Group Theory: Mathematical Expression of Symmetry in Physics

The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.

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Group Theory in Atomic and Molecular Physics

Symmetries in Science ◽

10.1007/978-1-4684-3833-8_10 ◽

1980 ◽

pp. 151-160 ◽

Cited By ~ 1

Author(s):

B. R. Judd

Keyword(s):

Group Theory ◽

Molecular Physics ◽

Atomic And Molecular Physics

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The Topological Origin of Quantum Randomness

Symmetry ◽

10.3390/sym13040581 ◽

2021 ◽

Vol 13 (4) ◽

pp. 581

Author(s):

Stefan Heusler ◽

Paul Schlummer ◽

Malte S. Ubben

Keyword(s):

High School ◽

Hilbert Space ◽

Group Theory ◽

Lorentz Group ◽

Quantum Physics ◽

Time Development ◽

Mathematical Structures ◽

Stochastic Nature ◽

Quantum Randomness ◽

And Topology

What is the origin of quantum randomness? Why does the deterministic, unitary time development in Hilbert space (the ‘4π-realm’) lead to a probabilistic behaviour of observables in space-time (the ‘2π-realm’)? We propose a simple topological model for quantum randomness. Following Kauffmann, we elaborate the mathematical structures that follow from a distinction(A,B) using group theory and topology. Crucially, the 2:1-mapping from SL(2,C) to the Lorentz group SO(3,1) turns out to be responsible for the stochastic nature of observables in quantum physics, as this 2:1-mapping breaks down during interactions. Entanglement leads to a change of topology, such that a distinction between A and B becomes impossible. In this sense, entanglement is the counterpart of a distinction (A,B). While the mathematical formalism involved in our argument based on virtual Dehn twists and torus splitting is non-trivial, the resulting haptic model is so simple that we think it might be suitable for undergraduate courses and maybe even for High school classes.

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Astronomical spectroscopy: an introduction to the atomic and molecular physics of astronomical spectroscopy

Contemporary Physics ◽

10.1080/00107514.2020.1853241 ◽

2021 ◽

pp. 1-1

Author(s):

David J. Miller

Keyword(s):

Molecular Physics ◽

Atomic And Molecular Physics

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Atomic and molecular physics using positron accumulation techniques – Summary and a look to the future

Nuclear Instruments and Methods in Physics Research Section B Beam Interactions with Materials and Atoms ◽

10.1016/s0168-583x(00)00057-4 ◽

2000 ◽

Vol 171 (1-2) ◽

pp. 2-16 ◽

Cited By ~ 14

Author(s):

C.M. Surko ◽

R.G. Greaves ◽

K. Iwata ◽

S.J. Gilbert

Keyword(s):

Molecular Physics ◽

The Future ◽

Atomic And Molecular Physics

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Thermodynamics of structured fluids. Hard science for soft materials

Pure and Applied Chemistry ◽

10.1351/pac200072101819 ◽

2000 ◽

Vol 72 (10) ◽

pp. 1819-1834 ◽

Cited By ~ 3

Author(s):

John M. Prausnitz

Keyword(s):

Self Assembly ◽

Theoretical Calculations ◽

Soft Materials ◽

Confined Space ◽

Molecular Physics ◽

Fluid Structure ◽

Light Emitting ◽

Structured Fluids ◽

Recent Developments ◽

Narrow Space

At liquid-like densities, molecules of complex fluids can assume a variety of structures (or positions) in space; when the molecules contain many atoms as, for example, in polymers, that variety becomes very large. Further, when confined to a narrow space, it is possible to achieve structures that are not normally observed. Thanks to recent advances in statistical mechanics and molecular physics, and thanks to increasingly fast computers, it is now possible to calculate a fluid's structure, that is, the positions of molecules at equilibrium under given conditions. Calculation of fluid structure is useful because thermodynamic properties depend strongly on that structure, leading to possible applications for new materials. Three examples illustrate some recent developments; each example is presented only schematically (with a minimum of equations) to indicate the physical basis of the mathematical description. The first example considers the effect of branching on self-assembly (micellization) of copolymers (with possible long-range applications in medicine). The second and third examples consider the effect of confinement on fluid structure: first, crystallization in a narrow, confined space to produce a desired crystal structure (with possible applications for light-emitting diodes) and second, suppression of micellization of a diblock copolymer in a thin film (with possible application in lithography). Whenever possible, theoretical calculations are compared with experimental results.

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