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The orbits are verified with Smale's alpha-criterion, which offers a rigorous certificate of existence. The billiard exhibits only a few families of nongeneric periodic orbits. For some families of ball configurations, Athreya, Burdzy, and Duarte have established the maximum higher sure for the number of pseudo-collisions, thereby demonstrating that the number of collisions is finite. Within the equivalent circuit image (ECP), this reduces to a binomial distribution within the number of loops of time machine. Here we develop a quantum version of the paradox, whereby a (semiclassical) wave packet evolves by means of a area containing a wormhole time machine. We apply the 2 foremost quantum theories of CTCs to our mannequin: Deutsch's mannequin (D-CTCs) and postselected teleportation (P-CTCs). The postselected teleportation prescription (P-CTCs) however predicts a pure-state solution through which the loop counts have binomial coefficient weights. We find that D-CTCs reproduce the classical solution multiplicity in the form of a combined state, while P-CTCs predict an equal superposition of the 2 trajectories, supporting a conjecture by Friedman et al.

In this article, we focus on billiard techniques of their many varieties and show how such a simple setup can reveal fundamental insights into the habits of nature at both classical and quantum scales. Here we introduce a new quantum formulation of a basic example, the place a billiard ball can travel alongside two attainable trajectories: one unperturbed and one, alongside a CTC, where it collides with its previous self. It consists of two quarter cylinders which are rotated with respect to each other by 90 degrees, and it is classically chaotic. In this undertaking, we do extensive simulations to review two specific configurations. Computer simulations present that the diffusion coefficient of this system is a highly irregular operate of the vibration frequency exhibiting pronounced maxima each time there are resonances between the vibration frequency and the common time of flight of a particle. Simulations counsel that in the long term, a lot of the power 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 talk about the mannequin in the continuum limit, with a selected concentrate on the varied methods one may make use of in order to guarantee convergence in the average number of clock evolutions.

We then talk about the model within the continuum restrict, with a particular deal with the varied methods one might employ in order to ensure convergence in the common number of clock evolutions. Abstract:We present a recreation impressed by research on the possible variety of billiard ball collisions in the whole Euclidean area. The other player tries to find initial conditions for the cue ball to maximise the variety of collisions. While typical collisions in billiards are practically completely elastic, with a restitution coefficient close to 1 and low friction, we discover three deviations from ideal elastic collisions: The non-elastic nature, the friction effects between the balls during collision, the friction between the ball and the desk. Pseudo-velocities change in accordance with the same guidelines as those for velocities of completely elastic collisions between shifting balls. Using this reality we deduce that for any area 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 movement. We look at the existence of integral of movement which is polynomial in velocities. Abstract:We consider billiard ball motion in a convex area of a relentless curvature surface influenced by the constant magnetic area.

View PDF Abstract:We consider billiard ball motion in a convex domain on a constant curvature surface influenced by the constant magnetic field. This result's a manifestation of the so-called Hopf rigidity phenomenon which was not too long ago obtained for classical billiards on fixed curvature surfaces. Abstract:Past research of the billiard-ball paradox, a problem involving an object that travels back in time along a closed timelike curve (CTC), sometimes concern themselves with entirely classical histories, whereby any trajectorial results associated with quantum mechanics can't manifest. That is achieved 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. That is completed by mapping all related paths on to a quantum circuit, wherein the distinction of the varied paths is facilitated by representing the billiard particle with a clock state. For this mannequin, we discover that Deutsch's prescription (D-CTCs) provides self-consistent solutions in the type of a mixed state composed of phrases which symbolize each doable configuration of the particle's evolution by means of the circuit. As an application, we characterize the attainable contact angles and exhibit an infinite family of real analytic non-spherical cylinders that float in neutral equilibrium at any orientation with fixed contact angles.