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

2. Maxwell Equations
Consider Maxwell eqns.
------(1)

3. Consider the antisymmetric tensor
----(2)
We define
-----(3)

4. Maxwell Eqns. can be written as
---(4)
And
---(5)
Ex.: Derive Eq. (1) from (4) and (5).

5. Four current density satisfy
Continuity Eq
----(6 )
We define field strength tensor in terms of
Four vector potential
---(6a)
Note :

6. Eq. (6a) can be used to obtain homogenous
Maxwell Eqns.
Non-homozenous Eqns can be obtained from
----(6b).

7. From 3rd Maxwell Eq, we can write
----(7)
Also, using above in 2nd Maxwell Eq
----(8)

8. From (8) we can write
---(9)
Using above in 1st Maxwell Eq, we get
----(10)

9. Also, from 4th Maxwell Eq
---(11)
Derive (10) and (11) from (6b).

10. Gauge Transformations
We define transformations
---(12)
Under which E and B will not change.
Takeing curl of 1st eq in above, we get
----(13)

11. From (13), we can write
---(14).
Since two potentials will give same E. Thus
-----(15)

12. From (15), we write
-----(16)
From above, we get
----(17)

13. From (12), (14) and (17), we can write
Transformations
-----(18)
Above Eqns are Gauge transformations of
2nd kind. Note that λ is function of x.

14. In four vector notations, we define
the gauge transformations as
------(19)
Field strength tensor and hence, E and B
Will not change under above transformations.
----(20)

15. To obtain E and B one should solve Eq. (10)
and (11). We choose some particular Guage
To simplify these Eqns.
Coulomb Gauge:
Here
----(21)
From (10) we get
---(22)

16. Or from Eq (6b), using μ = 0 and Eq (21)
Eq. (22) and (23) are same
Sol:
----(22b)
A0 is not independent.

17. From (11), using coulomb Gauge conditionOr
---(23)
Eq (22) is easy to solve but not (23).

18. Or From (6b) using spatial comonnet i.e. μ = j
Which is same as Eq (23).

19. Lorentz Gauge
------(24)
Or in four-vector notation
------(25)
Thus, from (10) and (11), we get
----(26)
-----(27)

20. Time Gauge
Axial Gauge