This chapter brings together Einstein’s 1907 vision of ‘quantization as selection problem’ (see Section 5.1) and our explorations of the possible and impossible (momentum) configurations in Chapter 4. The hierarchy is formulated in terms of the latter ones and consists of Newtonian systems, non-Newtonian classical systems, non-classical systems and non-mechanical systems. These explorations are assisted by an overview on selection problems in classical and quantum theory, by further reasonings for Einstein’s vision and by few explanations about alternative and harmony in Euler’s axioms of classical mechanics. Two fundamental conclusions for non-classical systems can be drawn already at this stage. (i), The notion of path as a point-wise relationship between the configurations and the momentum configurations looses its meaning. (ii), there are no longer coordinates describing the boundaries of a system in (momentum) like the classical turning points of an oscillator.
This chapter brings together Einstein’s 1907 vision of ‘quantization as selection problem’ (see Section 5.1) and our explorations of the possible and impossible (momentum) configurations in Chapter 4. The hierarchy is formulated in terms of the latter ones and consists of Newtonian systems, non-Newtonian classical systems, non-classical systems and non-mechanical systems. These explorations are assisted by an overview on selection problems in classical and quantum theory, by further reasonings for Einstein’s vision and by few explanations about alternative and harmony in Euler’s axioms of classical mechanics. Two fundamental conclusions for non-classical systems can be drawn already at this stage. (i), The notion of path as a point-wise relationship between the configurations and the momentum configurations looses its meaning. (ii), there are no longer coordinates describing the boundaries of a system in (momentum) like the classical turning points of an oscillator.
This chapter introduces the paradigm ‘quantization as selection problem’ by means of few historical remarks. They include Einstein’s 1907 reasoning, Schrödinger’s 1926 derivation of his wave equation, and Gödel’s theorem and Munchhausen’s trilemma. The latter ones concern questions, which can be posed, but not be answered within CM. Such questions transcendent CM. This book poses the question, how the mechanics of oscillators without turning points looks like. (“oscillators without turning points” means oscillators, which may assume configurations beyond the classical turning points.)
This chapter summarizes the main results of this book and draws some conclusions. Special attention is paid to the axiomatic basis, i.e., to Schrödinger’s requirements and Hertz’s program, which lead to the novel paradigm of quantization as selection problem (rather than eigenvalue problem). Both ones are realized by means of limiting (Schrödingerian weight) functions, which guarantee the validity of the energy conservation law in the classically forbidden regions of configuration and momentum configuration spaces (e.g, beyond the turning points of classical oscillators). Further topics are the separation of external and internal parameters as initiated by Euler, the differences between classical and quantum wave equations, causality and determinism, hidden variables, symmetry and field quantization.
The "quantum equation" should "carry the quantum conditions in itself" (Second Commun., p. 511); 2. There should be a special mathematical method for solving the stationary Schrödinger equation, which accounts for the non-classical character of the quantization problem, ie, which is different from the classical eigenvalue methods for calculating the (eigen)modes of strings, resonators and so on (ibid., p. 513);3. The derivation should uniquely decide, that the energy rather than the frequency values are discretized (ibid., pp.511, 519), since only the former means true quantization, while the latter corresponds to the classical discretization mentioned in requirement 2;4. The use of the classical expressions for the kinetic and potential energies should be justified (Fourth Commun., p. 113).Schrödinger's requirements 1 and 2 mean, that -contrary to the very title of the papersquantization is actually not an eigenvalue problem.For in the latter the discretization is imposed not by the differential equation itself, but by the boundary conditions, and this is the classical discretization for standing waves in organ pipes etc.I will fulfill all four requirements by treating quantization as a selection problem.The number of stationary states of a quantum system is smaller than that of a classical system [6].I will describe, (i), the selection of quantum systems out of the set of all mechanical systems and, (ii), the selection of their stationary states out of the continuum of classical stationary states.Earlier arguing [19][10] is improved and extended in several essential points. Elements of an Eulerian representation of classical mechanics Euler's axiomsLeonhard Euler [20][21][22][23][24][25] was the first to apply the calculus to all areas of mathematics and mechanics of his time, and he developed novel methods, such as the calculus of variations and topology.Moreover, he worked out an axiomatic of mechanics, where only Newton's Law (axiom) 1 concerning the conservation of stationary states is retained as an axiom, while Newton's Laws (axioms) 2 and 3 concerning the change of stationary states are treated as problems to be solved (for a detailed account, see [19][10][45]).This allows for introducing alternative equations of motion without loosing the contact to CM.The existence of stationary states is postulated in the following axioms (as in Newton's axioms, rotatory motion is not considered). Axiom E0: Every body is either resting or moving.This means, that the subsequent axioms E1 and E2 are not independent of each other; they exclude each other and, at once, they are in harmony with each other [22].
This chapter prepares a smooth way from Eulerian classical mechanics to quantum mechanics. Starting from Helmholtz’s foundation of the energy conservation law using Leibniz’s theorem as described in the foregoing Chapter 2, the configurations a system can (cannot) assume in a given stationary state are explored. The following constraints are taken into account: Newton-Euler’s exclusion principle, d’Alembertian constraints and constraints imposed by conservation laws. The set of possible and impossible momentum configurations is also considered, using Nemorarius’ theorem of the foregoing chapter.
In section 6.5 we looked at the problem of evaluating quantum mechanical path integrals in curvilinear co-ordinates. For the model considered there (a point particle in two dimensions) the choice of curvilinear co-ordinates was capricious, but the use of curvilinear co-ordinates is natural in describing constrained systems. For example, suppose the particle in two dimensions was forced to move on a ring of radius R . The problem of path roughness would then go away, since the scalar curvature would be constant along the path. However, we acquire a new problem in that the configuration space for the particle (the circle) is no longer simply connected – a notion that we shall define later. This, in turn, can induce a new term in the action. We stress that this term is in no way a replacement for that due to roughness. For one thing, it is of order ħ , rather than of order ħ 2 . Most importantly it is not unique , leading to an ambiguity in quantisation. This ambiguity is a general property of quantisation on non-simply-connected manifolds. In this chapter we shall sketch how such O( ħ ) terms (topological ‘charges’) arise naturally in the path integral formalism which, being an ‘integral’ over paths in the configuration space of the system, is ideally suited for handling non-trivial configuration spaces.