Accidental symmetry


In field theory

In physics, particularly in renormalization theory, an accidental symmetry is a symmetry which is present in a renormalizable theory only because the terms which break it have too high a dimension to appear in the Lagrangian.
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 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. The observed degeneracy boils down to the fact that the system has a D3 symmetry.
In quantum mechanics, calculations 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 symmetries. In the latter case, Noether's theorem also guarantees a conserved current. "Accidental" symmetry is the name given to observed degeneracies that are apparently not a consequence of symmetry.
The term is misleading as often the observed degeneracy is not accidental at all, and is a consequence of a 'hidden' symmetry which is not immediately obvious from the Hamiltonian in a given basis. The non relativistic Hydrogen atoms a good example of this - by construction, its Hamiltonian is invariant under the full rotation group in 3 dimensions, SO. A less obvious feature is that the Hamiltonian is also invariant under SO, the extension of SO to 4D, of which SO is a subgroup. This gives rise to the 'accidental' degeneracy observed in the Hydrogenic eigenspectrum.
As a more palatable example, consider the Hermitian matrix:
Although there is already some suggestive relationships between the matrix elements, it is not clear what the symmetry of this matrix is at first glance. However, it is easy to demonstrate that by a unitary transformation, this matrix is equivalent to:
Which can be verified directly by numerically diagonalising the sub-matrix formed by removing the first row and column. Rotating the basis defining this sub matrix using the resulting unitary brings the original matrix into the originally stated form. This matrix has a P4 permutation symmetry, which in this basis is much easier to see, and could constitute a 'hidden' symmetry. In this case, there are no degeneracies in the eigenspectrum. The technical reason for this is that each eigenstate transforms with respect to a different irreducible representation of P4. If one encountered a case where some group of eigenstates correspond to the same irreducible representation of the 'hidden' symmetry group, a degeneracy would be observed.
Although for this simple 4x4 matrix the symmetry could have been guessed, if the matrix was larger, it would have been more difficult to spot.