528 research outputs found
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
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
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 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
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
Universal Hidden Supersymmetry in Classical Mechanics and its Local Extension
We review here a path-integral approach to classical mechanics and explore
the geometrical meaning of this construction. In particular we bring to light a
universal hidden BRS invariance and its geometrical relevance for the Cartan
calculus on symplectic manifolds. Together with this BRS invariance we also
show the presence of a universal hidden genuine non-relativistic supersymmetry.
In an attempt to understand its geometry we make this susy local following the
analogous construction done for the supersymmetric quantum mechanics of Witten.Comment: 6 pages, latex, Volkov Memorial Proceeding
On Koopman-von Neumann Waves II
In this paper we continue the study, started in [1], of the operatorial
formulation of classical mechanics given by Koopman and von Neumann (KvN) in
the Thirties. In particular we show that the introduction of the KvN Hilbert
space of complex and square integrable "wave functions" requires an enlargement
of the set of the observables of ordinary classical mechanics. The possible
role and the meaning of these extra observables is briefly indicated in this
work. We also analyze the similarities and differences between non selective
measurements and two-slit experiments in classical and quantum mechanics.Comment: 18+1 pages, 1 figure, misprints fixe
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
Time and Geometric Quantization
In this paper we briefly review the functional version of the Koopman-von
Neumann operatorial approach to classical mechanics. We then show that its
quantization can be achieved by freezing to zero two Grassmannian partners of
time. This method of quantization presents many similarities with the one known
as Geometric Quantization.Comment: Talk given by EG at "Spacetime and Fundamental Interactions: Quantum
Aspects. A conference to honour A.P.Balachandran's 65th birthday
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