1. Symmetries of the nuclear Hamiltonian (exact or almost exact)
Translational invariance
Galilean invariance (or Lorentz invariance)
Rotational invariance
Time reversal
Parity (space reflection)
Charge independence and isobaric symmetry
Baryon and lepton number symmetry
Permutation between the two nucleons (imposed by the exclusion principle)
Continuous transformations
(appear to be universally valid)
Dynamical symmetries
apply in certain cases, provide useful coupling schemes
Chiral symmetry (broken by a quark condensate; valid for massless quarks)
SU(4) symmetry (Wigner supermultiplet)
SU(2) symmetry (seniority)
SU(3) symmetry (Elliott model)

2. Symmetries in quantum mechanics
(see "Symmetry in Physics", J.P. Elliott and P.G. Dawber, The Macmillan Press, London)
Wave equation for the Hamiltonian operator:
Group of transformations G whose elements G commute with H:
We say that H is invariant under G or totally symmetric with respect to the elements of G
What are the properties of ?
Reminder: representation of the group (the basis, a set of operators acting on the basis functions and corresponding to a symmetry group)
dimension of the representation
matrix representation
of the group
basis
If all matrices D can be put into a block-diagonal form, the representation is reducible

3. Hence is also an eigenfunction of H with eigenvalue Ek .
If Ek is nondegenerate
If Ek is n-fold degenerate, there are n partner functions
one-dimensional irrep of G
We can thus label the wave function fully as
n-dimensional irrep of G , except for accidental degeneracy
We can thus label the wave function fully as

4. Classification of eigenstates with respect to symmetry group
Wave functions for different energy levels Ek transform as basis functions of irreducible representations of the group G
If we know the properties of G, we can classify the wave functions
Further, the same group-theoretical structure will tell us about the spectroscopy of the system
Point groups: geometric symmetries that keep at least one point fixed
Lie groups: continuous transformation groups
D1: (dihedral) reflection group (2 element group: identity and single reflection)
Cn: cyclic n-fold rotation (C1 is a trivial group containing identity operation)
SO(3): group of rotations in 3D (isomorphic with SU(2))
Poincare group (Translations, Lorentz transformations)
What are generators and irreducible representations of SO(3)?

5. Scalars, Vectors, Tensors…
Orthogonal transformations U
preserve lengths of vectors and angles between them
map orthonormal bases to orthonormal bases
Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations).
det(U)=1 – usual (stiff) rotations (scalars, vectors,…)
det(U)=-1 – improper rotations (pseudo-scalars, axial vectors, …)
Improper rotation operation S4 in CH4

6. Translational Invariance
Total momentum (nucleons, mesons, photons, leptons, etc.)
Transformation generator
Time displacement

7. Rotations in 3D (space isotropy)
(+ cycl.)
Total angular momentum
Transformation generator
SU(2) group!
Rotational states of the system labeled by the total angular momentum quantum numbers JM
see examples at

8. Galilean (Lorentz) Invariance
In atomic nucleus v2/c2<0.1, i.e., kinematics is nonrelativistic
Such a separation can be done
for Galilean-invariant
interactions
Depends only on relative coordinates and velocities!

9. Relativistic generalization
no new conservation laws and quantum numbers!
Relativistic generalization
Center-of-mass coordinate cannot be introduced in a relativistically covariant manner
All powers of c.m. momentum are present
Unitary transformation contains gradient terms and spin-dependent pieces!

10. Space Reflection (Parity)
Parity is violated by weak interaction. The simplest of a parity-violated interaction requires a pseudoscalar field. If one assumes rotational invariance, the field looks like
(*)
Why not take instead?
The interaction (*) produces a very small parity mixing
J=
Parity-violating matrix elements are of the order of 0.1 eV. This leads to the mixing amplitude of the order of 10-7
J=+

11. Experimental test of parity violation
(Lee and Yang, 1956; Wu et al., 1957)
Parity violation in a beta decay of polarized 60Co: the emission of beta particles is greater in the direction opposite to that of the nuclear spin.
pseudoscalar

12. Time Reversal
T cannot be represented by an unitary operator. Unitary operations preserve algebraic relations between operators, while T changes the sign of commutation relations.
In order to save the commutation relations, on has to introduce:
antiunitary
takes complex conjugate of all c numbers
unitary

13. Time Reversal symmetry and nuclear reactions
normal and inverse
kinematics!

14. C - interchanges particles & antiparticles (charge conjugation)
Other Symmetries
C - interchanges particles & antiparticles (charge conjugation)
CP - violated in K0 decay (1964 Cronin & Fitch experiment)
CPT - follows from relativistic invariance
The CPT theorem appeared for the first time in the work of Julian Schwinger in 1951 to prove the connection between spin and statistics. In 1954, Lüders and Pauli derived more explicit proofs. At about the same time, and independently, this theorem was also proved by John Stewart Bell. These proofs are based on the principle of Lorentz invariance and the principle of locality in the interaction of quantum fields.
Since CP is violated, T has to be violated as well!

15. Isospin Symmetry Introduced 1932 by Heisenberg wave functions
Protons and neutrons have almost identical mass: Dm/m = 1.4x10-3
Low energy np scattering and pp scattering below E=5 MeV, after correcting for Coulomb effects, is equal within a few percent in the 1S scattering channel.
Energy spectra of “mirror” nuclei, (N,Z) and (Z,N), are almost identical.
up and down quarks are very similar in mass, and have the same strong interactions. Particles made of the same numbers of up and down quarks have similar masses and are grouped together.
wave functions
Pauli isospin matrices
SU(2) commutations
Using spin and isospin algebra, and Pauli principle, find two-nucleon wave functions.
Assume that the spatial part of the wave functions corresponds to an s-wave.

16. total isospin
Tz component conserved!
(charge conservation)
charge independence
T is conserved!
The concept of isospin symmetry can be broadened to an even larger symmetry group, now called flavor symmetry. Once the kaons and their property of strangeness became better understood, it started to become clear that these, too, seemed to be a part of an enlarged symmetry that contained isospin as a subgroup. The larger symmetry was named the Eightfold Way by Gell-Mann, and was recognized to correspond to the adjoint representation of SU(3). While isospin symmetry is broken slightly, SU(3) symmetry is badly broken, due to the much higher mass of the strange quark compared to the up and down.

17. Group of permutations Exchange operators - exchanges particles i and j
is hermitian and unitary:
eigenvalues of are
(identical particles cannot be distinguished)
For identical particles, measurements performed on quantum
states and have to yield identical results
A principle, supported by experiment

18. is a basis of one-dimensional representation of the permutation group
This principle implies that all many-body wave functions are eigenstates of
is a basis of one-dimensional representation of the permutation group
There are only two one-dimensional representations of the permutation group:
for all i,j - fully symmetric representation
for all i,j - fully antisymmetric representation
Consequently, systems of identical particles form two separate classes:
bosons (integer spins)
fermions (half-integer spins)
For spin-statistics theorem, see W. Pauli, Phys. Rev. 58, (1940)