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arXiv:2512.00247

On the Dynamics of Multiparticle Carroll-Schrdinger Quantum Systems

José Rojas, Melvin Arias

quant-phSubmitted Nov 28, 2025• Announced Dec 2, 2025• 0 citations
v1 · Nov 28, 2025v2 · Dec 2, 2025
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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.

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arXiv:1301.0807•Physical Review C•Apr 4, 2014•9 citations
We study the dynamics of multiparticle Carroll-Schrödinger (CS) quantum systems in 1+11{+}11+1 dimensions, where xxx acts as the evolution variable and ttt as the configuration coordinate. We derive the NNN-body theory on equal-xxx 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 xxx-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(x)U_{\!tot}(\mathbf x)Utot​(x), we show that the evolution is driven by the collective force ∑j∂xjU ⁣tot\sum_j\partial_{x_j}U_{\!tot}∑j​∂xj​​Utot​; 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′)g^{(2)}(t,t')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=−3/16β=-3/16β=−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.
Motivated by the temporal dynamics identified in the (1+1)(1+1)(1+1) Carroll-Schrödinger theory, we derive a post-Carrollian Schrödinger dynamics in flat Klein space with signature (2,2)(2,2)(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 (t1,t2)(t_1,t_2)(t1​,t2​) serves as the equal-radius configuration space. The additional time direction supplies an SO(2)SO(2)SO(2) temporal angular momentum JJJ, produces temporal vortex sectors, and gives the centrifugal contribution to the post-Carrollian momentum PPC=Eτ2/(2Meff)+J2/(2Meffτ2)P_{\mathrm{PC}}=E_τ^{2}/(2M_{\mathrm{eff}})+J^{2}/(2M_{\mathrm{eff}}τ^{2})PPC​=Eτ2​/(2Meff​)+J2/(2Meff​τ2) in the Hamilton-Jacobi limit. We determine the regular Bessel modes, Gaussian packets, oscillator spectrum, radial SU(1,1)SU(1,1)SU(1,1) tower, equal-rrr continuity equation, sch(2)\mathfrak{sch}(2)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)SO(2,1)SO(2,1)-symmetric Kleinian Schwarzschild exterior, where the Kleinian gravitational source produces a lensing-type angular deviation on the temporal plane.
Extreme kinematic limits of space-time symmetries reveal alternative forms of quantum behaviour. In the Carrollian limit the speed of light is taken to zero ([Formula: see text]): causal light cones collapse onto the time axis, spatial points become disconnected, and the usual notion of particle motion disappears. The corresponding quantum equation is the Carroll–Schrödinger equation—the structural mirror image of the ordinary Schrödinger equation, being first order in space and second order in time. We explore the mathematical connections between these two equations in one space and one time dimension. Using operator methods, we find conditions on external potentials that allow both equations to share solutions, and show that any Carrollian problem can be mapped to an equivalent Schrödinger problem via an explicit coordinate transformation. Probability densities and currents are related by removing the interaction potential and swapping space and time coordinates. An extreme relativistic boost combined with a classical approximation yields the Carrollian dispersion relation and conditions for coinciding classical trajectories. By formulating the dynamics on a Hilbert space of time rather than space, we prove that evolution in the spatial coordinate preserves probability. Closed-form solutions illustrate each result and together offer a practical framework for translating between the two descriptions.
Given a time-independent Hamiltonian H~\widetilde HH , one can construct a time-dependent Hamiltonian HtH_tHt​ by means of the gauge transformation Ht=UtH~Ut†−iUt∂tUt†H_t=U_t\kern1pt \widetilde H \kern1pt U^\dagger_t-i\kern1pt U_t\kern1pt\partial_t U_t^\daggerHt​=Ut​HUt†​−iUt​∂t​Ut†​ . Here UtU_tUt​ is the unitary transformation that relates the solutions of the corresponding Schrödinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as physical. We formulate sufficient conditions on UtU_tUt​ ensuring that HtH_tHt​ is physical as long as H~\widetilde HH is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians HtH_tHt​ and H~\widetilde HH such that the driven many-body dynamics governed by HtH_tHt​ can be reduced to the quench dynamics due to the time-independent H~\widetilde HH . We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where H~=H~t\widetilde H = \widetilde H_tH=Ht​ is time-dependent but dynamically integrable.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of NNN interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M)\text{MCTDHB}(M)MCTDHB(M), where MMM specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number MMM of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
Recently [G. Kaniadakis, Phys. Rev. A 55, 941 (1997)], we introduced a Schr\"odinger equation containing a complex nonlinearity $W(\ensuremath{\rho},\mathbf{j})+i\mathcal{W}(\ensuremath{\rho},\mathbf{j})$ which describes the collective interaction introduced by an exclusion-inclusion principle (EIP). The EIP does not affect $W(\ensuremath{\rho},\mathbf{j})$ and determines $\mathcal{W}(\ensuremath{\rho},\mathbf{j})$ univocally. In the above reference $W(\ensuremath{\rho},\mathbf{j})$ was deduced by means of a stochastic quantization approach, in this way obtaining a noncanonical quantum system. In this work we introduce a family of nonlinearities $W(\ensuremath{\rho},\mathbf{j})$ generating a family of nonlinear canonical quantum systems, and derive their Lagrangian and the Hamiltonian functions and the evolution equations of the fields. We derive also the Ehrenfest relations and study the soliton properties. The shape of the soliton, propagating in the system obeying the EIP, can be obtained by solving a first-order ordinary differential equation. We show that, in the case of soliton solutions, by means of a unitary transformation, the EIP potential is equivalent to a real algebraic nonlinear potential proportional to $\ensuremath{\kappa}{\ensuremath{\rho}}^{2}/(1+\ensuremath{\kappa}\ensuremath{\rho}).$
This article generalizes the notion of the local density of a many-body system to introduce collective coordinates as explicit degrees of freedom. It is shown that the energy of the system can be expressed as a functional of this object. The latter can in turn be factorized as the product of the square modulus of a collective wave function and a normalized collective-coordinate-dependent density. Energy minimization translates into a set of coupled equations, i.e., a local Schr\"odinger equation for the collective wave function and a set of Kohn-Sham equations for optimizing the normalized density at each point in the collective space. These equations reformulate the many-body problem exactly provided one is able to determine density- and collective-wave-function-dependent terms of the collective mass and potential which play a similar role to the exchange-correlation term in electronic Kohn-Sham density functional theory.