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Aspects of Information in Physics

Elemér Elad Rosinger

Published Dec 1, 2009• 1 citation
OpenAlex

Abstract

Two somewhat overlooked aspects of information, namely, total involvement and simultaneous presence, are presented, aspects that may be useful in theories of Physics.

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Textbooks often emphasize the philosophical aspects of Heisenberg's Principle calling it “The Uncertainty Principle”. However, as a physical principle there is nothing uncertain about it. In its “exact” form it is a useful relationship between physical parameters. It is an expression of the fact that particles of matter behave simultaneously in two modes: one that is particle like, and the other that is wave like. These two modes are intimately connected, and it is this connection that is expressed in the Heisenberg Principle, and more generally in quantum mechanics including Schrödinger's equation (Atkins and Friedman, 1997). From this viewpoint, the principle can be used to deduce some of the properties of atoms and molecules in a simple way. This method of analysis is quantitative, but it is limited to cases that involve only a single coordinate that is given in advance (Born, 1989). The method has been extended somewhat by Simons and Bloch (1973) to cover some cases involving the angular momentum quantum number. It is a rudimentary form of density functional theory. In a somewhat different form than the one used here, it has been applied to molecules by Parr and his collaborators (Borkman and Parr, 1968). Physical properties can never be measured exactly. Errors, however small, are always present in a measurement procedure. Therefore, measured values of variables are average values derived from a distribution of individual measurements.
The Dirac-Bergmann algorithm for the Hamiltonian analysis of constrained systems is a nice and powerful tool, widely used for quantization and non-perturbative counting of degrees of freedom. However, certain aspects of its application to systems with first-class constraints are often overlooked in the literature, which is unfortunate, as a naive treatment leads to incorrect results. In particular, when transitioning from the total to the extended Hamiltonian, the physical information encoded in the constrained modes is lost unless a suitable redefinition of gauge invariant quantities is made. An example of this is electrodynamics, in which the electric field gets an additional contribution to its longitudinal component in the form of the gradient of an arbitrary Lagrange multiplier. Moreover, Dirac's conjecture, the common claim that all first-class constraints are independent generators of gauge transformations, is somewhat misleading in the standard notion of gauge symmetry used in field theories. At the level of the total Hamiltonian, the true gauge generator is a specific combination of primary and secondary first-class constraints; in general, Dirac's conjecture holds only in the case of the extended Hamiltonian. The aim of the paper is primarily pedagogical. We review these issues, providing examples and general arguments. Also, we show that the aforementioned redefinition of gauge invariants within the extended Hamiltonian approach is equivalent to a form of the Stueckelberg trick applied to variables that are second-class with respect to the primary constraints.
These lecture notes aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how that physics is realized geometrically by the Ryu-Takayanagi formula. In order to make the notes somewhat self-contained for readers whose background is in high-energy theory, a brief introduction to the relevant aspects of quantum information theory is included.