Temporal-Plane Carroll--Schrödinger Dynamics and Vortex Sectors in (2,2) Klein Space

Abstract: Motivated by the temporal dynamics identified in the $(1+1)$ Carroll-Schrödinger theory, we derive a post-Carrollian Schrödinger 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 $$P_{\mathrm{PC}}=E_τ<sup>{2}/(2M_{\mathrm{eff}})+J<sup>{2}/(2M_{\mathrm{eff}}τ<sup>{2})$$ 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.

• The paper derives a post-Carrollian Schrödinger equation with spatial radius r as the evolution parameter, converting the two-dimensional temporal plane into an equal-r quantum configuration space with conserved norm and an explicit propagator.
• The theory predicts quantized temporal vortex circulation, centrifugal corrections, Gouy-phase interference rotation, oscillator and Landau spectra, and possible anyonic exchange phases through effective connection and punctured-plane sectors.
• The paper extends the construction to split-signature Einstein backgrounds, obtaining exact Kleinian Schwarzschild trajectories and a branch propagator whose temporal-plane deflection grows logarithmically with the optical baseline, while leaving gauge dynamics and self-adjoint extensions open.

The paper develops a post-Carrollian quantum dynamics on flat Klein space $\mathbb{K}^{2,2}$ with signature $(-,-,+,+)$, extending the $1+1$ dimensional Carroll–Schrödinger construction to a setting with two temporal directions. The central structural move is to promote the spatial radius $r$ of the double-polar decomposition to an evolution parameter while demoting the temporal two-plane $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$ to an equal-$r$ configuration space. This inversion supplies a conserved temporal angular momentum $J$, organizes modes into vortex sectors with quantized circulation, and generates a centrifugal correction to the post-Carrollian momentum that is absent in one temporal dimension. The paper then carries the same carrier-reduction logic onto split-signature Einstein backgrounds, specializing to an $SO(2,1)$-symmetric Kleinian Schwarzschild exterior where it derives exact classical trajectories and a branch propagator exhibiting lensing-type angular deflection on the temporal plane.

Derivation from the tachyonic Klein–Gordon equation

Starting from the tachyonic Klein–Gordon equation $(\Box_g+\mu^2)\phi=0$ in double-polar coordinates, with metric $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$, the authors factor out a radially symmetric spatial carrier $(-,-,+,+)$0 and impose the Carrollian contraction $(-,-,+,+)$1 together with $(-,-,+,+)$2. The limit eliminates the $(-,-,+,+)$3 dependence and second-order radial derivatives, yielding

$(-,-,+,+)$4

with effective mass $(-,-,+,+)$5. A key interpretive claim, established in an appendix, is that the scaling $(-,-,+,+)$6 is not arbitrary: for the family of contractions $(-,-,+,+)$7, $(-,-,+,+)$8, $(-,-,+,+)$9, keeping the post-Carrollian momentum $1+1$0 finite forces $1+1$1, hence $1+1$2 independently of how the contraction is distributed between $1+1$3 and $1+1$4. Under the half-density redefinition $1+1$5, the equation becomes the standard free two-dimensional Schrödinger equation on the temporal plane with $1+1$6 as time; the $1+1$7 term is identified as a measure artifact of the polar radial carrier, and its general form $1+1$8 is characterized through a linear carrier functional equation and an associated integral representation.

Vortex sectors, oscillator spectrum, and connection couplings

Separated solutions take the form $1+1$9 with $r$0 and $r$1; negative $r$2 yields evanescent modified-Bessel sectors outside the post-Carrollian momentum shell. Every mode with $r$3 is a temporal vortex: the temporal current gives quantized circulation $r$4, independent of $r$5 and topologically invariant, with vorticity concentrated at the temporal origin. The authors note that the $r$6 sector has deficiency indices $r$7 and admits a one-parameter family of self-adjoint extensions (logarithmic boundary behavior); they restrict to the regular extension throughout, so contact interactions at the temporal origin remain unexplored within this framework.

Gaussian packets spread according to $r$8 with temporal Rayleigh range $r$9, directly analogous to optical diffraction but with $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$0 playing the propagation role. A genuine harmonic confinement arises only through a momentum-sector coupling $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$1 (a quadratic shift of $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$2 merely dresses modes by a cubic phase), giving the isotropic planar oscillator spectrum

$(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$3

with degeneracy $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$4 at principal level $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$5, organized by circular ladder operators into $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$6 representations with Bargmann index $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$7. An effective flat $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$8 connection shifts $(t_1,t_2)=(c\tau\cos\psi,\,c\tau\sin\psi)$9 and lifts these degeneracies except at $r$0; the paper is explicit that this Aharonov–Bohm flux is an effective model coupling, not derived from a first-principles Carrollian electrodynamics. Nonzero-curvature connections yield Landau levels $r$1 with guiding-center algebra $r$2, and combining curvature with confinement produces a Fock–Darwin spectrum with chiral frequencies $r$3.

Two features distinguish this from a mere transcription of planar quantum mechanics. First, the Gouy phase $r$4 grades vortex sectors by the weight $r$5, which is simultaneously the oscillator spectral weight — both realized on subgroups of $r$6. A two-mode superposition therefore rotates its interference pattern by $r$7, an effect with no counterpart in the $r$8 theory. Second, the two-body relative problem lives on the punctured plane $r$9, whose fundamental group admits a $J$0 family of inequivalent quantizations: exchange of two identical excitations carries phase $J$1, interpolating between bosons ($J$2) and fermions ($J$3). The second temporal direction thus makes anyonic statistics available in a Carroll–Schrödinger system, though the realized holonomy depends on microscopic core regularization, which the paper does not specify.

Continuity equation, symmetry algebra, and Bargmann lift

Multiplying the reduced equation by $J$4 and subtracting the conjugate gives an equal-$J$5 continuity equation; under the half-density map the conserved norm is simply $J$6, making unitarity of the radial evolution manifest via a Bessel–Fourier spectral resolution. The symmetry algebra is computed explicitly in Cartesian temporal variables and closes on the nine-dimensional $J$7: temporal translations $J$8, Carroll boosts $J$9, rotation $SO(2,1)$0, evolution translation $SO(2,1)$1, dilatation $SO(2,1)$2, special conformal generator $SO(2,1)$3, and central charge $SO(2,1)$4. Setting $SO(2,1)$5 recovers the known $SO(2,1)$6 of the one-dimensional case, confirming consistency.

The Bargmann interpretation is geometrically clean: on the null metric $SO(2,1)$7, the fixed-mass ansatz reduces the massless wave equation exactly to the gauge-fixed Carroll–Schrödinger equation, with the Bargmann fiber coordinate $SO(2,1)$8 representing the post-Carrollian mass $SO(2,1)$9. Interactions enter through the Eisenhart deformation $(\Box_g+\mu^2)\phi=0$0, reproducing momentum-coupled potentials. All quadratic sectors — free spreading, oscillator, Landau, Fock–Darwin — are unified as metaplectic lifts $(\Box_g+\mu^2)\phi=0$1 of classical symplectic flows $(\Box_g+\mu^2)\phi=0$2, with the Rayleigh range interpreted as the diffraction length of a metaplectic shear.

Radial quantization and propagator

Equal-$(\Box_g+\mu^2)\phi=0$3 canonical quantization of the first-order action gives $(\Box_g+\mu^2)\phi=0$4 (with an $(\Box_g+\mu^2)\phi=0$5 factor for the ungauged field), and the retarded kernel is the exact two-dimensional Schrödinger propagator,

$(\Box_g+\mu^2)\phi=0$6

whose polar form follows from the Jacobi–Anger expansion. Because the reduced equation is Schrödinger-like in $(\Box_g+\mu^2)\phi=0$7, data on equal-$(\Box_g+\mu^2)\phi=0$8 slices propagate only toward larger $(\Box_g+\mu^2)\phi=0$9: the radial initial-value structure replaces the usual multi-time determinism concerns, which the authors acknowledge as the standard obstruction to well-posedness in multi-time formulations.

Hamilton–Jacobi limit and centrifugal structure

The WKB limit discards the $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$0 term (it is subleading in $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$1) and yields the Hamilton–Jacobi equation on the temporal plane, with separated action giving

$$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$2

which is precisely the expansion of the tachyonic mass shell after subtracting the carrier momentum $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$3. The centrifugal term is the qualitative new contribution relative to one temporal dimension. Jacobi's theorem gives $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$4 with turning radius $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$5, and the angular relation collapses to $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$6: despite the two-dimensional configuration space, free classical trajectories are straight lines in the temporal plane, selected by conservation of $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$7. The classically forbidden region $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$8 matches the Bessel node structure $$ds^2=-c^2d\tau^2-c^2\tau^2d\psi^2+dr^2+r^2d\theta^2$$9 near the origin, providing a concrete quantum-classical correspondence. Interacting Hamilton equations with momentum coupling $(-,-,+,+)$00 and energy shift $(-,-,+,+)$01 reduce to the one-dimensional results when $(-,-,+,+)$02 or $(-,-,+,+)$03.

Curved extension and Kleinian Schwarzschild lensing

On a split-signature Einstein vacuum, the carrier ansatz $(-,-,+,+)$04 with unit-spacelike eikonal $(-,-,+,+)$05 produces the local curved Carroll–Schrödinger equation

$(-,-,+,+)$06

where $(-,-,+,+)$07 fixes the spatial evolution direction and $(-,-,+,+)$08 is the temporal Laplacian of the chosen carrier branch. In a carrier-adapted patch the half-density transformation removes the measure term, and asymptotically radial conditions recover the flat equation with the $(-,-,+,+)$09 term from $(-,-,+,+)$10. The generalized post-Carrollian momentum $(-,-,+,+)$11 enforces the same $(-,-,+,+)$12 scaling.

Specializing to the $(-,-,+,+)$13-symmetric Kleinian Schwarzschild metric $(-,-,+,+)$14 with $(-,-,+,+)$15, the authors construct an $(-,-,+,+)$16-adapted carrier branch (shown to differ from the $(-,-,+,+)$17-invariant $(-,-,+,+)$18-adapted branch by a finite temporal phase $(-,-,+,+)$19) and derive the branch equation with geometric drift and measure terms. Classically, the Kleinian source introduces a radial drift proportional to $(-,-,+,+)$20 and rescales the kinetic term by $(-,-,+,+)$21, while $(-,-,+,+)$22 remains conserved. The key exact result is that $(-,-,+,+)$23 evolves affinely in the optical variable $(-,-,+,+)$24, which for $(-,-,+,+)$25 evaluates in closed form to $(-,-,+,+)$26 plus constants. To first order in $(-,-,+,+)$27 the temporal-plane trajectory acquires the logarithmic angular deflection

$(-,-,+,+)$28

which vanishes for radial temporal trajectories ($(-,-,+,+)$29) and grows logarithmically with baseline. Quantum mechanically, the dilation $(-,-,+,+)$30, the optical variable $(-,-,+,+)$31, and the rescaling $(-,-,+,+)$32 map the branch equation exactly to the flat free Schrödinger equation (the cancellation of first-order terms relies specifically on $(-,-,+,+)$33), yielding a closed-form branch propagator $(-,-,+,+)$34 with prefactors $(-,-,+,+)$35 and the optical phase $(-,-,+,+)$36.

Limitations and open questions

Several restrictions are stated plainly by the authors. The effective $(-,-,+,+)$37 connection sectors (Aharonov–Bohm flux, temporal "magnetic" curvature) are introduced as model couplings on the reduced configuration space rather than derived from a post-Carrollian electrodynamics; a first-principles gauge construction is left open. The anyonic holonomy requires removing the coincidence point of the relative coordinate, and the physical selection among self-adjoint extensions is deferred to unspecified microscopic interactions. The $(-,-,+,+)$38 non-regular extensions are excluded by fiat. On the gravitational side, the curved reduction is branch-dependent: the $(-,-,+,+)$39-adapted representative is directional, and the authors concede that the asymptotic radial limit may impose additional constraints on allowed Einstein-vacuum geometries beyond those verified for the Kleinian Schwarzschild branch, which is presented as illustrative rather than exhaustive. The multi-time well-posedness literature is invoked precisely because the construction sidesteps, rather than resolves, determinism questions by evolving along a single spatial parameter. Whether the temporal-plane lensing formula has an invariant formulation beyond the optical-path excess $(-,-,+,+)$40, and whether the logarithmic growth persists for strong-field baselines, are not addressed.

Conclusion

This work demonstrates that adding a second temporal dimension to the Carroll–Schrödinger program produces a structurally rich but internally consistent reduced theory: the equal-$(-,-,+,+)$41 Hilbert space is ordinary $(-,-,+,+)$42, the symmetry closes on $(-,-,+,+)$43 with a Bargmann realization identifying the fiber with $(-,-,+,+)$44, and the extra temporal direction manifests physically through quantized vortex circulation, Gouy-type dephasing graded by $(-,-,+,+)$45, centrifugal corrections to $(-,-,+,+)$46, and anyonic relative holonomy. The curved extension shows that the carrier-reduction mechanism survives on split-signature Einstein backgrounds, with the Kleinian Schwarzschild branch yielding exact affine temporal trajectories in the optical variable and a controlled logarithmic angular deflection. The main unresolved issues — a first-principles post-Carrollian gauge sector, the selection of self-adjoint extensions, and the generality of the curved branch construction — define the immediate technical agenda for this framework.