1. Chapter II
Klein Gordan Field
Lecture 1
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
Quantum Field Theory by Michio Kaku

2. Review of Simple Harmonic Oscillator
Hamitonian for Harmonic oscillator
------(1)
Introducing annihilation and creation operators
(2)

3. We can write
----(3)
Hamiltonian will be
----(4)

4. Commutation of Hamiltonian with creation
And annihilation operators
----(5)
Considering
-----(6)
We can write
-----(7)

5. Ground state energy
-----(8)
For excited states
-----(9)

6. Klein Gordon Field
Lagrangian density for real scalar field
------(1)
Momentum field density
-----(2)

7. Hamiltonian density
----(3)

8. Total Hamiltonian
(4)
Dimensions : (write (1) using and c)

9. Canonical quantization for scalar fields
(5)

10. Field Decomposition
We use plane wave solutions as complete set
of basis and expand field operator as
(6)

11. Satisfy Klein Gordon Eq
-----(7)
Using (6) in (7)
-----(8)

12. Fourier transform has non-vanishing component
only on the mass shell
-----(9)
We can write
(10)

13. Using (10) in (6),
(11)
Argument of Dirac delta vanished when
----(12)

14. We write Dirac delta function as
(13)

15. Using (13) in (11)
(14)

16. In 2nd term of Eq (14) use
We have
---(15)

17. Real field will be hermitian
(16)

18. We have
---(17)
And we write
------(18)

19. Note that
---(19)
And using
----(20)
Using above in (18),
-----(21)

20. Positive and negative energy part
----(21)
And thus
(22)