Geometric Formulation of Classical and Quantum Mechanics

Abstract

Geometry of symplectic and Poisson manifolds is well known to provide the adequate mathematical formulation of autonomous Hamiltonian mechanics. The literature on this subject is extensive.This book presents the advanced geometric formulation of classical and quantum non-relativistic mechanics in a general setting of time-dependent coordinate and reference frame transformations. This formulation of mechanics as like as that of classical field theory lies in the framework of general theory of dynamic systems, Lagrangian and Hamiltonian formalism on fibre bundles.Non-autonomous dynamic systems, Newtonian systems, Lagrangian and Hamiltonian non-relativistic mechanics, relativistic mechanics, quantum non-autonomous mechanics are considered.Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over the time axis R. Quantum nonrelativistic mechanics is phrased in the geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization. Relativistic mechanics is adequately formulated as particular classical string theory of onedimensional submanifolds.The concept of a connection is the central link throughout the book. Connections on a configuration space of non-relativistic mechanics describe reference frames. Holonomic connections on a velocity space define nonrelativistic dynamic equations. Hamiltonian connections in Hamiltonian non-relativistic mechanics define the Hamilton equations. Evolution of quantum systems is described in terms of algebraic connections. A connection on a prequantization bundle is the main ingredient in geometric quantization. v vi PrefaceThe book provides a detailed exposition of theory of partially integrable and superintegrable systems and their quantization, classical and quantum non-autonomous constraint systems, Lagrangian and Hamiltonian theory of Jacobi fields, classical and quantum mechanics with time-dependent parameters, the technique of non-adiabatic holonomy operators, formalism of instantwise quantization and quantization with respect to different reference frames.Our book addresses to a wide audience of theoreticians and mathematicians of undergraduate, post-graduate and researcher levels. It aims to be a guide to advanced geometric methods in classical and quantum mechanics.For the convenience of the reader, a few relevant mathematical topics are compiled in Appendixes, thus making our exposition self-contained.

Superintegrable systems on Poisson manifolds

“…The following two assertions clarify the structure of superintegrable systems [4,6,12,26]. Lemma 2.1: Given a symplectic manifold (Z, Ω), let π : Z → N be a fibred manifold such that, for any two functions f , f ′ constant on fibres of π, their Poisson bracket {f, f ′ } also is well.…”

“…We refer to [25,26] for a global analysis of superintegrable systems. Given a superintegrable system in accordance with Definition 2.1, the above mentioned generalization of the Mishchenko -Fomenko theorem to non-compact invariant submanifolds states the following [6,12,25,26]. Theorem 2.3: Let the Hamiltonian vector fields ϑ i (6.13) of the generating functions F i be complete, and let fibres of the fibred manifold F (2.2) be connected and mutually diffeomorphic.…”

“…(ii) The foliation S is a fibred manifold π S : Z → N whose fibres are mutually diffeomorphic and, being integral manifolds, are connected. Then the following hold [10,12,25,26].…”

“…(Theorems 2.5, 2.6 and 2.3, respectively) [6,10,12,25,26,30]. However, Definition 2.1 of a superintegrable (non-commutative completely integrable) system on a symplectic manifold is rather restrictive.…”

Graded Geometric Structures Underlying F-Theory Related Defect Theories

“…We shall formally address these issues in the following subsections. For some references on connections and covariant differentials see for example [42][43][44][45][46][47][48][49][50] while for issues regarding graded manifolds see [42][43][44][45][46][47]51]. …”