arXiv subjects

Andrey Alcala

Publications and source records attributed to Andrey Alcala.

4 recordsLinked to original sources

arXiv2026

Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle

We study the one-dimensional linear-potential system as an affine extension of the conformal triangle formed by the free particle, harmonic oscillator (HO) and inverted harmonic oscillator (IHO). Unlike these homogeneous quadratic Hamiltonians in the $sl(2,\mathbb R)$ sector, the linear-potential Hamiltonian involves the Heisenberg ideal of the Schr\"odinger algebra. Its direct relation to the free particle is a regular accelerated-frame transformation, developed here at the levels of the classical action, canonical transformation, wave-function intertwiner and propagator; its HO and IHO realizations instead arise through singular displaced-oscillator limits. The Airy energy eigenstates follow from a cubic-phase transform of free-particle momentum eigenstates, from condensation of highly excited harmonic-oscillator levels, and from the limit of a subdominant parabolic-cylinder scattering branch of the inverted oscillator. We also develop a planar extension in homogeneous crossed electric and magnetic fields, where uniform acceleration generates the electric interaction, while uniform rotation produces the Landau coupling and the centrifugal inverted-oscillator term; the guiding-center dynamics yields the Hall drift.

math-ph↗

arXiv2026

The Free Particle--Oscillator--Inverted Oscillator Triangle: Conformal Bridges, Metaplectic Rotations and $\mathfrak{osp}(1|2)$ Structure

We study the free particle (FP), the harmonic oscillator (HO) and the inverted harmonic oscillator (IHO) as parabolic, elliptic and hyperbolic realizations of one conformal/metaplectic structure, naturally extended to the superconformal algebra $\mathfrak{osp}(1|2)$. Since the corresponding self-adjoint Hamiltonians have different spectra, the relations between them are not ordinary unitary equivalences. They are instead bridge transformations between different realizations of the same conformal module. We show that the zero-energy Jordan states of the FP are mapped to HO bound states and to the two IHO Gamow families, while FP plane waves are mapped to HO coherent states and, after light-cone Mellin decomposition, to the IHO scattering data. The direct FP--IHO bridge is a real metaplectic quarter-rotation, in contrast with the stationary FP--HO conformal bridge, which is nonunitary in the Schr\"odinger representation but becomes unitary as a change of polarization to the Fock--Bargmann representation. The IHO transmission and reflection amplitudes are obtained as Fourier--Mellin connection coefficients, equivalently as Weber/Stokes connection data. We also describe the hyperbolic Cayley--Niederer map for the time-dependent Schr\"odinger equation, the Wigner/separatrix picture, and the coherent-state and Bogoliubov-transformation aspects of the construction. Some physical applications of the hyperbolic sector are briefly discussed, including quantum Hall saddle scattering, Schwinger-type production, Rindler/Unruh and near-horizon Hawking settings, and Berry--Keating/inverse-square structures.

hep-th↗

arXiv2026

Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence

We investigate the relation between the one--dimensional free particle and the harmonic oscillator from a unified viewpoint based on projective geometry, Cayley transformations, and the Schwarzian derivative. Treating time as a projective coordinate on $\mathbb {RP}^1$ clarifies the $SL(2,\mathbb R)\cong Sp(2,\mathbb R)$ conformal sector of the Schr\"odinger--Jacobi symmetry and provides a common framework for two seemingly different correspondences: the Cayley--Niederer (lens) map between the time--dependent Schr\"odinger equations and the conformal bridge transformation relating the stationary problems. We formulate these relations as canonical transformations on the extended phase space and as their metaplectic lifts, identifying the quantum Cayley map with the Bargmann transform. General time reparametrizations induce oscillator--type terms governed universally by the Schwarzian cocycle, connecting the present construction to broader appearances of Schwarzian dynamics.

hep-th↗

arXiv2023

Weak-strong duality of the non-commutative Landau problem induced by a two-vortex permutation, and conformal bridge transformation

A correspondence is established between the dynamics of the two-vortex system and the non-commutative Landau problem (NCLP) in its sub- (non-chiral), super- (chiral) and critical phases. As a result, a trivial permutation symmetry of the point vortices induces a weak-strong coupling duality in the NCLP. We show that quantum two-vortex systems with non-zero total vorticity can be generated by applying conformal bridge transformation to a two-dimensional quantum free particle or to a quantum vortex-antivortex system of zero total vorticity. The sub- and super-critical phases of the quantum NCLP are generated in a similar way from the 2D quantum free particle in a commutative or non-commutative plane. The composition of the inverse and direct transformations of the conformal bridge also makes it possible to link the non-chiral and chiral phases in each of these two systems.

hep-th↗