# Geometric phase

In physical systems, a geometric phase is a difference in phase (which describes the stages of a cyclic wave process ℭ𝔚𝔓), appearing during the cyclic wave process ℭ𝔚𝔓, resulting from the geometric properties of the parameter space of the Hamiltonian which governs the cyclic wave process ℭ𝔚𝔓, and being due to cyclic adiabatic processes affecting the cyclic wave process ℭ𝔚𝔓. The geometric phase depends solely (or almost solely) on the geometry of the circuitous evolution and phase state being transported through the circuitous evolution.

## Quotes

* The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)$^{2S}$. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles.
** Michael V. Berry and Jonathan M. Robbins, (1997). "Indistinguishability for quantum particles: Spin, statistics and the geometric phase". Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 453 (1963): 1771–1790. DOI:10.1098/rspa.1997.0096.
* One of the reasons for being interested in the geometric phase is that it connects a number of different areas.
** Michael Berry, (February 10, 2020)"Geometric phases and the separation of the world by Michael Berry". International Centre for Theoretical Sciences, YouTube. (quote at 2:57 of 1:29:53)
* Whenever a quantum system undergoes a cyclic evolution governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov–Bohm phase and the Pancharatnam and Berry phase, but both earlier and later manifestations exist. Although traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and become increasingly influential in many areas from condensed-matter physics and optics to high-energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review, we first introduce the Aharonov–Bohm effect as an important realization of the geometric phase. Then, we discuss in detail the broader meaning, consequences and realizations of the geometric phase, emphasizing the most important mathematical methods and experimental techniques used in the study of the geometric phase, in particular those related to recent works in optics and condensed-matter physics.
** Eliahu Cohen, Hugo Larocque, Frédéric Bouchard, Farshad Nejadsattari, Yuval Gefen, and Ebrahim Karimi, (2019). "Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond". Nature Reviews Physics 1 (7): 437–449. DOI:10.1038/s42254-019-0071-1.
* One of the simplest chemical exchange reactions involves a system of three hydrogen atoms: H+H$_2$→H$_2$+H. Surely, chemists have felt, one should be able to calculate the cross sections for this reaction from first principles. But the computations have not been easy. Only in the last six years or so have theorists, aided by efficient methodologies and access to supercomputers, been able to predict the cross sections in sufficient detail for comparison with experiments, which themselves have evolved in precision. The agreement has been good—well, almost. Small discrepancies, especially at higher total energies, stubbornly refused to yield to adjustments in either the calculations or the experiments. Now Yi‐Shuen Mark Wu and Aron Kuppermann of Caltech have erased these pesky discrepancies by including a topological effect known as the geometric phase. Michael Berry (University of Bristol) has called attention to the presence of this phase, which now bears his name, in a wide variety of physical systems.
** Barbara Goss Levi, (1993). "The Geometric Phase Shows up in Chemical Reactions". Physics Today 46 (3): 17–19. DOI:10.1063/1.2808832.
* The geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two-form through a surface in the system’s parameter space. We obtain the classical limit of this two-form in a form applicable to systems whose classical dynamics is chaotic. For integrable systems the expression is equivalent to the Hannay two-form. We discuss various properties of the classical two-form, derive semiclassical corrections to it (associated with classical periodic orbits), and consider implications for the semiclassical density of degeneracies.
** Jonathan M. Robbins and Michael V. Berry, (1992). "The geometric phase for chaotic systems". Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 436 (1898): 631–661. DOI:10.1098/rspa.1992.0039.
* Examples of geometric phases abound in many areas of physics. Many familiar problems that we do not ordinary associate with geometric phases may be phrased in terms of them. Often, the result is a clearer understanding of the structure of the problem, and of its solution.
** Alfred Shapere and Frank Wilczek, in: Shapere, Alfred, ed (July 1989). "Introduction and Overview". Geometric Phases in Physics. Advanced Series in Mathematical Physics, vol. 5. World Scientific. p. 3. ISBN 9789814507585.