1. HW 2:
1. What are the main subdisciplines of nuclear science? Which one(s) do you find most intriguing? Why?
2. Based on answer the following questions:
What is the neutron number of 160Yb? 
What are practical applications of antimatter?
What is the current temperature of the Universe (in Fahrenheit)?
Explain your answers.

2. Symmetry: the secret of nature
Wikipedia: A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.
“The role of symmetry in Fundamental physics”: David Gross
“Einstein’s great advance in 1905 was to put symmetry first, to regard the symmetry principle as the primary feature of nature that constrains the allowable dynamical laws. Thus the transformation properties of the electromagnetic field were not to be derived from Maxwell’s equations, as Lorentz did, but rather were consequences of relativistic invariance, and indeed largely dictate the form of Maxwell’s equations.”
“With the development of quantum mechanics in the 1920s symmetry principles came to play an even more fundamental role. In the latter half of the 20th century symmetry has been the most dominant concept in the exploration and formulation of the fundamental laws of physics. Today it serves as a guiding principle in the search for further unification and progress.”
2004 Nobel for asymptoptic freedom

3. The role of symmetries
Provide conservation laws and quantum numbers. (In 1918 Emmy Noether proved her famous theorem relating symmetry and conservation laws.)
Summarize the regularities of the laws that are independent of the specific dynamics/boundary conditions.
The theory of representations of continuous and discrete groups plays an important role deducing the consequences of symmetry in quantum mechanics.
Any quantum state can be written as a sum of states transforming according to irreducible representations of the symmetry group. These special states can be used to classify all the states of a system possessing symmetries and play a fundamental role in the analysis of such systems through quantum numbers and associated selection rules.
Much of the texture of the world is due to mechanisms of symmetry breaking.
Explicit symmetry breaking (the symmetry violation as a small correction and approximate conservation laws are present)
Spontaneous symmetry breaking (the laws of physics are symmetric but the state of the system is not). Example: crystals (translational invariance), magnetism (rotational invariance), supeconductivity (particle number). Thus for every broken global symmetry there exist fluctuations with very low energy. These appear as massless particles (Goldstone bosons).
2004 Nobel for asymptoptic freedom

4. 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)
Local symmetries (important for gauge theories)
Different transformations at different points of spacetime

5. 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 ?
Representation of the group (represents group elements as matrices so that the group operation can be represented by matrix multiplication)
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 irreducible

6. 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. Transformations generated by physical operators. The set of commutators between generators is closed. Its Casimir operator commuters with all the generators.
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)

7. Vector spaces: 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

8. Translational Invariance
Unitary transformations: U+=U-1
Under unitary transformation U, an operator A transforms as A’=UAU-1
Total momentum (nucleons, mesons, photons, leptons, etc.)
Transformation generator
For [X,Y] central, i.e., commuting with both X and Y:
Time displacement

9. Rotations in 3D (space isotropy)
a set of three angles (a vector) representing rotations along x,y,z
(+ cycl.)
Total angular momentum
Transformation generator
SO(3) or SU(2) group!
Rotational states of the system labeled by the total angular momentum quantum numbers JM
see examples of spectra at

10. 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!

11. 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!

12. Space Reflection (Parity)
HW: Which quantities/operators are invariant with respect to space reflection:
(a) Kinetic energy; (b) Projection of particle’s spin on its coordinate vector; (c) Mass
Explain how you figured it out!
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
(*)
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=+

13. Experimental test of parity violation
(Lee and Yang, 1956; Wu et al., 1957)
T1/2=5.2713(8) y, produced in nuclear reactors
Parity violation in a beta decay of polarized 60Co (Jp=5+): the emission of beta particles is greater in the direction opposite to that of the nuclear spin.
pseudoscalar

14. 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

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

17. Intrinsic parity Parity is a multiplicative quantum number
Quarks have intrinsic parity +1
The lighter baryons (qqq) have positive intrinsic parity. What about light antibaryons?
What about mesons?
In 1954, Chinowsky and Steinberger demonstrated that the pion has negative parity (is a pseudoscalar particle)

18. C - interchanges particles & antiparticles
Charge conjugation
C - interchanges particles & antiparticles
It reverses all the internal quantum numbers such as charge, lepton number, baryon number, and strangeness. It does not affect mass, energy, momentum or spin.
What are the eigenstates of charge conjugation?
C-parity or charge parity
⇒ photon, neutral pion…
What about positronium, neutrino?
Maxwell equations are invariant under C
C reverses the electric field
Photon has charge parity hC=-1
Is the following decay possible?

19. CP - violated in K0 decay (1964 Cronin & Fitch experiment)
Other Symmetries
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!

20. Since CP is violated, T has to be violated as well!
Atomic/neutron electric dipole moment: The violation of CP-symmetry is responsible for the fact that the Universe is dominated by matter over anti-matter
Neutron EDM searches
Closely spaced parity doublet gives rise to enhanced electric dipole moment
Large intrinsic Schiff moment
199Hg (Seattle, 1980’s – present)
225Ra (ANL, KVI)
Parker et al. 2015, d<5x10-22 e cm
223Rn at TRIUMF (E929)
FRIB
Widest search for octupole deformations
238U beam, beam dump recovery: 225Ra: 6x109/s
232Th beam: 225Ra: 5x1010/s, 223Rn: 1x109/s
1012/s with ISOL target FRIB upgrade
An international team of physicists has developed a shielding that dampens low frequency magnetic fields more than a million-fold. Using this mechanism, they have created a space that boasts the weakest magnetic field of our solar system.

21. HW: Using information from PDG. lbl. gov and nndc. bnl
HW: Using information from PDG.lbl.gov and nndc.bnl.gov determine whether the following decays/reactions are allowed by fundamental symmetries:
p0⟶m+e-
p +p ⟶g
Gamma decay of excited state of 16Ca at 6049 keV
Decay of meson h⟶gg
Decay of meson h⟶p0 +p0

22. 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

23. Still Waiting For Electron Decay…
total isospin
Tz component conserved!
(charge conservation)
Still Waiting For Electron Decay…
The current bound from Borexino is t>6.6×1028 yr: Phys. Rev. Lett. 115, (2015)
“The conservation of electric charge, suggested since the 19th century, is fundamental to the physics of the standard model as a direct consequence of Maxwell’s equations and the unbroken U(1) gauge symmetry of the electroweak theory. Despite the present undisputed validity of this law, experimental tests of charge conservation remain a way to search for physics beyond the standard model, and they deserve to be investigated with the highest possible sensitivity. An experimental search for the hypothetical charge nonconserving decay of the electron, which is the lightest known charged particle, into a neutrino and a photon is reported in this Letter. No presently viable theory predicts such a decay, and a large charge violation is excluded by the absence of macroscopic effects in matter.”

24. total isospin
Tz component conserved!
(charge conservation)
charge independence
T is conserved!
Using the NNDC website
find two examples of spectra of mirror nuclei. How good is isospin symmetry in those cases?

25. A principle, supported by experiment
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

26. 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)

27. 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.
XC: 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 (i.e., is symmetric).