805 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
. We prove that under this new set of transformations the
Hamiltonian , appearing in our path-integral, is an exact
scalar and the same for the Lagrangian. Despite this different transformation,
the variables 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 () of our
path-integral is the cotangent bundle to the {\it reversed-parity} tangent
bundle of the phase space of our system and it is indicated as
. 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 which is the
double cotangent bundle of phase-space.Comment: Title changed, appendix expanded, few misprints fixe
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
The Negative Dimensional Oscillator at Finite Temperature
We study the thermal behavior of the negative dimensional harmonic oscillator
of Dunne and Halliday that at zero temperature, due to a hidden BRST symmetry
of the classical harmonic oscillator, is shown to be equivalent to the
Grassmann oscillator of Finkelstein and Villasante. At finite temperature we
verify that although being described by Grassmann numbers the thermal behavior
of the negative dimensional oscillator is quite different from a Fermi system.Comment: 8 pages, IF/UFRJ/93/0
Bulges
We model the evolution of the galactic bulge and of the bulges of a selected
sample of external spiral galaxies, via the multiphase multizone evolution
model. We address a few questions concerning the role of the bulges within
galactic evolution schemes and the properties of bulge stellar populations. We
provide solutions to the problems of chemical abundances and spectral indices,
the two main observational constraints to bulge structure.Comment: 15 pages, 10 figures, to be published in MNRA
Non-Commutative Geometry, Multiscalars, and the Symbol Map
Starting from the concept of the universal exterior algebra in
non-commutative differential geometry we construct differential forms on the
quantum phase-space of an arbitrary system. They bear the same natural
relationship to quantum dynamics which ordinary tensor fields have with respect
to classical hamiltonian dynamics.Comment: 8 pages, late
Quantum Deformed Canonical Transformations, W_{\infty}-algebras and Unitary Transformations
We investigate the algebraic properties of the quantum counterpart of the
classical canonical transformations using the symbol-calculus approach to
quantum mechanics. In this framework we construct a set of pseudo-differential
operators which act on the symbols of operators, i.e., on functions defined
over phase-space. They act as operatorial left- and right- multiplication and
form a - algebra which contracts to its diagonal
subalgebra in the classical limit. We also describe the Gel'fand-Naimark-Segal
(GNS) construction in this language and show that the GNS representation-space
(a doubled Hilbert space) is closely related to the algebra of functions over
phase-space equipped with the star-product of the symbol-calculus.Comment: TeX file with phyzzs macro, 23 pages, no figure
Endogenous growth and wave-like business fluctuations
This paper argues that observed long lags in innovation implementation rationalize Schumpeter's statement that âwave-like fluctuations in business ... are the form economic development takes in the era of capitalism.â Adding implementation delays to an otherwise standard endogenous growth model with expanding product variety, the equilibrium path admits a Hopf bifurcation where consumption, R&D and output permanently fluctuate. This mechanism is quantitatively consistent with the observed medium-term movements of US aggregate output. In this framework, an optimal allocation may be restored at equilibrium by the mean of a procyclical subsidy, needed to generate additional consumption smoothing. Finally, a procyclical R&D subsidy rate designed to half consumption fluctuations will increase the growth rate from 2.4% to 3.4% with a 9.6% (compensation equivalent) increase in welfare
The Response Field and the Saddle Points of Quantum Mechanical Path Integrals
In quantum statistical mechanics, Moyal's equation governs the time evolution
of Wigner functions and of more general Weyl symbols that represent the density
matrix of arbitrary mixed states. A formal solution to Moyal's equation is
given by Marinov's path integral. In this paper we demonstrate that this path
integral can be regarded as the natural link between several conceptual,
geometric, and dynamical issues in quantum mechanics. A unifying perspective is
achieved by highlighting the pivotal role which the response field, one of the
integration variables in Marinov's integral, plays for pure states even. The
discussion focuses on how the integral's semiclassical approximation relates to
its strictly classical limit; unlike for Feynman type path integrals, the
latter is well defined in the Marinov case. The topics covered include a random
force representation of Marinov's integral based upon the concept of "Airy
averaging", a related discussion of positivity-violating Wigner functions
describing tunneling processes, and the role of the response field in
maintaining quantum coherence and enabling interference phenomena. The double
slit experiment for electrons and the Bohm-Aharonov effect are analyzed as
illustrative examples. Furthermore, a surprising relationship between the
instantons of the Marinov path integral over an analytically continued ("Wick
rotated") response field, and the complex instantons of Feynman-type integrals
is found. The latter play a prominent role in recent work towards a
Picard-Lefschetz theory applicable to oscillatory path integrals and the
resurgence program.Comment: 58 page
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