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Quantum Symmetry

Special Issue Information

Dear Colleagues,

There are several differences between classical and quantum theories that have an impact on the possible symmetries:
  • Physical states represented in Hilbert space rather than phase space.
  • Quantum mechanics defines symmetries as mappings between physical states that preserve transition amplitudes. (As Wigner proved, these symmetries can be represented in Hilbert space by unitary and anti-unitary operators.)
  • Quantum mechanics assigns complex numbers to these transition amplitudes.
  • The algebra of observables in quantum mechanics is non-commutative.
  • Quantum particles are indistinguishable.
  • Composite quantum systems are not represented by a Cartesian product structure, but by a linear tensor structure.
Quantum symmetries may also include gauge redundancies and dualities. Gauge redundancies can be understood as multiple representations of the same physical state. Dualities can be understood as isomorphisms holding between pairs of Hilbert spaces together with (canonical) operators. The possibilities for quantum symmetries are tightly constrained by the number of spacetime dimensions and by the dimensionality of the objects of the theory (including whether they are extensionless or structured). Quantum symmetries also refer to quantum groups, which aren't groups as such but algebras that reduce to groups in the limit as a deformation parameter (playing the part of Planck's constant) goes to 1 (returning multiplication to normal).

Contributions are invited on all aspects of quantum symmetries. Those that involve foundational issues or the intersection of theoretical physics and pure mathematics are especially welcomed. Possible themes (not ranked in order preference) include:

  • 2D Conformal Field Theory, Modular Invariance, Statistical Mechanics.
  • Dualities in Quantum Theories.
  • Mirror Symmetry in String Theory.
  • Emergent Quantum Symmetries, Symmetry Breaking, Effective Field Theory, Renormalization Group.
  • Hopf Algebras, Quantum Groups and Low Dimensional Physics.
  • Quantum Geometry (including Non-Commutative Geometry).
  • Spin-Statistics, Anyons, Fractional Quantum Hall Effect.
  • Connections between Quantum Symmetries and Spacetime/Object Dimensionality.
  • Quantum Symmetries in Computation.
  • Relationship between Classical and Quantum Symmetries.

Dr. Dean Rickles
Guest Editor

Keywords

  • quantum symmetry
  • S-duality
  • symmetry breaking
  • anyons
  • braid group
  • quantum groups
  • conformal field theory
  • modular invariance

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Symmetry - ISSN 2073-8994
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