Hilbert Space Structure in Classical Mechanics: (II)

Abstract

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.

45 pages, RevTex; Abstract and Introduction improved

Cited by in corpus (11)

• Operational Dynamic Modeling Transcending Quantum and Classical Mechanics
• The Boltzmann Equation in Classical and Quantum Field Theory
• Wigner phase space distribution as a wave function
• Efficient method to generate time evolution of the Wigner function for open quantum systems
• Fundamental Theory of Statistical Particle Dynamics
• Geometric Dequantization
• Field Theoretic Formulation of Kinetic theory: I. Basic Development
• Kramers equation and supersymmetry
• New Application of Functional Integrals to Classical Mechanics
• Operational Dynamical Modeling of spin 1/2 relativistic particles: the Dirac equation and its classical limit
• On the metric structure of time in classical and quantum mechanics