From metaphysical principles to dynamical laws

My thesis in this paper is: the modern concept of laws of motion—qua dynamical laws—emerges in 18th-century mechanics. The driving factor for it was the need to extend mechanics beyond the centroid theories of the late-1600s. The enabling result behind it was the rise of differential equations. In consequence, by the mid-1700s we see a deep shift in the form and status of laws of motion. The shift is among the critical inflection points where early modern mechanics turns into classical mechanics as we know it. Previously, laws of motion had been channels for truth and reference into mechanics. By 1750, the laws lose these features. Instead, now they just assert equalities between functions; and serve just to entail (differential) equations of motion for particular mechanical setups. This creates two philosophical problems. First, it’s unclear what counts as evidence for the laws of motion in the Enlightenment. Second, it’s a mystery whether these laws retain any notion of causality. That subverts the early-modern dictum that physics is a science of causes

The reorganization of secondary school mathematics

Thesis (M.A.)--Boston Universit

Statistical Geometry in Quantum Mechanics

A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the Hilbert space H. By consideration of the square-root density function we can regard M as a submanifold of the unit sphere in H. Therefore, H embodies the `state space' of the probability distributions, and the geometry of M can be described in terms of the embedding of in H. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramer-Rao and Bhattacharyya inequalities. The statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.Comment: 32 pages, LaTex file, Extended version to include quantum measurement theor

New variational and multisymplectic formulations of the Euler-Poincar\'e equation on the Virasoro-Bott group using the inverse map

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincar\'e equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm, and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with $2$-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.Comment: 19 page