Accidental symmetry

In field theory

In physics, particularly in renormalization theory, an accidental symmetry is a symmetry which is present in an effective field theory because the operators in the Lagrangian that violate this symmetry are irrelevant operators. Since the contribution by irrelevant operators at low energies is small, the low energy theory appears to have this symmetry.

In the Standard Model, the lepton number and the baryon number are accidental symmetries, while in lattice models, rotational invariance is accidental.

In Quantum Mechanics

The connection between symmetry and degeneracy (that is, the fact that apparently unrelated quantities turn out to be equal) is familiar in every day experience. Consider a simple example, where we draw three points on a plane, and calculate the distance between each of the three points. If the points are placed randomly, then in general all of these distances will be different. However, if the points are arranged so that a rotation by 120 degrees leaves the picture invariant, then the distances between them will all be equal (as this situation describes an equilateral triangle). The observed degeneracy boils down to the fact that the system has a D3 symmetry.

In quantum mechanics, calculations (at least formally) boil down to the diagonalization of Hermitian matrices - in particular, the Hamiltonian, or in the continuous case, the solution of linear differential equations. Again, observed degeneracies in the eigenspectrum are a consequence of discrete (or continuous) symmetries.