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The orbits are verified with Smale's alpha-criterion, which offers a rigorous certificate of existence. The billiard exhibits just a few households of nongeneric periodic orbits. For some households of ball configurations, Athreya, Burdzy, and Duarte have established the maximum upper certain for the variety of pseudo-collisions, thereby demonstrating that the variety of collisions is finite. Within the equivalent circuit picture (ECP), this reduces to a binomial distribution within the variety of loops of time machine. Here we develop a quantum model of the paradox, wherein a (semiclassical) wave packet evolves by way of a region containing a wormhole time machine. We apply the 2 foremost quantum theories of CTCs to our mannequin: Deutsch's model (D-CTCs) and postselected teleportation (P-CTCs). The postselected teleportation prescription (P-CTCs) alternatively predicts a pure-state answer by which the loop counts have binomial coefficient weights. We find that D-CTCs reproduce the classical solution multiplicity within the type of a combined state, whereas P-CTCs predict an equal superposition of the 2 trajectories, supporting a conjecture by Friedman et al.

In this text, we focus on billiard systems of their many types and present how such a simple setup can reveal elementary insights into the behavior of nature at both classical and quantum scales. Here we introduce a new quantum formulation of a basic instance, where a billiard ball can journey alongside two attainable trajectories: one unperturbed and one, alongside a CTC, the place it collides with its previous self. It consists of two quarter cylinders that are rotated with respect to each other by ninety levels, and it is classically chaotic. In this undertaking, we do intensive simulations to review two specific configurations. Computer simulations show that the diffusion coefficient of this system is a highly irregular operate of the vibration frequency exhibiting pronounced maxima every time there are resonances between the vibration frequency and the common time of flight of a particle. Simulations counsel that in the long run, most of the energy is concentrated close to the boundary. We prove that if the billiard map is completely integrable then the boundary curve is essentially a circle. We then focus on the model within the continuum restrict, with a specific concentrate on the assorted methods one may employ in order to ensure convergence in the average number of clock evolutions.

We then discuss the model in the continuum limit, with a selected give attention to the varied strategies one could employ in order to guarantee convergence in the common number of clock evolutions. Abstract:We present a game inspired by analysis on the doable variety of billiard ball collisions in the whole Euclidean space. The other participant tries to seek out initial situations for the cue ball to maximize the variety of collisions. While typical collisions in billiards are practically completely elastic, with a restitution coefficient near 1 and low friction, we discover three deviations from preferrred elastic collisions: The non-elastic nature, the friction results between the balls during collision, the friction between the ball and the table. Pseudo-velocities change based on the identical guidelines as those for velocities of completely elastic collisions between transferring balls. Using this reality we deduce that for any domain totally different from spherical disc for all but finitely many values of the magnitude of the magnetic field billiard movement does not have Polynomial in velocities integral of motion. We examine the existence of integral of movement which is polynomial in velocities. Abstract:We consider billiard ball movement in a convex area of a continuing curvature surface influenced by the constant magnetic field.

View PDF Abstract:We consider billiard ball movement in a convex area on a constant curvature floor influenced by the fixed magnetic subject. This result is a manifestation of the so-referred to as Hopf rigidity phenomenon which was lately obtained for classical billiards on fixed curvature surfaces. Abstract:Past research of the billiard-ball paradox, a problem involving an object that travels again in time alongside a closed timelike curve (CTC), sometimes concern themselves with fully classical histories, whereby any trajectorial results related to quantum mechanics can not manifest. That is accomplished by mapping all relevant paths on to a quantum circuit, wherein the distinction of the various paths is facilitated by representing the billiard particle with a clock state. That is accomplished by mapping all relevant paths on to a quantum circuit, in which the distinction of the various paths is facilitated by representing the billiard particle with a clock state. For this model, we discover that Deutsch's prescription (D-CTCs) supplies self-constant options within the type of a mixed state composed of phrases which represent every potential configuration of the particle's evolution by the circuit. As an application, we characterize the attainable contact angles and exhibit an infinite household of real analytic non-round cylinders that float in neutral equilibrium at any orientation with fixed contact angles.