Hilbert Space Structure in Classical Mechanics: (II)

In this paper we analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper no. (I), it is possible to endow the associated Hilbert space with a positive definite scalar product and to describe the dynamics via a Hermitian Hamiltonian. The drawback of this formulation is that higher forms do not appear automatically and that the description of chaotic systems may need a further extension of the Hilbert space.Comment: 45 pages, RevTex; Abstract and Introduction improve

On the "Universal" N=2 Supersymmetry of Classical Mechanics

In this paper we continue the study of the geometrical features of a functional approach to classical mechanics proposed some time ago. In particular we try to shed some light on a N=2 "universal" supersymmetry which seems to have an interesting interplay with the concept of ergodicity of the system. To study the geometry better we make this susy local and clarify pedagogically several issues present in the literature. Secondly, in order to prepare the ground for a better understanding of its relation to ergodicity, we study the system on constant energy surfaces. We find that the procedure of constraining the system on these surfaces injects in it some local grassmannian invariances and reduces the N=2 global susy to an N=1.Comment: few misprints fixed with respect to Int.Jour.Mod.Phys.A vol 16, no15 (2001) 270

A New Supersymmetric Extension of Conformal Mechanics

In this paper a new supersymmetric extension of conformal mechanics is put forward. The beauty of this extension is that all variables have a clear geometrical meaning and the super-Hamiltonian turns out to be the Lie-derivative of the Hamiltonian flow of standard conformal mechanics. In this paper we also provide a supersymmetric extension of the other conformal generators of the theory and find their "square-roots". The whole superalgebra of these charges is then analyzed in details. We conclude the paper by showing that, using superfields, a constraint can be built which provides the exact solution of the system.Comment: 11 pages, no figure

Geometric Dequantization

Dequantization is a set of rules which turn quantum mechanics (QM) into classical mechanics (CM). It is not the WKB limit of QM. In this paper we show that, by extending time to a 3-dimensional "supertime", we can dequantize the system in the sense of turning the Feynman path integral version of QM into the functional counterpart of the Koopman-von Neumann operatorial approach to CM. Somehow this procedure is the inverse of geometric quantization and we present it in three different polarizations: the Schroedinger, the momentum and the coherent states ones.Comment: 50+1 pages, Late

A New Superconformal Mechanics

In this paper we propose a new supersymmetric extension of conformal mechanics. The Grassmannian variables that we introduce are the basis of the forms and of the vector-fields built over the symplectic space of the original system. Our supersymmetric Hamiltonian itself turns out to have a clear geometrical meaning being the Lie-derivative of the Hamiltonian flow of conformal mechanics. Using superfields we derive a constraint which gives the exact solution of the supersymmetric system in a way analogous to the constraint in configuration space which solved the original non-supersymmetric model. Besides the supersymmetric extension of the original Hamiltonian, we also provide the extension of the other conformal generators present in the original system. These extensions have also a supersymmetric character being the square of some Grassmannian charge. We build the whole superalgebra of these charges and analyze their closure. The representation of the even part of this superalgebra on the odd part turns out to be integer and not spinorial in character.Comment: Superfield re-define

Grover's algorithm on a Feynman computer

We present an implementation of Grover's algorithm in the framework of Feynman's cursor model of a quantum computer. The cursor degrees of freedom act as a quantum clocking mechanism, and allow Grover's algorithm to be performed using a single, time-independent Hamiltonian. We examine issues of locality and resource usage in implementing such a Hamiltonian. In the familiar language of Heisenberg spin-spin coupling, the clocking mechanism appears as an excitation of a basically linear chain of spins, with occasional controlled jumps that allow for motion on a planar graph: in this sense our model implements the idea of "timing" a quantum algorithm using a continuous-time random walk. In this context we examine some consequences of the entanglement between the states of the input/output register and the states of the quantum clock

Scale symmetry in classical and quantum mechanics

In this paper we address again the issue of the scale anomaly in quantum mechanical models with inverse square potential. In particular we examine the interplay between the classical and quantum aspects of the system using in both cases an operatorial approach.Comment: 11 pages, Late

Typicality vs. probability in trajectory-based formulations of quantum mechanics

Bohmian mechanics represents the universe as a set of paths with a probability measure defined on it. The way in which a mathematical model of this kind can explain the observed phenomena of the universe is examined in general. It is shown that the explanation does not make use of the full probability measure, but rather of a suitable set function deriving from it, which defines relative typicality between single-time cylinder sets. Such a set function can also be derived directly from the standard quantum formalism, without the need of an underlying probability measure. The key concept for this derivation is the {\it quantum typicality rule}, which can be considered as a generalization of the Born rule. The result is a new formulation of quantum mechanics, in which particles follow definite trajectories, but which is only based on the standard formalism of quantum mechanics.Comment: 24 pages, no figures. To appear in Foundation of Physic

Quantum search by measurement

We propose a quantum algorithm for solving combinatorial search problems that uses only a sequence of measurements. The algorithm is similar in spirit to quantum computation by adiabatic evolution, in that the goal is to remain in the ground state of a time-varying Hamiltonian. Indeed, we show that the running times of the two algorithms are closely related. We also show how to achieve the quadratic speedup for Grover's unstructured search problem with only two measurements. Finally, we discuss some similarities and differences between the adiabatic and measurement algorithms.Comment: 8 pages, 2 figure

The classical supersymmetric Coulomb problem

After setting up a general model for supersymmetric classical mechanics in more than one dimension we describe systems with centrally symmetric potentials and their Poisson algebra. We then apply this information to the investigation and solution of the supersymmetric Coulomb problem, specified by an 1/|x| repulsive bosonic potential.Comment: 25 pages, 2 figures; reference added, some minor modification