1. Chapter IV
Gauge Field
Lecture 2
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
Advanced Quantum Mechanics by Schwabl

2. Lagrangian Density
------(1)
Where
From (1)
------(2)

3. Also,
------(3)
Thus, Eq, of motion will be
------(4)
Momentum conjugate to field
----(5)

4. From (5), we get
----(6)
And
(7)
It is not possible to apply canonical
quantization to all four components of
radiation field.

5. Exercise: Another Lagrangian density for
E.M. Field is
(8)
Derive Eq of motion from above
Show that Eq (1) and (8) are differ by total
divergence

6. Free Electromagnetic field and quantization
For free field
---(9)
Sol of Poisson Eq will be
= 0
----(10)

7. For E.M field we have
---(11)
Lagrangian density for free field
---(12)
Here we used

8. Hamiltonian density for free e.m. Field
(13)