arXiv subjects

Melvin Arias

Publications and source records attributed to Melvin Arias.

6 recordsLinked to original sources

arXiv2026

Temporal-Plane Carroll--Schr\"odinger Dynamics and Vortex Sectors in (2,2) Klein Space

Motivated by the temporal dynamics identified in the $(1+1)$ Carroll-Schr\"odinger theory, we derive a post-Carrollian Schr\"odinger dynamics in flat Klein space with signature $(2,2)$. Starting from the tachyonic Klein-Gordon equation in double-polar coordinates and removing a spacelike carrier, the spatial radius behaves as an effective evolution parameter, whereas the temporal two-plane $(t_1,t_2)$ serves as the equal-radius configuration space. The additional time direction supplies an $SO(2)$ temporal angular momentum $J$, produces temporal vortex sectors, and gives the centrifugal contribution to the post-Carrollian momentum $P_{\mathrm{PC}}=E_{\tau}^{2}/(2M_{\mathrm{eff}})+J^{2}/(2M_{\mathrm{eff}}\tau^{2})$ in the Hamilton-Jacobi limit. We determine the regular Bessel modes, Gaussian packets, oscillator spectrum, radial $SU(1,1)$ tower, equal-$r$ continuity equation, $\mathfrak{sch}(2)$ symmetry algebra, radial-ordered propagator, and the metaplectic organization of the quadratic sectors. Effective flat connections on the temporal configuration plane give Aharonov-Bohm, Landau, and Fock-Darwin analogues, while the two-body relative sector admits anyonic boundary conditions on the punctured temporal plane. As a curved extension, we derive a branch-dependent carrier reduction and apply it to an illustrative $SO(2,1)$-symmetric Kleinian Schwarzschild exterior, where the Kleinian gravitational source produces a lensing-type angular deviation on the temporal plane.

quant-ph↗

arXiv2025

Probability Conservation, Liouville Measure, and the Symplectic Origin of Hamiltonian Dynamics

Liouville's theorem -- the preservation of phase-space volume -- is often presented as a corollary of Hamilton's canonical equations. Here we adopt an ensemble-first viewpoint in which the starting point is local probability conservation on phase space. For a probability density $ρ$ on a $2N$-dimensional symplectic manifold $(\mathcal{M},ω)$, probability transport is expressed intrinsically with respect to the Liouville volume form $Ω=ω^N/N!$ through a continuity equation defined by the $Ω$-divergence. For Hamiltonian evolution, specified by $ι_{X_H}ω=\mathrm{d}H$, Cartan's identity implies $\mathcal{L}_{X_H}ω=0$ and hence $\mathcal{L}_{X_H}Ω=0$, so the Hamiltonian flow is incompressible in the Liouville sense and the continuity law reduces to Liouville's equation. In canonical coordinates this reproduces Hamilton's equations. In particular, the canonical Poisson-bracket relations $\{q^i,p_j\}=δ^i_{\ j}$ provide the kinematic input that fixes the evolution of observables and underlies the canonical form of the continuity equation. The same organization clarifies the distinction between conservation of total probability and preservation of fine-grained information measures (Gibbs--Shannon entropy), which holds specifically for Liouville-measure-preserving dynamics.

physics.ed-ph↗

arXiv2025

On the Dynamics of Multiparticle Carroll-Schrdinger Quantum Systems

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 $U_{\!tot}(\mathbf x)$, we show that the evolution is driven by the collective force $\sum_j\partial_{x_j}U_{\!tot}$; 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.

quant-ph↗

arXiv2025

An Algebraic-Recursive Approach to Generate Higher-Order Symmetry Operators for Schrödinger and Klein-Gordon equations

This article explores an algebraic-recursive approach to construct differential operators that commute with a central operator $\hat{H}$ in quantum mechanics. Starting from the Schrödinger equation for a free particle, the work derives first-order symmetry generators, such as translations, rotations, and boosts, and examines their algebraic basis encompassing Lie and Jordan algebras. The analysis is then extended to higher-order operators, demonstrating how they can be constructed from the first-order ones through algebraic operations and Lie algebra simplification. This methodology is applied to the Klein-Gordon equation in Minkowski space-time, yielding relativistic symmetry operators. Furthermore, we defined an approximation to fractional symmetry operators of the Schrodinger equation, and a perturbative approach is employed for a case where the commutation is more general, illustrated with a one-dimensional harmonic oscillator and the fourth-order Klein-Gordon equation. The results include a general formula for the number of operators as a function of the order and the dimension of the algebraic basis, providing a reduced-form development of the differential higher-order centralizers' basis.

quant-ph↗

arXiv2025

On the Schrödinger and Carroll Schrödinger Equations: Dualities and Applications

We investigate precise structural relations between the standard Schrödinger equation and its Carrollian analogue-the Carroll-Schrödinger equation-in 1+1 dimensions, with emphasis on dualities, potential maps, and solution behavior. Our contributions proceed in the order of the paper: (i) we encode both dynamics with operators $H$ and $F$ under external potentials and explore conditions for obtaining the same type of solutions within both formalisms; (ii) we construct a potential-dependent reparametrization $x = δ(t)$ mapping the space-independent Carroll equation to the time-independent Schrödinger equation, and derive a Schwarzian relation that specifies the map $δ$ for any static $V_{sch}$ (with harmonic, Coulomb-like, and free examples); (iii) we relate conserved densities and currents by removing $V_{car}$ through a gauge transform followed by a coordinate inversion, establishing equivalence of the continuity equations; (iv) we obtain a Carrollian dispersion relation from an ultra-boost of the energy-momentum two-vector and also derive the classical limit of the Carroll wave equation via the Hamilton-Jacobi formalism; (v) we place Carroll dynamics on an equal-$x$ Hilbert space $L^2(R_t)$, prove unitary $x$-evolution, and illustrate dynamics with an exactly solvable Gaussian packet and finite-time quantization for time-localized perturbations; and (vi) for general $V(x; t)$ we perform a gauge reduction to an interaction momentum and set up a controlled Dyson expansion about solvable time profiles.

quant-ph↗

arXiv2025

On the Derivation of Equations of Motion from Symmetries in Quantum-Mechanical Systems via Heisenberg's Uncertainty

We propose the construction of equations of motion based on symmetries in quantum-mechanical systems, using Heisenberg's uncertainty principle as a minimal foundation. From canonical operators, two spaces of conjugate operators are constructed, along with a third space derived from the former, which includes the ``Symmetry-Dilation'' operator. When this operator commutes with the main equation of motion, it defines the set of observables compatible with a complete basis of operators (symmetry generators), organized into a Lie algebra dependent on Heisenberg's uncertainty principle within Minkowski spacetime. Furthermore, by requiring the dilation operator to commute with the central operator, the wavefunction is constrained, thereby constructing known structures. Specific cases are derived -- relativistic, non-relativistic, and a lesser-studied case: ``ultra-relativistic (Carroll-Schrödinger)''. Our work may open new avenues for understanding and classifying symmetries in quantum mechanics, as well as offer an alternative method for deriving equations of motion and applying them to complex scenarios involving exotic particles.

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