Lecture 9: Feb 21st 2012
Reading: Griffiths chapter 4
Homework due Thursday Feb 23rd: Griffiths: 4.8, 4.10, 4.11,
4.29(delayed till next week)
1) Symmetries
Many symmetries we observe in nature are associated with conservation laws.
Some Examples:
Time invariance: Conservation of energy
Translation invariance: Momentum
Rotational Invariance: Angular Momentum
Gauge Transformation: Charge
Time invariance and energy conservation is interwoven in the way we treat quantum
mechanics using a Hamiltonian or more typically a Lagrangian. The Schrödinger
equation contains this explicitly (with a time independent potential).
d 2 Y(x,t)
dY(x,t)
+ U(x)Y(x,t) = i
2
2m dx
dt
2
HY(x,t) = i
dY(x,t)
= EY(x,t)
dt
Solutions of the Schrödinger equation
Y(x,t) = y(x)e iwt
Has a constant as a function of time, or conserved property  that we identified with the
energy: E = w
This example also illustrates another point. For each conserved property you should be
able to write down an operator, in this case H, the Hamiltonian. The wave equation
solutions will be eigenstates of this operator with an eigenvalue representing the value of
the conserved property for that eigenstate.
Next consider an example of gauge transformation invariance.
Gauge transformation:   ei and anti-particle y , e-i y
If gauge invariance is a symmetry that leaves interactions in nature unchanged then it
should leave both probabilities and the Lagrangian unchanged. Probabilities are
calculated using the wave functions as y y , or yy for an antiparticle (think bra-ket like
notation), are clearly unchanged.
Consider a relativistic Lagrangian chosen so that the Euler Lagrange equations reproduce
the Klein Gordon equation, which is the simplest quantinization of the relativistic energy
and momentum relationship.
( )
L= ¶my (¶ my ) + m 2yy
This is clearly left unchanged by the gauge transformation. We will find that the
conserved property associated with the gauge transformation is the electric charge and the
details of the force can be studied by studying the properties of the gauge transformation.
Each of the forces has it’s own group of gauge transformation and conserved charges.
Later we will demonstrate conservation of charge by considering the effect of the gauge
transformation on the Euler Lagrange equations.
2) Symmetries, group theory
Often these kinds of symmetries are categorized by group theory. The gauge
transformation is a unitary transformation in one dimension, U(1). The transformation
can also be represented as a matrix, just a constant in one dimension. There are many
group transformations we encounter in particle physics
U(n) Unitary transformation in n dimensions, nxn matrix, UU+=I, U+ ajoint or transpose
conjugate.
SU(n) Special unitary transformation in n dimensions, nxn matrix with determinant 1.
O(n) Orthogonal transformation in n dimensions, nxn matrix. Os transpose and inverse
are the same.
SO(n) Special orthogonal transformation in n dimensions, nxn real orthogonal matrix
with determinant 1.
Examples of these in nature
U(1) Gauge symmetry and conservation of charge
SU(2) spin and isospin symmetry and conservation
SO(3) rotational symmetry and conservation of angular momentum.
The operations involving SO(3) and SU(2) matrices are the same except for a minus sign.
Thus the same in probabilities, which is why we combine them later.
3) Angular momentum and spin
The quantum numbers for a system with spherical symmetry are the radial quantum
number which is associated with the energy of the state, n, total angular momentum, l,
and the projection of the angular momentum on the z axis, ml. Note that total angular
operator L2 yields an angular momentum of l(l+1)hbar2. The Lz quantum number gives
projection mlhbar
These quantum numbers run from l=0,1…n-1 and ml=-l,…-1,0,1…l. The last quantum
number can be thought of as running from the angular momentum pointing along the z
axis to pointing along the negative z axis.
The spin quantum number can be integer or half integer, integer for bosons and half
integer for fermions. Similarly ms runs from =-s,-s+1,…,s-1,s, skipping 0 for fermions.
Usually spin quantum numbers for individual particles are only s=0,1/2,1. Though we
believe that the graviton is probably spin 2.
The spin states, quantum numbers and operators can be represented as vectors and
matrices.
Lets write the eigenvectors of spin up and down particles as orthogonal vectors. A real
particle can be in some arbitrary combination of these until we measure it at which point
it will have some probability to be in either state.
æ1ö æ0ö
c ± = ç ÷,ç ÷
è0ø è1ø
æa ö
æ1ö æ0ö
ç ÷ = aç ÷ + b ç ÷
èb ø
è0ø è1ø
We can then write two operators that the  vectors will be eigenvectors of and have
specific eigenvalues.
æ 1 0 ö
Sz = ç
÷
2 è 0 -1 ø
S 2 = Sx2 + Sy2 + Sz2 =
3
4
2
æ 1 0 ö
æ 0 1 ö
æ 0 -i ö
ç
÷, Sx = ç
÷, Sy = ç
÷
2è 1 0 ø
2è i 0 ø
è 0 1 ø
with eigenvalues Sz : ±mz = ±
1
, S2 : s(s +1)
2
2
1 1
= ( +1)
2 2
2
=
3 2
,
4
Note: even the arbitrary combination is still in an eigenstate of S2.
Lets expand on the connection between spin SU(2) and and angular momentum SO(3).
Consider rotations. Rotations can be represented by 3x3 matrices which are applied to
three vectors. Consider rotations around the Y axis. These rotations will leave total
angular momentum and angular momentum projection unaffected.
An arbitrary rotation is:
æ cosa 0 -sin a
ç
Ry = ç 0
1
0
ç sin a 0 cosa
è
ö
÷
÷
÷
ø
Consider the effect of two rotations of 180 degree on an angular momentum vector along
the z axis
æ -1 0 0 öæ 0 ö æ 0 ö æ -1 0 0 öæ 0 ö æ 0 ö
ç
֍
÷ ç
֍
֍
÷ ç
÷
ç 0 0 0 ÷ç 0 ÷ = ç 0 ÷, ç 0 0 0 ÷ç 0 ÷ = ç 0 ÷
ç 0 0 -1 ÷ç 1 ÷ ç -1 ÷ ç 0 0 -1 ÷ç -1 ÷ ç 1 ÷
è
øè
ø è
øè
øè
ø è
ø
Similarly for SU(2): Sy is projection operator is comes from a 180 rotation of the
æ
æ
-iq /2 ö
-ip /2 ö
æ
ö
÷ Sy = ç 0 e
÷ = ç 0 -i ÷
arbitrary y rotation matrix Ry = çç 0 e
ç eip /2
iq /2
0 ÷ø
0 ÷ø 2 è i 0 ø
è e
è
The Sz operator will rotate a z projection eigenvector ms =1/2 to ms =-1/2 and two
applications will rotate you to the original.
æ 0 -i ö æ 0 -i öæ 1 ö æ 0 ö æ 0 -i öæ 0 ö æ 1 ö
Sy = ç
÷, ç
֍
÷=ç
÷, ç
֍
÷=ç
÷
2 è i 0 ø è i 0 øè 0 ø è i ø è i 0 ø è i ø è 0 ø
These operations have the same effect or the same symmetry under rotation. Thus they
can be thought of having a common source and can be combined.
Combination angular momentum is an operation that can be very important for various
interactions. For instance, if you bind two quarks together into a meson you start with
two particles with spin angular momentums and combine them into one particle with a
total spin/angular momentum. The initial can be in eigenstates of the two separate spins
while the final state can be in an eigenstate of the combined spin. To find out what final
states we produce with what probabilities we need to understand how to express one set
of eigenstates as a linear combination of the other eigenstates. This sort of problem may
happen involving just spins or possibly orbital angular momentums as well. For instance
if a photon impacts the 2 quark system and is absorbed it’s spin angular momentum of 1
would be added to the system possibly bumping it up to a higher orbital angular
momentum state just as in an atom. In the case of particles composed of quarks these
spin and orbital angular momentum states are so energetic that we refer to them as
different particles.
To add angular momentums or spins:
1) The z projection components simply add m=m1+m2. The projection of the
spin/angular momentum along an axis is the directly “measureable” quantum
property. The two particles can be in a definite state of m and we can measure
this property before and after and we will observe that it is conserved.
2) The j( or l/s) components may be aligned j=j1+j2, opposite, j=|j1-j2| or anywhere
in between, j=|j1-j2|,|j1-j2|+1…j1+j2. There can be several values of j that can
accommodate the value m for the z production of the angular momentum
The coefficients that determine the relative probability of each allowed state are called
the Clebsch-Gordan coefficients and can be looked up in the particle databook. They can
also be derived by starting with the top state, which is trivial, using step down operators.
Take the case of combining two spin ½ quarks.
|s,m>=|s1,m1>|s2,m2>
|1,1>=|1/2,1/2>|1/2,1/2>
|1,0>=(1/2)(|1/2,-1/2>|1/2,1/2>+|1/2,-1/2>|1/2,1/2>)
|1,-1>=|1/2,-1/2>|1/2,-1/2>
vector mesons – symmetric wave function – example s=1 0 m = 776 MeV/c2
|0,0>=(1/2)(|1/2,-1/2>|1/2,1/2>-|1/2,-1/2>|1/2,1/2>)
pseudoscalar mesons – antisymmetric wavefunction ,example s=0 0 m = 136 MeV/c2
Note, just as with the particle in a box the symmetric state is usually higher energy. Here,
we can measure indirectly, though the mass, which total spin angular momentum state the
particle is in.
Quark states can also have orbital angular momentum. Example s=0 l=1 B m = 1229
MeV/c2
Consequences
For the 0 since the two quarks are in a ms=0 state they can decays to two spin 1 photons,
but the spins have to be oppositely aligned. This rate dominates.
Similarly the rho can decay into two pions if in the ms=0 state. This is a strong decay so
it dominates. If not in a ms=0(spins not aligned) the decay can still happen. However,
there is going to be an angular distribution involved. Decay is favored along the axis
where the ms=0. You end up with pions with relative l=1,ml=1 angular momentum to
conserve total angular momentum.
4) Isospin
Isospin is a spin like quantity that was observed to be conserved in interactions involving
hadrons. Isospin actually has to do with quark flavor and symmetries between how
different quarks interact via the strong force. The essential symmetry is that up and down
quarks are nearly the same mass and should interact via the strong force the same way.
This symmetry leads to a conserved quantum number, the Isospin. This relationship can
be extended to the strange quark, but not perfectly because though the interactions
between the strange quark and the strong force should not be any different the strange
quark is considerably more massive so that not all interactions, example pair production,
are the same.
Define I and I3
I is going to lead to 2I+1 states delineated by I3=Q-1/2(A+S) which goes from –I to I in
integer steps. Q = charge, A = baryon number and S=strangeness
For the light quarks this comes from assigning the quarks isospin, I3, quantum numbers u,
½ and d -½ and the inverse for the antiquarks.
u: |½ ,½>, d: |½ ,-½>, u : |½ ,-½>, d : |½ ,½>,
For the pions |I,I3> I = ½ + ½ = 1, I3 = -1, 0, 1
Also considering how the pions are added up
Pi+: u d :
|1,1> = |½ ,½>|½ ,½>
Pi0: u u or d d :|1,0> = (1/2)(|½ ,½>|½ ,-½> + |½ ,½>|½ ,-½>)
Pi-= d u :
|1,-1> = |½ ,-½>|½ ,-½>
There should be a singlet as well.  |I,I3> I = ½ - ½ = 1, I3 = 0
: the system breaks down somewhat at this point since we should have been including
the strange quark. The  is part strange and the states are a bit complex since isospin
with the strange isn’t perfect. See chapter 5 if you are interested