Algebras of observables with continuous representations of symmetry groups

Kow-Lung Chang

The Postulates and Principles of Quantum Mechanics Commutability and Compatibility of Quantum Observables Non-Commuting Operators and Uncertainty Principle Quantum Dynamics and Schrodinger Equation Time Evolution Operator and Propagator Symmetry, Transformation and Continuous Groups Lie Groups and Lie Algebras O(3) Group, SU(2) Group and Angular Momentum Dynamical Symmetry and Hydrogen Atom Lorentz Group and Group Representation Chiral Transformation and Spinor Algebra Gamma Matrix Algebra and Discrete Symmetry in Dirac Spinor System.

Cheng-Hao He, Yi-Zhuang You, Xiao Yan Xu

Symmetry plays a central role in modern physics, from classifying quantum states to characterizing phases of matter through spontaneous symmetry breaking. In interacting fermionic systems with multiple internal degrees of freedom, however, determining the full continuous symmetry group and classifying possible order parameters remain challenging. In this work, we present a systematic framework for analyzing continuous symmetries and identifying candidate order parameters in such systems. By mapping the Hamiltonian to a Majorana representation, we obtain the generators of continuous symmetries from the Lie algebra of operators that commute with the Hamiltonian. We then identify the structure of this Lie algebra using the theory of semisimple Lie algebras. Building on representation theory, we further develop a systematic method for exhaustively enumerating candidate order parameters. By decomposing the exterior-power representations induced by the symmetry algebra on the Majorana space and incorporating discrete lattice symmetries, we classify these order parameters according to the symmetries they break. (Abridged. Please see the PDF manuscript for the complete abstract and specific model applications.)

Alexander Semenovich Holevo, Juha-Pekka Pellonpää

Serguei B. Isakov, Jon Magne Leinaas, Jan Myrheim +2 more

Vasily Evgenevich Tarasov

Shun-Jin Wang, Cao Jian-min, A. Weiguny

The continuous-variable representation of the dynamic group is discussed in general terms, based on the differential of the group manifold. The unified formalism of various continuous-variable representations of the dynamic group is presented, and the essential common features of different continuous-variable representation theories are exploited from the general mathematical-physical point of view. The right-hand isomorphic representation and the left-hand anti-isomorphic representation of the dynamic group, and their relationship and physical implication are discussed in detail. Specialization of the general formalism to the function spaces of the group, based on both the Hilbert space of quantum-mechanical states and the von Neumann space of the quantum-statistical states (density matrices), leads to the results of both the generator coordinate approach to the dynamic-group representation and the generalized quantum characteristic function method. Techniques for calculating the continuous-variable representation of the group algebra are proposed and illustrated with the Lipkin SU(2), the Elliott SU(3), and its extension Sp(6) symmetries as examples. The results obtained for these dynamic groups are instructive for both physicists and mathematicians.