Complete Hamiltonian Description of Wave-Like Features in Classical and Quantum Physics

The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system describing in terms of ray trajectories, for a stationary refractive medium, a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the geometrical optics (“eikonal”) approximation, which is contained as a simple limiting case. Due to the fact, moreover, that the time independent Schrödinger equation is itself a Helmholtz-like equation, the same mathematics holding for a classical optical beam turns out to apply to a quantum particle beam moving in a stationary force field, and leads to a system of Hamiltonian equations providing exact and deterministic particle trajectories and dynamical laws, and containing the laws of Classical Mechanics in the eikonal limit.     

Vagueness, ignorance, and margins for error

We argue that the epistemic theory of vagueness cannot adequately justify its key tenet-that vague predicates have precisely bounded extensions, of which we are necessarily ignorant. Nor can the theory adequately account for our ignorance of the truth values of borderline cases. Furthermore, we argue that Williamson’s promising attempt to explicate our understanding of vague language on the model of a certain sort of “inexact knowledge” is at best incomplete, since certain forms of vagueness do not fit Williamson’s model, and in fact fit an alternative model. Finally, we point out that a certain kind of irremediable inexactitude postulated by physics need not be-and is not commonly-interpreted as epistemic. Thus, there are aspects of contemporary science that do not accord well with the epistemicist outlook.     

Nonconservation of Energy and Loss of Determinism I. Infinitely Many Colliding Balls

An infinite number of elastically colliding balls is considered in a classical, and then in a relativistic setting. Energy and momentum are not necessarily conserved globally, even though each collision does separately conserve them. This result holds in particular when the total mass of all the balls is finite, and even when the spatial extent and temporal duration of the process are also finite. Further, the process is shown to be indeterministic: there is an arbitrary parameter in the general solution that corresponds to the injection of an arbitrary amount of energy (classically), or energy-momentum (relativistically), into the system at the point of accumulation of the locations of the balls. Specific examples are given that illustrate these counter-intuitive results, including one in which all the balls move with the same velocity after every collision has taken place.     

A remark on the role of indeterminism and non-locality in the violation of Bell’s inequalities

Diederik Aerts was the first in the eighties to develop a concrete example of a macroscopic “classical” entity violating Bell’s inequalities (BI). In more recent years, he also developed a macroscopic model in which the amount of non-locality and indeterminism can be continuously varied, and used it to show that by increasing non-locality one increases the degree of violation of BI, whereas by increasing indeterminism one decreases the degree of violation of BI. In this article we introduce and analyze a different macroscopic model in which the amount of non-locality and indeterminism can also be parameterized, and therefore varied, and find that, in accordance with the model of Aerts, an increase of non-locality does produce a stronger violation of BI. However, differently from his model, we also find that, depending on the initial state in which the system is prepared, an increase of indeterminism can either strengthen or weaken the degree of violation of BI.