass appear as emergent quantities resulting from relations between states.
An operational approach to quantum mechanics has been developed in the past decades in our group in Brussels. A similar approach is taken in this work, making use of the extra operational depth offered by this approach, to show that the construction of spacetime is specific to each observer. What is usually referred to as the block universe then emerges by noting that parts of the past and future are also contained in the present, but without the limitations that a four-dimensional block universe usually implies, of a reality in which change would be impossible. In our operational approach, reality remains dynamic, with free choice playing a central role in its conceptualization. We also show that, when operationally analyzed, the theory implies that objects move not only in space, but also and especially in time, and more generally in spacetime, with their rest mass being a measure of their kinetic time energy. We therefore claim that Einstein’s relativity revolution has not been fully realized, since most physicists are not ready to accept these consequences of the theory although the formula’s showing them, when operationally analyzed, are in every relativity textbook. In particular, when relativistic motion is revisited as a genuine four-dimensional motion, it becomes possible to reinterpret the parameter c associated with the coordinate speed of light, which becomes the magnitude of the four-velocity of all material entities. We observe that this four-dimensional motion in Minkowski space can also be better understood if placed in the broader perspective of quantum mechanics, provided that non-locality is interpreted as non-spatiality, thus indicating the existence of an underlying non-spatial reality, the nature of which could be conceptual, consistent with the conceptuality interpretation of quantum mechanics. This hypothesis is reinforced by noting that when observers, or experiencers, as they will be referred to in this article, are described by acknowledging their cognitive nature of entities moving in a semantic space, Minkowski metric emerges in a natural way.
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.
Can spatial distance and proper time be defined operationally — by how a system leaks information to its environment — rather than postulated as primitive geometric quantities? This paper answers affirmatively for both, within Markovian open quantum dynamics, building on the survival-expansion framework of the companion paper "Local Curvature of Quantum Information" (doi:10.5281/zenodo.18792424). Spatial distance is defined as the excess irreversibility generated by superposing two states: when a system is superposed over two locations, the environment learns which at a rate growing with their separation. This distance is proven to be a genuine metric — it satisfies the triangle inequality via an isometric embedding into environmental "shadow space", recovers the Euclidean metric of R³ for independent position channels, and becomes undefined before decoherence even for orthogonal states, so that geometry and classicality emerge together. Proper time is defined as elapsed background time minus the minimal dissipation needed to traverse a worldline. The reversed triangle inequality of relativity (the twin paradox) then follows automatically from the subadditivity of optimal costs, and the exact Minkowski interval corresponds to a dissipation rate that saturates at high transport velocity: the light cone as a bound on irreversibility production. Both Lorentzian signatures are traced to a single convex-analytic root: spatial distance is directly a subadditive object (a norm), while proper time is an additive duration minus a subadditive optimal cost, hence superadditive. In an exactly solvable driven damped-oscillator model, the information deficit coincides identically with Clausius entropy production, the minimal transport deficit obeys Λ∗=2kBTbγmTΔx2, and the emergent light speed is c2=γmκkBTb=κD, with D the Einstein diffusion coefficient. The universality of the emergent light cone is shown to be equivalent to a Compton-clock scaling of the dissipation–time conversion rate (κ=ℏ2mc2), with the quantum–classical crossover condition kBTb=2ℏγ arising as an internal consistency requirement rather than a postulate. A junction-cost theorem establishes that velocity discontinuities incur strictly positive excess dissipation, so the turning twin ages strictly less even at equal speeds. All results are verified numerically on lattice decoherence models, exact linear–quadratic optimal control, and Fock-space computations; the verification scripts accompany the manuscript. Locality of the environmental couplings is assumed, not derived, and the construction has analog-gravity status: a Lorentzian interval in a preferred frame defined by the bath.
This Volume III develops a complete geometric formulation of quantum mechanics, demonstrating that quantum phenomena emerge from the very structure of spacetime. We demonstrate that Einstein's field equations—when interpreted in the spectral domain and taking phase coherence into account—already contain the entire structure of quantization. Quantum mechanics appears not as an axiom, but as a geometric consequence. Based on the experimental and mathematical results consolidated in Volumes~0, I, and II, the present work shows that quantization need not be postulated: it arises inevitably from three fundamental geometric elements—proper frequency, parallel transport, and phase coherence. The action appears as the phase accumulated along the worldline, and its discretization into coherence windows, derived without external hypotheses, naturally leads to the condition Sn=nh and to all associated quantum phenomenology. We derive the Schrödinger equation as the limit of short windows, obtain the commutator [x,p]=iℏ from the incompatibility between spatiotemporal synchronizations, and reinterpret interference, propagators, and decoherence as spectral properties of parallel transport. The Hilbert space arises as the spectral space of the metric, observables correspond to generators of isometric deformations, while the collapse is understood as the geometric selection of coherent subspaces. The formalism extends to relativistic particles, spin, quantum statistics, the Dirac equation, harmonic oscillators, and field quantization, where spectral modes of the metric assume the role of particles: photons as harmonic 1-forms, weak modes as connections of 2-forms, gluons as 3-forms, and the Higgs as the fundamental scalar curvature mode. The reinterpretation of the Standard Model thus becomes a direct consequence of spectral geometry. The Volume culminates in the Central Theorem —Geometry = Quantization— showing that the entire quantum structure is a manifestation of the spectral granularity of spacetime. We conclude with experimental predictions and falsification criteria, as well as the conceptual framework necessary for Volume~IV, dedicated to the complete quantization of the metric within the same paradigm.
A construction of the real 4D Minkowski space-time starting from quantum harmonic oscillators is proposed. First, a 2D spinor space and its dual are derived from the standard commutation relations obeyed by the ladder operators of two independent 1D harmonic oscillators. The complex 4D Minkowvski vector space V is then constructed from these spinor space. The flat, real 4D Minkowski manifold is finally built as an approximate description of a manifold of unitary operators constructed from V. Lorentz invariance is recovered and several possible extensions are discussed, which connections to quantum optics and condensed matter physics.