1. Canonical Quantization
Chapter I
Canonical Quantization
Lecture 3
Books Recommended:
Field Quantization by W. Griener
Lectures on Quantum Field Theory by Ashok Das

2. We start from classical mechanics:
Quantization
We start from classical mechanics:
For two dynamical quantities A and B, Poission bracket
----(1)
Time derivative of A
-----(2)
Where, we used Hamilton Eqs

3. Note that, for coordinates
----(3)
And
------(4)
In going from CM to QM, we use
---(5)

4. We now defined variation in functional
(6)
Functional derivative
It tells us how the functional changes with the change in field at x.

5. Momenta canonical to field
(7)
Note that L is total Lagrangian
Also
------(8)
Hamilton Eqs
-----(9)
H is total Hamiltonian.

6. We define the Poission bracket for functionalas
---(10)
Time dependence of Functional is defined as
----(11)
Where (9) is used.

7. Considering function as functional, we define
--(12)
Functional derivative of above will be
(13)

8. Eq. (13) is true can be verified using
definition (6),
----(14)
Using (13) in above we get left side.
Also, we have
---(15)
-----(16)

9. We can now write
--(17)
---(18)

10. Poission bracket relations
(19)
And
--(20)
Above relations are equal time possion bracket
Relations.

11. We may also use following notations later
-----(21)
In above we used space time coordinate,
Whose 0th component is time.

12. Single particle classical mechanics is
Quantized by replacing the Poisson bracket
among the coordinates by corresponding
Commutation relations and operators.
This is called first quantization.
In moving to the Quantum field theory, we
quantize the classical fields and treat them as
operators. Poisson bracket among the fields
given inn Eq (21) Is replaced by corresponding
Commutation Relations. This is called second
quantization.

13. Thus, we write for QFT, commutation relations
---(22)
Note that for Bose particles commutation
Relations are used whereas as we will see
Later for Dirac field anti-commutation relations
Are used. Above procedure of quantization is
Called Canonical quantization