GesamtSymmetrie

GesamtSymmetrie, literally “total symmetry” in German, is a concept used to describe the complete set of transformations that leave a system invariant. In mathematics and physics, symmetry is formalized by a group G acting on the objects or states of interest. GesamtSymmetrie denotes the maximal, exact symmetry group of an idealized model, including external (spatial) and internal (gauge, flavor) symmetries, as well as possible discrete or continuous dynamical symmetries.

In concrete terms, external symmetries encompass translations, rotations, reflections, and their continuous counterparts; internal symmetries act on internal degrees of freedom and may form groups such as U(1), SU(n), or permutation groups. The total symmetry group is often described as a product or semidirect product of these subgroups, though interactions between sectors can modify the group structure.

Recognizing the GesamtSymmetrie of a system guides its analysis. Noether’s theorem links continuous symmetries to conserved quantities. Spontaneous or explicit symmetry breaking reduces the GesamtSymmetrie to a proper subgroup, producing phenomena such as energy level splitting and phase transitions.

Examples include molecular point groups with high symmetry, such as methane with Td symmetry, and crystalline solids with cubic Oh symmetry. In particle physics, the combined gauge and spacetime symmetries contribute to the overall symmetry structure, with the Standard Model gauge group playing a central role alongside Lorentz invariance, though real systems often exhibit only approximate or broken GesamtSymmetrie.

In practice, determining the GesamtSymmetrie of a system informs representation-theoretic analysis, selection rules for transitions, and the degeneracy patterns of states.