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 $P_{\mathrm{PC}}=E_τ^{2}/(2M_{\mathrm{eff}})+J^{2}/(2M_{\mathrm{eff}}τ^{2})$ 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.
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 (t1,t2) 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 PPC=Eτ2/(2Meff)+J2/(2Meffτ2) in the Hamilton-Jacobi limit. We determine the regular Bessel modes, Gaussian packets, oscillator spectrum, radial SU(1,1) tower, equal-r continuity equation, 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.
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 Utot(x), we show that the evolution is driven by the collective force ∑j∂xjUtot; 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.
Non-Hermitian PT symmetric Klein-Gordon fields in Lorentz-violating wormholes spacetime background
We investigate a PT-symmetric Klein--Gordon (KG) oscillator in a (3+1)-dimensional Lorentz-violating (LV) traversable wormhole spacetime. The geometry is characterized by a smooth throat of radius a and a constant lapse function, A(x)=1, so that no Killing horizons occur and the two asymptotic regions at x→±∞ remain connected through the throat. The Lorentz violation modifies the radial geometry and the oscillator scale according to Ω=Ω/1−ζ. A nonminimal non-Hermitian coupling, Ft(x)=iΩx, leads to a PT-symmetric KG oscillator with a regular effective potential at the throat and a quadratic confinement at large ∣x∣. The radial equation can be mapped to the confluent Heun equation, and polynomial truncation gives a conditionally exactly solvable sector in which the oscillator frequency, throat radius, LV parameter, angular momentum, and radial quantum number satisfy algebraic constraints. For the degree-1 sector, these conditions restrict the allowed angular momenta and correlate them with the LV parameter. The resulting bound-state energies are real and discrete, with exact particle--antiparticle symmetry, E−=−E+, while the corresponding wave functions are regular and localized on the two-sided wormhole geometry. These results show how Lorentz violation and the PT-symmetric coupling modify the relativistic bound-state spectrum through the geometry of the wormhole.
Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.
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.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
I use a time-dependent Lagrangian formalism and a variational trial function to study the dynamics of a two-component vortex in a spin-orbit coupled Bose-Einstein condensate (BEC). For a single-component BEC, various experiments have validated this theoretical approach, for example a thermal quench that yields a quantized vortex in roughly 25% of trials. To be definite, I assume the specific spin-orbit form used in recent NIST experiments, which introduces a spatial asymmetry because of the external Raman laser beams. I here generalize this formalism to include a two-component order parameter that has quantized circulation in each component but not necessarily with the same circulation. For example a singly quantized vortex in just one component yields a BEC analog of the half-quantum vortex familiar in 3He-A and in p-wave chiral superconductors. This and other unusual two-component vortices have both periodic trajectories and unbounded trajectories that leave the condensate, depending on the initial conditions. The optimized phase of the order parameter induces a term in the particle current that cancels the contribution from the vector potential, leaving pure circulating current around the vortex.