1. Chapter IV
Gauge Field
Lecture 3
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
Advanced Quantum Mechanics by Schwabl

2. Field Decomposition
As we discussed earlier
(1)
Thus, we can write
----(2)
Field Eq
(3)

3. For ν = 0
(4)
And for ν = i
----(5)
Which is like mass-less KG Eq.
Sol will be
----(6)
Where
---(7)

4. Using coulomb Gauge
----(8)
For two polarization direction
-----(9)

5. We can write following expression for the
decomposition for field
----(10)
where

6. Exercise:
Using inverse Fourier transform, find
(11)
----(12)

7. Quantization of field
----(13)
Above equations are not consistent with
Transverse character of vector potential

8. We define following quantization condition
which is consistent with the transverse
condition.
---(14)
Where transverse delta function is defined as
(15)

9. Note that
---(16)
In momentum space

10. Exercise: Using the expression for fields
in LHS of (14) prove that it gives RHS.
See Schwabl book.
Commutators for annihilation and creation
Operators
---(17)

11. Normal ordered Hamiltonian and momentum
------(18)
----(19)

12. We can write
---(20)
---(21)