1,781 research outputs found
On the Koopman von Neumann waves. II
No full textIn this paper we continue the study, started in \cite{waves1}, 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
Mechanical similarity as a generalization of scale symmetry
No full textIn this paper we study the symmetry known (Landau and Lifshits 1976 Course of Theoretical Physics Vol.1: Mechanics (Oxford: Pergamon } as mechanical similarity (LMS) and present for any monomial potential. We analyze it in the framework of the Koopman-von Neumann formulation of classical mechanics and prove that in this framework the LMS can be given a canonical implementation. We also show that the LMS is a generalization of the scale symmetry which is present only for the inverse square potential. Finally we study the main obstructions which one encounters in implementing the LMS at the quantum mechanical level
Scale symmetry in classical and quantum mechanics
No full textIn this paper we address again the issue of the scale anomaly in quantum mechanical models with inverse square potential. In particular we examine the interplay between the classical and quantum aspects of the system using in both cases an operatorial approach
A new look at the Schouten-Nijenhuis, Frolicher-Nijenhuis and Nijenhuis-Richardson brackets for symplectic spaces
No full textQuantization as a dimensional reduction phenomenon
No full textClassical mechanics, in the operatorial formulation of Koopman and von Neumann, can be written also in a functional form. This form two grassmannian 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
Koopman-von Neumann formulation of classical Yang-Mills theories: I
No full textIn 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
Chiral Anomalies via classical and quantum functional methods
No full textIn 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
On the metric structure of time in classical and quantum mechanics
No full textIn this paper we show that, via an extension of time, some metric structures naturally appear in both classical and quantum mechanics when both are formulated via path-integrals. We calculate the various Ricci scalar and curvatures associated to these metric and prove that they can be chosen to be zero in classical mechanics while this is not possible in quantum mechanics
Geometric dequantization
No full textDequantization 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 Schr\"odinger, the momentum and the coherent states ones
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