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

Introduction to the show: ident: Mathematical Physics. The day in mathematical physics: rigorous models of physical systems.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle".

Mira: Here's the gist of the paper. 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) This paper determines how the linear-potential (LP) system fits into this picture Algebraic positioning within the conformal structure The free particle, harmonic oscillator and inverted harmonic oscillator are organized by the standard quadratic conformal generators Their basic dynamical integrals are PF = p + F t, XF = q − t/m p − F 2/2m t squared, and they generate translations and Galilean boosts The linear-potential Hamiltonian generates q˙ = p/m and p˙ = −F This suggests that HLP is a quadratic free-particle element supplemented by a linear Heisenberg–Galilei generator, rather than another quadratic representative of sl(2, R) The natural algebraic setting is consequently the non-semisimple Schrödinger algebra, equivalently the one-dimensional Jacobi algebra Extension via affine and singular limits The relation to the oscillator sectors is summarized by Hκ,F = p 2/2m + mκ squared q 2/2 + F q = p 2/2m + mκ squared Q squared − F 2/4mκ, Q = q + F/mκ, κ ≠ 0 For F = 0, the choices κ = 0, ω2, −omega2 reproduce FP, HO and IHO, respectively The limit κ → 0 is regular in the original family Hκ,F, where it yields HLP However, its HO and IHO realizations require controlled singular displaced-oscillator limits Spectral transformations and Airy states The LP Airy eigenstates arise from a cubic-phase transform of FP momentum eigenstates, from spectral condensation of highly excited displaced-HO states, and from the limit of a subdominant parabolic-cylinder scattering branch of the inverted oscillator The same Airy eigenstate is therefore reached through three complementary constructions: a regular cubic-phase change of spectral representation from FP plane waves, a singular condensation limit of highly excited displaced-HO states, and, as shown in Appendix A, the limit of a subdominant IHO parabolic-cylinder scattering branch Planar extension and guiding-center dynamics A planar extension in homogeneous crossed electric and magnetic fields gives a further interpretation A uniformly accelerated and rotating frame produces the electric interaction, the Coriolis/Landau coupling and the centrifugal IHO term; a compensating harmonic term isolates the Landau sector In guiding-center variables the electric field acts linearly on a noncommutative Heisenberg plane, generates the Hall drift, and supplements the Landau CBT by a displacement, a uniform drift and a scalar phase The quantum counterpart of the classical Hall drift is vD = −eE/mωc = −cE/B The LP system is an affine–Heisenberg companion of the FP–HO–IHO conformal triangle, not a fourth homogeneous quadratic vertex The crossed-field extension provides a planar realization of the same distinction The electric field acts linearly on the noncommuting guidingcenter plane, produces the Hall drift, and supplements the Landau CBT by a displacement, drift and scalar phase The IHO–LP spectral limit At the Hamiltonian, action and propagator levels, the IHO–LP limit follows from the HO construction by the analytic continuation ω 7→ σiomega, σ = ±1 Spectrally, normalizable Hermite states are replaced by generalized parabolic-cylinder scattering states and an appropriate Stokes branch must be selected The two signs select conjugate complex rotations and Stokes sectors This is an analytic continuation of the differential equation, not a unitary continuation of a normalized HO state The IHO–LP spectral edge The dominant barrier branch tends locally to Bi Unlike the HO–LP limit, no condensation of discrete levels is involved: the IHO spectrum is continuous for every omega > 0 The two dashed spectral links therefore share the same local Airy turning-point normal form but differ globally: the HO route is a discrete-to-continuous large-level limit, whereas the IHO route is a continuous-scattering limit with vanishing transmission The finite turning point becomes the LP turning point while the IHO forbidden interval tends to the LP forbidden half-line q > E/F Relativistic and geometric extensions Rindler kinematics provides a relativistic completion of the regular FP–LP bridge Newton–Hooke mechanics arises from dS/AdS contractions and null reductions of homogeneous pp waves Accelerated and rotating frames in (A)dS may therefore provide a relativistic parent of Hκ,F, separating curvature κ from affine acceleration F/m The ordinary Galilean crossed-field problem arises from the magnetic-dominated contraction, so no transition at E = B occurs within the nonrelativistic system considered here Future research directions The guiding-center formulation points naturally to the subcritical, supercritical and critical phases of the noncommutative Landau problem, related by CBT and weak–strong duality It remains to determine how the electric-field affine intertwiner and Hall drift are realized across these phases A related relativistic direction starts from the identity of classical evolution between a charge in a constant (2 + 1)-dimensional electromagnetic field and the higher-derivative particle with torsion Geometric quantization gives internal discrete-series representations D±α of the universal cover of SL(2, R), while the physical constraint selects a discrete massive mass–spin tower and generalized massless and tachyonic states Clarifying the relation of this relativistic threefold structure to the Landau/HO, LP/Airy and IHO sectors remains open The ordinary Galilean crossed-field problem arises from the magnetic-dominated contraction, so no transition at E = B occurs within the nonrelativistic system considered here Acknowledgments The work was partially supported by the FONDECYT Project 1242046 The IHO–LP spectral limit follows from the HO construction by the analytic continuation ω 7→ σiomega, σ = ±1 The two signs select conjugate complex rotations and Stokes sectors This is an analytic continuation of the differential equation, not a unitary continuation of a normalized HO state For F > 0, the potential Vomega(q) = F q − momega squared q 2/2 has turning points q−(E, omega) = E/F + O(omega 2), q+(E, omega) = 2F/mω squared − E/F + O(ω 2), so that q− → E/F, while q+ → +∞ The slope at the finite turning point is somega = V′omega(q−) = √F squared − 2momega 2E → F Consequently, on fixed turning-point intervals, ξomega → κF (q − E/F) and the parabolic-cylinder equation reduces to the Airy equation The LP eigenstate selected in (5.3) is obtained from the IHO branch that is subdominant in the forbidden region The dominant barrier branch tends locally to Bi Unlike the HO–LP limit, no condensation of discrete levels is involved: the IHO spectrum is continuous for every omega > 0 The two dashed spectral links therefore share the same local Airy turning-point normal form but differ globally: the HO route is a discrete-to-continuous large-level limit, whereas the IHO route is a continuous-scattering limit with vanishing transmission References

1: U.

Kai: Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv.

Mira: Phys.

Kai: Acta 45 (1972) 802–810

2: C.

Paper summary: Mira: To unpack that a bit more. The paper claims this linear-potential Hamiltonian sits in the non-semisimple Schrödinger algebra, which they call the one-dimensional Jacobi algebra >

Kai: They are saying the free particle to linear potential link is a regular Heisenberg–Weyl displacement with its Bargmann phase, while the projective sl(two R) action is inherited through dynamical integrals > <ref:2607.20711#pg1>

Lev: So they're essentially showing that this structure isn't just about finding a new Hamiltonian; it’s about understanding how the symmetries of these different systems interact in a non-semisimple setting >

Mira: They construct the regular FP–LP bridge at the Hamiltonian and action levels, using a canonical transformation and wave-function intertwiner, and also at the propagator level >

Kai: And they analyze spectral transformations to see how it differs from what you get when you just look at fixed-F harmonic oscillator or inverted oscillator limits >

Lev: I'm curious about that comparison. Does this paper give us any concrete numbers on how much the spectra diverge when we move from a standard quadratic system to this affine–Heisenberg one >

Mira: The paper also focuses on the Airy energy eigenstates, showing they can come from three different places: a cubic-phase transform of free-particle momentum eigenstates, condensation of highly excited harmonic-oscillator levels, and the limit of a subdominant parabolic-cylinder scattering branch of the inverted oscillator >

Kai: That sounds like a lot to track. So if I'm building something, does this mean I have to worry about these three different ways to get that Airy state happening in my physical setup >

Lev: It suggests that whatever physical system you are modeling, you might need to consider these three complementary constructions when looking for those specific stationary spectral states >

Mira: And they mention a planar extension in homogeneous crossed electric and magnetic fields gives another interpretation of this structure >

Conclusion: Kai: So wrapping up the "Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle," the authors are really emphasizing that this linear potential system is an affine–Heisenberg companion to the free particle, harmonic oscillator, and inverted harmonic oscillator triangle >

Mira: They’re making it clear that it’s not just a fourth quadratic vertex in sl(two R), but something sitting in a different algebraic setting entirely called the non-semisimple Schrödinger algebra > <ref:2607.20711#pg1>

Lev: For me, the implication is that if you're working on quantum error correction or any system where you need to map dynamics onto a structure with these specific symmetries, you can’t just use standard quadratic tools alone >

Kai: Right. The key distinction they make is that the free particle to linear potential link comes from a regular accelerated-frame transformation, which is different from the standard way we usually connect those systems >

Mira: And the planar extension in crossed electric and magnetic fields gives us another way to visualize this relationship, showing how guiding-center dynamics with an electric field acts linearly on a noncommutative plane >

Lev: So for the practical side, it’s suggesting that understanding this affine intertwiner is crucial if you want to figure out how the Hall drift or other effects manifest in a physical system under those conditions >