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Elementary Particles and Interactions — Overview
Published in K Grotz, H V Klapdor, S S Wilson, The Weak Interaction in Nuclear, Particle and Astrophysics, 2020
K Grotz, H V Klapdor, S S Wilson
One can also distinguish between discrete and continuous symmetry operations. Continuous symmetry transformations may be parameterised with real numbers (e.g. a phase), whilst discrete transformations are parameterised by integers. Examples of discrete symmetry operations include the spatial reflection through the origin or parity operation and charge conjugation. Invariance under these operations leads to multiplicative quantum numbers, i.e. the quantum number of a system of particles is the product of the quantum numbers of the individual particles. Similarly, continuous symmetry operations lead to additive quantum numbers, for example the electric charge. We shall now describe some of the important quantum numbers in more detail.
Emergence of a new symmetry class for Bogoliubov–de Gennes (BdG) Hamiltonians: expanding 10-fold symmetry classes
Published in Phase Transitions, 2020
A symmetry is a transformation that leaves the physical system invariant. These transformations include translation, reflection, rotation, scaling, etc. One of the most important implications of symmetry in physics is the existence of conservation laws. For every global continuous symmetry, there exists an associated conserved quantity [1]. In quantum mechanics, symmetry transformation can be represented on the Hilbert space of physical states by an operator that is either linear and unitary or anti-linear and anti-unitary [2]. Any symmetry operator acts on these states and transforms them to new states. These symmetry operators can be classified as continuous (rotation, translation) and discrete (parity, lattice translations, time reversal). Continuous symmetry transformations give rise to the conservation of probabilities and discrete symmetry transformations give rise to the quantum numbers. Another important implication of symmetry in quantum mechanics is the symmetry on exchanging identical particles [3].