561 research outputs found

### Addenda and corrections to work done on the path-integral approach to classical mechanics

In this paper we continue the study of the path-integral formulation of
classical mechanics and in particular we better clarify, with respect to
previous papers, the geometrical meaning of the variables entering this
formulation. With respect to the first paper with the same title, we {\it
correct} here the set of transformations for the auxiliary variables
$\lambda_{a}$. We prove that under this new set of transformations the
Hamiltonian ${\widetilde{\cal H}}$, appearing in our path-integral, is an exact
scalar and the same for the Lagrangian. Despite this different transformation,
the variables $\lambda_{a}$ maintain the same operatorial meaning as before but
on a different functional space. Cleared up this point we then show that the
space spanned by the whole set of variables ($\phi, c, \lambda,\bar c$) of our
path-integral is the cotangent bundle to the {\it reversed-parity} tangent
bundle of the phase space ${\cal M}$ of our system and it is indicated as
$T^{\star}(\Pi T{\cal M})$. In case the reader feel uneasy with this strange
{\it Grassmannian} double bundle, we show in this paper that it is possible to
build a different path-integral made only of {\it bosonic} variables. These
turn out to be the coordinates of $T^{\star}(T^{\star}{\cal M})$ which is the
double cotangent bundle of phase-space.Comment: Title changed, appendix expanded, few misprints fixe

### 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

### 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

### 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

### 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

### Diagrammar In Classical Scalar Field Theory

In this paper we analyze perturbatively a g phi^4 classical field theory with
and without temperature. In order to do that, we make use of a path-integral
approach developed some time ago for classical theories. It turns out that the
diagrams appearing at the classical level are many more than at the quantum
level due to the presence of extra auxiliary fields in the classical formalism.
We shall show that several of those diagrams cancel against each other due to a
universal supersymmetry present in the classical path integral mentioned above.
The same supersymmetry allows the introduction of super-fields and
super-diagrams which considerably simplify the calculations and make the
classical perturbative calculations almost "identical" formally to the quantum
ones. Using the super-diagrams technique we develop the classical perturbation
theory up to third order. We conclude the paper with a perturbative check of
the fluctuation-dissipation theorem.Comment: 67 pages. Improvements inserted in the third order calculation

### 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 Proposal for a Differential Calculus in Quantum Mechanics

In this paper, using the Weyl-Wigner-Moyal formalism for quantum mechanics,
we develop a {\it quantum-deformed} exterior calculus on the phase-space of an
arbitrary hamiltonian system. Introducing additional bosonic and fermionic
coordinates we construct a super-manifold which is closely related to the
tangent and cotangent bundle over phase-space. Scalar functions on the
super-manifold become equivalent to differential forms on the standard
phase-space. The algebra of these functions is equipped with a Moyal super-star
product which deforms the pointwise product of the classical tensor calculus.
We use the Moyal bracket algebra in order to derive a set of quantum-deformed
rules for the exterior derivative, Lie derivative, contraction, and similar
operations of the Cartan calculus.Comment: TeX file with phyzzx macro, 43 pages, no figure

### Entanglement, Superselection Rules and Supersymmetric Quantum Mechanics

In this paper we show that the energy eigenstates of supersymmetric quantum
mechanics (SUSYQM) with non definite "fermion" number are entangled states.
They are "physical states" of the model provided that observables with odd
number of spin variables are allowed in the theory like it happens in the
Jaynes-Cummings model. Those states generalize the so called "spin spring"
states of the Jaynes-Cummings model which have played an important role in the
study of entanglement.Comment: 2 words added in the title, a section (IV) added in the text, a new
author joined the projec

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