دانلود کتاب مکانیک فضای فاز کلاسیک و کوانتومی

Mechanics is not matter in motion. It is geometric structure in possibility. For more than a century, the principal mode of representation for mechanical theory has been via higher dimensional possibility spaces. Primary amongst these are phase spaces. This Element is about the structure of these spaces and the manner in which this structure affords representations of mechanical systems. Our intention is to set out the core mathematical and physical content of classical and quantum phase space mechanics as a staging post for future foundational investigation. Whilst we confine ourselves to phase space representations in finite dimensions, the content of this Element is relevant to the foundations of essentially all areas of modern physics. Each of classical mechanics, statistical mechanics, quantum theory, and general relativity admits phase space formulations and, as such, foundational questions from each of these areas can be reformulated in phase space terms. Moreover, the phase space representation is perhaps uniquely positioned to provide formal means to explore the connection between physical theories, most significantly the complex of relationships between open and closed, and statistical and quantum theories which is one of the primary motivation for our study. From a more historical perspective, it is worth remarking that the transition to the study of physical theory in terms of phase spaces can be understood to mark third revolutionary change in physical theory, foundationally as important and almost coincident with the quantum and relativity revolutions, but vastly less studied by historians and philosophers of science. This is the transformation from the quantitive to the qualitative methods in the study of dynamical systems that ran from the late nineteenth century to the mid twentieth century and had as its crowning achievement the celebrated Kolmogorov–Arnold–Moser (KAM) theorem which establishes, under certain conditions, the persistence under small perturbations of quasi-periodic motions in ‘most’ initial states of a Hamiltonian dynamical systems (Arnold, Kozlov, & Neishtadt, 2006, §6.3). The quantitive to qualitative revolution was initiated by a series of results concerning the stability of the solar system in the last decade of the nineteenth century by Henri Poincaré and brought together in his monumental Les Méthodes Nouvelles de la Mécanique Céleste (New Methods of Celestial Mechanics) published in three volumes between 1892 and 1899. The essence of Poincaré’s qualitative approach was a set of new methods for the study of features of sets of solutions to a mechanical problem in terms of properties of flows on phase spaces. As noted by Abraham and Marsden in the introduction to their textbook, this qualitative approach then immediately suggests a connection between physical properties such as stability, and the geometric structure of phase space: [Poincaré] visualised a dynamical system as a field of vectors on a phase space, in which a solution is a smooth curve tangent at each of its points to the vector at that point. The qualitative theory is based on geometric properties of the phase portrait: the family of solution curves, which fill up the entire phase space. For questions such as stability, it is necessary to study the entire phase portrait including the behaviour for all values of the time parameter. Thus it was essential to consider the entire phase space at once as a geometric object . . . [The] special geometric structure pertaining to the occurrence of phase variables in canonical conjugate pairs [is] a symplectic structure. (Abraham & Marsden, 1980, p.xviii) One of the key stepping stones to Poincaré’s stability results was his demonstration of the generic existence of an integral invariant, equivalent to symplectic volume form, under the dynamics (Goroff, 1993, p. I79). This result was earlier proved by both Liouville (1838) and Boltzmann (1871) and is now known as Liouville’s theorem. It is the first and arguably most basic result of qualitative mechanics. Liouville’s theorem and its failure will be one of the three interconnected themes that run throughout this Element. The other two themes are, first, more generally, the conceptualisation of phase spaces as structured possibility spaces and, second, more specifically, the connection between the notions of ‘open’ and ‘closed’ mechanical systems, dissipation and decoherence, and the idea of probability and quasi-probability flows on phase spaces as incompressible fluid flows. Whilst the first topic has been subject to a limited discussion in the philosophical literature, the second is a novel intervention, and, as such this Element can be understood, in part, as a short research monograph setting out a novel interpretative stance on these topics. Our main focus is didactic not dialectic. This Element is primarily intended for researchers and graduate students in the philosophy and foundations of physics with an interest in the conceptual foundations of phase space formulations of mechanics. As such, our primary goal is a pedagogical and expository one. An array of formal and conceptual machinery for the analysis of phase space mechanics is introduced and it is hoped that a suitable platform for further studies and research on the topic has been provided.