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 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 $W_{\infty}\times W_{\infty}$- 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

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

Endogenous growth and wave-like business fluctuations

This paper argues that observed long lags in innovation implementation rationalize Schumpeter's statement that “wave-like fluctuations in business ... are the form economic development takes in the era of capitalism.” 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