Symmetry Results
Symmetry result cards summarize the theorem-level statements that connect transformations with conserved quantities, selection rules, degeneracies, and particle statistics.
| Result | Main role | Key assumption to check |
|---|---|---|
| Noether Theorem in Quantum Mechanics | Continuous symmetry generators are conserved when they commute with the Hamiltonian | unitary symmetry and no explicit time dependence |
| Wigner–Eckart Theorem | Rotational symmetry factors matrix elements into angular and reduced parts | spherical tensor convention |
| Kramers Degeneracy | Time-reversal symmetry with gives paired states | antiunitary symmetry and odd time-reversal square |
| Spin-Statistics Preview | Relativistic QFT links spin to bosonic or fermionic statistics | locality, relativistic invariance, positive energy |
Shared Caveats
Section titled “Shared Caveats”- A symmetry result is only as precise as the transformation law being used.
- Projective phases, antiunitarity, domains, and degeneracies can change the naive statement.
- Nonrelativistic quantum mechanics uses the symmetrization postulate; the full spin-statistics theorem belongs to relativistic quantum field theory.
References
Section titled “References”- E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
- J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
- A. Messiah, Quantum Mechanics, Dover, 1999.
- R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press, 2000.