Since the appearance of Berry's seminal paper in 1984, geometric phases have been discovered in virtually all fields of physics. Here we address molecules and solids, and we limit our scope to the Berry's phases of the many-electron wavefunction. Many advances have occurred in very recent years relating to the theory of such phases and their observable consequences. After discussing the basic features of Berry's phases in a generic quantum system, we specialize to selected examples taken from molecular physics and condensed matter physics; in each of these cases, a Berry's phase of the electronic wavefunction leads to measurable effects.

https://iopscience.iop.org/article/10.1088/0953-8984/12/9/201/pdf

Manifestations of Berry's phase in molecules and condensed matter

The “gravitational memory effect” due to an exact plane wave provides us with an elementary description of the diffeomorphisms associated with the analogue of “soft gravitons for this nonasymptotically flat system. We explain how the presence of the latter may be detected by observing the motion of freely falling particles or other forms of gravitational wave detection. Numerical calculations confirm the relevance of the first, second and third time integrals of the Riemann tensor pointed out earlier. Solutions for various profiles are constructed. It is also shown how to extend our treatment to Einstein-Maxwell plane waves and a midisuperspace quantization is given.

Soft gravitons and the memory effect for plane gravitational waves

This paper addresses issues surrounding the concept of geometric phase or "anholonomy". Certain physical phenomena apparently require for their explanation and understanding, reference to toplogocial/geometric features of some abstract space of parameters. These issues are related to the question of how gauge structures are to be interpreted and whether or not the debate over their "reality" is really going to be fruitful.

/pdf/falling-cats-parallel-parking-and-polarized-light-3kv0hbyjez.pdf

Falling Cats, Parallel Parking, and Polarized Light

The problem of whether or not the equations of motion of a quantum system determine the commutation relations was posed by E.P.Wigner in 1950. A similar problem (known as ”The Inverse Problem in the Calculus of Variations”) was posed in a classical setting as back as in 1887 by H.Helmoltz and has received great attention also in recent times. The aim of this paper is to discuss how these two apparently unrelated problems can actually be discussed in a somewhat unified framework. After reviewing briefly the Inverse Problem and the existence of alternative structures for classical systems, we discuss the geometric structures that are intrinsically present in Quantum Mechanics, starting from finite-level systems and then moving to a more general setting by using the Weyl-Wigner approach, showing how this approach can accomodate in an almost natural way the existence of alternative structures in Quantum Mechanics as well.

/pdf/from-the-equations-of-motion-to-the-canonical-commutation-1en1bawmtb.pdf

From the equations of motion to the canonical commutation relations

The problem of whether or not the equations of motion of a quantum system determine the commutation relations was posed by E.P.Wigner in 1950. A similar problem (known as "The Inverse Problem in the Calculus of Variations") was posed in a classical setting as back as in 1887 by H.Helmoltz and has received great attention also in recent times. The aim of this paper is to discuss how these two apparently unrelated problems can actually be discussed in a somewhat unified framework. After reviewing briefly the Inverse Problem and the existence of alternative structures for classical systems, we discuss the geometric structures that are intrinsically present in Quantum Mechanics, starting from finite-level systems and then moving to a more general setting by using the Weyl-Wigner approach, showing how this approach can accomodate in an almost natural way the existence of alternative structures in Quantum Mechanics as well.

From the Equations of Motion to the Canonical Commutation Relations

An Elementary Introduction to Algebraic Topology..- Topological Methods in Classical Field Theory..- Inequivalent Quantizations in Multiply Connected Spaces. Braid Groups and Anyons..- Topics in Chern-Simons Physics..- A Short Introduction to: Connections on (1) Bundles and Berry's Phase..- Electrons in A Magnetic Field, and A Cursory Look at the Quantum Hall Effect..

Giuseppe Morandi

Papers

The Role of Topology in Classical and Quantum Physics