group theory and quantum mechanics

Lie groups and Lie algebras provide the mathematical framework for symmetries in quantum mechanics. In practice this means studying their unitary representation theory to see how quantum states and observables transform under symmetry operations. These works emphasize that unitary (often strongly continuous) representations and the related algebraic structure capture the core symmetry-based relations and the mathematical subtleties that arise in foundational treatments.

The literature connects group-based formalisms to quantum-gravity ideas, notably group field theory and related frameworks. Specific approaches cited include matrix models, tensor models, spin foam models and loop quantum gravity, which use group-theoretic and combinatorial structures to model quantum geometry.

These works treat the construction of strongly continuous unitary representations of the Poincaré and related Lorentz groups and illustrate them with spinor and field examples. Typical topics include Weyl and Dirac spinors and fields, the Dirac equation, and Maxwell fields — showing how spin and field equations arise from group-representation principles.

Unitary representations are presented as the central mathematical tool for describing how quantum systems transform under symmetry groups. Emphasis is placed on strongly continuous representations in relativistic quantum field theory and on unitary representation theory in foundational and structural treatments of quantum systems.

Page counts in this group range from 138 to 668 pages. Formats listed include Hardcover and Paperback.

Beyond group theory, the works cover algebraic groups, quantum groups and crystal bases, affine Lie algebras, affine Hecke algebras, algebraic geometry, and topics related to representations of finite reductive and complex reflection groups.