P780.02 Spring 2003 L4
Richard Kass
Conservation Laws
When something doesn’t happen there is usually a reason!
n
\ pe- or p
\ ne+ v or  
\ e
Read:
M&S Chapters 2, 4, and 5.1,
That something is a conservation law !
A conserved quantity is related to a symmetry in the Lagrangian that
describes the interaction. (“Noether’s Theorem”)
A symmetry is associated with a transformation that leaves the Lagrangian invariant.
time invariance leads to energy conservation
translation invariance leads to linear momentum conservation
rotational invariance leads to angular momentum conservation
Familiar Conserved Quantities
Strong EM Weak
Quantity
Comments
energy
sacred
linear momentum
ang. momentum
Y
Y
Y
Y
Y
Y
Y
Y
Y
sacred
sacred
P780.02 Spring 2003 L4
Other Conserved Quantities
Quantity
Strong EM Weak
Baryon number
Y
Y
Y
Lepton number(s)
Y
Y
Y
top
Y
Y
N
strangeness
Y
Y
N
charm
Y
Y
N
bottom
Y
Y
N
Isospin
Y
N
N
Charge conjugation (C)
Y
Y
N
Parity (P)
Y
Y
N
CP or Time (T)
Y
Y
y/n
CPT
Y
Y
Y
G Parity
Y
N
N
Richard Kass
Comments
no p+ 0
no -e- Nobel 88, 94, 02
discovered 1995
discovered 1947
discovered 1974, Nobel 1976
discovered 1977
proton = neutron (mumd)
particle  anti-particle
Nobel prize 1957
small No, Nobel prize 1980
sacred
works for pions only
Neutrino oscillations give first evidence of lepton # violation!
These experiments were designed to look for baryon # violation!!
Classic example of strangeness violation in a decay: p- (S=-1 S=0)
Very subtle example of CP violation:
expect: Kolong +0BUT
Kolong +-(1 part in 103)
P780.02 Spring 2003 L4
Some Reaction Examples
Richard Kass
Problem 2.1 (M&S page 43):
a) Consider the reaction vu+p++n
What force is involved here? Since neutrinos are involved, it must be WEAK interaction.
Is this reaction allowed or forbidden?
Consider quantities conserved by weak interaction: lepton #, baryon #, q, E, p, L, etc.
muon lepton number of vu=1, +=-1 (particle Vs. anti-particle)
Reaction not allowed!
b) Consider the reaction ve+pe-+++p
Must be weak interaction since neutrino is involved.
conserves all weak interaction quantities
Reaction is allowed
c) Consider the reaction e-+++ (anti-ve)
Must be weak interaction since neutrino is involved.
conserves electron lepton #, but not baryon # (1  0)
Reaction is not allowed
d) Consider the reaction K+-+0+ (anti-v)
Must be weak interaction since neutrino is involved.
conserves all weak interaction (e.g. muon lepton #) quantities
Reaction is allowed
P780.02 Spring 2003 L4
More Reaction Examples
Richard Kass
Let’s consider the following reactions to see if they are allowed:
K   su
a) K+p ++ b) K-p n
c) K0+K   su
First we should figure out which forces are involved in the reaction.
0
K
 sd
All three reactions involve only strongly interacting particles (no leptons)
so it is natural to consider the strong interaction first.
a)
b)
c)
Not possible via strong interaction since strangeness is violated (1-1)
Ok via strong interaction (e.g. strangeness –1-1)
Not possible via strong interaction since strangeness is violated (1 0)
If a reaction is possible through the strong force then it will happen that way!
Next, consider if reactions a) and c) could occur through the electromagnetic interaction.
Since there are no photons involved in this reaction (initial or final state) we can neglect EM.
Also, EM conserves strangeness.
Next, consider if reactions a) and c) could occur through the weak interaction.
Here we must distinguish between interactions (collisions) as in a) and decays as in c).
The probability of an interaction (e.g. a) involving only baryons and mesons occurring through
the weak interactions is so small that we neglect it.
Reaction c) is a decay. Many particles decay via the weak interaction through strangeness
changing decays, so this can (and does) occur via the weak interaction.
To summarize:
a) Not possible via weak interaction
c) OK via weak interaction
Don’t even bother to consider Gravity!
P780.02 Spring 2003 L4
Richard Kass
Conserved Quantities and Symmetries
Every conservation law corresponds to an invariance of the
Hamiltonian (or Lagrangian) of the system under some transformation.
We call these invariances symmetries.
There are 2 types of transformations: continuous and discontinuous
Continuous  give additive conservation laws
x  x+dx or    +d 
examples of conserved quantities:
electric charge
momentum
baryon #
Discontinuous  give multiplicative conservation laws
parity transformation: x, y, z  (-x), (-y), (-z)
charge conjugation (particleantiparticle): e-  e+
examples of conserved quantities:
parity (in strong and EM)
charge conjugation (in strong and EM)
parity and charge conjugation (strong, EM, almost always in weak)
P780.02 Spring 2003 L4
Richard Kass
Conserved Quantities and Symmetries
Example of classical mechanics and momentum conservation.
In general a system can be described by the following Hamiltonian:
H=H(pi,qi,t) with pi=momentum coordinate, qi=space coordinate, t=time
Consider the variation of H due to a translation qi only.
3 H
3 H
H
dH  
dqi  
dpi 
dt
t
i 1 qi
i 1 pi
For our example dpi=dt=0 so we have:
3 H
dH  
dqi
i 1 qi
Using Hamilton’s canonical equations:
qi 
H
pi
We can rewrite dH as:
p i  
H
qi
with p i 
dpi
dt
3
H
dH  
dqi   p i dqi
i 1 qi
i 1
3
If H is invariant under a translation (dq) then by definition we must have:
3 H
3
dH  
dqi   p i dqi  0

q
i 1
i 1
i
This can only be true if: 3
d 3
 p i  0 or
i 1
 pi  0
dt i 1
Thus each p component is constant in time and momentum is conserved.
P780.02 Spring 2003 L4
Richard Kass
Conserved Quantities and Quantum Mechanics
In quantum mechanics quantities whose operators commute with the
Hamiltonian are conserved.
Recall: the expectation value of an operator Q is:
Q   *Qdx with    ( x , t ) and Q  Q( x , x , t )
How does <Q> change with time?
d
d
*
Q

*
Q    Qdx  
Qdx   *
dx   *Q
dx
dt
dt
t
t
t
Recall Schrodinger’s equation:

*
i
 H and  i
 * H 
t
t
H+= H*T= hermitian conjugate of H
Substituting the Schrodinger equation into the time derivative of Q gives:
d
d
1
Q
1
Q   *Qdx    *H Qdx   *
dx   *QHdx
dt
dt
i
t
i
Since H is hermitian (H+= H) we can rewrite the above as:
d
Q 1
Q   * (
 [Q, H ])dx
dt
t i
So if Q/t=0 and [Q,H]=0 then <Q> is conserved.
P780.02 Spring 2003 L4
Richard Kass
Conservation of electric charge and gauge invariance
Conservation of electric charge: SQi=SQf
Evidence for conservation of electric charge: Consider reaction e-ve which
violates charge conservation but not lepton number or any other quantum number.
If the above transition occurs in nature then we should see x-rays from atomic
transitions. The absence of such x-rays leads to the limit:
te > 2x1022 years
There is a connection between charge conservation, gauge invariance,
and quantum field theory.
Recall Maxwell’s Equations are invariant under a gauge transformation:
vector potential : A  A  
scalar potential :    
1 
c t
A Lagrangian that is invariant under a transformation U=ei is said
to be gauge invariant.
There are two types of gauge transformations:
local: =(x,t)
global: =constant, independent of (x,t)
Maxwell’s EQs are
locally gauge invariant
Conservation of electric charge is the result of global gauge invariance
Photon is massless due to local gauge invariance
P780.02 Spring 2003 L4
Richard Kass
Gauge invariance, Group Theory, and Stuff
Consider a transformation (U) that acts on a wavefunction (y):
yUy
Let U be a continuous transformation then U is of the form:
U=ei
 is an operator.
If  is a hermitian operator (=*T) then U is a unitary transformation:
U=ei U+=(ei)*T= e-i*T = e-i  UU+= ei e-i =1
Note: U is not a hermitian operator since UU+
In the language of group theory  is said to be the generator of U
There are 4 properties that define a group:
1) closure: if A and B are members of the group then so is AB
2) identity: for all members of the set I exists such that IA=A
3) Inverse: the set must contain an inverse for every element in the set AA-1=I
4) Associativity: if A,B,C are members of the group then A(BC)=(AB)C
If  = (1, 2, 3,..) then the transformation is “Abelian” if:
U(1)U(2) = U(2)U(1) i.e. the operators commute
If the operators do not commute then the group is non-Abelian.
The transformation with only one  forms the unitary abelian group U(1)
The Pauli (spin) matrices generate the non-Abelian group SU(2)
0 1
0  i 
1 0 
 x  
  y  
  z  

1
0
i
0
0

1






S= “special”= unit determinant
U=unitary
n=dimension (e.g.2)
P780.02 Spring 2003 L4
Richard Kass
Global Gauge Invariance and Charge Conservation
The relativistic Lagrangian for a free electron is:
L  iy u  u y  myy
  c 1
This Lagrangian gives
the Dirac equation:
y is the electron field (a 4 component spinor)
i u  u y  mcy  0
m is the electrons mass
u= “gamma” matrices, four (u=0,1,2,3) 4x4 matrices that satisfy uv+ vu =2guv
u= (0, 1, 2, 3)= (/t, /x, /y, /z)
Let’s apply a global gauge transformation to L
y  ei y y   ye i
L  iy  u  u y  my y
L  iye i  u  u e i y  mye i e i y
L  iye i e i  u  u y  mye i e i y
since λ is a constant
L  iy  u  u y  m y y  L
By Noether’s Theorem there must be a conserved quantity associated
with this symmetry!
P780.02 Spring 2003 L4
The Dirac Equation on One Page
Richard Kass
The Dirac equation: i u  u   mc  0

1 
2 
3 
i[ 



]  mc  0
t
x
y
z
1 0  i  0
  
    i
0

1


 
0
0
1

0
i[
0

0
0
0
 0


1 0
0    0
0

0  1 0  t  0  1


0 0  1
1 0
0
0
0 1
 0 0 0  i
 0




1 0    0 0 i 0    0


0 0  x  0 i 0 0  y   1




0 0 

i
0
0
0


 0
0
1


0 0  1 
0
]

mc
0
0 0 0  z


1 0 0
0
0 1
i 

0 
0 0 0
0

 
1 0 0
0


0
0 1 0

 
0 0 1
0
A solution (one of 4, two with +E, two with -E) to the Dirac equation is:
1




0


cp z
2

 exp[ (i / )( Et  p  r ]  U exp[ (i / )( Et  p  r ]
  (| E |  mc ) / c
2
 E  Mc 
 c( p x  ip y ) 


 E  Mc 2 
The function U is a (two-component) SPINOR and satisfies the following equation:
(  u pu  mc)U  0
Spinors are most commonly used in physics to describe spin 1/2 objects. For example:
1
0
  represents spin up (  /2) while   represents spin down (  / 2)
0
1
Spinors also have the property that they change sign under a 3600 rotation!
P780.02 Spring 2003 L4
Richard Kass
Global Gauge Invariance and Charge Conservation
We need to find the quantity that is conserved by our symmetry.
In general if a Lagrangian density, L=L(, xu) with  a field, is invariant
under a transformation we have:
L  0 
L
L

 

 xu


xu
Result from
field theory
i
For our global gauge transformation we have:    e   (1  i )
 

       (1  i )    i and 

 i
xu xu
xu
Plugging this result into the equation above we get (after some algebra…)

L
L
  L

L
L  0 
 

 
(
 xu   xu





xu
xu



) 
xu


The first term is zero by the Euler-Lagrange equation.
The second term gives us a continuity equation.


 L







 xu 
E-L equation in 1D
L d L
 ( )0
x dt x
P780.02 Spring 2003 L4
Richard Kass
Global Gauge Invariance and Charge Conservation
The continuity equation is:





 J u
  L
  L


 
i 
0
xu     xu   

x

u
 xu 
 xu

L
with J  i



xu
u
Result from
quantum field
theory
Recall that in classical E&M the (charge/current) continuity equation has the form:


 J  0
t
(J0, J1, J2, J3) =(, Jx, Jy, Jz)
Also, recall that the Schrodinger equation give a conserved (probability) current:

y
2 2
*
i

 y  Vy    cy y and J  ic[y*y  (y* )y]
t
2m
If we use the Dirac Lagrangian in the above equation for L we find:
Conserved quantity
J u   y  u y
This is just the relativistic electromagnetic current density for an electron.
The electric charge is just the zeroth component of the 4-vector:
0 
Q   J dx
Therefore, if there are no current sources or sinks (J=0) charge is conserved as:
J u
0
xu
J 0

0
t
P780.02 Spring 2003 L4
Richard Kass
Local Gauge Invariance and Physics
Some consequences of local gauge invariance:
a) For QED local gauge invariance implies that the photon is massless.
b) In theories with local gauge invariance a conserved quantum number implies
a long range field.
e.g. electric and magnetic field
However, there are other quantum numbers that are similar to electric charge
(e.g. lepton number, baryon number) that don’t seem to have a long range force
associated with them!
Perhaps these are not exact symmetries!
 evidence for neutrino oscillation implies lepton number violation.
c) Theories with local gauge invariance can be renormalizable, i.e. can use
perturbation theory to calculate decay rates, cross sections, etc.
Strong, Weak and EM theories are described by local gauge theories.
U(1) local gauge invariance first discussed by Weyl in 1919
SU(2) local gauge invariance discussed by Yang&Mills in 1954 (electro-weak)
ye it(x,t)y t is represented by the 2x2 Pauli matrices (non-Abelian)
SU(3) local gauge invariance used to describe strong interaction (QCD) in 1970’s
ye it(x,t)y t is represented by the 3x3 matrices of SU(3) (non-Abelian)
P780.02 Spring 2003 L4
Richard Kass
Local Gauge Invariance and QED
Consider the case of local gauge invariance, =(x,t) with transformation:
y  e

i ( x ,t )
y y   ye

 i ( x ,t )
The relativistic Lagrangian for a free electron is NOT invariant under this transformation.
L  iy u  u y  myy
  c 1
The derivative in the Lagrangian introduces an extra term:
y  e

i ( x ,t )
 ye
u

i ( x ,t )

y [i( x , t )]
u
We can MAKE a Lagrangian that is locally gauge invariant by adding
an extra piece to the free electron Lagrangian that will cancel the
derivative term.
We need to add a vector field Au which transforms under a gauge transformation as:
AuAu+u(x,t) with (x,t)=-q(x,t) (for electron q=-|e|)
The new, locally gauge invariant Lagrangian is:
L  iy u  u y  myy 
1 uv
F Fuv  q y  u yAu
16
P780.02 Spring 2003 L4
Richard Kass
The Locally Gauge Invariance QED Lagrangian
L  iy u  u y  myy 
1 uv
F Fuv  q y  u yAu
16
Several important things to note about the above Lagrangian:
1) Au is the field associated with the photon.
2) The mass of the photon must be zero or else there would be a term in the Lagrangian
of the form:
mAuAu
However, AuAu is not gauge invariant!
3) Fuv=uAv-vAu and represents the kinetic energy term of the photon.
Au

4) The photon and electron interact via the last term in the Lagrangian.
This is sometimes called a current interaction since:
eu
u
q y u yA  J u A
Ju
eIn order to do QED calculations we apply perturbation theory
(via Feynman diagrams) to JuAu term.
5) The symmetry group involved here is unitary and has one parameter  U(1)