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

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

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

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

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

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

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

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

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

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

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

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

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.

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)

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)

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

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

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

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

Time Gauge Axial Gauge