Physics 535 lecture notes: - 14 Oct 18th, 2007
No homework for next week.
Midterm Exam Thursday Oct 25th. Covering Griffiths 1-4, 6 Perkins 2, Lecture Notes
through Tuesday Oct 16.
All homework returned Tuesday. Solutions will be on the web site this weekend.
1) Features of quantum field theories
For the toy theory we developed we used scalar particles and presented a set of rules that
would allow the calculation of lifetimes and scattering cross sections. This simple theory
was actually quite successful at calculating the order of magnitude of decay lifetimes and
simple scattering problems. These rules were consistent the qualitative features of the
forces and interactions in terms of diagrammatic representation of interactions, coupling
constants, and modified coupling constants for heavy force carriers.
However, to more fully understand these quantum field theories we have to develop a
treatment from basic concepts. The results of this treatment should have the following
features.
a) The system should be described by a wave equation, which we will develop by
quantizing the total energy to derive the Hamiltonian.
b) The particles involved should be described by wave functions that are the
solutions or eigenstates of a wave equation.
c) These solutions should intrinsically include the ideas of spin, total angular
momentum, parity, charge conjugation and other conserved properties. These
properties will be described by operators and the wave functions will be
simultaneous eigenstates of these operators and the Hamiltonian. They will
commute with the Hamiltonian.
d) These solutions should intrinsically include the idea of antiparticles.
e) We will treat scattering and decay perturbatively. The default wave functions will
be wave functions in free space with no potential. The potential of the fields will
perturb the system, scattering the particles from the initial to the final state
<i|V|f>. <i|V|f> is the matrix element.
f) To get this treatments we will quantize the classical potential of the fields.
Electromagnetic in this case.
g) The quantized potential will give us the propagator. Higher order terms will
involve multiple propagators. The propagator will be identified with the force
carrying photon in the case of electromagnetic scattering. Note that the
propagator of the photon will also intrinsically carry spin information.
h) Particles, electrons, can also be thought of a probability fields and we will derive
a propagator for them as well.
i) With this formulation the interactions can be represented diagrammatically with a
set of rules to translate the diagram into the matrix element <i|V|f>. Higher order
diagrams are just multiple transitions which will happen at lower probability.
.
2) Review how the Schrödinger wave equation was developed.
Start with the expression for the total energy
E = |p|2/2m +V
Quantize by replacing E and p with the momentum and energy operators. These
operators when they operate on the eigenstate wave function solutions will return the
momentum and total energy of the wave function.
E  i  / t
p  -i 
  
then
 
2

i

  V
t
2m
or in free space

i
2



t
2m
with solutions, in 1D


(x,t)   (x) (t)  Neikxeit , p= k, E= ,
2
2m
k2   
p2
E
2m
3) Klein-Gordon equation



We want to do the same thing for a relativistic
theory starting with the total energy.
We will neglect any contribution from a potential since we hope to treat the effect of a
potential perturbatively later on.
E2 = p2 + m2
or more compactly
pp - m2 = 0
Introducing a relativistic notation where there will be a sum over the indices  = 1 to 4
and a negative sign for the lower indices in order to give the proper relativistic invariant
expression.
Quantizing
p  i  ,   =  /  t,  /  x … , 
results in the Klein-Gordon equation
  
   
- 2     - m2 = 0

or
  


2
m2
2





2
t 2
With a solution:
 (x,t)  Ne ip xiEt

Note that this is the one of the simplest solutions. A more general solution includes a
function of the four momentum u(p).
This solution also doesn’t include the features we want for relativistic particles such as
spin. Though it does give negative energy solutions, which will be identified with
antiparticles.
However, it will let us investigate the normalization of the relativistic wave equation.
One can define a continuity equation as the change in time of the probability density per
unit volume is equal to the current of probability out of the volume:

  j  0
t

Consider multiplying the Klein Gordon -i* and subtracting the complex conjugate of the
equation multiplied by -i
Then you get
  * 
 * 
*
*
i




 i(    ) 0
t  t
t 
the first term is probability density  and the second the probability current out of the
volume.

Calculating  = 2E|N|2
One needs the additional factor of 2E as well as the normal normalization factor as we
introduced in our toy model. Also the factor makes things Lorentz invariant. As space in
contracts the E goes up making the probability density greater.
4) Dirac Equation
Now consider a more general solution. The Dirac equation was developed while trying
to simplify the Klein-Gordon equation by factoring it into two first order differential
equations.
Consider the original energy momentum equation.
pp - m2 = 0, and wave equation -     - m2 = 0
To factorize this we need to solve the problem
  
(kpk + m)(lpl - m) = klpkpl +m(k - k) pk - m2 = 0
for the 8 constants k and l
For this to be true clearly kl = 1,-1 for k=l and 0 for some combination kl terms to get
back to pp, and k = l to get rid of the term linear in p.
A solution to this problem is for the l to be 4x4 matrices. If you anticipate that for the
simplest massive particle, the electron, you would like to have solutions for spin ½ and ½ electrons and for antiparticle positrons then this seems natural. The wave equation
 (x) solutions are four component vectors with four orthogonal eigenvector solutions to
the wave equation, which involves the four by four matrices.

I 0  i  0  i 
  
,
,    i
0 
0 I

0
0 1
,
1 0
 1  
0 i
1 0 
3
,   
,
i 0 
0 1
 2  
with these definitions then

(0)2 
= 1,
(i)2 = -1,



and  -  = 0 for   , or {,} = 2g,
(kpk + m)(lpl - m) = 0
Quantizing. The four vector solutions can come from either term giving 0 when
multiplied by the wave function.
i    - m = 0 or i    + m = 0
 
The solutions to these equations are similar to Schrödinger equation or the Klein-Gordon
equation solutions
(in four vector notation).
 
 (x)  ae
(i / )x  p 
u( p)
where u(p) is a four vector

Inserting this solution into the Dirac equation
(i / )x  p 
  p ae
(i / )x  p 
u( p)  mae
u( p)  0
or

(p - mc)u(p) = 0, or the positive m solution
p is the four momentum now and doesn’t need to be quantized again.
The solutions are:
uA 
u( p)1  N  ,
uB 
1
uA   ,
0
uB 
1
1
1  pz 
( p   ) 

,
Em
0 E  m px  ipy 
0
uA  
 ,
1
uB 
0
1
1 px  ipy 
( p   ) 

…
Em
1 E  m  pz 
N=(|E|+m)
uA 
u( p)1  N  ,
uB 

The other two solutions have negative energy and momentum (-p) and are identified
with positive energy antiparticles.



ui+ui= 2|E|, the normalization constant we used in the toy theory. Also for different
solutions ui+uj = 0, they are orthogonal.
Since the later two solutions have been identified with negative energy, or anitparticles
we typically rewrite (kpk + m)(lpl - m)u(p) = 0, Hu = 0 as H = kpk + 0m. This could
have been done from the beginning as solutions to linearizing the Klein-Gordon equation,
but would not have been as intuitive for deducing the identity of the second set of
eigenvectors.
I 0 
0 =  0  
,
0 I

0
 i   i

 i 

0 
If we pick p such that the particle is moving in z then these wave equations are
eigenvectors of

 z 0 
Sz  

2 0  z 

and can clearly be seen from the solution above. In general the wavefunctions are
eigenvectors of the helicity operator
  pˆ
0 
Sz  pˆ  

  pˆ 
2  0
and of J=L+S using quantized L= rxp

Sz  pˆ , S, L and H can all be written as four vectors. Therefore the easiest way to
demonstrate conserved properties is the check the commutation relationship [H,J] = 0

Next time the photon, neutrino and the electromagnetic field.