2,782 research outputs found

### Can We Look at The Quantisation Rules as Constraints?

In this paper we explore the idea of looking at the Dirac quantisation
conditions as $\hbar$-dependent constraints on the tangent bundle to
phase-space. Starting from the path-integral version of classical mechanics and
using the natural Poisson brackets structure present in the cotangent bundle to
the tangent bundle of phase- space, we handle the above constraints using the
standard theory of Dirac for constrained systems. The hope is to obtain, as
total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets
the Moyal ones. Unfortunately the program fails indicating that something is
missing. We put forward at the end some ideas for future work which may
overcome this failure.Comment: 4-pages, late

### 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

### 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

### 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

### Koopman-von Neumann Formulation of Classical Yang-Mills Theories: I

In this paper we present the Koopman-von Neumann (KvN) formulation of
classical non-Abelian gauge field theories. In particular we shall explore the
functional (or classical path integral) counterpart of the KvN method. In the
quantum path integral quantization of Yang-Mills theories concepts like
gauge-fixing and Faddeev-Popov determinant appear in a quite natural way. We
will prove that these same objects are needed also in this classical path
integral formulation for Yang-Mills theories. We shall also explore the
classical path integral counterpart of the BFV formalism and build all the
associated universal and gauge charges. These last are quite different from the
analog quantum ones and we shall show the relation between the two. This paper
lays the foundation of this formalism which, due to the many auxiliary fields
present, is rather heavy. Applications to specific topics outlined in the paper
will appear in later publications.Comment: 46 pages, Late

### 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

### Classical and quantum mechanics via supermetrics in time

Koopman-von Neumann in the 30's gave an operatorial formululation of
Classical Mechanics. It was shown later on that this formulation could also be
written in a path-integral form. We will label this functional approach as CPI
(for classical path-integral) to distinguish it from the quantum mechanical
one, which we will indicate with QPI. In the CPI two Grassmannian partners of
time make their natural appearance and in this manner time becomes something
like a three dimensional supermanifold. Next we introduce a metric in this
supermanifold and show that a particular choice of the supermetric reproduces
the CPI while a different one gives the QPI.Comment: To appear in the proceedings of the conference held in Trieste in
October 2008 with title: "Theoretical and Experimental aspects of the spin
statistics connection and related symmetries

### New Application of Functional Integrals to Classical Mechanics

In this paper a new functional integral representation for classical dynamics
is introduced. It is achieved by rewriting the Liouville picture in terms of
bosonic creation-annihilation operators and utilizing the standard derivation
of functional integrals for dynamical quantities in the coherent states
representation. This results in a new class of functional integrals which are
exactly solvable and can be found explicitly when the underlying classical
systems are integrable.Comment: 13 page

### NEWTON's trajectories versus MOND's trajectories

MOND dynamics consists of a modification of the acceleration with respect to
the one provided by Newtonian mechanics. In this paper we investigate whether
it can be derived from a velocity-dependent deformation of the coordinates of
the systems. The conclusion is that it cannot be derived this way because of
the intrinsic non-local character in time of the MOND procedure. This is a
feature pointed out some time ago already by Milgrom himself.Comment: Improved the abstract, the conclusions and inserted some further new
reference

### Cartan-Calculus and its Generalizations via a Path-Integral Approach to Classical Mechanics

In this paper we review the recently proposed path-integral counterpart of
the Koopman-von Neumann operatorial approach to classical Hamiltonian
mechanics. We identify in particular the geometrical variables entering this
formulation and show that they are essentially a basis of the cotangent bundle
to the tangent bundle to phase-space. In this space we introduce an extended
Poisson brackets structure which allows us to re-do all the usual Cartan
calculus on symplectic manifolds via these brackets. We also briefly sketch how
the Schouten-Nijenhuis, the Fr\"olicher- Nijenhuis and the Nijenhuis-Richardson
brackets look in our formalism.Comment: 6 pages, amste

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