Download Notes

Sec. 1 Classical Free Field Theory
1. The theory to describe particle physics needs to combine the principles of both Quantum
Mechanics and Special Relativity. The usual framework of QM by Schrodinger Wave Equation
is non-relativistic and hence doesn’t work in this case. In this formalism, the space position is
treated as an operator x̂ , while time is treated as a parameter t, ie a number. It doesn’t allow a
Lorentz symmetry that mixes time and space.
2. However, Special Relativity was born from the classical theory of Electromagnetic Field. So a
field theory is much easier to be made Lorentz invariant. So let’s speculate that the theory to
describe fundamental particles is a field theory but quantized. In the following, first, we work

out the simplest classical field theory-scalar field  ( x , t ) in the next three pages from Peskin’s
book on quantum field theory.

3. A scalar field  ( x , t ) is a number function of space and time. Note that space and time are
treated equally as parameters. Its dynamics can be determined from its action, or Lagrangian.
The beginning of section 2-2 on P.2 describe how to derive its equation of motion, also called
Euler Equation (2.3) by the least action principle from the Lagrangian L. This formalism, just
like that of particles, can be also rewritten in terms of Hamiltonian. It is done on P.3 following
(2.3)
4. For free field, its wave equation is linear so that superposition principle applies. For linear wave

equation, the Lagrangian must be quadratic in the field  ( x , t ) . The Lagrangian must also be a
Lorentz scalar so that the theory is Lorentz invariant. Add one more requirement that the
Lagrangian contains at most two space-time derivatives. We can conclude the most general
Lagrangian is written as in Eq. (2.6) on P.3. Derive its Euler Eq. and we get (2.7). It’s called
Klein-Gorden Equation.
5. To solve this equation classically, we employ the plane wave solutions as e
 ik  x 
. The general
solution is linear combinations of the plane wave solutions, as at the bottom of P.7. From the
1
derivation on P.7, you can find the dispersion relation between angular wavenumber k and
angular frequency ω or equivalently the relation between momentum and energy. Look it’s
exactly the relation for a relativistic particle! This indicate a close relation between particle and
field. The strange thing is that there is a positive solution energy plus a negative energy solution.
Why it’s so will become clear when the field is quantized.
Note that the integration is Lorenz Invariant. The Lorenz invariance of the Lagrangian
density will guarantee the Lorenz invariance of Action and hence EOM.
2
3
4
5
6
To solve the KG Equation, we first expand the field in terms of a Fourier Series:
The time dependence can be determined by plugging the series into KG Eq.
2
Every Fourier Component behaves like a SHO with ω:  p  E p  p  m2 . KG Field
is just a collection of SHO’s. Each SHO is characterized by its k or p “momentum”.
The frequency ω or “energy” of the SHO is just that of a relativistic particle with
mass m.
A general solution is a linear superposition of all plane waves.


 
 
d3p
d3p
iEt  ip x
 e


c p  eiEtipx 
a

p
3
3

2 
2 

3
 
 
d p
d 3 p'
iEt  ip x
 e


c p '  eiEtip'x 

a

p
3
3

2 
2 
 ( x)  


Here we changed the variable: p'   p . But in the integration, the variable is just a
dummy and hence can be renamed as p:
 ( x)  


d3p
a

p
e

d3p
a

p
 e ip x  b p  e ip x
2 3
2 
3
 
iEt  ip x
 

d 3 p'
2 3
 
b


p'
 
 e iEtip ' x


The solution of KG Equation can be written as:

 
d3p
a p  eipx  bp  eipx 
 x   
3
2 
2

with E p  p  m2 . If the field is real,    . Hence bp  a p . The real solution of
KG Equation can be written as:

 
d3p
 x   
a p  e ipx  a p  e ipx
3
2 

7

Sec. 2 Quantization of Free Field Theory
1. The Lagrangian form of the classical field theory can be written in Hamiltonian
form so that it’s easier to be quantized:
1
1
1

2
H   d 3 x   2     m 2 2 
2
2
2

2. As explained, field theory is actually a collection of SHOs. Simple harmonic
oscillator can be quantized using the raising and lowering operators a and a†:
(briefly reviewed on p20 after Eq. (2.22), and certainly explained in your quantum
mechanics textbook)
a, a   1

 
1

H   aa  
2,

n  a
n
0
3. The scalar field theory could be quantized like in quantum mechanics. It means the
fields become operators. For operators, we need to specify their commutation
relation. Using Canonical Quantization procedure as in QM:
 ( x),   y   i (3) ( x  y)
4. The scalar field when Fourier transformed constitutes a series of SHO. And the
quantization of scalar field as specified in 3. is equivalent to quantizing the SHO’s:


d3p
 (x)  
(2 ) 3
a
H 

p
1
2 p
a
 

p
 
e ip x  a p e ip x


 
3
, a p '  2   (3)  p  p'

d3p
2 
3


 p  a p a p 


1

a p , a p ' 
2

5. The energy eigenstate states can be constructed like in SHO and the energy
eigenvalues look exactly like particle states with various particle numbers. This is
also true for momentum. See the paragragh after Eq. (2.33) on P. 9. So field, when
quantized, become particles. Even better, these are relativistic particles.
8
For field, the space coordinates x correspond to the indices to specify degrees of
freedom. So the above commutation relation can be easily generalized to the field
operator ϕ(x) and its conjugate momentum π(x)：
 ( x, t ),   y, t   i (3) ( x  y),
 ( x, t ),   y, t   0,  ( x, t ),   y, t   0
These relations are equivalent to corresponding commutation relations between the a
operators. It can be shown that the above relation implies:
 
3
a p , a p '  2   3  p  p'


and vise versa:
This is exactly commutation relation of the a operators for a series of quantum simple
9
harmonic oscillator (SHO). For a quantum SHO:
The spectrum of SHO is like a ladder.
The operator a+ can be used to raise the energy by one quantum while the operator a
can be used to lower the energy by one quantum:
The operator a+ is called Raising Operator while the operator a Lowering Operator.
It has been shown above that Field theory consists of a series of SHO. Hence
Quantum Field Theory is just a series of quantum SHO.
Now the Hamiltonian of QFT can be written in terms of a’s:
10


d3p
 (x)  
(2 ) 3
a 
 n

p
1
2 p
a
 

p
 
e ip x  a p e ip x

 



 
0  p, n P p, n  np p, n H p, n  nE p p, n
11
12
Dirac Field
13
14