1. Canonical Quantization
Chapter I
Canonical Quantization
Lecture 4
Books Recommended:
Advanced Quantum Mechanics by Schwabl
Quantum Field Theory by Michio Kaku (2nd chapter for understanding Lorentz group)

2. Noether Theorem: Relate symmetry to
conservation laws.
Continuous transformations which leave the
action unchanged lead to conservation laws.
e.g. ,
Temporal invariance: conservation of energy
Translational invariance : four momentum
conservation
Rotational invariance: Angular momentum

3. Consider infinitesimal Lorentz transformations
------(1)
Corresponds to same point in space-time
viewed from two Inertial frames.
Anti symmetric tensor
----(2)

4. causes infinitesimal translation.
Anti- symmetric in μ and ν . These
depends upon the transformation properties
of fields. For example, for spinors

5. Invariance under transformation (1) means
(3)
Variation in field will be
----(4)
Total variation in field
----(5)

6. Eq. (4) and (5) are related as
------(6)

7. Total variation in the Lagrangian density
will vanish as per Eq. (3) i.e.,
(7)

8. First term of Eq (7) will be written as
-----(8)
Where in 2nd line Euler Lagrange Eq is used.
In last line Eq (6) is used.

9. Also, we can write
(9)
Using (8) and (9) in (7), we get
-----(10)
Where
-----(11)

10. Eq. (10) is continuity eq and is a statement of
Noether theorem.
is called Noether current.
Corresponding Noether charge
-----(12)
We can write
----(13)

11. Eq. (13) will vanish i.e., charge will conserved
if field vanish sufficiently rapidly at
infinity.
(14)

12. Space-time translation
---(15)
From (1), we get
(16)
And thus,
(17)

13. Thus, Noether theorem (Eq. (11)) reduces to
-----(18)
And thus, From (10) and (18),
----(19)
For
----(20)
which is four momentum.

14. For
(21)
In above for zero compoent ie. , μ = 0,
spatial momentum will be conserved and
For spatial component stress tensor will be
conserved i.e.,

15. Rotations
Here,
(22)
and
----(23)
From (11),
---(24)

16. From (24), using antisymmetrization property
(25)
Where,
(26)

17. From (25), we write using continuity Eq
------(27)
From above, we write six quantities
-----(28)

18. For spatial component, we obtain angular
momentum operator
-----(29)
Here, 2nd and 3rd contain vector product of
coordinate and space momentum density.
First term is interpreted as intrinsic angular
Momentum.

19. Gauge Transformation
(Related to internal symmetry)
Under internal symmetry fields transform
But space time point do not.
Consider Gauge transformations
----(30)

20. As space time point do not transform and
therefore, from (1,4),
------(31)
And also from (5),
----(32)

21. Four current density (From Noether Theorem)
(33)

22. Noether Charge
----(34)
Quantized form

23. We calculate