Classical Mechanics in Hilbert Space , Part 2
نویسنده
چکیده
We continue from Part 1. We will illustrate the general theory of Hamiltonian mechanics in the Lie group formalism. We then obtain the Hamiltonian formalism in the Hilbert spaces of square integrable functions on the symplectic spaces. We illustrate this general theory with several concrete examples, two of which are the representations of the Lorentz group and the Poincaré group with interactions.
منابع مشابه
$L_{p;r} $ spaces: Cauchy Singular Integral, Hardy Classes and Riemann-Hilbert Problem in this Framework
In the present work the space $L_{p;r} $ which is continuously embedded into $L_{p} $ is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is...
متن کامل2 00 2 Classical and Quantum Nambu Mechanics
The classical and quantum features of Nambu mechanics are analyzed and fundamental issues are resolved. The classical theory is reviewed and developed utilizing varied examples. The quantum theory is discussed in a parallel presentation, and illustrated with detailed specific cases. Quantization is carried out with standard Hilbert space methods. With the proper physical interpretation, obtaine...
متن کاملFrom Classical to Quantum Mechanics:
From Classical to Quantum Mechanics: " How to translate physical ideas into mathematical language " Abstract In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts:-General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators and so on...)-Quantum Mechanics properly ...
متن کاملOn Koopman-von Neumann Waves
In this paper we study the classical Hilbert space introduced by Koopman and von Neumann in their operatorial formulation of classical mechanics. In particular we show that the states of this Hilbert space do not spread, differently than what happens in quantum mechanics. The role of the phases associated to these classical ”wave functions” is analyzed in details. In this framework we also perf...
متن کاملQuantum Mechanics as an Approximation to Classical Mechanics in Hilbert Space
Classical mechanics is formulated in Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket, and a quasidensity operator. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Classical mechanics can now be viewed as a deformation of quantum mechanics. The forms of sem...
متن کامل