Quantization as a dimensional reduction phenomenon

Classical mechanics, in the operatorial formulation of Koopman and von Neumann, can be written also in a functional form. In this form two Grassmann partners of time make their natural appearance extending in this manner time to a three dimensional supermanifold. Quantization is then achieved by a process of dimensional reduction of this supermanifold. We prove that this procedure is equivalent to the well-known method of geometric quantization.Comment: 19 pages, Talk given by EG at the conference "On the Present Status of Quantum Mechanics", Mali Losinj, Croatia, September 2005. New results are contained in the last part of the pape

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

Functional Approach to Classical Yang-Mills Theories

Sometime ago it was shown that the operatorial approach to classical mechanics, pioneered in the 30's by Koopman and von Neumann, can have a functional version. In this talk we will extend this functional approach to the case of classical field theories and in particular to the Yang-Mills ones. We shall show that the issues of gauge-fixing and Faddeev-Popov determinant arise also in this classical formalism.Comment: 4 pages, Contribution to the Proceedings of the International Meeting "Quantum Gravity and Spectral Geometry" (Naples, July 2-7, 2001

A New Look at the Schouten-Nijenhuis, Fr\"olicher-Nijenhuis and Nijenhuis-Richardson Brackets for Symplectic Spaces

In this paper we re-express the Schouten-Nijenhuis, the Fr\"olicher-Nijenhuis and the Nijenhuis-Richardson brackets on a symplectic space using the extended Poisson brackets structure present in the path-integral formulation of classical mechanics.Comment: 27+1 pages, Latex, no figure

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

A New Quantization Map

In this paper we find a simple rule to reproduce the algebra of quantum observables using only the commutators and operators which appear in the Koopman-von Neumann (KvN) formulation of classical mechanics. The usual Hilbert space of quantum mechanics becomes embedded in the KvN Hilbert space: in particular it turns out to be the subspace on which the quantum positions Q and momenta P act irreducibly.Comment: 12 pages, 1 figure, Late

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

Path-dependent equations and viscosity solutions in infinite dimension

Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [15, 9]) and viscosity solutions (see e.g. [16]). In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.Comment: To appear in the Annals of Probabilit

Optimal policy and consumption smoothing effects in the time-to-build AK model

In this paper the dynamic programming approach is exploited in order to identify the closed loop policy function, and the consumption smoothing mechanism in an endogenous growth model with time to build, linear technology and irreversibility constraint in investment. Moreover the link among the time to build parameter, the real interest rate, and the magnitude of the smoothing effect is deeply investigated and compared with what happens in a vintage capital model characterized by the same technology and utility function. Finally we have analyzed the effect of time to build on the speed of convergence of the main aggregate variables.Time-to-build, AK model, Dynamic programming, optimal