On the Dynamics of Multiparticle Carroll-Schrdinger Quantum Systems

TL;DR

The paper develops a multiparticle Carroll–Schrödinger framework in $1{+}1$ dimensions by exchanging the roles of space and time, deriving equal-$x$ dynamics from a relativistic multi-time Klein–Gordon limit. Temporal interactions are introduced via minimal coupling to temporal energy operators and re-expressed as explicit time potentials through $x$-dependent gauge transformations, while a complementary spatial viewpoint reveals ultralocal Carrollian dynamics with translation-invariant internal forces canceling in the collective drive. A coordinate duality based on Schwarzian derivatives maps separable Schrödinger potentials to time-only Carroll potentials, and exchange symmetry in time leads to temporal Hanbury Brown–Twiss diagnostics. In second quantization, a short-memory limit yields a temporal derivative cubic–quintic NLSE with a fixed coefficient $\beta=-3/16$, indicating a universal temporal Carroll gas regime. Finally, a Carrollian density-functional framework is outlined via a current-density mapping and Kohn–Sham construction, providing a structured, if formal, bridge to functional descriptions of time-domain quantum correlations in Carrollian systems. These results open avenues for exploring Carrollian quantum dynamics, coherence, and effective theories, with potential links to Carroll fluids and cosmological models.

Abstract

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.

Figures (4)

• Figure 1: Carroll evolution of the two–oscillator model in the $(x_1,x_2)$ plane.
• Figure 2: Spatial profiles of the one–body observables at fixed times.
• Figure 3: Temporal profiles of the one–body observables at fixed $x_1$.
• Figure 4: Numerical evolution of the interaction-driven Carroll gas density $|\psi(X,T)|^2$ for the derived parameters (cubic $-i$, quintic $\beta=-3/16$). The simulation demonstrates that the pulse remains strictly localized as it evolves along $X$. The dynamics are governed by the interplay between the derivative nonlinearity, which sustains the solitary wave structure, and the repulsive quintic term, which saturates the amplitude and ensures stability against collapse.