It is argued that awareness of the distinction between dynamical and variational symmetries is crucial to understanding the significance of Noether's 1918 work. Specific attention is paid, by way of a number of striking examples, to Noether's first theorem, which establishes a correlation between dynamical symmetries and conservation principles.

Einstein’s unpublished 1927 deterministic trajectory interpretation of quantum mechanics is critically examined, in particular with regard to the reason given by Einstein for rejecting his theory. It is shown that the aspect Einstein found objectionable—the mutual dependence of the motions of particles when the (many-body) wavefunction factorises—is a generic attribute of his theory but that this feature may be removed by modifying Einstein’s method in either of two ways: using a suggestion of Grommer or, in a physically important special case, (...) using a simpler technique. It is emphasized though that the presence or absence of the interdependence property does not determine the acceptability of a trajectory theory. It is shown that there are other grounds for rejecting Einstein’s theory (and the two modified theories), to do with its domain of applicability and compatibility with empirical predictions. That Einstein’s reason for rejection is not a priori grounds for discarding a trajectory theory is demonstrated by reference to an alternative deterministic trajectory theory that displays similar particle interdependence yet is compatible with quantum predictions. (shrink)

The aim of this paper is to illustrate four properties of the non-relativistic limits of relativistic theories: that a massless relativistic field may have a meaningful non-relativistic limt, that a relativistic field may have more than one non-relativistic limit, that coupled relativistic systems may be "more relativistic" than their uncoupled counterparts, and that the properties of the non-relativistic limit of a dynamical equation may differ from those obtained when the limiting equation is based directly on exact Galilean kinematics. These properties (...) are demonstrated through an examination of the non-relativistic limit of the familiar equations of first-quantized QED, i.e., the Dirac and Maxwell equations. The conditions under which each set of equations admits non-relativistic limits are given, particular attention being given to a gauge-invariant formulation of the limiting process especially as it applied to the electromagnetic potentials. The difference between the properties of a limiting theory and an exactly Galilean covariant theory based on the same dynamical equation is demonstrated by examination of the Pauli equation. (shrink)

It is known that the Schrödinger equation may be derived from a hydrodynamic model in which the Lagrangian position coordinates of a continuum of particles represent the quantum state. Using Routh’s method of ignorable coordinates it is shown that the quantum potential energy of particle interaction that represents quantum effects in this model may be regarded as the kinetic energy of additional ‘concealed’ freedoms. The method brings an alternative perspective to Planck’s constant, which plays the role of a hidden variable, (...) and to the canonical quantization procedure, since what is termed ‘kinetic energy’ in quantum mechanics may be regarded literally as energy due to motion. (shrink)

Internal global symmetries exist for the free non-relativistic Schrodinger particle, whose associated Noether charges---the space integrals of the wavefunction and the wavefunction multiplied by the spatial coordinate---are exhibited. Analogous symmetries in classical electromagnetism are also demonstrated.

A wide range of anatomical features are shared by all vertebrates, but absent in our closest invertebrate relatives. The origin of vertebrate embryogenesis must have involved the evolution of new regulatory pathways to control the development of new features, but how did this occur? Mutations affecting regulatory genes, including those containing homeobox sequences, may have been important: for example, perhaps gene duplications allowed recruitment of genes to new roles. Here I ask whether comparative data on the genomic organization and expression (...) patterns of homeobox genes support this hypothesis. I propose a model in which duplications of particular homeobox genes, followed by the acquisition of gene‐specific secondary expression domains, allowed the evolution of the neural crest, extensive organogenesis and craniofacial morphogenesis. Specific details of the model are amenable to testing by extension of this comparative approach to molecular embryology. (shrink)

We observe that Schrödinger’s equation may be written as two real coupled Hamilton–Jacobi (HJ)-like equations, each involving a quantum potential. Developing our established programme of representing the quantum state through exact free-standing deterministic trajectory models, it is shown how quantum evolution may be treated as the autonomous propagation of two coupled congruences. The wavefunction at a point is derived from two action functions, each generated by a single trajectory. The model shows that conservation as expressed through a continuity equation is (...) not a necessary component of a trajectory theory of state. Probability is determined by the difference in the action functions, not by the congruence densities. The theory also illustrates how time-reversal symmetry may be implemented through the collective behaviour of elements that individually disobey the conventional transformation (T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{T}$$\end{document}) of displacement (scalar) and velocity (reversal). We prove that an integral curve of the linear superposition of two vectors can be derived algebraically from the integral curves of one of the constituent vectors labelled by integral curves associated with the other constituent. A corollary establishes relations between displacement functions in diverse trajectory models, including where the functions obey different symmetry transformations. This is illustrated by showing that a (T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{T}$$\end{document}-obeying) de Broglie-Bohm trajectory is a sequence of points on the (non-T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{T}$$\end{document}) HJ trajectories, and vice versa. (shrink)

Hidden variables are usually presented as potential completions of the quantum description. We describe an alternative role for these entities, as aids to calculation in quantum mechanics. This is illustrated by the computation of the time-dependence of a massless relativistic spinor field obeying Weyl’s equation from a single-valued continuum of deterministic trajectories (the “hidden variables”). This is achieved by generalizing the exact method of state construction proposed previously for spin 0 systems to a general Riemannian manifold from which the spinor (...) construction is extracted as a special case. The trajectories form a non-covariant structure and the Lorentz covariance of the spinor field theory emerges as a kind of collective effect. The method makes manifest the spin 1/2 analogue of the quantum potential that is tacit in Weyl’s equation. This implies a novel definition of the “classical limit” of Weyl’s equation. (shrink)