Extreme kinematic limits of space-time symmetries reveal alternative forms of quantum behaviour. In the Carrollian limit the speed of light is taken to zero ([Formula: see text]): causal light cones collapse onto the time axis, spatial points become disconnected, and the usual notion of particle motion disappears. The corresponding quantum equation is the Carroll–Schrödinger equation—the structural mirror image of the ordinary Schrödinger equation, being first order in space and second order in time. We explore the mathematical connections between these two equations in one space and one time dimension. Using operator methods, we find conditions on external potentials that allow both equations to share solutions, and show that any Carrollian problem can be mapped to an equivalent Schrödinger problem via an explicit coordinate transformation. Probability densities and currents are related by removing the interaction potential and swapping space and time coordinates. An extreme relativistic boost combined with a classical approximation yields the Carrollian dispersion relation and conditions for coinciding classical trajectories. By formulating the dynamics on a Hilbert space of time rather than space, we prove that evolution in the spatial coordinate preserves probability. Closed-form solutions illustrate each result and together offer a practical framework for translating between the two descriptions.
Extreme kinematic limits of space-time symmetries reveal alternative forms of quantum behaviour. In the Carrollian limit the speed of light is taken to zero ([Formula: see text]): causal light cones collapse onto the time axis, spatial points become disconnected, and the usual notion of particle motion disappears. The corresponding quantum equation is the Carroll–Schrödinger equation—the structural mirror image of the ordinary Schrödinger equation, being first order in space and second order in time. We explore the mathematical connections between these two equations in one space and one time dimension. Using operator methods, we find conditions on external potentials that allow both equations to share solutions, and show that any Carrollian problem can be mapped to an equivalent Schrödinger problem via an explicit coordinate transformation. Probability densities and currents are related by removing the interaction potential and swapping space and time coordinates. An extreme relativistic boost combined with a classical approximation yields the Carrollian dispersion relation and conditions for coinciding classical trajectories. By formulating the dynamics on a Hilbert space of time rather than space, we prove that evolution in the spatial coordinate preserves probability. Closed-form solutions illustrate each result and together offer a practical framework for translating between the two descriptions.
We study the dynamics of multiparticle Carroll-Schrödinger (CS) quantum systems in 1+1 dimensions, where x acts as the evolution variable and t as the configuration coordinate. We derive the N-body theory on equal-x slices as the Carrollian limit of a relativistic multi-time Klein-Gordon model, introducing temporal interactions via minimal coupling to the temporal energy operators. An x-dependent gauge transformation maps this to an equivalent description with explicit many-body potentials, illustrated by a temporal coupled-oscillator model that exhibits synchronization. Adopting a complementary spatial viewpoint with a static potential Utot(x), we show that the evolution is driven by the collective force ∑j∂xjUtot; for any translation-invariant interaction (such as a regularized Coulomb potential), these internal forces cancel, rendering the collective dynamics free and highlighting Carrollian ultralocality. We also construct a coordinate duality mapping separable Schrödinger Hamiltonians to CS generators via Schwarzian derivatives. Exchange symmetry is formulated in the time domain, yielding temporal bunching for bosons and antibunching for fermions via the second-order coherence function g(2)(t,t′). In second quantization, the contact limit yields a temporal derivative cubic--quintic nonlinear Schrödinger equation with a theoretically fixed nonlinearity coefficient β=−3/16. Finally, by coupling canonical pairs to external scalar and gauge fields, we establish an isomorphism with one-dimensional current-density functional theory, outlining a Carrollian Hohenberg-Kohn mapping and Kohn-Sham scheme.
Understanding whether the mathematical structure of quantum mechanics is fundamental or emergent remains a central question in the foundations of physics. In particular, the special role played by time in quantum theory, appearing as an external evolution parameter rather than a dynamical observable, suggests that the formalism itself may arise from deeper structural considerations. In this work, we investigate the emergence of quantum mechanical formalism from classical wave dynamics by adopting a dimensional framework in which time is treated as a +1 evolution parameter relative to the dimensions through which physical phenomena (fields or disturbances of a field) propagate and interact. Within this perspective, different fields may evolve with respect to different effective dimensions, while remaining embedded in a common higher-dimensional space, allowing time to acquire a relational and context-dependent role. This means that in our proposed model, time is not a fixed dimension which is experienced the same way for every field or field interaction of any dimensionality. In that sense, time for one physical phenomenon can behave as space for a higher dimensional physical phenomenon, whose time is a different +1 dimension. The central objective of this paper is to determine how a higher-dimensional deterministic field can be consistently represented by a lower-dimensional description that lacks direct access to its full set of evolution parameters and evolves through a spatial (for the higher-dimensional field) dimension. To this end, we introduce a general projection framework in which a higher-dimensional field is mapped to a reduced field through an interaction-based recording process. Crucially, we do not assume the form of this mapping a priori. Instead, we impose the requirement that it preserve the maximum amount of physically accessible information. In particular, we demand the faithful encoding of phase relations, interference structure, and spectral composition, including the relative contributions of different Fourier modes and their superposition. We first demonstrate, within a purely classical 3+1-dimensional wave framework that these constraints severely restrict the admissible form of the reduced description and naturally lead to complex amplitudes, linear superposition, Hilbert space structure, and canonical operator relations. This analysis provides an intuitive and mathematically explicit route to quantum-like descriptions without assuming quantum postulates. We then generalize the construction to a 4+1-dimensional framework, introducing an additional evolution parameter and showing that under the same information-preserving constraints, the Schrödinger equation appears as an effective low-energy description of the reduced dynamics, while a relativistic dispersion relation emerges simultaneously through the encoding of the hidden evolution parameter as an invariant frequency scale. In this way, both quantum mechanical and relativistic structures arise from the same underlying requirement: the consistent and information-preserving representation of higher-dimensional wave propagation in a lower-dimensional observational framework. The results suggest that the formal structure of quantum mechanics need not be postulated a priori, but may instead be understood as the unique mathematical language required to encode the observable remnant of a higher-dimensional deterministic dynamics under strict constraints of symmetry, invariance, and information preservation.
This paper derives the non-relativistic time-dependent Schrödinger equation as the logically unavoidable dynamical law for de Broglie matter waves, extending the reconstruction begun in the companion work “The Logical Origin of the de Broglie Matter Wave and its Physical Meaning (the Born Rule)” (Chan, 2026). Starting from the same minimal premises—continuous position in real space, Fourier conjugacy (and the logical uncertainty principle), translation invariance via Noether’s theorem, and probability conservation via statistical energy flux—the Schrödinger equation follows deductively once norm preservation and classical non-relativistic correspondence are enforced. The derivation proceeds in clear stages: Plane-wave solutions for free particles and constant potentials, Quadratic dispersion relation from classical kinematics, Exact local plane-wave structure (defined via phase gradients) that must hold at every point for any smooth wave function, The unique linear differential operator of order ≤ 2 that consistently “glues” these local structures across smooth external potentials V(x,t) while preserving the L²-norm (unitarity). No canonical quantization, operator promotion, commutation relations, or postulated wave equations are used. The Born rule is taken as already established from the prior paper and serves only to constrain unitary evolution. The Schrödinger equation thus emerges not as an empirical postulate or convenient ansatz, but as the mathematically and symmetrically forced description of indivisible particles propagating in continuous, translationally invariant (or smoothly perturbed) space. Domain of validity and natural limitations (non-relativistic kinematics, smooth potentials, single-particle/mean-field regime, closed-system unitarity) are explicitly delineated.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
The Schrödinger equation iℏ ∂ₜψ = Ĥψ, with non‑relativistic Hamiltonian Ĥ = ℏ²|p̂|²/(2m) + V(x̂), has been quantum mechanics’s central dynamical statement for nearly a century. It governs atomic spectra, molecular bonding, semiconductors, neutron interferometry, and countless other empirical settings. The question is why it has the form it does. In standard QM, the Schrödinger equation is a postulate. The Hilbert space is given. The momentum operator p̂ = −iℏ∇ is given. The kinetic form |p̂|²/(2m) is inherited directly from classical mechanics — quantum theory simply adopts the classical T = p²/(2m) structure without deeper justification. The first‑order time derivative is chosen because empirically it works. Each piece is accepted because the resulting equation matches experiment. This walkthrough shows how the Event Density framework derives the Schrödinger equation, in its full standard form, from a smaller set of structural commitments about what reality is made of at the substrate level. Every element — the Hilbert space, the momentum operator, linearity, first‑order time structure, the kinetic‑plus‑potential form, the factor of 1/(2m) — emerges as a forced consequence rather than a separate postulate. The derivation runs through Stone’s theorem applied twice — once on spatial translation (forcing p̂ = −iℏ∇) and once on time translation (forcing existence and uniqueness of a self‑adjoint generator with automatic linearity and first‑order structure) — closed by Galilean Lie algebra integration that identifies the time‑translation generator with the kinetic‑plus‑potential operator. The factor of 1/(2m) emerges as the chain‑rule Jacobian of the Galilean commutator [Ĥ, K̂ᵢ] = −iℏ p̂ᵢ, not as a borrowing from classical mechanics. The first‑order time derivative is a consequence of operator exponentials, not a chosen feature.
In this work, we introduce a novel framework for quantum dynamics based on a second-order evolution equation, ∂2/∂t2∣Ψ⟩=K∣Ψ⟩ , with K=i(αD+βV) , where the operators D and V satisfy the Heisenberg algebra [D,V]=dI in d -dimensional space. Unlike conventional quantum mechanics governed by the first-order Schrödinger equation, this formulation emerges from a unified dynamical structure rooted in phase-space symmetry, naturally accommodating non-Hermitian interactions while preserving unitary evolution through the introduction of an η -inner product and a dual-state mapping. We demonstrate that this theory shares an SL(2,R) symmetry with the standard quantum formalism, revealing that both the Schrödinger and our second-order dynamics arise as distinct realizations within the same symplectic framework — the former as a quadratic generator (energy conservation), the latter as a linear generator (driving-dissipation balance). Extending the theory to four dimensions via Dirac matrices yields a covariant, spinor-valued wave equation whose stationary solutions carry instanton topological charge, π3(S3)=Z , thereby unifying the topological structures from one to four spatial dimensions under the homotopy classification πd−1(Sd−1)=Z . This approach not only generalizes non-Hermitian quantum mechanics beyond PT -symmetric potentials but also establishes a direct bridge between quantum dynamics, symplectic geometry, and topological field theory.