Quantum Symmetries: Structures, Dynamics, and Algebra
A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".
Deadline for manuscript submissions: 30 June 2026 | Viewed by 28
Special Issue Editor
Special Issue Information
Dear Colleagues,
Quantum symmetries play a foundational role in the mathematical structure of various domains within quantum physics, including quantum mechanics, relativistic quantum mechanics, quantum field theory, the Standard Model, Grand Unified Theories (GUTs), and condensed matter physics. Beyond their connection to conservation laws, quantum symmetries provide essential restrictions that help formulate robust and physically consistent theories.
Quantum theory and particle physics are fundamentally built on symmetries involving the following:
- Unitary operator groups, which govern the evolution of quantum systems.
- Hermitian operator groups, which describe quantum states and observables.
The SU(2) symmetry is particularly significant in describing quantum spin and the algebra of rotational transformations of quantum states. In non-relativistic quantum mechanics, Lie groups and their associated generators are commonly used to represent continuous symmetries, such as rotations and phase shifts. These groups define how quantum systems transform smoothly under such operations.
In high-energy physics, gauge symmetry groups such as U(1), SU(2), and SU(3) encode the fundamental interactions within the Standard Model and its possible unifications in GUTs. In addition to continuous symmetries, discrete symmetries such as charge conjugation (C), parity (P), and time reversal (T) are critical in governing quantum dynamics and are central to the CPT theorem.
In the relativistic quantum regime, the Lorentz group and the Poincaré group define the symmetries and algebraic structure of spacetime and relativistic fields.
Beyond these well-established frameworks, emerging areas of research explore more exotic forms of symmetry, including the following:
- Supersymmetry (SUSY), which relates bosons and fermions.
- Hopf algebra, critical in quantum field theory, condensed matter physics, high-energy physics, and string theory.
- Conformal symmetry, important in locally scale-invariant systems.
- Topological quantum symmetries, such as braiding and fusion rules in anyonic systems used in topological quantum computing.
- Noncommutative quantum groups, which generalize classical symmetry concepts to noncommutative spaces.
- Deformed quantum symmetries and algebra (e.g., q-deformed quantum theories and deformed Lie algebras).
Dr. Timothy Ganesan
Guest Editor
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Keywords
- Lie groups and Lie algebra
- unitary groups, U(n)
- Lorentz and Poincaré groups
- gauge symmetries and invariance
- GUT symmetry
- CPT symmetries
- spin symmetry
- supersymmetry (SUSY)
- noncommutative geometry
- braid groups and its algebra
- anyonic statistics and fusion rules
- conformal symmetries
- spin symmetry
- deformed quantum symmetries
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