Analytical Mechanics for Relativity and Quantum Mechanics

Abstract

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac’s formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics, such as the development of quantum information technology and the problem of quantising the gravitational field, require a rethinking of the quantum-classical connection.

A Unified Framework for Relativity and Curvilinear-Time Newtonian Mechanics

“…which is in agreement with [7]. Although this presentation of Newtonian mechanics is valuable in its own right, it constitutes, from our perspective, a blind alley, because L C4 does not contain any free function that can be re-interpreted, in relativity, as a Lagrange multiplier.…”

“…At this stage, we have demonstrated how classical and relativistic mechanics can be unified under the aegis of the Lagrangian L of (4.1). It is important to realise, however, that no unification would have occurred (in our sense), had one started from existing four-dimensional reformulations [6,7] of Newtonian mechanics with t as a coordinate of space-time. The corresponding Lagrangian L C4 is extremely simple to obtain from our framework: it suffices to replace, in the classical Lagrangian L C of (2.11), the temporal multiplier e by its value e = t coming from the definition (3.3) of t. The result reads…”

“…Moreover, the Lagrangian and the Hamiltonian formalisms can also, in full generality, be extended [6,7] to incorporate t as a coordinate. In all such extensions, it then becomes necessary to introduce another parameter, say λ, to describe the trajectories in the larger space.…”

Symplectic Reduction and the Problem of Time in Nonrelativistic Mechanics

“…Moreover, although general relativity does feature an infinite set of Hamiltonian constraints, which together play a role similar to the single Hamiltonian constraint we have encountered in nonrelativistic timeless theory, these Hamiltonian constraints are non-trivially related both to each other and a second set of constraints (called the momentum constraints) which arise in the theory. A full exploration of the algebra (which in fact fails to be a Lie algebra) constituted by the various constraints is a necessary precondition of a meaningful analysis of the symmetries of general relativity and goes beyond the level of our current discussion 32 .…”

Nonautonomous Hamiltonian quantum systems, operator equations, and representations of the Bender–Dunne Weyl-ordered basis under time-dependent canonical transformations

“…In practice, this means to deal with extended canonical transformations (q, p, t) → (Q, P, τ ) that mix position and momentum (coordinates or operators, reliant on the dynamical regime) through a time dependent phase-space map implemented by a redefinition of the time variable see e.g. [33,34] (for a discussion treating with time-dependent quadratic systems, see also [35]). As we already said, the basic Q, P structure that enters in the analysis of the non-autonomous Hamiltonians in which we are interested is linear in the original canonical pair (q, p).…”