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

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

Least-action principle and path-integral for classical mechanics

In this paper we show how the equations of motion of a superfield, which makes its appearance in a path-integral approach to classical mechanics, can be derived without the need of the least-action principleComment: to appear IN PHYS.REV.D (Brief Report

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

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

Quantum mechanics over a q-deformed (0+1)-dimensional superspace

We built up a explicit realization of (0+1)-dimensional q-deformed superspace coordinates as operators on standard superspace. A q-generalization of supersymmetric transformations is obtained, enabling us to introduce scalar superfields and a q-supersymmetric action. We consider a functional integral based on this action. Integration is implemented, at the level of the coordinates and at the level of the fields, as traces over the corresponding representation spaces. Evaluation of these traces lead us to standard functional integrals. The generation of a mass term for the fermion field leads, at this level, to an explicitely broken version of supersymmetric quantum mechanics.Comment: 11 pages, Late

Non equilibrium statistical physics with fictitious time

Problems in non equilibrium statistical physics are characterized by the absence of a fluctuation dissipation theorem. The usual analytic route for treating these vast class of problems is to use response fields in addition to the real fields that are pertinent to a given problem. This line of argument was introduced by Martin, Siggia and Rose. We show that instead of using the response field, one can, following the stochastic quantization of Parisi and Wu, introduce a fictitious time. In this extra dimension a fluctuation dissipation theorem is built in and provides a different outlook to problems in non equilibrium statistical physics.Comment: 4 page

Optimal portfolio choice with path dependent labor income: the infinite horizon case

We consider an infinite horizon portfolio problem with borrowing constraints, in which an agentreceives labor income which adjusts to financial market shocks in a path dependent way. Thispath-dependency is the novelty of the model, and leads to an infinite dimensional stochasticoptimal control problem. We solve the problem completely, and find explicitly the optimalcontrols in feedback form. This is possible because we are able to find an explicit solutionto the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if stateconstraints are present. To the best of our knowledge, this is the first infinite dimensionalgeneralization of Merton’s optimal portfolio problem for which explicit solutions can be found.The explicit solution allows us to study the properties of optimal strategies and discuss theirfinancial implications

Chiral Anomalies via Classical and Quantum Functional Methods

In the quantum path integral formulation of a field theory model an anomaly arises when the functional measure is not invariant under a symmetry transformation of the Lagrangian. In this paper, generalizing previous work done on the point particle, we show that even at the classical level we can give a path integral formulation for any field theory model. Since classical mechanics cannot be affected by anomalies, the measure of the classical path integral of a field theory must be invariant under the symmetry. The classical path integral measure contains the fields of the quantum one plus some extra auxiliary ones. So, at the classical level, there must be a sort of "cancellation" of the quantum anomaly between the original fields and the auxiliary ones. In this paper we prove in detail how this occurs for the chiral anomaly.Comment: 26 pages, Latex, misprint fixed, a dedication include